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A parametric study of the axisymmetric loading of an anisotropically elastic limb model by a tourniquet Hodgson, Antony John 1986

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A PARAMETRIC STUDY OF THE AXISYMMETRIC LOADING OF AN ANISOTROPICALLY ELASTIC LIMB MODEL BY A TOURNIQUET By Antony John Hodgson BASc, The University of British Columbia, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1986 c ) 8 Antony John Hodgson, 1986 IN In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of hA eorNQir4<^i ELrxgjfve£-r'lf\g The University of B r i t i s h Columbia 1956 M a i n M a l l Vancouver, Canada V6T 1Y3 Date acJb&bojr 8 , 1 3 8 6 ABSTRACT The c l i n i c a l use of tourniquets has an associated nerve palsy i n c i -dence r a t e of approximately 0.1%. The mechanism of damage i s thought to be some form of mechanical deformation. No previous work, has q u a n t i f i e d such a mechanism. In t h i s work, an a x i a l l y compressive damage mechanism i s proposed and supported by an order-of-magnitude a n a l y s i s comparing i t with other proposed mechanisms. E a r l i e r attempts to model the a p p l i c a -t i o n of a tourniquet to a limb have e i t h e r not modelled the a p p l i e d pressure adequately or have not made c l e a r the connection between the mechanical properties being modelled and the induced damage. The model proposed i n t h i s paper incoporates a n i s o t r o p i c e l a s t i c parameters i n a st r e s s function formulation f o r a l i n e a r l y e l a s t i c m a t erial with small deformations assumed. An a n a l y t i c a l s o l u t i o n technique i s developed f o r rectangular regions. The s o l u t i o n technique was applied to a c l o s e - t o -i s o t r o p i c case (the technique degenerates when pure i s o t r o p y i s assumed) and the r e s u l t s were compared with an experimental i n v e s t i g a t i o n . Good agreement was obtained with most of the important features noted i n the experimental l i t e r a t u r e , and a region of high negative a x i a l s t r a i n was found to c o r r e l a t e w e l l with observations of the l o c a t i o n s and charac-t e r i s t i c s of the nerve l e s i o n s . A s e n s i t i v i t y a n a l y s i s showed that the s o l u t i o n was r e l a t i v e l y i n s e n s i t i v e to the a n i s o t r o p i c parameters. Because of t h i s i n s e n s i t i v i t y and the c o r r e l a t i o n of the regions of negative a x i a l s t r a i n with the regions of damage, the model was used to suggest improvements i n current tourniquet cuff designs. I t was found that a smoother ap p l i e d pressure d i s t r i b u t i o n ( s i n u s o i d a l as oposed to - i i -rectangular), a wider c u f f , and an e l i m i n a t i o n of shear s t r e s s between the cuff and the skin a l l contributed to a decrease i n the maximum nega-t i v e a x i a l s t r a i n . The l a s t e f f e c t was not q u a n t i f i e d , but reductions of 50-70% could be achieved by taking the f i r s t two measures. - i i i -TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES v i i i AUTHOR'S PREFACE x i ACKNOWLEDGEMENTS x i i CHAPTER 1 INTRODUCTION 1 1.1 Tourniquet-Induced Paralysis During C l i n i c a l Application 1 1.2 Nerve Physiology Background 5 1.3 Historical Overview of Proposed Damage Mechanisms 9 1.3.1 Forms of Reported Nerve Damage 9 1.3.2 Historical Evidence for Theory of Ischaemia 9 1.3.3 Support for Theory of Mechanical Damage 11 1.3.4 Oschoa et a l . Proposed mechanisms for Mechanical Damage 17 1.3.5 Quantitative Evidence for Axial Compression Mechanism 19 CHAPTER 2 TWO EARLIER MODELS 24 2.1 One-Dimensional Isotropic Model 24 2.1.1 Advantages and Limitations of G r i f f i t h s and Heywood's Model 27 2.2 Two-Dimensional Axisymmetric Isotropic Model 28 2.2.1 Verification of Model Against Experimental Work .. 29 2.2.2 Application of Model to Surface Octahedral Shear Stress Calculations 31 2.2.3 Advantages and Limitations of Auerbach's Model ... 34 CHAPTER 3 MODELLING 36 3.1 D i f f i c u l t i e s in Modelling 37 3.2 Simplifying Assumptions 38 3.2.1 Ju s t i f i c a t i o n of Assumptions 38 3.3 Mathematical Development of Model 42 3.3.1 Moment Equilibrium 44 3.3.2 Force Equilibrium 44 3.3.3 Strain-Displacement Relations 46 3.3.4 El a s t i c Relationships 47 3.3.4.1 Isotropic E l a s t i c Relationships 47 - iv -TABLE OF CONTENTS (Continued) Page 3.3.4.2 Anisotropic E l a s t i c Relationships 49 3.3.4.2.1 Definitions of E l a s t i c Constants 49 3.3.4.2.2 Constraints on Choice of El a s t i c Constants .. 51 3.3.5 Compatibility Equations 52 3.3.5.1 Expressed in Terms of Strain Components 52 3.3.5.2 Expressed i n Terms of Stress Components 53 3.3.6 The Use of a Stress Function 54 3.3.6.1 Isotropic Stress Function 54 3.3.6.2 Anisotropic Stress Fucntion 55 3.3.6.2.1 Determination of Anisotropic Coefficients 56 3.3.6.2.2 Determination of Displacement Expressions 57 3.3.7 Geometric Parameters 59 3.3.8 Non-Dimensionalization 59 3.3.9 Method of Solution 60 3.3.9.1 Separation of Equation 60 3.3.9.2 Solution to Axial Equation 62 3.3.9.3 Solution to Radial Equation 63 3.3.9.4 Application of Boundary Conditions 66 3.3.9.4.1 Permissible Values for the Separation Constant 68 3.3.9.5 General Solution 70 CHAPTER 4 RESULTS AND DISCUSSION 73 4.1 Rationale for Test Selection 73 4.1.1 Approximation of Loading Distribution 73 4.1.2 Domain of Reasonable Solutions 74 4.1.3 Selection of Test Cases 77 4.2 General Characteristics of Solution 79 4.2.1 Comparison with Thomson and Doupe 79 4.2.1.1 Rectangular or Sinusoidal — Which Most Likely? 79 4.2.1.2 Free or Sticky — Which Most Likely? 84 4.2.2 General Characteristics of Induced Stresses and Strains 85 4.2.2.1 Radial Stresses 92 4.2.2.2 Axial Stresses 92 4.2.2.3 Axial Strains 95 4.3 Factors Affecting Regions of Negative Axial Strain 97 4.3.1 Shape of Loading Distribution 99 4.3.2 Width of Cuff 99 4.3.3 Thickness of Limb 100 - v -TABLE OF CONTENTS (Continued) Page 4.4 Sensitivity of Solution to El a s t i c Parameters 102 4.4.1 Sensitivity of Solution to E r 102 4.4.2 Sensitivity of Solution to j» 102 4.4.3 Sensitivity of Solution to VP 106 4.4.4 Sensitivity of Solution to G r I l l CHAPTER 5 DISCUSSION AND DIRECTIONS FOR FUTURE WORK 114 5.1 Validity of Damage Mechanism Estimates 115 5.2 Cr i t e r i a for Improved Model 117 5.2.1 Connection Between Axial Strain F i e l i d and Proposed Damage Mechanism 117 5.2.2 Comparison of Predictions with Observations 124 5.3 Cuff Design Recommendations 124 5.3.1 Other Factors Affecting Use as Design Tool 124 5.3.1.1 Neglect of Shear Stress Applied at Outer Surface 124 5.3.1.2 Presence of Large Strains 125 5.3.1.3 Limited Form of Anisotropy Available for Calculations 125 5.3.1.3.1 Sensitivity of Solution to Changes i n Anisotropy 126 5.3.1.3.2 D i f f i c u l t y in Ascertaining 2-D Material Properties 127 5.3.1.4 D i f f i c u l t i e s i n Solution of Fiel d Equation .. 127 5.3.2 General Design Recommendations 128 CHAPTER 6 CONCLUSIONS 132 BIBLIOGRAPHY 135 APPENDICES: A - MECHANISMS OF NERVE IMPULSE TRANSMISSION 137 B - COMPARISON OF PROPOSED DAMAGE MECHANISMS 146 C - DETAILED MATHEMATICAL DEVELOPMENT OF MODEL 157 D - COMPLETE SET OF AXIAL STRAIN PLOTS FROM ALL TESTS RUN .. 201 E - REDUCTION OF COMPLEX SOLUTIONS TO REAL SOLUTIONS 208 - v i -LIST OF TABLES Page Table 1 Typical parameters used in simulation 77 2 Names and parameters used in each test of model 80 3 Maximum axial strain for different cuff width ratios, limb thickness and loading distributions 98 - v i i -LIST OF FIGURES Page Figure 1 T y p i c a l points of a p p l i c a t i o n f o r a tourniquet (with apologies to Leonardo Da V i n c i ) 2 2 Diagrammatic representation of a neuron 6 3 Components of an axon 8 4 Examples of invagination ranging from mild to severe 13 5 D i r e c t i o n of Nodal Displacement - 14 6 Histogram of Proportion of Damaged Fibers as a Function of P o s i t i o n R e l a t i v e to Cuff 14 7 Relat i v e p o s i t i o n s of undeformed and overlapped, deformed myelin sheaths f o r order-of-magnitude c a l c u l a t i o n s 21 8 Stresses induced i n an i n f i n i t e l y long c y l i n d e r of i s o t r o p i c , l i n e a r l y e l a s t i c m a t e r i a l around a r i g i d core by an axisymmetric r a d i a l pressure 25 9 Isobaric plots of h y d r o s t a t i c pressure under a t o u r n i -quet as determined expderimentally by Thomson & Doupe (top) and numerically by Auerbach (bottom) 30 10 Comparison of subcutaneous octahedral shear stresses f o r tourniquets with d i f f e r e n t edge rounding r a d i i 33 11 Superposition of l e s i o n l o c a t i o n histogram and octahedral shear s t r e s s d i s t r i b u t i o n 33 12 Geometric parameters i n the tourniquet/limb system 43 13 Stresses on elemental volume 45 14 Extension of limb with negative loading 71 15 Fourier approximation to a rectangular loading d i s t r i b u t i o n f o r c = 0.4, 0.6 and 0.8 75 16 Fourier approximation to a s i n u s o i d a l loading d i s t r i b u t i o n f o r c = 0.4, 0.6 and 0.8 76 17 Comparison of predicted h y d r o s t a t i c and transverse pressure maps with Thomson & Doupe's find i n g s 81 - v i i i -LIST OF FIGURES (Continued) Page 18 Comparison of predicted h y d r o s t a t i c pressure maps with Thomson & Doupe's f i n d i n g s 82 19 Comparison of predicted transverse pressure maps with Thomson & Doupe's fi n d i n g s 86 20 Contour plots f o r te s t RSAA 87 21 Contour plo t s f o r t e s t RSAA 88 22 Contour plots f o r test RSAA 89 23 Comparison of c i r c u m f e r e n t i a l s t r e s s and r a d i a l displacement p l o t s from t e s t s RSAA and RSAA*2 90 24 Comparison of r a d i a l and c i r c u m f e r e n t i a l s t r a i n p l o t s from t e s t s RSAA and RSAA*2 91 25 Comparison of Auerbach's p r e d i c t i o n s of hy d r o s t a t i c pressure l e v e l s with r a d i a l s t r e s s plots, from t e s t s RFAW and RSAA 93 26 Comparison of a x i a l s t r e s s plots from tests p l o t s SSAA, RSAA, RFAW and SFAA 94 27 Comparison of a x i a l s t r a i n plots from te s t s RSAA, SSAA and RFAA 96 28 Comparison of r a d i a l and a x i a l s t r e s s plots from te s t s RSAA and RSAA*2 103 29 Comparison of a x i a l s t r a i n and hy d r o s t a t i c pressure p l o t s from t e s t s RSAA and RSAA*2 104 30 Comparison of r a d i a l and a x i a l s t r e s s plots from t e s t s SSAA and SSAA*1 105 31 Comparison of a x i a l s t r a i n and hyd r o s t a t i c pressure p l o t s from t e s t s SSAA and SSAA*1 106 32 Comparison of r a d i a l and a x i a l s t r e s s plots from t e s t s SSAA*1 and SSAA*4 107 33 Comparison of a x i a l s t r a i n and hyd r o s t a t i c pressure p l o t s from t e s t s SSAA*1 and SSAA*4 108 34 Comparison of r a d i a l and a x i a l s t r e s s plots from t e s t s SSAA*1 and SSAA*5 109 - i x -LIST OF FIGURES (Continued) Page 35 Comparison of a x i a l s t r a i n and hy d r o s t a t i c pressure p l o t s from t e s t s SSAA*1 and SSAA*5 110 36 Comparison of a x i a l and r a d i a l stress plots from t e s t s SSAA*1 and SSAA*6 112 37 Comparison of a x i a l s t r a i n and hy d r o s t a t i c pressure p l o t s from t e s t s SSAA*1 and SSAA*6 113 38 Expansion of threshold s t r a i n l e v e l with in c r e a s i n g a p p l i e d pressure 119 A l Compositional d i f f e r e n c e s between c e l l i n t e r i o r and e x t e r i o r 139 A2 Increasing p o t e n t i a l gradient opposes d i f f u s i o n of KT1" ions due to concentration gradient 141 A3 Time course of graded p o t e n t i a l 143 A4 Time course of a c t i o n p o t e n t i a l 143 CI Stresses on elemental volume 159 C2 E l a s t i c a l l y equivalent coordinate systems 170 E l RSNN, RSNA, RSNW 209 E2 RSAN, RSAA, RSAW 210 E3 RSWN, RSWA, RSWW 211 E4 SSMM. SSMA. SSMW 212 E5 SSAN, SSAA, SSAW 213 E6 SSWN, SSWA, SSWW 214 E7 RFNN, RFNA, RFNW 215 E8 RFAN, RFAA, RFAW 216 E9 RFWN, RFWA, RFWW 217 E10 SFNN, SFNA, SFNW 218 E l l SFAN, SFAA, SFAW 219 E12 SFWN, SFWA, SFWW 220 E13 SSAA*1, RSAA*2, SFAA*3 221 E14 SSAA*4, SSAA*5, SSAA*6 222 - x — AUTHOR'S PREFACE The problem of tourniquet-induced nerve damage i s one which spans the f i e l d s of medicine and engineering. Both l i f e s c i e n t i s t s and engineers, therefore, should be i n t e r e s t e d i n t h i s study. I have endeavoured to make t h i s t h e s i s r e l a t i v e l y s e l f - c o n t a i n e d because I am aware that many of the people who might use i t w i l l lack a background i n one of the two areas and w i l l not want to spend a great deal of time acquiring that background elsewhere before reading t h i s t h e s i s . I have therefore arranged the t h e s i s so that s u b s t a n t i a l p o r t i o n s of the body can be read by e i t h e r l i f e s c i e n t i s t s or engineers. Because of the a n a l y t i c a l nature of the study, engineers w i l l be more at home i n a l l but the f i r s t chapter, but l i f e s c i e n t i s t s w i l l f i n d much of value and i n t e r e s t i n Chapters 1, 2, 5 and 6. A more d e t a i l e d mathematical development i s contained i n an appendix to help those who f i n d Chapter 3 somewhat d i s j o i n t e d . I do not wish anyone to f e e l that t h i s t h e s i s i s unapproachable because of i t s s i z e . Enough meat i s contained i n the roughly 50 pages ( i n c l u d i n g p i c t u r e s ) of Chapter 1, 2, 5 and 6 to ensure a good under-standing of the pr e d i c t i o n s and l i m i t a t i o n s of the model. Understanding the d e t a i l s can come l a t e r . I hope you enjoy your reading. - x i -ACKNOWLEDGEMENTS I would l i k e to thank above a l l the Creator for g i v i n g me so much to wonder about, and so much joy i n my own discovery of what He has made. To Him I o f f e r t h i s t h e s i s i n humble se r v i c e as my echo of His cre a t i o n . I am very g r a t e f u l to my wife, E l i z a b e t h , f o r k i w i f r u i t and hugs, for packing and housework, for love and encouragement, for earning money and f o r f i n d i n g ways to keep our l i v e s i n t e r e s t i n g even when we didn't r e a l l y have the time. I have been glad of my time with Dr. Edward Hauptmann; he has co n t i n u a l l y reminded me of what I am hoping to achieve through t h i s work and has helped me to keep t h i s p r o j e c t i n perspective. I o f f e r my s p e c i a l thanks to Dr. A. Cayford i n the Department of Mathematics at U.B.C. f o r s e v e r a l d i s c u s s i o n s on the nature of f o u r t h order, biharmonic-like, p a r t i a l d i f f e r e n t i a l equations; perhaps one day one of us w i l l understand them. I owe thanks to my "computer consultant" and f r i e n d , B r i a n Konesky, for helping me i n my f i r s t crude experiment i n p a r a l l e l processing: simultaneously running f i v e copies of my a n a l y s i s program on f i v e separate microcomputers i n order to generate the masses of data I would need to t r a n s f e r to the mainframe before the data l i b r a r y closed f o r the f i n a l weekend before t h i s t hesis had to be submitted. F i n a l l y , of course, my thanks to the Natural Sciences and Engineering Research Council and the U n i v e r s i t y of B r i t i s h Columbia f o r my food and t r a n s p o r t a t i o n and clothes and Christmas presents and any-thing else money has bought me i n the past two years ( i n c l u d i n g t u i t i o n ) . - x i i -CHAPTER 1 INTRODUCTION 1.1 Tourniquet-Induced P a r a l y s i s During C l i n i c a l A p p l i c a t i o n When we are i n the h o s p i t a l , we a l l hope that our ailments can be e a s i l y treated and that there w i l l no s i d e - e f f e c t s or complications. C e r t a i n l y one device that minimizes the p r o b a b i l i t y of complications i s the pneumatic tourniquet. I t i s used almost u n i v e r s a l l y i n limb surgery (see Figure 1) because i t provides a bloodless f i e l d i n which the surgeon can work, thus making s u r g i c a l techniques e a s i e r to carry out. In turn, operations take l e s s time and the induced trauma i s generally le s s than i t would be i f a tourniquet were not used (McEwen, 1981). Unfortunately, there are infrequent but p e r s i s t e n t reports of s o f t -t i s s u e damage and p a r a l y s i s caused by using a pneumatic tourniquet i n limb surgery. A study done by the Biomedical Engineering Department at the Vancouver General H o s p i t a l uncovered 15 incidents of complications a t t r i b u t a b l e to the use of a pneumatic tourniquet during an 18 month period i n which approximately 10,000 procedures were performed (McEwen, 1981). Dr. McEwen at VGH believes that t h i s apparent incidence rate of 0.15% i s lower than the true incidence rate f o r two reasons: 1) the symptoms may be d i f f i c u l t to a t t r i b u t e to the use of a pneumatic tourniquet because they are generally t r a n s i e n t , r e v e r s i b l e , and easy to confuse with other s i d e - e f f e c t s a r i s i n g from the s u r g i c a l procedure, and, 2) p a r t i c u l a r i n c i d e n t s may not be reported because of a concern about l e g a l l i a b i l i t y (McEwen, 1981). The most discomforting i n j u r y which can be Induced by a pneumatic tourniquet i s nerve damage r e s u l t i n g i n temporary p a r a l y s i s . Of the 15 2. Figure 1. T y p i c a l points of a p p l i c a t i o n for a tourniquet (with apologies to Leonardo Da V i n c i ) 3. cases mentioned above, s i x were cases of nerve i n j u r y or p a r a l y s i s , while the other nine were s o f t t i s s u e i n j u r i e s such as pinching, b l i s t e r i n g or b r u i s i n g , or s w e l l i n g due to o c c l u s i o n of the veins with-out a corresponding o c c l u s i o n of the a r t e r i e s . The apparent rate of pneumatic tourniquet-Induced p a r a l y s i s i s , therefore, 0.06%, although for the reasons discussed above, the true rate i s probably c l o s e r to 0.1%. Given that there are over one m i l l i o n procedures performed annually i n North America using a pneumatic tourniquet (McEwen, 1981), an estimated one thousand people annually would s u f f e r a s i g n i f i c a n t degree of nerve damage or p a r a l y s i s . The desire to minimize or eliminate t h i s s u f f e r i n g , then, i s the motivation f o r the work of t h i s t h e s i s . Since the problem of tourniquet-induced p a r a l y s i s has been known for decades, physicians and researchers have developed many em p i r i c a l c o r r e l a t i o n s between the c o n t r i b u t i n g f a c t o r s which are under the c o n t r o l of the surgeon or anaesthetist and the r e s u l t i n g extent of damage. The major f a c t o r s which have been i n v e s t i g a t e d have been the cuff pressure, the duration of a p p l i c a t i o n , and the temperature of the limb. In general, a lower c u f f pressure, a shorter a p p l i c a t i o n , and a lower limb temperature r e s u l t In a l e s s e r extent of damage (Bruner, 1970). There are, of course, l i m i t a t i o n s to the lowering and shortening of these v a r i a b l e s . If the cuff pressure i s too low, a r t e r i a l blood w i l l be able to enter the limb, but the veins, which are at a lower pressure, w i l l not allow the blood to leave. This i s known as venous congestion and r e s u l t s i n p a i n f u l and damaging sw e l l i n g . The time of a p p l i c a t i o n must be long enough for the surgery to be completed properly, and the temperature must not become so low that the t i s s u e s are damaged. 4. The most important of these f a c t o r s appears to be the a p p l i e d c u f f pressure. In the s i x cases of nerve damage or p a r a l y s i s i n v e s t i g a t e d i n the study at Vancouver General H o s p i t a l , a l l of the i n j u r i e s were a t t r i b u t e d to o v e r p r e s s u r i z a t i o n r e s u l t i n g from malfunctioning or in a c c u r a t e l y c a l i b r a t e d pressure gauges. A major e f f o r t of the Biomedical Engineering Department at VGH, aimed at minimizing the p o s s i b i l i t y of such i n c i d e n t s , has been the development of a microprocessor-controlled tourniquet which "tracks" the body's maximum a r t e r i a l blood pressure and maintains the c u f f pressure at the minimum e f f e c t i v e l e v e l . The routine use of such a device i s expected to markedly reduce the incidence of nerve damge (McEwen, 1981). In s p i t e of the development of such an advanced tourniquet, i t s developers do not regard i t as the ultimate device; they expect that " i t should also f a c i l i t a t e f u r t h e r advances, such as improvements i n cuff design" (McEwen, 1981). Indeed, i t i s t h i s aspect of the problem which has been most neglected because i t i s the most d i f f i c u l t to optimize e m p i r i c a l l y . Without an adequate understanding of the mechanisms of damage and the ways i n which the cuff parameters (such as the width and the pressure d i s t r i b u t i o n ) can be a l t e r e d to aggrevate or minimize the e f f e c t s of these mechanisms, i t i s impossible to know how to modify the extant c u f f designs. Consequently, the performance of pneumatic tourniquets w i l l not be improved. The r e s t of t h i s i n t r o d u c t i o n includes a b r i e f overview of nerve physiology as preparation for a d i s c u s s i o n of the proposed mechanisms f o r tourniquet-induced nerve damage. This i s followed by a presentation of two attempts to understand and model the connections between the cuff parameters and the induced nerve damage. 5. 1.2 Nerve Physiology Background 1 Although there are many types of nerves (or neurons) i n the body, the ones which are of i n t e r e s t here are the pe r i p h e r a l neurons, those neurons which transmit information between the s p i n a l column and the body's ex t r e m i t i e s . Like a l l other neurons, the p e r i p h e r a l neurons have three b a s i c parts (see Figure 2): 1) the dendrites, which are a network of c e l l outgrowths which receive information from other c e l l s ; 2) the body, which contains most of the s u b c e l l u l a r organelles responsible f o r c a r r y -ing out the metabolic processes; and, 3) the axon, which i s a long s i n g l e outgrowth (sometimes up to one meter i n length) which connects to other neurons' d e n d r i t i c s t r u c t u r e s or to end organs such as muscles. The p r i n c i p l e feature which d i s t i n g u i s h e s p e r i p h e r a l neurons from most other neurons i s the length of the axon; i n p e r i p h e r a l neurons, the axon i s t y p i c a l l y s e v e r a l tens of centimeters long while the axons of other neurons are generally much shorter. The axons of pe r i p h e r a l neurons d i f f e r from one another i n two main ways: 1) s i z e ; and 2) the presence or absence of a f a t t y sheath known as myelin, which functions both as an a i d to nerve impulse transmission and as s t r u c t u r a l support f o r the neuron. The main motor neurons are often quite large, with diameters on the order of 15 ym, and they u s u a l l y have a myelin sheath. The sensory neurons, on the other hand, are generally smaller, with diameters of l e s s than 5 um, and they are often unsheathed. Acknowledgement i s due to the book Human Physiology, by Vander, Sherman and Luciano, as the source f o r many of the concepts explained i n t h i s s e c t i o n . 6. Figure 2. Diagrammatic representation of a neuron 7. The axon i s the part of a p e r i p h e r a l neuron which i s of greatest i n t e r e s t i n t h i s study because i t w i l l t y p i c a l l y extend from the s p i n a l cord to a point d i s t a l to ( f u r t h e r away from the body trunk) the a p p l i c a t i o n point of the tourniquet; i t i s , therefore, the only part of most p e r i p h e r a l neurons which l i e s underneath the tourniquet. The axon can be imagined to be a long f l u i d - f i l l e d f l e x i b l e tube (see Figure 3); the f l u i d i s c a l l e d axoplasm and the tube i s formed from the outer membrane of the c e l l . A network of c e l l s runs along the outside of the axon and supports the neuron i n various ways. Some of the l a r g e r neurons have the aforementioned myelin sheath. When t h i s i s present, i t does not extend uninterrupted down the length of the axon, but at regular i n t e r v a l s the myelin thins and exposes the neuron membrane to the e x t r a c e l l u l a r f l u i d . These points of exposure are known as the nodes of Ranvier and the area immediately surrounding these nodes i s c a l l e d the paranodal region. The area between the nodes i s c a l l e d the internode. In the paranodal region, the axon narrows so that the c r o s s - s e c t i o n a l area of the axoplasm i s reduced. As w i l l become apparent, t h i s s t r u c t u r e plays a s i g n i f i c a n t r o l e i n the transmission of nerve impulses. In f a c t , the nodes of Ranvier can be considered to be the "booster s t a t i o n s " where a nerve s i g n a l Is re-energized. An appendix d e t a i l i n g the mechanisms of impulse transmission i s included f o r those readers who wish to understand how the nodes of Ranvier a i d i n the transmission of nerve impulses. This understanding should make i t e a s i e r to understand the following d i s c u s s i o n of proposed damage mechanisms. i n t e r n o d a l r e g i o n — v p a r a n o d a l r e g i o n y-q ^ m y e l i n s h e a t h - ^ a x o p l a s m n o d e o f R a n v i e r - n a r r o w i n g n o d e Figure 3. Components of an axon 9. 1.3 H i s t o r i c a l Overview of Proposed Damage Mechanisms 1.3.1 Forms of Reported Nerve Damage Ever since Harvey Cushing developed the pneumatic tourniquet i n 1904, there have been reports of nerve damage. The degree and form of t h i s damage depends on the s e v e r i t y of the applied compression. In cases of severe compression, the nerve f i b e r s may be crushed, r e s u l t i n g i n degeneration of the p o r t i o n of the nerve d i s t a l to the point of crushing. Recovery, i f i t occurs, takes many months (Denny-Brown, 1944a). At the other end of the scale, mild compression can r e s u l t i n a slowing of the conduction speed or perhaps a complete block of conduc-t i o n under the tourniquet which reverses i t s e l f immediately upon release. The slowing or block under these conditions was a t t r i b u t e d , not to crushing of the nerve f i b e r s , but to asphyxia (lowered concentra-t i o n of oxygen i n the blood)(Lewis, 1931). If p a r a l y s i s i s to occur at a l l , i t i s obviously d e s i r a b l e to have i t reverse i t s e l f as soon as poss i b l e . Unfortunately, the pressures necessary to ensure a bloodless f i e l d f o r surgery are t y p i c a l l y high enough to cause l a s t i n g nerve damage i n a s i g n i f i c a n t number of cases. 1.3.2 H i s t o r i c a l Evidence f o r Theory of Ischaemia There have been two t r a i n s of thought since the 1930s as to the cause of tourniquet-induced nerve damage: 1) ischaemia (reduced or eliminated blood supply); and, 2) d i r e c t mechanical damage. For many years the theory that oxygen s t a r v a t i o n caused the i n j u r y p r e v a i l e d , b o l s t e r e d by three important pieces of evidence. The f i r s t was the work of Lewis, P i c k e r i n g , and Rothschild mentioned above i n which they demonstrated that asphyxia was responsible 1 0 . f o r conduction block i n cases of mild compression. They supposed that the l e s i o n caused by the c l i n i c a l a p p l i c a t i o n of tourniquets r e s u l t e d from the same mechanism (Lewis, 1931). Then, i n 1936, the theory of mechanical damage received a blow from experiments performed by Grundfest i n which he subjected an excised f r o g nerve to high pressures i n an oxygenated pressure chamber. He demon-strated that h y d r o s t a t i c pressures on the order of one thousand atmos-pheres (approximately two thousand times c l i n i c a l l e v e l s ) were required to induce a conduction block i n the absence of anoxic conditions (Grundfest, 1936). The t h i r d important piece of evidence came from work done by Denny-Brown and Brenner i n 1944. They used a tourniquet to produce l e v e l s of compression which were greater than those i n v e s t i g a t e d by Lewis, but which were s t i l l i n the c l i n i c a l range. This l e v e l of compression was associated with a l o c a l conduction block, with the I n t e g r i t y of the nerve preserved d i s t a l to the s i t e of a p p l i c a t i o n . The major change i n the nerve under the c u f f was demyelination, even though the nerve i t s e l f was not crushed, nor was the c o n t i n u i t y of the axoplasm in t e r r u p t e d (Denny-Brown, 1944b). The researchers thought that t h i s l e s i o n was s i m i l a r to that described by Lewis et a l . and consequently a t t r i b u t e d i t to a lack of oxygen due to r e s t r i c t e d blood flow. I t seems more l i k e l y that the preservation of d i s t a l e x c i t a b i l i t y i s evidence that the damage i s caused by d i r e c t pressure because the d i s t a l p o r t i o n of the nerve i s rendered as ischaemic as the part under the c u f f . Nonetheless, i n the l i g h t of Grundfest's r e s u l t s , these researchers f e l t that Ischaemia more adequately explained the l e s i o n s which they found. 11. These three pieces of evidence, then, provided p o s i t i v e reasons f o r supposing that the damage was caused by ischaemia and suggested that the theory of mechanical damage was u n l i k e l y . The theory of damage by ischaemia, therefore, p r e v a i l e d u n t i l the 1970s when a study by Ochoa, Fowler, and G i l l i a t t demonstrated the presence of a nerve l e s i o n i n a limb which had undergone moderate compression ( c l i n i c a l l e v e l s ) which was d i s t i n c t from the l e s i o n s described by e a r l i e r workers. Ochoa et  a l . f e l t that the l e s i o n which they produced was compatible with the theory of ischaemia (Ochoa, 1972). 1.3.3 Support f o r Theory of Mechanical Damage Ochoa's experiment involved the a p p l i c a t i o n of a tourniquet to the one of the lower limbs of a number of female baboons. The tourniquets were i n f l a t e d to e i t h e r 500 or 1000 mmHg and l e f t i n place f o r 1 to 3 hours. The baboons were k i l l e d for anatomical study at times ranging from a few minutes to s i x months a f t e r the tourniquets were released. Nerve samples were then taken from the region of the cuff and subjected to various a n a l y t i c a l processes i n c l u d i n g l i g h t and e l e c t r o n microscopy. This allowed the anatomical changes over time as the animals recovered to be analyzed. The most s t r i k i n g r e s u l t of t h e i r work was the discovery of a c h a r a c t e r i s t i c l e s i o n which preceded the demyelination noted by e a r l i e r workers. This l e s i o n occurred almost s o l e l y i n large myelinated f i b e r s (diameters greater than 5 ym) and was characterized by a p a r t i a l Invagination ( i n t r u s i o n ) of one si d e of the node of Ranvier i n t o the other. The degree of in v a g i n a t i o n ranged from mild to severe, with the most extreme example being a displacement of the node of Ranvier 12. approximately 15 axon diameters i n t o the adjoining internode (see Figure 4). Even i n mild cases of inva g i n a t i o n , the nodal gap separating the in t e r n o d a l myelin was o b l i t e r a t e d and the membrane separating the myelin from the e x t r a c e l l u l a r f l u i d was ruptured. The most i n t e r e s t i n g c h a r a c t e r i s t i c of the damage, i n terms of helping to resolve the question of the operative mechanism, was that any given a x i a l point, the d i r e c t i o n of displacement was i d e n t i c a l f o r a l l a f f e c t e d f i b e r s ; the node of Ranvier was displaced away from the center of the c u f f and toward the uncompressed t i s s u e (see Figure 5). This strongly suggests a mechanical cause, not ischaemia. Ochoa et a l . noted s e v e r a l other items of importance. As mentioned above, the nodal displacement was for the most part r e s t r i c t e d to the l a r g e r myelinated f i b e r s ; the smaller myelinated and unmyelinated f i b e r s were generally spared from any damage unless the applied pressure was qui t e considerable (on the order of 1000 mmHg). This observation i s s i g n i f i c a n t i n l i g h t of work by other researchers which demonstrates that pain sensation i s preserved even when motor a b i l i t y i s eliminated (Fowler, 1972). Since the motor neurons tend to be the la r g e s t nerve f i b e r s i n the body, and the sensory neurons tend to be smaller, the correspondence between f u n c t i o n a l impairment (motor a b i l i t y , but not sensation) and the presence of the l e s i o n s strongly affirms the hypo-thesis that these l e s i o n s are the cause of p a r a l y s i s . If i t can be convincingly argued that these l e s i o n s are mechanically induced, then, together with the f a c t that a l l nerves under the c u f f , whether small or large, s u f f e r the same degree of ischaemia, i t can be concluded that mechanical damage i s the most l i k e l y explanation. 13. Figure 4 . Examples of i n v a g i n a t i o n ranging from mild to severe. Note: a l l axon diameters equal. 14. proximal d i s t a l Figure 5. D i r e c t i o n of Nodal Displacement 15-io-5 -Figure 6. Histogram of Proportion of Damaged Fibers as a Function of P o s i t i o n R e l a t i v e to Cuff 15. The d i s t r i b u t i o n of the l e s i o n s under the c u f f a l s o shows an i n t e r e s t i n g pattern. Ochoa et_ a l . took several transverse sections at various a x i a l p o s i t i o n s along the limb and counted the number of damaged nerve f i b e r s (diameters >5 ym) showing evidence of the c h a r a c t e r i s t i c l e s i o n . The r e s u l t s , expressed as a percentage of the t o t a l number of large f i b e r s , are presented i n the histogram i n Figure 6. The most obvious r e s u l t i s that the damage i s almost e n t i r e l y l i m i t e d to a 1 cm region centered under each of the endpoints of the c u f f , with a s i g n i -f i c a n t l y greater proportion of the f i b e r s being damaged at the proximal edge. I t i s important to r e a l i z e that f i n d i n g 15% of the f i b e r s damaged at any p a r t i c u l a r transvers s e c t i o n implies that a much greater percent-age of the nerve f i b e r s are a c t u a l l y damaged. This i s true because the inv a g i n a t i o n does not extend throughout the e n t i r e i n t e r n o d a l segment. I f , f o r example, the average nodal displacement was 15% of the t y p i c a l i n t e r n o d a l distance, then i n a transverse s e c t i o n , 85% of the f i b e r s would appear to be normal and only 15% would appear to be damaged. Ochoa £t al_. b e l i e v e that t h i s i s a reasonable estimate; i f i t were accurate, then f i n d i n g 15% of the f i b e r s damaged i n any transverse s e c t i o n would i n d i c a t e that c l o s e to 100% of the f i b e r s are damaged i n that general v i c i n i t y . This b e l i e f i s supported by t h e i r observation that " a l l the nodes on the large myelinated f i b e r s were abnormal i n the most severely a f f e c t e d parts of compressed nerves" when the nerves were removed f o r microscopic examination. Two other observations are worth noting here, before going on to discuss the mechanism which Ochoa et a l . propose for the damage. These observations concern the e f f e c t s of pressure and time on the micro-st r u c t u r e of the nerves. The e f f e c t of increased pressure was what 16. might be expected; the extent of i n v a g i n a t i o n and the displacement of the nodes was increased, and the length of the a f f e c t e d zone at the edges of the c u f f s was increased. In a d d i t i o n , the tendency f o r the myelin sheath to rupture increased. This observation i s i n t e r e s t i n g because, although the p a t h o l o g i c a l changes d i f f e r e d only i n degree, the impulse conduction at low pressures was only slowed, but a higher pressures was blocked. Since both the high and the low pressures were s u f f i c i e n t to cut o f f the blood flow (induce haemostasis), the degree of ischaemia was no d i f f e r e n t , yet the conduction c h a r a c t e r i s t i c s of the nerve were markedly d i f f e r e n t . At the very l e a s t , t h i s suggests that mechanical pressures i s a c o n t r i b u t i n g f a c t o r to the nerve damage. The e f f e c t of a longer duration of a p p l i c a t i o n of the tourniquet was not what was expected. Since the l i t e r a t u r e abounds with studies c o r r e l a t i n g the c l i n i c a l s e v e r i t y of the damge with the duration of a p p l i c a t i o n , Ochoa e_t ad. expected to f i n d more severe l e s i o n s i n animals subjected to longer a p p l i c a t i o n s of the tourniquet. To t e s t t h i s , they subjected both lower limbs of one p a r t i c u l a r baboon to a tournqiuet; one tourniquet was l e f t on for one hour, the other f o r three hours. When the nerves were examined, they found that the maximum nodal displacement on both sides were roughly equivalent ( i . e . , l e s i o n s e v e r i t y was roughly e q u i v a l e n t ) , but that a greater number of nodes were found to be a f f e c t e d i n the limb compressed for three hours. I t i s not c l e a r from Ochoa's paper whether the "greater number of nodes" r e f e r r e d to a greater number of a f f e c t e d nodes i n each damaged neuron, or whether there was a greater number of damaged neurons each having roughly the same number of a f f e c t e d nodes. In l i g h t of the previous d i s c u s s i o n concerning the p r o b a b i l i t y that almost 100% of the neurons 17. were damaged, i t seems l i k e l y that he means that a greater number of nodes per neuron become a f f e c t e d with increasing time. 1.3.4 Ochoa et a l . ' s Proposed Mechanisms f o r Mechanical Damage Several c h a r a c t e r i s t i c s of the l e s i o n s i n v e s t i g a t e d by Ochoa et a l . make the l i k e l i h o o d of ischaemia as the casual mechanism quite small. In p a r t i c u l a r , the p r e s e r v a t i o n of the d i s t a l e x c i t a b i l i t y of the motor neurons, the sparing of the smaller nerve f i b e r s , the cuff-edge l o c a t i o n of the l e s i o n s , and, most importantly, the displacement of anatomical structures (the nodes of Ranvier) away from the cuff speak loudly against the p o s t u l a t i o n of ischaemia. These same c h a r a c t e r i s t i c s , with the possible exception of the sparing of smaller f i b e r s , i n t u i t i v e l y lend themselves to a mechanical explanation. The f a c t that a l l the important features of the damage occur i n the region of the c u f f ( p a r t i c u l a r l y at the c u f f edges) would seem to imply a mechanical connection with the tourniquet. Ochoa et a l . believed t h i s a l s o , and proposed a mechanism by which the damage could occur. They postulated that when compression was a p p l i e d to the nerve, the axoplasm (the viscous, c o l l o i d a l substance f i l l i n g the neuron) was forced away from the s i t e of compression i n the same manner that water i n a hose would be If i t were crushed. When the axoplasm encounteres the narrowing of the channel at the nodes of Ranvier, i t s f r e e movement would be hindered. If the v i s c o s i t y were high enough, the node would act as a plug and be forced Into the a d j o i n i n g segment. This would be the o r i g i n of the i n v a g i n a t i o n . This expiation also accounts f o r the sparing of smaller f i b e r s ; since smaller myelinated f i b e r s do not have a 18. narrowing at the nodes of Ranvier, the movement of the axoplasm would not be impeded and the neuron would not s u f f e r damage. Ochoa et a l . also point out that s i nce i t takes a higher s t r e s s to d i s p l a c e f l u i d from a smaller tube, movement of the axoplasm i n the smaller f i b e r s may not even occur. There i s no proof as yet one way or the other. Ochoa et a l . also attempt to expiation the concentration of the l e s i o n s at the edge of the c u f f i n terms of the pressure gradient there. C i t i n g Grundfest's experiment with the frog nerve i n the pressure chamber, they claim that i n the absence of pressure gradients, no axo-plasmic motion would occur. The reason f o r the l e s i o n concentration at the edges of the c u f f i s that the pressure gradient i s highest there. Ochoa et a l . did not f e e l j u s t i f i e d i n proposing the nature of the connection between mechanical damage sustained by the nerve and the r e s u l t i n g conduction block. They suggested that something to do with the o b l i t e r a t i o n of the nodal gap or changes i n the membrane perme-a b i l i t y was probably responsible. They also point out that Rasminsky and Sears (1972) demonstrated that myelin damage could r e s u l t i n conduc-t i o n block, although the exact mechanism was not understood by them e i t h e r . Even though we may have a l i m i t e d knowledge of the physiology of nerves, i t i s possible to make a couple of speculations. If one or more nodal gaps are o b l i t e r a t e d along a neuron, the a c t i o n p o t e n t i a l at the l a s t f u n c t i o n a l node or Ranvier may be too weak to generate a threshold p o t e n t i a l at the next f u n c t i o n a l node; t h i s i s more l i k e l y when several successive nodes are damaged. If the membrane i n the para-nodal region i s ruptured and exposed to the e x t r a c e l l u l a r space, the semi-permeable nature of the membrane would be eliminated, thus destroy-ing the a b i l i t y of the nerve to generate an a c t i o n p o t e n t i a l . Whatever 19. the r e a l reason i s , the c o r r e l a t i o n between the appearance of t h i s c h a r a c t e r i s t i c l e s i o n and the impairment of nerve f u n c t i o n i s very high. The mechanism proposed by Ochoa tit a l . explains s e v e r a l important features of the observed l e s i o n s , but i t has d i f f i c u l t i e s i n explaining c e r t a i n other ones. Ochoa's mechanism c e r t a i n l y explains the occurrence of invaginating l e s i o n s located under the edges of the cuff and d i r e c t e d away from the c u f f , as w e l l as the sparing of the smaller f i b e r s . It does not, however, adequately explain how a longer a p p l i c a t i o n time causes an increase i n the number of nodes a f f e c t e d . If the damage occurs, as they suggest, as the r e s u l t of axoplasmic movement, then a l l damage should stop as soon as the tourniquet i s f u l l y i n f l a t e d and the axoplasm i s expulsed from under the c u f f . In other words, the damage should be almost immediate and, hence, should not increase over time. This, of course, c o n t r a d i c t s the evidence. 1.3.5 Quantitative Evidence f o r A x i a l Compression Mechanism There i s al s o some q u a n t i t a t i v e evidence that the mechanism that Ochoa et_ a l . propose f o r the inva g i n a t i o n of one myelin segment into the adj o i n i n g one i s not adequate. The d e t a i l s are l e f t to an appendix (see Appendix B), but the s a l i e n t features of an order-of-magnitude a n a l y s i s of the proposed mechanisms are presented here. Three possible modes of inva g i n a t i o n were considered, two of which were suggested by Ochoa et al_. As has been mentioned, Ochoa et a l . f e l t that the high pressure gradients i n the region of the edge of the cuff were important i n determining the d i s t r i b u t i o n of the l e s i o n s . They also alluded to the p o s s i b i l i t y that the viscous shear forces a r i s i n g 20. from the flow of axoplasm through the narrowing at the nodes of Ranvier could drag the myelin sheath on one side of the node in t o the sheath on the other sid e . Accordingly, c a l c u l a t i o n s have been done to estimate the magnitude of the forces which could be a t t r i b u t a b l e to these mechanisms. The e f f e c t of d i r e c t compressive s t r e s s has a l s o been examined because the appearance of the invaginated nodes of Ranvier suggests some form of column c o l l a p s e . The f a c t that myelin has some s t r u c t u r a l strength r e i n f o r c e s t h i s s u s p i c i o n . For a l l three c a l c u l a t i o n s , i t was assumed that the myelin-encased axon behaved as a f l u i d - f i l l e d e l a s t i c tube, with the myelin providing the only s t r u c t u r a l support. The d e r i v a t i o n ws done f o r a r b i t r a r y d i a -meters, although s p e c i f i c values of q u a n t i t i e s of i n t e r e s t were c a l c u l a -ted f o r dimensions t y p i c a l of motor neurons and sensory neurons. For Young's modulus, a value of 15,000 Pa was used, which i s a value f o r s o f t t i s s u e ; admittedly, t h i s f i g u r e may be s i g n i f i c a n t l y i n e r r o r , although i t does not seem to be overly relevant i n t h i s a n a l y s i s because the a n a l y s i s i s a comparison of d i f f e r e n t mechanisms. I t was a l s o assumed that when the sheaths were overlapped, the changes i n the w a l l thickness were n e g l i g i b l e and that the i n t e r f a c e between the two tubes was located at the midthickness point of the undeformed tube (see Figure 7). Two mechanisms r e s i s t i n g i n v a g i n a t i o n were considered: 1) the d i s t o r t i o n energy required to compress one sheath and expand the other; and, 2) the f r i c t i o n a l f o r c e r e s i s t i n g the s l i d i n g of one myelin sheath i n s i d e the other. I t was quite a r b i t r a r i l y assumed that the c o e f f i c i e n t of f r i c t i o n was 0.3. With t h i s assumption, the energy required to over-3-C Figure 7. R e l a t i v e p o s i t i o n s of undeformed and overlapped, deformed myelin sheaths f o r order-of-magnitude c a l c u l a t i o n s some f r i c t i o n was found to be 30 to 1000 times greater than the energy of d i s t o r t i o n . The energy of d i s t o r t i o n was therefore ignored. When the shear force a c t i n g on the walls of the inv a g i n a t i n g portion of the myelin sheath was c a l c u l a t e d , the force tending to cause inv a g i n a t i o n was on the order of 1-6% of the value r e s u l t i n g from the f r i c t i o n a l a n a l y s i s above. The next procedure was to c a l c u l a t e what the e f f e c t of the pressure drop across the invaginating p o r t i o n of the sheath would be i f i t were act i n g on the annular region caused by the narrowing of the f l u i d c a v i t y ; the r e s u l t i n g force was on the order of 0.1-1% of the force required to overcome f r i c t i o n . Some preliminary r e s u l t s from the the s i s work suggested that i t was possible that an a x i a l l y compressive zone existed i n the region near the cuf f edges and that the magnitude of such stresses would be on the order of 10% of the cuff pressure. Using t h i s f i g u r e , i t was possible to determine that the compressive s t r e s s could r e s u l t i n a nodal d i s p l a c e -ment of 200 ym for larger neurons, which i s c e r t a i n l y the same order of magnitude as the observed displacements. For the smaller sensory neurons, the nodal displacement was predicted to be only 20 ym, which supports Ochoa's observation that smaller nerves were spared while larger ones were damaged. As a r e s u l t of t h i s a n a l y s i s , i t i s po s s i b l e to co n f i d e n t l y a s s e r t that the compressive stress mechanism i s l i k e l y to be the dominant one i n generating these invaginated l e s i o n s . The other postulated mechanisms seem to have e f f e c t s approximately two orders of magnitude lower than those of the compressive mechanism. So then, Ochoa's work provides greater confidence that the nerve damage under a pneumatic tourniquet i s t r u l y due to mechanical deforma-t i o n , not to ischaemia, and the order-of-magnitude work done suggests that i t would be p r o f i t a b l e to look for zones of a x i a l comrpession i n the t i s s u e compressed by a tourniquet. Before d i s c u s s i n g the s p e c i f i c s of the proposed model, i t w i l l help to look at two e a r l i e r attempts to simulate the compression of a limb with a tourniquet. 2 4 . CHAPTER 2  TWO EARLIER MODELS 2.1 One-Dimensional I s o t r o p i c Model P r i o r to the 1970s, the p r e v a i l i n g opinion was that the cause of tourniquet-induced nerve damage was ischaemia, although some experiments showed that mechanical pressure could conceivably cause such damage (Bentley, 1943). Not u n t i l Ochoa et a l . ' s work i n 1972 did the hypo-th e s i s of mechanical damage become a serious contender. In 1973, G r i f f i t h s and Heywood published a paper i n which they made the f i r s t attempt to apply t h e o r e t i c a l s t r e s s models to the problem of tourniquet-induced nerve damage ( G r i f f i t h s , 1973). Their primary concern was to provide an a n a l y s i s which would support the hypothesis of mechanical damage and to explain the v u l n e r a b i l i t y of the r a d i a l nerve i n the arm (a nerve l y i n g c l o s e to the bone); they were not overly concerned with the redesign of the cuff or with looking too c a r e f u l l y at the mechanism of damage. Because of t h i s purpose, the model they proposed was not very complicated; i n f a c t , they l i m i t e d themselves to a one-dimensional a n a l y s i s . They suggested that, as a f i r s t step, the limb could be treated as an i n f i n i t e l y long c y l i n d e r with a r i g i d c y l i n d r i c a l core. They then considered what the s t r e s s d i s t r i b u t i o n would be l i k e i f the underlying tis s u e were considered to behave as, f i r s t l y , a f l u i d , and secondly, an i s o t r o p i c e l a s t i c s o l i d (see Figure 8). If the underlying tissues were e s s e n t i a l l y f l u i d - l i k e , G r i f f i t h s and Heywood asserted that the a p p l i e d pressure would be transmitted uniformly throughout the t i s s u e , r e s u l t i n g i n a h y d r o s tatic s t r e s s state Figure 8. Stresses induced i n an i n f i n i t e l y long c y l i n d e r of i s o t r o p i c , l i n e a r l y e l a s t i c m a t e r i a l around a r i g i d core by an axisymmetric r a d i a l pressure 26. at a l l points. While they f e l t that t h i s r e s u l t c e r t a i n l y r e i n f o r c e d the c l i n i c a l assumption that the applied tourniquet pressure was echoed at a l l points i n the limb, " i t d i d not give any i n s i g h t i n t o the vulner-a b i l i t y of the r a d i a l nerve. They then considered the s t r e s s d i s t r i b u t i o n a r i s i n g from the applied pressure when the limb was treated as an i s o t r o p i c e l a s t i c s o l i d . They used the equations f o r the thick c y l i n d e r problem described i n Timoshenko's textbook (Timoshenko, 1956) and applied boundary conditions of a known ext e r n a l pressure and zero r a d i a l displacement at the inner boundary. Their a n a l y s i s demonstrated that the r a d i a l pressure increased as the bone was approached and that the circumferen-t i a l pressure decreased i n the same d i r e c t i o n . This f i n d i n g explained, to t h e i r s a t i s f a c t i o n , the v u l n e r a b i l i t y of the r a d i a l nerve, but was seen by them to be contrary to experimental evidence which demonstrates that the pressure under the c u f f remains r e l a t i v e l y constant down to the bone. They also showed that the p r i n c i p a l stresses could be resolved i n t o a combination of d i r e c t s t r e s s and shear s t r e s s , unlike the hydro-s t a t i c case; they claimed that the presence of these shear stresses was harmful, but d i d not propose a mechanism of damage. It seems l i k e l y that they were mistaken i n t h e i r b e l i e f that the t h e o r e t i c a l r e s u l t s contradicted e m p i r i c a l evidence concerning the constancy of pressure under the tourniquet. It could e a s i l y be that the devices used to measure the sub-cuff pressures measure e i t h e r the hydro-s t a t i c pressure or the average of the p r i n c i p a l stresses i n the trans-verse plane. If t h i s i s so, then the t h e o r e t i c a l a n a l y s i s agrees with the experiments because the sum of the r a d i a l and c i r c u m f e r e n t i a l stresses i s a constant. 27. 2.1.1 Advantages and L i m i t a t i o n s of G r i f f i t h s and Heywood's Model The model proposed by G r i f f i t h s and Heywood was important f o r two reasons. F i r s t , i t was the f i r s t a p p l i c a t i o n of theory to the problem under d i s c u s s i o n , and as such, i t provides a s t a r t i n g point f o r others who wish to extend the a n a l y s i s . Second, i t probably does give a reasonable p i c t u r e of the str e s s f i e l d at the mid-point of a r e l a t i v e l y wide c u f f and can thereby serve as an i n t u i t i v e check f o r the pre d i c -tions of l a t e r models. Their model does have some severe l i m i t a t i o n s and d e f i c i e n c i e s , however. The r e s t r i c t i o n of the model to one dimension means that i t cannot explain the concentration of damage at the edge of the c u f f ; t h i s r e s t r i c t i o n also implies that the only loading parameter which can be incorporated i n t o the model i s the applied pressure, which means that the only possible design conclusion, "lower pressure i s better", i s almost t a u t o l o g i c a l . The model e x h i b i t s no time dependent phenomena, so i t cannot explain the observation that a greater number of nodes are damaged as the duration of a p p l i c a t i o n increases. F i n a l l y , because the observed damage i s obviously dependent on a x i a l p o s i t i o n as well as r a d i a l p o s i t i o n , the r e s t r i c t i o n to one dimension means that the model cannot be used to explain how the nerve damage occurs; nor, i n the absence of a connection between a one dimensional s t r e s s f i e l d and the mechanism of damage, i s i t able to explain the sparing of the smaller f i b e r s or the c h a r a c t e r i s t i c invagination. Because of these severe l i m i t a t i o n s , then, any inferences made from the model's predictions and applied to the r e a l s i t u a t i o n w i l l be tenuous at best. 28. 2.2 Two-DimensIonal Axisymmetric I s o t r o p i c Model In 1984, Auerbach developed a f i n i t e element an a l y s i s of a limb under a tourniquet (Auerbach, 1984). His approach was q u i t e d i f f e r e n t from that of G r i f f i t h s and Heywood. Auerbach recognized that the weight of evidence supported the hypothesis that nerve damage was caused mechanically, so he set out to develop an a n a l y t i c a l t o o l to be used i n the improvement of c u f f designs. He treated the limb as a c o n i c a l mass of e l a s t i c m a terial surrounding a c y l i n d r i c a l , r i g i d core. By v i r t u e of t h i s axisymmetry, h i s model was two-dimensional. The e l a s t i c m a terial was considered to be l i n e a r l y e l a s t i c , homogeneous, and i s o t r o p i c . His model a l s o i m p l i c i t l y assumed that the s t r a i n s would be small even i f the deformations turned out to be l a r g e 1 . He took the e l a s t i c constants (Young's modulus and Poisson's r a t i o ) from a paper by Chow and O d e l l (1978) on stresses developed i n so f t t i s s u e s . Auerbach's strategy i n developing t h i s model was to make the simplest possible assumptions whie s t i l l r e t a i n i n g the e s s e n t i a l features found experimentally. He would then t e s t the model against some empirical r e s u l t s to see how c l o s e l y the predictions of his model c o r r e l a t e d with those r e s u l t s . If the c o r r e l a t i o n were good, he would consider h i s model to be u s e f u l i n the redesign of the tourniquet. 1 It seems that i n the p a r t i c u l a r s i t u a t i o n which i s under considera-t i o n , i t i s impossible to have large deformations with small s t r a i n s . My i n t e r p r e t a t i o n of Auerbach's comment i s that h i s model trea t s the E u l e r i a n and Lagrangian coordinate systems as being coincident, but does not l e t that deter him from using h i s model for s i t u a t i o n s i n which t h i s assumption becomes l e s s v a l i d ( i . e . , large deformations). 29. 2.2.1 V e r i f i c a t i o n of Model Against Experimental Work. The model he developed was f i r s t v e r i f i e d against the a n a l y t i c a l s o l u t i o n of a thick, c y l i n d e r problem; the model's p r e d i c t i o n s exactly matched those of theory. Auerbach then tested h i s model against an experimental i n v e s t i g a t i o n by Thomson and Doupe (1949) i n which the l a t t e r researchers measured the h y d r o s t a t i c pressure 3 at s e v e r a l points under c u f f s of d i f f e r e n t widths. The term, h y d r o s t a t i c pressure, r e f e r s to the mean value of the three normal stresses on an elemental volume. In s o l i d mechanics, t h i s i s u s u a l l y r e f e r r e d to as the mean normal s t r e s s ; i n the l i t e r a t u r e on tourniquets and t h e i r induced s t r e s s e s , however, t h i s value i s almost i n v a r i a b l y c a l l e d the h y d r o s t a t i c pressure so I w i l l use t h i s term. Their experiment w i l l be described i n greater d e t a i l l a t e r , as i t has a l s o been used here as a t e s t case, but the s a l i e n t features of the pressure f i e l d they measured were a zone of high pressure under the middle of the c u f f , a zone of low pressure outside the c u f f , and a zone of intermediate pressure just i n s i d e the cuff edge which seemed to widen with i n c r e a s i n g depth (see Figure 9). Auerbach's Thomson and Doupe claimed to have measured the h y d r o s t a t i c presusre, but they apparently used "a hollow needle which communicated with a pressure-sensing device", according to Auerbach. Thomson and Doupe merely r e f e r to using a m o d i f i c a t i o n of the apparatus of Wells e_t a l . (1938). Without knowing the design of the pressure probe, i t i s impossible to a s s e r t with confidence what pressure was a c t u a l l y being measured, e s p e c i a l l y i f the t i s s u e was behaving l i k e an e l a s t i c s o l i d rather than a f l u i d . If the t i s s u e behaved as an e l a s t i c s o l i d , then the stresses varied with d i r e c t i o n and, i f the probe was also d i r e c -t i o n a l , then there may have been l i t t l e correspondence between the measured and the true h y d r o s t a t i c pressures. It i s quite p o s s i b l e , i n f a c t , that they were measuring stresses i n a plane normal to the bone, that i s , e i t h e r a or a Q . The mean of these two stresses w i l l be r o c a l l e d the transverse pressure. I I 1 ! 1 t I 6 - 5 4 - 3 Z | O Distance from cu f f edge (cm) Figure 9. Isobaric plots of hydrostatic pressure under a tourniquet as determined expderimentally by Thomson & Doupe (top) and numerically by Auerbach (bottom). Note: Values are % of c u f f pressure. 31. model a l s o showed t h i s general tendency, although i n the high pressure zone, where Thomson and Doupe's observations showed l i t t l e change with depth, Auerbach's model predicted a 10% to 20% increase i n pressure with depth (see Figure 9). Auerbach's model, furthermore, predicted that the distance over which the pressure would drop from 100% to 0% of the applied value would be much shorter than the distance observed by Thomson and Doupe. Nonetheless, Auerbach f e l t that the s i m i l a r i t y was s u f f i c i e n t to allow him to simulate the a p p l i c a t i o n of a tourniquet to a thigh and derive some q u a l i t a t i v e l y v a l i d r e s u l t s . 2.2.2 A p p l i c a t i o n of Model to Surface Octahedral Shear Stress  C a l c u l a t i o n s Auerbach's simulations concentrated on determining the e f f e c t that the rounding at the edge of the i n f l a t a b l e part of the tourniquet had on the octahedral shear s t r e s s (the maximum shear s t r e s s i n three dimensions). Perhaps not s u r p r i s i n g l y , the greater the radius of round-ing, the l e s s the shear s t r e s s i n the t i s s u e immediately under the edge of the tourniquet. What may be s u r p r i s i n g , though, i s that t h i s e f f e c t i s not a l l that s i g n i f i c a n t . The octahedral shear s t r e s s i n the mid-section of the tourniquet was r e l a t i v e l y constant for a l l r a d i i considered; i n a d d i t i o n , the s t r e s s l e v e l dropped sharply i n a l l cases a f t e r the edge of the cuff was reached and approached zero within 1 or 2 cm beyond the edge (see Figure 10). At the edge, where one would expect the greatest discrepancies because of the d i f f e r e n t r a d i i of curvature, the s t r e s s l e v e l s only r i s e between 10% and 15% from the l e v e l under the tourniquet's mid-section. Only when a zero radius of curvature i s assumed f o r the tourniquet does the s t r e s s l e v e l r i s e by 60%. 3 2 . Aucherbach i n t e r p r e t s these r e s u l t s as being a suggestion that i t i s important to ensure that the tourniquet edges are rounded. In saying t h i s , he i s i m p l i c i t l y connecting the octahedral shear s t r e s s with the damage found by Ochoa est a l . Further on i n h i s di s c u s s i o n , he e x p l i c i t l y states that h i s r e s u l t s are consistent with Ochoa's e x p e r i -ment because the octahedral shear stress i s greatest at the cuff edges, l e s s i n the center, and zero beyond the edges; he claims that t h i s Is the same as the pattern of l e s i o n s that Ochoa found: the most damage under the c u f f edge, l e s s i n the mid-section, and minimal damage elsewhere. There i s one major d i f f e r e n c e betwen these two patterns, however. Ochoa found almost no damage i n the mid-section of the tourniquet; the damage was almost completely r e s t r i c t e d to a narrow region centered around the edge of the c u f f . Since Auerbach's f i g u r e s show only a 15% r i s e i n the octahedral shear stress l e v e l s at the ege of the c u f f , and that only at the outer radius of the limb, i t i s d i f f i c u l t to understand how t h i s small d i f f e r e n c e could give r i s e to such a well-defined region of damage (see Figure 11) In add i t i o n , Auerbach provides no reasons other than t h i s observed pattern for accepting his hypothesis that the d i s t r i b u t i o n of the octahedral shear stresses i s causally r e l a t e d to the l e s i o n s . In f a c t , since the changes i n the octahedral shear s t r e s s l e v e l s at the surface seem to be so gr e a t l y influenced by the rounding of the edge of the tourniquet, t h i s would be treated as evidence of a form of s t r e s s concentration due to the rapid change of loading i n the a x i a l d i r e c t i o n , and one would expect that i t s e f f e c t would die out deeper i n t o the t i s s u e ; any high l e v e l s of shear s t r e s s would be r e s t r i c t e d to the surface. Because of t h i s , the mechanism of damage 3 3 . r = O mrn Distance from cuff center (cm) Figure 10. Comparison of subcutaneous octahedral shear stresses f o r tourniquets with d i f f e r e n t edge rounding r a d i i Figure 11. Superposition of l e s i o n l o c a t i o n histogram and octahedral shear s t r e s s d i s t r i b u t i o n 3 4 . assumed by Auerbach cannot e x p l a i n the r e l a t i v e s u s c e p t i b i l i t y of the deeply embedded r a d i a l nerve which was such a concern of G r i f f i t h s and Heywood. 2.2.3 Advantages and L i m i t a t i o n s of Auerbach's Model Auerbach's model, though, does represent a s i g n i f i c a n t advance over G r i f f i t h s and Heywood's model. It accounts for the (at l e a s t ) two-dimensional nature of the problem, and a l s o provides f o r two-dimensional modelling of the applied s t r e s s due to the tourniquet; t h i s makes i t a useful t o o l f o r c u f f redesign. His model i s not i n t r i n s i c a l l y unable to account for such things as the sparing of the smaller f i b e r s or the cause of the c h a r a c t e r i s t i c l e s i o n s ; i n h i s paper, however, Auerbach does not look c l o s e l y at these questions, nor does he make e x p l i c i t the way i n which a higher octahedral shear s t r e s s w i l l cause the ki n d of damage observed by Ochoa et a l . At l e a s t two problems with Auerbach's model remain. F i r s t , the assumption of isotropy i s open to question, as Auerbach himself admits: "The muscle l a y e r i s c e r t a i n l y not i s o t r o p i c " . Second, because the model has no time dependent parameters, i t i s incapable of addressing the question of why a greater number of nodes are a f f e c t e d with i n c r e a s -ing duration of a p p l i c a t i o n of the tourniquet. The model proposed i n t h i s t h e s i s w i l l include a n i s o t r o p i c e l a s t i c properties and, though i t w i l l not e x p l i c i t l y contain time-dependent terms, i t w i l l f a c i l i t a t e s p e c u l a t i o n concerning the probable e f f e c t s of longer a p p l i c a t i o n s and w i l l provide d i r e c t i o n for future work. Perhaps most importantly, by attempting to i d e n t i f y and quantify the stresses required to cause the kind of nerve damage found by Ochoa et a l . , the connection between the tourniquet design/load parameters and the induced damage w i l l become e a s i l y understood, and a new, less dangerous tourniquet w i l l be able be designed. 36. CHAPTER 3  MODELLING Ochoa et a l . ' s work produced very strong evidence that tourniquet-induced nerve damage was caused mechanically. In response to that work, G r i f f i t h s & Heywood and Auerbach proposed mechanical models which they hoped would be of use In r e s o l v i n g questions which up u n t i l that point had only been addressed emprically; these questions were p r i m a r i l y o p e r a t i o n a l l y p r a c t i c a l ones ("What pressure should the tourniquet be kept at?", "How long should i t be kept on?"). Once the p o s s i b i l i t y of redesigning the c u f f i s r a i s e d , however, the questions which the model should address become more focussed on the mechanism of damage and i t s r e l a t i o n to the design of the c u f f ("What mechanisms might be p l a u s i b l y considered to be responsible for the damage observed by Ochoa et a l . ? " , "Can we develop a model which p r e d i c t s damage by the proposed mechanism i n the regions where damage i s observed?"). It has been shown that the two models which have been proposed were d e f i c i e n t or l i m i t e d In c e r t a i n ways. G r i f f i t h s and Heywood's model i s a c t u a l l y unable to answer any of the above questions. Auerbach's model, though i t i s not i n t r i n s i c a l l y unable to answer c e r t a i n operational and design questions, was not used to do so. Where he made some design recommendation on the basis of h i s model, he did not explain the causal connection between the model's p r e d i c t i o n s and the p h y s i o l o g i c a l damage. Before we attempt to improve on these models, then, we need to consider what damage mechanisms are most l i k e l y , what the most important charac-t e r i s t i c s of the s i t u a t i o n we want to model are, what s i m p l i f i c a t i o n s can be j u s t i f i e d , and whether a model with such s i m p l i f i c a t i o n s w i l l be t r a c t a b l e . The o v e r a l l goal of t h i s work i s to reduce the incidence of tourniquet-induced nerve damage. We suspect that the operative mechanism i s i n v a g i n a t i o n of the nodes of Ranvier caused by a x i a l l y compressive s t r e s s under the edges of the tourniquet; t h i s s u s p i c i o n i s supported by an order-of-magnitude a n a l y s i s which shows that t h i s e f f e c t i s roughly two orders of magnitude greater than e i t h e r of the other proposed mechanisms. We want, therefore, to model the s t r e s s and s t r a i n d i s t r i b u t i o n under the tourniquet i n order to v e r i f y or disprove t h i s hypothesis, or to suggest an a l t e r n a t i v e hypothesis. 3.1 D i f f i c u l t i e s i n Modelling The s i t u a t i o n which we want to model i s quite complicated. In the f i r s t place, a human limb i s composed of many d i f f e r e n t kinds of t i s s u e i n c l u d i n g bone, muscle, f a t , skin, connective t i s s u e s , nerves, blood vessels, etc. There are a l s o many f l u i d s such as blood and i n t e r s t i t i a l f l u i d s which are capable of being expulsed from the tiss u e under compression. In add i t i o n , some ti s s u e s (predominantly muscle) are subject to nervous c o n t r o l , which makes t h e i r properties v a r i a b l e . Tissues are, i n general, f a r from being the l i n e a r l y e l a s t i c bodies we would want them to be. They ex h i b i t a n i s o t r o p i c , nonlinear, v i s c o e l a s t i c / p l a s t i c behaviour and are capable of undergoing f i n i t e deformations. Not only are t h e i r properties complicated, but they tend to be grouped together i n non-symmetrical ways. Limbs are generally c o n i c a l rather than prismatic, and have d i s t i n c t masses of t i s s u e , such as the biceps and t r i c e p s , so that they are d e f i n i t e l y not axisymmetric (see Figure 11). It i s obviously impossible at the present time to take account of a l l the d i f f e r e n t c h a r a c t e r i s t i c s i n t h i s s i t u a t i o n . We must make some 38. s i m p l i f i c a t i o n s i n order to make the problem t r a c t a b l e , yet we must also take care not to s i m p l i f y i t so much that the a p p l i c a b i l i t y of the model cannot be j u s t i f i e d , as was the case with G r i f f i t h s and Heywood's model. 3.2 S i m p l i f y i n g Assumptions The proposed model involves the f o l l o w i n g s i m p l i f i c a t i o n s : 1) Treat the limb as a r i g i d bone surrounded by a s i n g l e type of t i s s u e ; 2) Ignore the nervous c o n t r o l of the t i s s u e s ; 3) Assume the limb i s axisymmetric; 4) Assume the deformations which the limb undergoes are i n f i n i t e s i m a l ; 5) Ignore the expulsable f l u i d s . Consider the t i s s u e to be a non-porous/solid; 6) Assume that the t i s s u e i s l i n e a r l y e l a s t i c and that i t e x h i b i t s no v i s c o e l a s t i c i t y ; 7) L i m i t the form of anisotropy i n the t i s s u e ' s e l a s t i c p r o p e r t i e s . 3.2.1 J u s t i f i c a t i o n of Assumptions The assumption that the bone can be treated as r i g i d i s reasonable because the expected stresses are on the order of 50 kPa (atmospheric pressure = 100 kPa) . These str e s s e s are low compared with the e l a s t i c modulus of bone (17,200 MPa for the femur according to Yamada (1970)). The assumption that a v a r i e t y of t i s s u e s can be treated as a s i n g l e type i s most reasonable when a large percentage of the limb mass i s composed of a s i n g l e type of t i s s u e . Since such t i s s u e s as skin, nerves and vascular ti s s u e represent a small f r a c t i o n of the limb mass, i t would seem reasonable to disregard t h e i r c ontributions to the gross mechanical properties of the limb, at l e a s t as a f i r s t approximation. The most abundant t i s s u e types are muscle and f a t , which tend to l i e i n two d i s t i n c t l a y e r s . A person who i s not s i g n i f i c a n t l y overweight w i l l have a preponderance of muscle t i s s u e over f a t on t h e i r limbs, so, f o r t h i s kind of person, the assumption i s probably reasonable. The e f f e c t s of nervous c o n t r o l of the t i s s u e s are ignored because the complex feedback mechanisms which would a f f e c t the t i s s u e properties are not w e l l understood and because, even i f they were w e l l understood, they would be very d i f f i c u l t to quantify and model. If these e f f e c t s were to be taken i n t o account, however, a reasonable f i r s t approximation to the e f f e c t s of muscle innervation would be to consider the stresses induced i n a body i n a s t a t e of i n i t i a l s t r e s s because the body's muscles are r a r e l y completely relaxed; they are almost always i n a state of tension. The limb has been assumed to be axisymmetric p r i m a r i l y f o r reasons of t r a c t a b i l i t y ; i t i s the most reasonable assumption which w i l l reduce the number of space v a r i a b l e s from three to two. Auerbach points out that Ochoa's work provides some support for the conjecture that t h i s assumption i s not s e r i o u s l y l i m i t i n g . Auerbach suggests that i f deformations i n the c i r c u m f e r e n t i a l d i r e c t i o n were responsible for the observed nerve damage, such damage would have been observed at a l l points under the cuff and not s o l e l y at the edges. 1 This argument i s not convincing because such a deformation may be a necessary but not a s u f f i c i e n t f a c t o r i n the causation of damage. In lay i n g aside t h i s f a c t o r , we should regard i t as possibly relevant. 40. The assumptions concerning small deformations and non-porosity can be discussed together. While i t i s perhaps true that a s i g n i f i c a n t amount of f l u i d w i l l be expulsed when a tourniquet i s a p p l i e d to a limb, such f l u i d w i l l be expulsed almost immediately. Since the s e v e r i t y of nerve damage increases with increased tourniquet a p p l i c a t i o n time, we might suppose that the damage-causing mechanism becomes operative once the f l u i d i s expulsed. Once t h i s expulsion occurs, i t becomes more reasonable to assume that the limb t i s s u e behaves as a s o l i d m a t e r i a l . We should note that t h i s f l u i d expulsion e f f e c t can be q u i t e minimal; Thomson and Doupe report that the arm g i r t h of a man with a f i r m arm was reduced by only 6% (reducing the c r o s s - s e c t i o n a l area by roughly 15%) when a tourniquet was applied (Thomson and Doupe 1949). Since t h i s reduction i n g i r t h i s due both to the expulsion of f l u i d and compression of the t i s s u e s , the maximum e f f e c t of e i t h e r of these two causes i s the 6% reduction. A reduction of t h i s order implies that the small deforma-t i o n assumption i s not l i k e l y to lead to s i g n i f i c a n t errors at c l i n i c a l pressures. There i s some evidence to support our assumption of l i n e a r beha-viour of the t i s s u e . Ziegert and Lewis measured the loading properties of the s o f t t i s s u e surrounding the t i b i a and found that a five-minute preloading r e s u l t e d i n " e s s e n t i a l l y l i n e a r " load-displacement r e l a t i o n s (Ziegert and Lewis, 1978). Even i f the t i s s u e i n the s i t u a t i o n we are concerned about does not e x h i b i t p e r f e c t l y l i n e a r e l a s t i c p r o p e r t i e s , the model we create may give us u s e f u l q u a l i t a t i v e information. The assumption that the t i s s u e exhibits no v i s c o e l a s t i c behaviour can be relaxed with l i t t l e decrease i n t r a c t a b i l i t y I f the v i s c o e l a s t i c i t y i s assumed to be l i n e a r . The s t a t i c case was proposed because i t would lead to some i n s i g h t into the stress state i n the t i s s u e soon a f t e r the tourniquet was a p p l i e d . If the v i s c o e l a s t i c r e l a -t ionships can be considered l i n e a r , then the v i s c o e l a s t i c equations can be transformed through Laplace transforms i n t o equations which are i d e n t i c a l i n form to those a r i s i n g from the s t a t i c case. The s o l u t i o n to the l i n e a r v i s c o e l a s t i c case, then, simply requires some transform steps beyond the steps required to solve the l i n e a r s t a t i c case. F i n a l l y , the s o r t of l i m i t a t i o n s which are reasonable to impose on the form of anisotropy permitted i n the model are considered. Since the model being considered i s two-dimensional, i t would probably not be u s e f u l to consider more than two p r i n c i p a l d i r e c t i o n s f or the e l a s t i c p r o p e r t i e s . Given the d i f f i c u l t y i n experimentally determining two-dimensional e l a s t i c properties i n t i s s u e , t h i s i s not a serious l i m i t a -t i o n ; experimentally determined three-dimensional t i s s u e p r o p e r t i e s are simply not a v a i l a b l e . Since muscle tis s u e i n the limbs can be v i s u a l i z e d as a bundle of p a r a l l e l f i b r e s , the t i s s u e can be considered to be s i m i l a r to e l a s t i c m a t erial which i s r e i n f o r c e d with impregnated f i b r e s running p a r a l l e l to one another. The d i r e c t i o n i n which the f i b r e s run can be considered the p r i n c i p a l d i r e c t i o n and corresponds to the l o n g i t u d i n a l axis of the limb, while a l l d i r e c t i o n s i n the transverse plane can be considered equivalent and correspond to the r a d i a l and c i r c u m f e r e n t i a l d i r e c t i o n s i n the limb. The requirements for a model which i s a reasonable representation of the r e a l system and which w i l l go beyond work which has already been done have been discussed. The mathematics underlying the proposed model 42. are now o u t l i n e d and the s o l u t i o n method of the r e s u l t i n g equations i s described. 3.3 Mathematical Development of Model The f o l l o w i n g development assumes some f a m i l i a r i t y with s o l i d mechanics p r i n c i p l e s . If the reader finds some of the steps somewhat d i s j o i n t e d , he or she may consult Appendix C which contains a step-by-step development of the concepts and reasoning summarized i n t h i s s e c t i o n . In t h i s s e c t i o n , the c h a r a c t e r i s t i c s of the proposed model are b r i e f l y reviewed, the q u a n t i t i e s to be investigated are mentioned, seve r a l b a s i c r e l a t i o n s h i p s between these q u a n t i t i e s are presented, a r e l a t i v e l y large number of simultaneous equations are condensed i n t o a s i n g l e equation, and the method of s o l u t i o n i s described. The limb-tourniquet system w i l l be treated as being comprised of a r i g i d c y l i n d r i c a l core surrounded by a l i n e a r l y e l a s t i c , a n i s o t r o p i c material subjected to axisymmetric r a d i a l and shear forces. The anisotropy w i l l be l i m i t e d i n form such that the e l a s t i c p r o p e r t i e s p a r a l l e l to the axis of the limb can d i f f e r from those i n any d i r e c t i o n perpendicular to that a x i s . The coordinate system used w i l l be standard c y l i n d r i c a l coordinates, with the midpoint of the pressure cuff c o i n c i -dent with z=0 (see Figure 12). Any r e s u l t i n g deformations w i l l be considered to be i n f i n i t e s i m a l . The q u a n t i t i e s of i n t e r e s t are the s t r e s s e s , s t r a i n s and d i s p l a c e -ments i n the limb. For the most part, i t w i l l be important to know what these q u a n t i t i e s are when align e d with the coordinate axes, although f o r Figure 12. Geometric parameters i n the tourniquet/limb system 44. c e r t a i n q u a n t i t i e s , such as the h y d r o s t a t i c pressure or the maximum shear st r e s s at a point, t h i s alignment w i l l not be necessary. The model which w i l l be developed w i l l be used to i n v e s t i g a t e the dependence of these s t r e s s , s t r a i n and displacement f i e l d s on the e l a s t i c p r o p e r t i e s of the t i s s u e and on the geometric parameters desc r i b i n g the loading of the limb. The e l a s t i c properties are described i n more d e t a i l l a t e r , but the geometric parameters include three measures of the limb and three measures of the loading imposed by the tourniquet. The limb measures are the diameter of the bone ( a ) , the thickness of the limb (b), and the length of the limb (I) (see Figure 12). The loading imposed by the tourniquet depends on the width of the pressure cuff ( c ) , the maximum pressures applied by the cuff (P ), and the max shape of the pressure d i s t r i b u t i o n applied to the limb ( T ) . 3.3.1 Moment E q u i l i b r i u m A standard notation for the stresses i s adopted (see Figure 13). Although nine components can be i d e n t i f i e d , t h i s number can be reduced to s i x by applying the requirement that the net moment about any of the coordinate axes must be zero f o r the s t a t i c s i t u a t i o n ( i n f a c t , t h i s conclusion i s more general, but i t s present form i s s u f f i c i e n t f o r t h i s development). These moment equi l i b r i u m equations show that the comple-mentary stresses must be equal: i . e . T = T ( s i m i l a r l y for the other rz zr two complementary shear stress p a i r s ) . 3.3.2 Force E q u i l i b r i u m Force balances i n each of the coordinate d i r e c t i o n s lead to two d i f f e r e n t i a l equations of equilibrium which msut be s a t i s f i e d by any set Figure 13. Stresses on elemental volume 46. of s t r e s s f i e l d s which are s o l u t i o n s to an e l a s t o s t a t i c problem. Because of the assumption of axisymmetry, a l l d e r i v a t i v e s with respect to 6 are zero; the f o r c e balance i n the c i r c u m f e r e n t i a l d i r e c t i o n , therefore, leads simply to an i d e n t i t y . The r a d i a l and a x i a l force balances, on the other hand, lead to two d i f f e r e n t i a l equations: Radial e q u i l i b r i u m : 3a a - a . 3T _JL + J L _ ! + _ E L = o , 3r r 3z * A x i a l e q u i l i b r i u m : 3o T 3T + _ « + = o . 3z r 3r 3.3.3 S t r a i n ~ Displacement Relations When the limb i s deformed, each point i n the t i s s u e takes up a new l o c a t i o n which i s given by the coordinates of the o r i g i n a l l o c a t i o n plus the displacement i n each of the relevant coordinate d i r e c t i o n s (the assumption of axisymmetry implies that the coordinate 9 remains unchanged upon deformation). The displacement i n the r a d i a l d i r e c t i o n i s given by u, i n the c i r c u m f e r e n t i a l d i r e c t i o n by v, and i n the a x i a l d i r e c t i o n by w. The new l o c a t i o n of a point can therefore be given by the mapping ( r ' , 9 ' , z') (r+u, 9+v, z+w) . If the s t r a i n s are defined as the elongation of an element due to deformation d i v i d e d by i t s undeformed length, or, i n the case of shear deformation, as the average angular change between two adjacent elements o r i g i n a l l y at r i g h t angles to one another, c e r t a i n r e l a t i o n s h i p s between the s t r a i n s and the displacements can be determined. These r e l a t i o n s h i p s can be summarized as = 3u 1 3v u e r 3r ' e8 r 36 r ' 3w 1_ , 3w jhju £ z 3z ; e r z 2 S r 9z ;* The generation of a s t r a i n i n the c i r c u m f e r e n t i a l d i r e c t i o n as a r e s u l t of r a d i a l displacement occurs because the c i r c u l a r element which was o r i g i n a l l y at radius r becomes la r g e r when displaced to radius r + u. Note that even though the two s t r a i n s e . and e . can be defined i n r8 zo terms of the two displacement components, they w i l l be zero because v = 0 and a l l d e r i v a t i v e s with respect to 0 are zero when axisymmetry i s assumed. Note too that because of the symmetry i n the i n d i c e s i n the d e f i n i t i o n of e , e = e rz zr rz 3.3.4 E l a s t i c Relationships 3.3.4.1 I s o t r o p i c E l a s t i c Relationships Three kinds of q u a n t i t i e s have been defined so f a r i n t h i s develop-ment: st r e s s e s , s t r a i n s , and displacements. Several r e l a t i o n s h i p s have been presented which connect the s t r a i n s and displacements; the e l a s t i c r e l a t i o n s h i p s , which connect the stresses and s t r a i n s , w i l l now be presented. The simplest p o s s i b l e r e l a t i o n s h i p between the streses and the st r a i n s they induce i s a l i n e a r one; i . e . , the stress i s r e l a t e d to the 4 8 . s t r a i n through an equation of the form: a = Ee. In t h i s equation, the parameter E i s known as Young's modulus and i s a measure of the s t i f f n e s s of the m a t e r i a l . If a m a t e r i a l i s i s o t r o p i c , then E w i l l not change i f the s t r e s s i s applied i n d i f f e r e n t d i r e c t i o n s ; i f E does change with the d i r e c t i o n of a p p l i c a t i o n of the s t r e s s , the m a t e r i a l i s c a l l e d a n i s o t r o p i c . I s o t r o p i c materials require one more parameter ( c a l l e d Poisson's r a t i o ) to f u l l y describe t h e i r e l a s t i c behavior. Consider a uniform rod of i s o t r o p i c m a t e r i a l undergoing uniform tension. The rod's diameter w i l l decrease by a c e r t a i n amount. Poisson's r a t i o , symbolized by v, i s defined as the negative of the r a t i o of the transverse s t r a i n to the a x i a l s t r a i n . This r a t i o can range from zero, which i n d i c a t e s an i n f i n i t e l y compressible m a t e r i a l , to 0.5, which i n d i c a t e s an incompressible m a t e r i a l . The s t r a i n s due to stresses i n the d i f f e r e n t p r i n c i p a l d i r e c t i o n s can be superimposed because the s t r e s s - s t r a i n r e l a t i o n s h i p s are l i n e a r . For an a r b i t r a r y state of s t r e s s , then, the s t r a i n s are given by e z r 9 x rz e 2G rz Note that the i s o t r o p i c shear modulus, G, i s defined as G = E 2(l+v)' 49. 3.3.4.2 An i s o t r o p i c E l a s t i c Relationships If the extension r e s u l t i n g from the a p p l i c a t i o n of a f i x e d s t r e s s changes with d i r e c t i o n , then the m a t e r i a l i s c a l l e d a n i s o t r o p i c . Unlike i s o t r o p i c materials, an a n i s o t r o p i c material requires more than two e l a s t i c parameters to s p e c i f y the s t r a i n s f o r an a r b i t r a r y s t r e s s s t a t e ; the number of such parameters depends on the form of the anisotropy. In the completely general case, where each of the s i x s t r a i n components depends on a l i n e a r combination of the product of each of the s i x s t r e s s components with a d i f f e r e n t e l a s t i c parameters, 36 d i f f e r e n t parameters can be required. By imposing conditions of symmetry on these equations, the number of parameters can be reduced. With the form of anisotropy permitted i n t h i s model, only f i v e d i f f e r e n t e l a s t i c constants are required (the process of reduction i s developed i n Appendix C). 3.3.4.2.1 D e f i n i t i o n of E l a s t i c Constants These f i v e constants w i l l be defined i n p h y s i c a l meaningful ways. If a sample of the t i s s u e i s subjected to tension aligned p a r a l l e l to the axis of the limb, then E i s defined as the r a t i o of the a p p l i e d stress to the a x i a l s t r a i n Induced. A quantity analogous to Poisson's r a t i o can.be defined f o r t h i s s i t u a t i o n ; v i s here defined as the r a t i o of the contraction i n the transverse plane to the extension i n the a x i a l d i r e c t i o n . If the t i s s u e i s then subjected to tension i n any transverse d i r e c t i o n , then E 2 i s defined as the r a t i o of the applied s t r e s s to the s t r a i n developed i n the d i r e c t i o n of the l i n e of a c t i o n of the s t r e s s . The parameter, E i s defined as the r a t i o of E„ to E. In t h i s same r 2 t e s t , v i s defined as the r a t i o of contraction i n the transverse plane, 50. perpendicular to the l i n e of a c t i o n of the a p p l i e d s t r e s s , and i s given by = v 2/v. The f i f t h constant comes from a consideration of the e f f e c t of shear i n any plane containing a l i n e p a r a l l e l to the limb a x i s . A shearing s t r e s s , x, w i l l induce a s t r a i n , e, which are r e l a t e d by x = 2G,.e , rz 1* rz where the subscript 1* i n d i c a t e s any plane containing the limb a x i s . I the i s o t r o p i c case, the shear modulus, G, was r e l a t e d to Young's modulu and Poisson's r a t i o through the following: G - E/2(l+v). An analogous nondimensionalized parameter can be defined as G = G,*/G. r 1* The f i v e constants which w i l l be used are the following: E, E^, v, v and G . With these d e f i n i t i o n s established, the r e l a t i o n s h i p s r r r between the s t r e s s e s and s t r a i n s can be put i n t o matrix form i n the following manner (disregarding those shear stress and shear s t r a i n components which have 9 as a su b s c r i p t because they w i l l be zero f o r reasons discussed e a r l i e r ) , rz 1 E 1 -v -v -v 1/E r -vv /E r -u -vv /E r r 1/E r 0 0 0 0 0 0 (l+v)/G x rz 3.3.4.2.2 Constraints on Choice of E l a s t i c Constants Although these f i v e c o e f f i c i e n t s are a r b i t r a r y , c e r t a i n p h y s i c a l c o n s t r a i n t s l i m i t the choice of p h y s i c a l l y meaningful sets of c o e f f i -c i e n t s . The primary c o n s t r a i n t i s that h y d r o s t a t i c compression should r e s u l t i n volumetric c o n t r a c t i o n rather than expansion. The h y d r o s t a t i c pressure 1 at a point i s given by the average of the three d i r e c t s t r e s s components at that point, The r a t i o of the change i n volume of an element upon deformation to the volume of the undeformed element i s given by ( i g n o r i n g second order terms) This r a t i o can be expressed i n terms of stresses by making use of the s t r e s s / s t r a i n r e l a t i o n s . If t h i s i s done, the r a t i o i s given by Suppose that only the a x i a l stress i s non-zero. In t h i s case, the volume change r a t i o i s given by AV V = e + e „ + e r 9 z [ l - v ( E +v ]I . r r AV V l-2v E a z or the mean normal s t r e s s 52 . If t h i s r a t i o i s to be greater than zero, v must be l e s s than 0.5. If v = 0, then the material i s i n f i n i t e l y extensible; i f v = 0.5, then the mat e r i a l i s incompressible. Th i s method can be ap p l i e d to non-zero r a d i a l and c i r c u m f e r e n t i a l stresses as w e l l . The r e s u l t i n g r e s t r i c t i o n i s E +v r r The o v e r a l l set of r e s t r i c t i o n s can be summarized by the following r e l a t i o n s , 1 o < v t < ^ _ . ' E +v ' r r 3.3.5 C o m p a t i b i l i t y Equations The four s t r a i n components have been defined i n terms of the two displacement components. Because of t h i s , the four s t r a i n components cannot be s p e c i f i e d a r b i t r a r i l y ; they must s a t i s f y c e r t a i n r e l a t i o n s h i p s known as c o m p a t i b i l i t y euqations i n order to lead to a consistent p a i r of displacement f i e l d s . 3.3.5.1 Expressed i n Terms of S t r a i n Components Since there are two more s t r a i n components than displacement components, there w i l l be two c o m p a t i b i l i t y equations. These equations can be determined from the strain-displacement r e l a t i o n s , which are summarized here by The f i r s t c o m p a t i b i l i t y equation comes from e l i m i n a t i n g u from the f i r s t two r e l a t i o n s + e = e Q + r - — . r 9 3r The second equation comes from comparing second order d e r i v a t i v e s of the f i r s t , t h i r d and fourth r e l a t i o n s , 3 2e 3 2e 3 2e L+ £ = 2 -3 z 2 3r2 3 r 3 z 3.3.5.2 Expressed i n Terms of Stress Components The above equations are the c o m p a t i b i l i t y equations expressed i n terms of the s t r a i n components. These r e l a t i o n s can be given i n terms of the stresses i f the e l a s t i c constant matrix i s used to s u b s t i t u t e the stress components for the s t r a i n s . The compatibilty equations i n terms of stresses are therefore given by (adopting the comma notation f o r d i f f e r e n t i a t i o n ) 1+VV „ VV 0" r , , N 3 , r , 9. 1 — ( a r + ° 9 ) = r 3 7 ( " V 0 z - — a r + E~>» r r r a vv (-vo- + ~ =r^  a ) + (a - v(o +a )) z E E y . z z z r o . r r r r G r z . r z r 54. These equations permit the s o l u t i o n of e l a s t o s t a t i c problems s o l e l y i n terms of the s t r e s s e s ; as long as the set of s t r e s s f i e l d s s a t i s f i e s the two c o m p a t i b i l i t y equations, the two equilibrium equations, and the boundary conditions of the problem, the s t r a i n s and displacements can be derived from the s t r e s s e s i n a straightforward manner. 3.3.6 The Use of a Stress Function The development thus f a r has succeeded i n reducing the large number of simultaneous equations to four stress equations. This set of equations can be reduced to a s i n g l e equation by employing a s t r e s s function, a s i n g l e function which y i e l d s a l l the s t r e s s components upon d i f f e r e n t i a t i o n . 3.3.6.1 I s o t r o p i c Stress Function For an i s o t r o p i c material, one p o s s i b l e d e f i n i t i o n of the s t r e s s components i n terms of a s t r e s s function <j> i s given by «*-!>•,„ + 7 * , r + . ( v * , r r + ^ T * , r + v * , « \ « » ((2-vH + 4> r + (l-v)d. ),z , 55. With t h i s set of d e f i n i t i o n s , the f i r s t of the e q u i l i b r i u m equations and both of the c o m p a t i b i l i t y equations are automatically s a t i s f i e d . The other e q u i l i b r i u m equation y i e l d s the biharmonic equation, = V 2V 24> = 0 , where , r r r ,r f 2 > o H > z z (Note: the t h i r d term drops out because of axisymmetry.) Any s t r e s s f u n c t i o n which s a t i s f i e s t h i s f i e l d equation i s a v a l i d s o l u t i o n to the problem because a l l relevant equations are embedded i n i t s s t r u c t u r e . 3.3.6.2 A n i s o t r o p i c Stress Function An appropriate d e f i n i t i o n of the s t r e s s components i n terms of a stress function for an a n i s o t r o p i c material can be determined by making the above d e f i n i t i o n f o r i s o t r o p i c materials more general. If the structure of the d e f i n i t i o n s i s assumed to remain the same, but the c o e f f i c i e n t s are allowed to d i f f e r , the following set of d e f i n i t i o n s r e s u l t s , 56. (J = (g 4 + — <)> +i<)) ) , z , z ° T , r r r ,r T , z z ' ' T =(j<|> + - <fr + Jt $ ) rz J , r r r ,r ,zz ,r If the assumption i s made that the same three equations which the i s o t r o p i c d e f i n i t i o n s s a t i s f y automatically are s a t i s f i e d automatically by these d e f i n i t i o n s , s u f f i c i e n t constraints are imposed on the c o e f f i -c i e n t s a through 1 that they can be defined s t r i c t l y i n terms of the f i v e e l a s t i c parameters. One of these c o e f f i c i e n t s i s a r b i t r a r y because the f i e l d equation i s l i n e a r ; i f a st r e s s function $ s a t i s f i e s the f i e l d equation, so w i l l K<J>, where K i s a constant. By the use of an appro-p r i a t e s c a l i n g f a c t o r , any of the i n f i n i t e number of acceptable f i e l d equations can be made to s a t i s f y the boundary conditions of the problem. 3.3.6.2.1 Determination of A n i s o t r o p i c C o e f f i c i e n t s If the c o e f f i c i e n t s are s p e c i f i e d i n such a way that, when the anisotropy i s reduced to isotropy, the c o e f f i c i e n t s w i l l match those i n the d e f i n i t i o n s f o r the i s o t r o p i c case, then one p o s s i b l e set of c o e f f i -cients expressed i n terms of the f i v e e l a s t i c parameters i s l - v 2 E _ . , r a = e = - i = - l = - rr— , J 1+vv r F 2 E (1+v) b = d = rrr~ [ £ VE - V ] , 1+vv L G r r J r r c = f = -i = v , 57. u 2(l+v) 8 = h = G (1+vv ) " V ' r r 1-vv r The f i e l d equation w i l l be given by 2 d> ij) <j> $ + l r r r , r r + ^ + a ( + _ 1 r z z ) + _ 0 , , r r r r r f 2 r3 ,rrzz r ,zzzz where a - -2 [i±!L- v(l+vv )] , 1-V2 E r r r and 1 - v 2 v 2 S = • E (1 - v 2 E ) r r The stresses have already been defined i n terms of the s t r e s s function; the s t r a i n s are r e a d i l y computed by a simple a p p l i c a t i o n of the e l a s t i c constant matrix. The boundary conditions for e l a s t o s t a t i c problems are almost i n v a r i a b l y given i n terms of stresses or d i s p l a c e -ment, so i t i s u s e f u l to derive the r e l a t i o n s h i p s f or displacements i n terms of the s t r e s s f u n c t i o n . 3.3.6.2.2 Determination of Displacement Expressions The r a d i a l displacement, u, comes d i r e c t l y from the r e l a t i o n : u = re . The a x i a l displacement r e l a t i o n comes from a consideration of the 9 two strain-displacement r e l a t i o n s 58. e z 3oj 3z 2e rz 3r + 3z * 3OJ , 3u Solving these two r e l a t i o n s for UJ gives - / e z dz + R(r) = J ( 2 e r z - |J)dr + Z(z) , where R and Z are functions of r and z r e s p e c t i v e l y . Once u has been expressed i n terms of the s t r e s s f u n c t i o n , i t can be substituted for i n the second i n t e g r a l and each of the i n t e g r a l s can then be evaluated. When t h i s i s done, i t i s apparent that the two i n t e g r a l s are i d e n t i c a l . This implies that R(r) = Z(z), which i s only p o s s i b l e i f R(r) = Z(z) = K, a constant. Since K i s a r b i t r a r y , i t can be taken to be zero. These displacements can therefore be expressed as u rz 01 E , r r r ' E , zz where r r and 2 ( l + v ) ( l - v 2 E r ) G (1+vv ) \ r and 1-vv r 5 9 . 3.3.7 Geometric Parameters The geometric parameters of i n t e r e s t f a l l i n t o two clas s e s : those which describe the limb and those which describe the loading imposed by the tourniquet. The limb, being c y l i n d r i c a l , can be described by four parameters: the length, I, the bone diameter, a, the t i s s u e thickness, t, and the limb diameter, b (see Figure 12). One of these parameters i s redundant, since t = (b-a)/2. The tourniquet loading can al s o be described by three non-redundant parameters i f r a d i a l loading only i s considered. These parameters are the maximum app l i e d pressure, P , the width of the c u f f , c, and the max shape paratmeter, r, which describes the loading d i s t i b u t i o n i n the a x i a l d i r e c t i o n . 3.3.8 Non-Dimensionalization Non-dimensionalization can be used to reduce the number of var i a b l e s and to make the r e l a t i v e importance of various parameters more apparent. The v a r i a b l e s and c o e f f i c i e n t s of i n t e r e s t i n t h i s problem f a l l into three groups: the e l a s t i c parameters, the st r e s s e s , s t r a i n s and displacements, and the geometric parameters which describe the limb and the loading imposed by the tourniquet. Many of the these parameters are already dimensionless; those which are not have u n i t s of e i t h e r length, pressure, or some combination of these two. This suggests that a reference length and reference s t r e s s or pressure can be chosen to reduce a l l v a r i a b l e s and parameters to non-dimensional form. The reference pressure w i l l be taken to be the maximum r a d i a l stress applied by the tourniquet, except i n the case where shear 6 0 . stresses are being considered i n i s o l a t i o n from d i r e c t r a d i a l s t r e s s ; i n the l a t t e r case, the maximum applied shear stress w i l l be taken to be the reference pressure. The reference length w i l l be taken to be the limb length, I. With these d e f i n i t i o n s f o r the reference pressure and length, a l l the relevant parameters can be rendered dimensionless. The e l a s t i c modulus i n the a x i a l d i r e c t i o n , E, together with the s t r e s s components, can be divided by the reference pressure to y i e l d dimensionless para-meters. The displacements and geometrical parameters can be d i v i d e d by the reference length, and the stress function i t s e l f , which has units of Pa»m 3 can be d i v i d e d by P £ 3 . The dimensionless s t r e s s f u n c t i o n can max be redefined as P I max 3 For reasons of s i m p l i c i t y , a l l future references to the v a r i a b l e s and c o e f f i c i e n t s w i l l not be subscripted to i n d i c a t e t h e i r dimensionless nature; t h i s dimensionlessness w i l l be understood. 3.3.9 Method of S o l u t i o n (Body) The a n i s o t r o p i c f i e l d equation which i s to be solved i s given by iii + — \i) + + - ib + a + 3 * = 0 , r r r r r T , r r r r 2 r3 r » r z z ,rrzz ,zzzz 3.3.9.1 Separation of Equation This expression can be obtained by the a p p l i c a t i o n of two d i f f e r e n t i a l operators; 61. where a • y + x » and 3 = YX • 1 This equation i s s i m i l a r to the equation found i n the i s o t r o p i c case. i n the case of i s o t r o p y , y - x = 1 and the f i e l d equation reduces to V 2 V 2 i p = vS|> = 0 where The i s o t r o p i c equation can be p a r t i a l l y solved by f i n d i n g solutions to V 2 ip = 0, because i f V 2 t = 0, then V 2 ( V 2 i p ) = V 2(0) = 0. The same reasoning a p p l i e s to the s o l u t i o n of the a n i s o t r o p i c f i e l d equation. Consider, therefore, the s o l u t i o n to the equation This i s a separable equation, so lp can be taken as i|» = RZ, where R and Z are functions of r and z r e s p e c t i v e l y . S u b s t i t u t i n g and d i v i d i n g through by <|> gives 1 Y and x can be e i t h e r r e a l or complex. I f they are complex, then they must be complex conjugates, since a and $ are r e a l . V 2 f 0 62. R" . R' Z" . or | ~ + ~ - Y|- = K ^ ( a constant) , since the l e f t hand side and r i g h t hand side of the f i r s t r e l a t i o n are functions of d i f f e r e n t v a r i a b l e s . The problem can now be reduced to f i n d i n g the solutions to the following ordinary d i f f e r e n t i a l equations rR" + R' - K rR = 0 , Y K Z _Y_ Y + ^ L _ = 0 3.3.9.2 S o l u t i o n to A x i a l Equation The s o l u t i o n to this second equation i s straightforward. If Z pz , „., o pz e^ , then Z = p*e r . p 2 ePZ + _! = o , pz K e F which i s s a t i s f i e d f o r a l l z only i f p Y Since Z(z) Is defined on -«><z<°°, and since Z(z) should not become unbounded i n e i t h e r the p o s i t i v e or negative far f i e l d , p must be imaginary. This w i l l r e s u l t i n a s i n u s o i d a l z f u n c t i o n which may or may not be symmetrical about z=0. It w i l l be u s e f u l to define a new A p o s i t i v e r e a l v a r i a b l e , X , such that p = ± iX = ± i / — . Since X i s f » Y » Y Y Y 63. r e a l and p o s i t i v e , K /y must be a l s o . With t h i s d e f i n i t i o n f o r X , the Y Y general s o l u t i o n to the second order ODE i n z i s given by Z = A cos X z + B s i n X z , Y Y where A and B are constants to be determined by the boundary conditio n s . 3.3.9.3 S o l u t i o n to R a d i a l Equation The s o l u t i o n to the ODE i n r i s somewhat more complicated because i t has a singular point at r = 0. This implies that at most one s o l u t i o n can be obtained by a s e r i e s s u b s t i t u t i o n . The second s o l u t i o n must be obtained by more so p h i s t i c a t e d means. To obtain the f i r s t s o l u t i o n , consider f i r s t the equation xX" + X' + xX = 0. If the s o l u t i o n X i s defined as a power s e r i e s X = Z a x 1 1 , n=0 then t h i s d e f i n i t i o n can be substituted into the d i f f e r e n t i a l equations and the necessary r e l a t i o n s between the c o e f f i c i e n t s , a^, can be deduced. The s o l u t i o n which i s obtained by t h i s process has a Q a r b i t r a r y , a^ = 0 and a l l other c o e f f i c i e n t s obtained r e c u r s i v e l y by the an-2 r e l a t i o n a = . Since every second term i s non-zero, the power n o s e r i e s can be replaced by one i n m, where m = n/2, and the re c u r s i v e d e f i n i t i o n can be employed to define each c o e f f i c i e n t d i r e c t l y . If t h i s i s done, the s e r i e s f o r X i s given by 64 . 0 0 . ,. m 2m x = j i z I 2 _ * _ m=0 2 Z m (m!) 2 This s e r i e s i s known as Bessel's function of the f i r s t kind of zero order, and i s given the designation J Q ( x ) . This s e r i e s can be demonstrated to be convergent f o r a l l values of x, r e a l or complex, by an a p p l i c a t i o n of the r a t i o t e s t . Consider a - x 02m, i n 2 2(m+1) 0 i , I m+1 . _ 2 (m!)^ x = , x >2 1 2m 1 02(m+l), ..,.,2 2m l2(m+l)J ' m+oo a x 2 (m+1)! x m This r a t i o approaches zero as m approaches i n f i n i t y , regardless of the value of x, r e a l or complex; the s e r i e s J Q ( x ) , therefore, converges f o r a l l x. The second s o l u t i o n to the d i f f e r e n t i a l equation can be determined by assuming that i t has the form Y = V(x) J Q ( x ) . By s u b s t i t u t i n g t h i s f u n c t i o n i n t o the d i f f e r e n t i a l equation, i t cn be shown that xV"J Q + V'(2 x j Q» + J) + V ( x J 0 " + J Q ' + x J Q ) = 0 Since X J Q " + J Q ' X J Q = 0> as has already been shown, i f the s u b s t i t u -t i o n v = V i s made, the above equation reduces to 2J« x J Q v' + (2xJ 0'+J Q)v = 0 ; v' = -(-j°.+ -)v . o This equation i s r e a d i l y solved; 65. J 0 X J 0 J x v(x) = e = e e 1 1 j 2 x 2  J 0 X J 0 The second s o l u t i o n can then be determined from the r e l a t i o n x Y = V ( X ) J q ( X ) = J v dx J Q ( x ) Y(x) = J (x) / X - 4 X -xJ 2 X J 0 If t h i s Integral i s evaluated i n terms of i t s s e r i e s representation, the r e s u l t i n g expression f o r Y(x) i s °» . .m+1 2m m Y(x) = J 0 ( x ) *nx + Z [ ( ~ ^ *2 \ n] ' m=l 2 (m!) n=l The convergence of the se r i e s summation i n the expression for Y(x) can be demonstrated by the r a t i o t e s t , as was the case with J Q ( x ) a x 2 ( m + 1 ) -2m, , , 2 2(m+l) ,, I nrfl 1 _ 2 (m!) x 1 2m 1 0 2(m+l), , . . , 2 2i m-n» a x 2 (m+1)! x m n=l m+1 1 I n n=l m 1 Z n n=l 66. This second s o l u t i o n i s given the designation Y Q(x) and i s known as Bessel's function of the second kind of zero order. The two f u n c t i o n s , J Q ( x ) a n ( * ^ o ^ x ^ ' a r e s ° l u t : i - o n s t 0 t n e equation xX" + X' + xX = 0 ; the equation of i n t e r e s t , however, is rR" + R' - K^rR = 0 where K = A 2 y. I t can be shown that R(r) = J.(i/K~~ r) i s a y y x ' 0 y s o l u t i o n to t h i s equation by s u b s t i t u t i n g t h i s d e f i n i t i o n i n t o the d i f f e r e n t i a l equation. -K r J " ( i / K ~ r ) + i / T " J »(i/K~r) - K r J n ( i / K ~ r ) y 0 y y 0 Y Y " Y = i / K ~ ( i / K ~ r J " ( i / K - r ) + J . ' C i / F ' r ) + i / K ~ r J . ( i / K ~ r ) Y Y ^ Y ^ Y Y " Y = lrtT ( x J 0 " ( x ) + J Q ' ( x ) + x J 0 ( x ) ) = 0 , where x = i/K r . Y This f i n a l step i s e s t a b l i s h e d by noting that the expression i n brackets i s zero by the d e f i n i t i o n of J Q ( x ) . This proof can be repeated for Y 0 ( i / K - r ) . 3 . 3 . 9 . 4 A p p l i c a t i o n of Boundary Conditions A s o l u t i o n to an a n i s o t r o p i c e l s t o s t a t i c problem which has boundary conditions applied at the inner and outer r a d i i can be constructed from the s o l u t i o n s to the second order p a r t i a l d i f f e r e n t i a l equations. If the values K and K are assumed to be known, then the general s o l u t i o n Y X to be f i e l d equation i s : 6 7 . i|> = F + F , Y X where F - (A cosA z + B sinA z ) ( J n ( i / F " r ) + C Y.(i/K~"r) , Y Y Y Y Y ° Y Y ° Y and F = (A cosA z + B sinA z)(J„(ivT~r) + C Y n ( i / l T r ) ) . X X X X X ° X X 0 X A ,B ,C ,A ,B and C are the a r b i t r a r y constants which are determined Y Y Y X X X by the boundary conditions. Since J q and Y q can be complex functions of r, depending on the values for and K^, the a r b i t r a r y constants must also be complex i n general. The complexity of the problem can be reduced by assuming that the loading imparted by the tourniquet i s symmetrical with respect to z+0. Since the normal s t r e s s , a /P , i s given by r max 3 b a / P = — (a <J> + - i> + c i|> ) , r max 3z ,rr r , r T , z z / ' i t can be rewritten i n terms of the general s o l u t i o n to the f i e l d equation as a / P = ~ [a (F + F ) + - (F + F ) + c(F + F )1 r max 3z L Y.rr x> r r r Y>r X»r Y,zz X« z z This expression w i l l be of the form: a / P = G (r) A sinA z + H (r) B cosA z r max Y Y Y Y Y Y + G (r) A sinA z + H (r) B cosA z X X X X X X 68. For t h i s expression to be even i n z, the sine terms must be zero. A Y and A are therefore zero. X The general s o l u t i o n f o r the s t r e s s f u n c t i o n i s therefore • = B sinA z ( J n ( i / r " r ) + C Y n(i / T T r ) ) Y y u Y y \) Y + B sinA z ( J n(i / J T r ) + C Y n(i/1T"r) X X 0 Y X 0 X The constants i n t h i s equation are determined by the boundary conditions, which, f o r the purposes of t h i s a n a l y s i s , w i l l be of the following forms: a) a /P and T /P s p e c i f i e d on r = r r max rz max o b) u = 0 on r = r c) T = 0 or a) = 0 on r = r. rz i d) fa dA = 0 over the ends of the limb model. 1 ' z ( r . = bone radius, r = limb radius) 1 o 3.3.9.4.1 Permissible Values f o r the Separation Constant The second boundary cond i t i o n can be used to f i n d a r e l a t i o n s h i p between A and A . The expression for the r a d i a l displacement i s u = Y X p \x — — i|> . At r = r , t h i s displacement equals zero; the boundary Ji % r z i condition can therefore be taken as f r z = 0 at r = r . . This equation has the form 1 This can be s a t i s f i e d by superimposing a tension or compression on the s o l u t i o n determined by the other boundary conditions. 69. i> = K B cosA z + K B cosA z = 0 > r z Y Y Y X X X where K ,K ,B and B are not, i n general, zero. This equation can be Y X Y X rewritten as cos A z cosA z X — = K (a constant) which can only be s a t i s f i e d for a l l z i f A = A . Since these two must Y X be equal, a new v a r i a b l e , A, can be defined such that A = A = A . If X Although an a r b i t r a r y value f o r A w i l l r e s u l t i n a s o l u t i o n to the f i e l d equations only c e r t a i n values of A are u s e f u l i n s a t i s f y i n g the boundary condit i o n s . Consider the limb being modelled. The shape of the loading d i s t r i b u t i o n can be quite general; a u s e f u l approach, therefore, i s to decompose the loading d i s t r i b u t i o n i n t o i t s F o u r i e r components so that a a <=° (z) = •=—I- E a cos P v " 2 ' \ n ^ a L ' max n=l 2 L °r where a n = — / (z) cos — — dz and L equals the h a l f - l e n g t h of the o max limb. Note that i n the above equation z i s dimensional. The term a Q/2 i s of d i f f e r e n t form than the other terms and w i l l lead to a d i f f e r e n t form of the s t r e s s function, i|>, than the other terms. This d i f f i c u l t y can be avoided by considering the limb to be extended by h a l f i t s length i n both d i r e c t i o n s and by assuming that 7 0 . these extensions are loaded by the negative of the loading around z = 0 (see Figure 14). Note that L i s redefined as the length of the limb. If t h i s i s done, the term a Q w i l l be zero, as w i l l a l l the terms with n even. The loading w i l l then be decomposed as a oo r I \ r n i I Z P (z) - £ * n cos - j - , max n=l where 4 1 / 2 Q r a = — / (z) cos —-— dz i f n odd n Li r Li o max = 0 i f n even 3 . 3 . 9 . 5 General S o l u t i o n If the general s o l u t i o n f o r a p a r t i c u l a r value of X i s used to generate an expression for o /P , the expression w i l l have the form r max a rr-^— (z) = (H (r) B + H (r) B ) cos Xz P Y Y X X max Since t h i s expression must match a term i n the Fourier decomposition, X must equal mr/L. From t h i s , the values f o r K and K can be determined Y X (adding the s u b s c r i p t n as a reminder of t h e i r dependence on n) V = Y (r ) 2 ; K x n • *(r)2 • The s o l u t i o n for the general loading s i t u a t i o n can therefore be written as Figure 14. Extension of limb with negative loading 72. 00 = I s i n X z (P J . ( i / K ~ r ) + Q Y n(i/K~~r) , n yn 0 y yn 0 y n=l + P J (i/T~r) + Q Y n ( i / K ~ r ) ) , X « ° X Xn 0 V x where P and Q are determined by the boundary conditions and are d i f f e r e n t f o r each value of n. R e c a l l that the constants P , P , Q and X Y X are complex because J q and Y q are complex i n general. It i s p o s s i b l e to take l i n e a r combinations of the four s o l u t i o n functions i n such a way that four independent r e a l s o l u t i o n functions are obtained. In t h i s case, the constants are a l s o r e a l (see Appendix D f o r d e t a i l s ) . This s o l u t i o n w i l l now be applied to a test case before being used i n a parametric study. 73. CHAPTER 4  RESULTS AND DISCUSSIONS Rationale f o r Test S e l e c t i o n 4.1.1 Approximation of Loading D i s t r i b u t i o n The mathematical technique developed i n the previous chapter for the s o l u t i o n of the f i e l d equation was applied to s e v e r a l t e s t cases. Since the exact shape of the pressure d i s t r i b u t i o n applied by a t o u r n i -quet to a limb i s unknown, two p o s s i b l e d i s t r i b u t i o n s were considered: a rectangular d i s t r i b u t i o n and a s i n u s o i d a l d i s t r i b u t i o n . The e l a s t o -s t a t i c problem was solved by decomposing these d i s t r i b u t i o n s i n t o t h e i r Fourier components, solving i n d i v i d u a l l y for each component, and summing the r e s u l t s . Obviously, only a f i n i t e number of terms i n the F o u r i e r decomposition could be solved f o r , so the s o l u t i o n obtained i s an approximation to the s o l u t i o n of the problem with the desired loading d i s t r i b u t i o n . The approximation i s better f or the s i n u s o i d a l d i s t r i b u -t i o n because that d i s t r i b u t i o n i s much smoother than the rectangular d i s t r i b u t i o n ; there i s no sharp drop i n the applied pressure at the edge of the c u f f . The rectangular approximation was, f o r two reasons, taken to be that d i s t r i b u t i o n given by the sum of the f i r s t 13 terms i n the Fou r i e r decomposition. The most important reason f o r l i m i t i n g the sum was an observation that the solutions obtained when higher order approximations were made exhi b i t e d large deviations from the lower order s o l u t i o n s ; these deviations appeared to be i n s t a b i l i t i e s rather than refinements to the s o l u t i o n . Time co n s t r a i n t s unfortunately prevented the pursuit of the cause of t h i s behaviour. The second reason for l i m i t i n g the sum was that 13 terms gives a reasonable approximation to a rectangular loading d i s t r i b u t i o n f o r a l l three cuff width r a t i o s tested (see Figure 15) i n that there i s a r e l a t i v e l y steep drop i n the app l i e d pressure near the cuff edge. Even so, "wobbles" on the order of 10% of the maximum appl i e d pressure remain, so i n analyzing the r e s u l t s , some care should be taken i n i n t e r p r e t i n g extreme values. The s i n u s o i d a l approximation was al s o taken as the sum of the f i r s t seven to t h i r t e e n terms of the Fourier decomposition, depending on the width of the c u f f . As can be seen from Figure 16, t h i s approximation i s almost i n d i s t i n g u i s h a b l e from the pure s i n u s o i d a l d i s t r i b u t i o n . • 4.1.2 Domain of Reasonable Solutions At the o r i g i n a l time of w r i t i n g t h i s t h e s i s , I t was not r e a l i z e d that the a r b i t r a r y constants used to t a i l o r the general s o l u t i o n to the boundary conditions could be complex, nor was the development o u t l i n e d i n Appendix D well understood. As a r e s u l t , the e a r l i e r path taken l e d to the d i s q u a l i f i c a t i o n of c e r t a i n r e s u l t s on the grounds that they were p h y s i c a l l y impossible, when, i n f a c t , a s l i g h t l y d i f f e r e n t procedure would have l e d to p h y s i c a l l y meaningful r e s u l t s . At t h i s stage, unfortunately, the dismissed cases cannot be reconsidered, although t h i s would be a f r u i t f u l course f o r f u r t h e r i n q u i r y . The d i s q u a l i f i e d cases f e l l i nto an e a s i l y defineable c l a s s . The constants y and x i n t n e separated equations depend on the values of a and B i n the f i e l d equation, which depend i n turn on the e l a s t i c constants. For c e r t a i n values of a and y and x are complex conjugates. In these cases, the e a r l i e r method of s o l u t i o n f a i l e d to 7 5 . Fourier Loading Approximation Figure 15. Fourier approximation to a rectangular loading d i s t r i b u t i o n f o r c = 0.4, 0.6 and 0.8 Fourier Loading Approximation V2-, Axial Distance from Tourniquet Center Figure 16. Fourier approximation to a s i n u s o i d a l loading d i s t r i b u t i o n f o r c = 0.4, 0.6 and 0.8 77. give reasonable s o l u t i o n s . Only when Y and x were r e a l were reasonable r e s u l t s obtained. 4.1.3 S e l e c t i o n of Test Cases Unfortunately, the e l a s t i c parameters which would most accurately describe the behaviour of muscle t i s s u e lead to complex values f o r Y and X * For example, i n r e a l muscle t i s s u e , the e l a s t i c modulus i n the transverse d i r e c t i o n i s lower than i n the a x i a l d i r e c t i o n , which causes Y and x t o D e complex. Because the e a r l i e r method of s o l u t i o n would f a i l i n t h i s k i n d of case, the i s o t r o p i c problem w i l l be described i n d e t a i l as the " t y p i c a l " problem; i t i s the c l o s e s t representation of the r e a l s i t u a t i o n which can be solved f o r using t h i s f o r m u l a t i o n . 1 T y p i c a l values for some of the dimensions and q u a n t i t i e s of i n t e r e s t i n t h i s problem are l i s t e d i n Table 1. Table 1. T y p i c a l Parameters Used i n Simulation Limb Length : 20 cm. Bone Diameter 2 cm. Normalized : 0.1 Limb Diameter 7 cm. Normalized : 0.35 Cuff Width 12 cm. Normalized : 0.6 Applied Pressure : 30 kPa Young's Modulus, E 15 kPa 1 A c t u a l l y , an approximation to the i s o t r o p i c problem w i l l be computed. The formulation c a l l s f o r separating the f i e l d equation i n t o two d i f f e r e n t second order equations and sol v i n g each of these. In the i s o t r o p i c case, the two d i f f e r e n t equations degenerate i n t o a s i n g l e equation. To use t h i s formulation, then, the e l a s t i c modulus i n the transverse plane w i l l be taken to be 1.01 times the modulus i n the a x i a l d i r e c t i o n . 78. In a d d i t i o n to these t y p i c a l values, the shape of the loading d i s t r i b u -t i o n and the choice of the boundary conditions to be applied at the bone/tissue i n t e r f a c e 1 must be s p e c i f i e d . T h i r t y - s i x simulations were run to determine the e f f e c t on the str e s s and s t r a i n f i e l d s of changes i n the thickness of the limb, the width of the c u f f , the shape of the tourniquet loading, and the assump-t i o n of " s t i c k i n e s s " or free movement at the bone t i s s u e i n t e r f a c e . The Poisson's r a t i o f o r these t e s t s was set at 0.45; Chow and O d e l l (1978) suggest a value of 0.49 for soft t i s s u e , which i n d i c a t e s almost complete i n c o m p r e s s i b i l i t y , but a value of 0.45 was taken i n case any of the solutions became unstable as the c o m p r e s s i b i l i t y approached zero. This measure was purely precautionary. The parameter d e s c r i b i n g the diameter of the bone was held constant for the s e r i e s of tests as i t was f e l t to be r e l a t i v e l y constant from one limb to another. These t e s t s are described i n the next two sections. Six further tests were run to determine the s e n s i t i v i t y of the a n a l y s i s to changes i n the e l a s t i c constants. These t e s t s were a l l ascribed codes f o r ease of reference. The codes are four l e t t e r s long, such as SFNA; the f i r s t l e t t e r r e f e r s to S i n u s o i d a l or Rectangular loading, the second to Sticky or Free at the i n t e r f a c e , the t h i r d to the thickness of the limb, and the fourth to the width of the c u f f . For these l a s t two l e t t e r s , permissible codes are 1 The t i s s u e at the l e v e l of the bone may reasonably be supposed to e i t h e r s l i d e f r e e l y i n the a x i a l d i r e c t i o n or to be r e s t r i c t e d from such motion by f r i c t i o n a l f o r c e s . Although combinations of such behaviour may be expected, a c o n s i d e r a t i o n of these two extremes w i l l be u s e f u l i n understanding t h e i r r e l a t i v e importance. 79. Narrow, Average, and Wide. The code SFNA therefore r e f e r s to s i n u s o i d a l loading on a narrow limb with an average width tourniquet applied; i n ad d i t i o n , the t i s s u e i s assumed to be able to move f r e e l y over the bone. (Note: a "*" follows the code i f the e l a s t i c constants are not i s o t r o -p i c ) . The a c t u a l parameter values f o r each t e s t are shown i n Table 2. 4.2 General C h a r a c t e r i s t i c s of S o l u t i o n 4.2.1 Comparison with Thomson and Doupe's Experiment Since i t i s unclear whether the tourniquet loading i s more c l o s e l y approximated by the s i n u s o i d a l or the rectangular loading d i s t r i b u t i o n , as well as whether the t i s s u e at the bone/tissue i n t e r f a c e should be assumed to be f r e e to s l i d e or not, a comparison of the h y d r o s t a t i c pressures 1 generated i n the limb for severeal d i f f e r e n t simulations was compared with the r e s u l t s from Thomson and Doupe (see Figures 17 and 18). 4.2.1.1 S i n u s o i d a l or Rectangular - Which Most L i k e l y ? Compare f i r s t of a l l the r e s u l t s from t e s t s RSAA and SSAA. The rectangular loading test shows a maximum normalized h y d r o s t a t i c pressure of approximately 1.0 along the bone underneath the c u f f , whereas the maximum pressure seen i n the s i n u s o i d a l t e s t i s 0.85 underneath the center of the c u f f . This i s somewhat out of agreement with Thomson & Doupe's observation that the pressure equals the applied value d i r e c t l y 1 The s o l i d mechanics term i s mean normal s t r e s s ; " h y drostatic pressure" occurs i n the l i t e r a t u r e r e l a t e d to tourniquets, so i t i s used here. 80. Table 2. Names and parameters used i n each t e s t of model Test Limb Cuff V E r G r Loading Sticky Code Dia. Width Shape or Free RFNN 0.25 0.4 0.45 1 1.01 1 R F RFNA 0.6 RFNW 0.8 RFAN 0.35 0.4 RFAA 0.6 RFAW 0.8 RFWN 0.50 0.4 RFWA 0.6 RFWW 0.8 RSNN 0.25 0.4 0.45 1 1.01 1 R S RSAA 0.6 RSNW 0.8 RSAN 0.35 0.4 RSAA 0.6 RSAW 0.8 RSWN 0.50 0.4 RSWA 0.6 RSWW 0.8 SFNN 0.25 0.4 0.45 1 1.01 1 S F SFNA 0.6 SFNW 0.8 SFAN 0.35 0.4 SFAA 0.6 SFAW 0.8 SFWN 0.50 0.4 SFWA 0.6 SFWW 0.8 SSNN 0.25 0.4 0.45 1 1.01 1 S S SSNA 0.6 SSNW 0.8 SSAN 0.35 0.4 SSAA 0.6 SSAW 0.8 SSWN 0.50 0.4 SSWA 0.6 SSWW 0.8 SSAA*1 0.35 0.6 0.25 1 2 1 S S RSAA*2 0.45 1 2 1 R S SFAA*3 0.25 1 2 1 S F SSAA*4 0.25 0.5 2 1 S S SSAA*5 0.25 2 2 1 S s SSAA*6 0.25 1 2 2 S s Transverse Pressure R$AA TC Figure 17. Comparison of predicted h y d r o s t a t i c and transverse pressure maps with Thomson & Doupe's findings Figure 18. Comparison of predicted hydrostatic pressure maps with Thomson & Doupe's fi n d i n g s 83. underneath the c u f f , but, i f the e a r l i e r d i s c u s s i o n concerning the p o s s i b i l i t y that Thomson and Doupe a c t u a l l y measured the transverse pressure i s v a l i d (see Section 2.1.1), then i t can be seen from the contour map of the transverse pressure from the t e s t RSAA that there i s both a remarkable s i m i l a r i t y between the general shape of the contours and a s i g n i f i c a n t correspondence between the a c t u a l v a l u e s . 1 Aside from the values of the contours, the general shapes of the transverse pressure map and the h y d r o s t a t i c pressure map are very s i m i l a r . Because of t h i s correspondence i n shape, i t i s reasonable to i n f e r that the stress l e v e l s from a transverse pressure map for the case of s i n u s o i d a l loading would also be more i n agreement with Thomson and Doupe's experiment; t h i s map was unfortunately not generated with the other ones because i t s s i g n i f i c a n c e was not a n t i c i p a t e d . Another important observation concerning the two h y d r o s t a t i c pressure maps from the RSAA and SSAA t e s t s i s the spacing of the contour l i n e s i n the v i c i n i t y of the edge of the c u f f . In the rectangular loading case, the contour l i n e s are q u i t e c l o s e l y packed, whereas the s i n u s o i d a l case, they are spread out over approximately twice the same distance. This l a t t e r spreading matches Thomson and Doupe's observations. Because of t h i s match i n the packing of the contour l i n e s , one might be i n c l i n e d to claim that the s i n u s o i d a l loading d i s t r i b u t i o n accurately r e f l e c t s the true loading; on the other hand, 1 Note that i n the wide region under the c u f f i n the RSAA transverse pressure p l o t there are s e v e r a l -1.00 contours. Remembering that the loading approximation contains "wobbles", i t i s reasonable to suppose that the e n t i r e sub-cuff region has an e s s e n t i a l l y constant transverse pressure l e v e l . 84. the a c t u a l values f o r the contours r e s u l t i n g from s i n u s o i d a l loading ar roughly 15% lower than those r e s u l t i n g from rectangular loading, which correspond w e l l with the values reported by Thomson and Doupe. It i s probably safe to claim that the r e a l loading d i s t r i b u t i o n l i e s between these two estimates. 4.2.1.2 Free or S t i c k y - Which Most L i k e l y ? Consider now the two t e s t s RSAA and RFNW and the h y d r o s t a t i c pressure maps a r i s i n g from them (see Figure 18). Assuming for the moment that the narrow limb and wide tourniquet do not r e a l l y a f f e c t th shape of the contours, i t i s apparent that i t i s important whether the t i s s u e is" f r e e or constrained at the bone/tissue i n t e r f a c e . If i t i s free to move, then the contour l i n e s appear par a b o l i c ; they do not get close together at greater depth. With movement r e s t r i c t e d , on the othe hand, the hydrostatic pressure contours do come closer together i n the lower part of the t i s s u e . The experimental evidence of Thomson & Doupe i s not conclusive i n resol v i n g t h i s c o n f l i c t ; an argument could be made e i t h e r way because, while Thomson & Doupe's midrange h y d r o s t a t i c pressure contours show the bowing at mid-depth, l i k e the RSAA plo t , the 1.0 contour l i n e simply comes down on a diagonal, much l i k e the RFNW p l o t . If not only the shape of the contour l i n e s i s considered, but the magnitude of the contours as well, then a further argument can be made that the hypothesis of co n s t r a i n t at the bone/tissue i n t e r f a c e i s co r r e c t . Thomson & Doupe report observed hydrostatic pressures equal t the maximum applied pressure i n the t i s s u e d i r e c t l y under the mid-point of the c u f f . It has been explained how the RSAA plot of the transverse 85. pressure can.be seen to be i n accord with t h i s observation. The RFNW plot of the transverse pressure (see Figure 19) shows that under the midregion of the c u f f , the pressure i s 15-20% lower than the pressure observed by Thomson and Doupe. If the transverse pressure i s indeed c l o s e r than the h y d r o s t a t i c pressure to what Thomson and Doupe a c t u a l l y measured, then the RFNW plot shows values s i g n i f i c a n t l y lower than those which are expected. Since the values from the RSAA t e s t are almost exactly what would be expected, t h i s i s evidence to support the view that the t i s s u e i s constrained from motion along the bone. Even i f Thomson and Doupe did measure the hy d r o s t a t i c pressure, the values from the RSAA t s e t are s i g n i f i c a n t l y c l o s e r to t h e i r observations than the values from the RFNW t e s t . So then, the " t y p i c a l " s i t u a t i o n , the model of the current use of tourniquets, i s one i n which the loading imposed by the tourniquet can be described as being somewhere between rectangular and s i n u s o i d a l (perhaps a region of r e l a t i v e l y constant pressure across the midregion of the c u f f bounded by s i n u s o i d a l drops to zero near the edges) and i n which the tissues i s seen to be not free to move a x i a l l y at the bone/tissue i n t e r f a c e . 4.2.2 General C h a r a c t e r i s t i c s of Induced Stresses and St r a i n s A f u l l s e r i e s of contour maps f o r the induced s t r e s s e s , d i s p l a c e -ments, s t r a i n s , and pressures for the test RSAA i s c o l l a t e d i n Figures 20 to 22. The hy d r o s t a t i c and transverse pressures have been discussed already, so the general features of some of the other maps w i l l now be Transverse Pressure Figure 19. Comparison of predicted transverse pressure maps with Thomson & Doupe's f i n d i n g s 87. Radial S tress z Axial Stress z Circumferential S t ress z Shear Stress Figure 20. Contour plots f o r test RSAA Radial Displacement Figure 21. Contour plo t s for test RSAA Circumferential Strain Figure 22. Contour plots f o r test RSAA Circumferential S t ress R S A A CM Circumferential S t ress R S A A * 2 Radial Displacement R S A A .5 Z Radial Displacement R S A A * 2 Figure 23. Comparison of c i r c u m f e r e n t i a l s t r e s s and r a d i a l displacement p l o t s from t e s t s RSAA and RSAA*2 91. Radial Strain R S A A Circumferential Strain RSAA Figure 24. Comparison of r a d i a l and c i r c u m f e r e n t i a l s t r a i n p l o t s from t e s t s RSAA and RSAA*2 92. discussed. The most important v a r i a b l e s are the r a d i a l s t r e s s , the a x i a l s t r e s s , and the a x i a l s t r a i n . 1 4.2.2.1 R a d i a l Stresses Roughly speaking, the r a d i a l s t r e s s at a p a r t i c u l a r a x i a l l o c a t i o n does not change with depth; the l a r g e s t changes i n the a x i a l d i r e c t i o n . This tendency, however, i s strongly a f f e c t e d by the assumption of " s t i c k i n e s s " ; i f the t i s s u e i s free to move (see te s t RFAW i n Figure 25), then the r a d i a l s t r e s s increases with depth, reaching roughly 125% of the surface value at the l e v e l of the bone. I n t e r e s t i n g l y , t h i s i s the same so r t of pattern predicted by Auerbach's model, although he was discussing the change i n the hydrostatic pressure with depth. 4.2.2.2 A x i a l Stresses The a x i a l s t r e s s i s al s o strongly a f f e c t e d by the assumption of s t i c k i n e s s or free motion. In the " s t i c k y " t e s t , RSA, the a x i a l s t r e s s i s compressive at almost a l l points i n the t i s s u e (see Figure 26); i n 1 The c i r c u m f e r e n t i a l s t r e s s , r a d i a l displacement, r a d i a l s t r a i n , and c i r c u m f e r e n t i a l s t r a i n f o r both the RSAA t e s t and The RSAA*2 t e s t which has Er set equal to 2 are compared i n Figures 23 and 24. Note the s i g n i f i c a n t d i f f e r n e c e s i n each p a i r of contour maps, most notably i n the r a d i a l displacements and s t r a i n s . A l l other pa i r s of plots are s i m i l a r to one another, so i t i s hypothesized that the d e v i a t i o n f r o n the i s o t r o p i c case which i s required to use the s o l u t i o n method developed i n t h i s paper i s small enough to cause some i l l - c o n d i t i o n i n g i n the part of the s o l u t i o n process where the four a r b i t r a r y constants are s e l e c t e d . Because of the close correspondence between a l l the other pairs of maps, and becuase the e l a s t i c parameters f o r the t e s t RSAA* shoudl not lead to i l l - c o n d i t i o n i n g , i t i s f u r t h e r supposed that the i l l - c o n d i t i o n i n g i n the q u a s i - i s o t r o p i c case most severely a f f e c t s the c a l c u l a t i o n of the four v a r i a b l e s shown i n Figures 23 and 24. 93. Figure 25. Comparison of Auerbach's p r e d i c t i o n s of hy d r o s t a t i c pressure l e v e l s with r a d i a l s t r e s s p l o t s from t e s t s RFAW and RSAA 94. Axial Stress S S A A Axial Stress R S A A Figure 26. Comparison of a x i a l s t r e s s plots from tests p l o t s SSAA, RSAA, RFAW and SFAA 95. addition, the magnitude of the compressive s t r e s s under the cu f f i s between 40% and 90% of the maximum applied pressure, although the contour l i n e s are qu i t e d i f f e r e n t from the r a d i a l s t r e s s contours. If the t i s s u e i s free to move, as i n the tes t RFAW, then the maximum compressive s t r e s s i s only 30% of the maximum pressure. There are two d i s t i n c t regions i n t h i s contour map, one compressive and the other t e n s i l e , which form a "saddle" region. The region of compressive a x i a l stress i s centered around the edge of the c u f f ; on the in s i d e of the c u f f edge, the s t r e s s i s most negative near the skin, while on the outside, the stress i s most negative near the bone. The appearance of these regions of compressive a x i a l s t r e s s i s s i m i l a r when s i n u s o i d a l loading i s applied. 4.2.2.3 A x i a l S t r a i n s The a x i a l s t r a i n f i e l d i s probably the f i e l d of greatest i n t e r e s t i n t h i s i n v e s t i g a t i o n . It has been hypothesized that a x i a l compressive of the p e r i p h e r a l neuron i s responsible f o r the c l i n i c a l p a r a l y s i s . If negative a x i a l s t r a i n s can be demonstrated, that w i l l be evidence for the p l a u s i b i l i t y of t h i s hypothesis. Thi s i s , i n f a c t , what i s found. If the assumption of " s t i c k i n e s s " i s made, and i t i s a reasonable assumption for the reasons discussed e a r l i e r , then f o r both rectangular and s i n u s o i d a l loading d i s t r i b u t i o n s , a region of negative a x i a l s t r a i n i s found e i t h e r just i n s i d e the edge of the c u f f , i n the case of s i n u s o i d a l loading, or j u s t outside the edge of the c u f f , i n the case of rectangular loading (see Figure 27). The magnitude of the s t r a i n s i s qu i t e high, on the order of 0.15 to 0.2. Figure 27. Comparison of a x i a l s t r a i n plots from t e s t s RSAA, SSAA and RFAA 97. This can be compared with r a d i a l s t r a i n s of 0.2 and c i r c u m f e r e n t i a l s t r a i n s of 0.05 for the same case. The t e s t s c a r r i e d out under the assumption of f r e e movement at the bone/tissue i n t e r f a c e also show a region of negative a x i a l s t r a i n just outside of the c u f f edge, but the magnitude of the s t r a i n i s much lower, on the order of 0.05. This f i n d i n g implies that i f i t i s discovered that there i s f r e e movement of the t i s s u e s at the bone, the hypothesis of damage by a x i a l compression i n the manner described e a r l i e r i s l e s s l i k e l y . 4.3 Factors A f f e c t i n g Regions of Negative A x i a l S t r a i n The maximum negative a x i a l s t r a i n s f o r the t e s t s run assuming const r a i n t at the bone/tissue i n t e r f a c e have been estimated from the contour p l t o s and are summarized i n Table 3. The fol l o w i n g three sections w i l l r e f e r to t h i s t a b l e . 1 Note: the f u l l set of a x i a l s t r a i n contour p l o t s f o r a l l t e s t s run are contained i n Appendix E. 1 Since f u n c t i o n a l nerve damage i s p a r t l y a t t r i b u t a b l e to the damage at any p a r t i c u l a r node of Ranvier and p a r t l y to the number of nodes af f e c t e d , measures other than the maximum negative a x i a l s t r a i n could be considered to be i n d i c a t o r s of damage (/ezdA over the region of negative a x i a l s t r a i n , for example). The maximum negative a x i a l s t r a i n i s chosen because othe existence of such a value implies a region of negative values; one such value could not occur i n i s o l a t i o n . ble 3. Maximum a x i a l s t r a i n f o r d i f f e r e n t c u f f width r a t i o s , limb thickness and loading d i s t r i b u t i o n s Rectangular Loading Limb Thickness Narrow Average Wide Narrow Cuff 0.25 0.25 0.17 Average Cuff 0.23 0.20 0.15 Wide Cuff 0.20 0.13 0.07 Si n u s o i d a l Loading Limb Thickness Narrow Average Wide Narrow Cuff 0.14 0.16 0.13 Average Cuff 0.12 0.14 0.12 Wide Cuff 0.09 0.10 0.07 99. 4.3.1 Shape of Loading D i s t r i b u t i o n The shape of the loading d i s t r i b u t i o n on the maximum value of the a x i a l s t r a i n i s quite s i g n i f i c a n t , e s p e c i a l l y for t h i n to average limbs. For t h i n limbs, the maximum s t r a i n induced udner a rectangularly d i s t r i -buted load i s roughly twice the maximum s t r a i n under a s i n u s o i d a l load. For normal, or average, limbs, t h i s r a t i o i s roughly 1.5, and f o r thick limbs, t h i s r a t i o i s roughly 1.25. The shape of the loading d i s t r i b u t i o n has two e f f e c t s on the shape of the region of negative a x i a l s t r a i n . The f i r s t i s the p o s i t i o n of the center (most negative part) of the region r e l a t i v e to the edge of the c u f f . With rectangular loading, the center of the region i s appro-ximately h a l f the t i s s u e thickness out from the edge of the c u f f . With s i n u s o i d a l loading, the center i s us u a l l y h a l f the ti s s u e thickness i n from the c u f f edge. This obseration i s s i g n i f i c a n t i n that, i f the e a r l i e r argument that r e a l tourniquets have a loading d i s t i b r u t i o n midway between a rectangular and a s i n u s o i d a l d i s t r i b u t i o n i s true, then the region of negative a x i a l s t r a i n would be centered under the edge of the c u f f , i n p e r f e c t accord with Ochoa's observations. The second e f f e c t i s on the aspect r a t i o of the region; the rectan-gular loading r e s u l t s i n a roughly square region of negative a x i a l s t r a i n , whereas the s i n u s o i d a l loading r e s u l t s i n elongated regions with a length roughly 1.5-2 times the t i s s u e thickness. A l l the regions of negative a x i a l s t r a i n are r e l a t i v e l y narrower i n thicker limbs. 4.3.2 Width of Cuff In a l l cases, a wider c u f f lowered the maximum s t r a i n . The e f f e c t was most pronounced for thick limbs; i n going from a narrow cuff to a 100. wide c u f f , the maximum s t r a i n was reduced by roughly 50%. In t h i n limbs, the reduction was between 20% and 30%. There was a smaller drop i n going from a narrow c u f f to a normal c u f f than i n going from a normal cuff to a wide cu f f ; the l a t t e r drop was u s u a l l y twice the former. This observation needs to be tempered by some con s i d e r a t i o n of the f a c t that the limb i s of f i n i t e length, and with a wide cuff applied, the edge of the c u f f i s near to the end of the limb. This s i t u a t i o n w i l l l i k e l y be quite d i f f e r e n t from the s i t u a t i o n i n which an equally wide cuff i s a p p l i e d to a limb of i n f i n i t e extent. It i s most important to note that a r e a l limb tapers near the end; t h i s tapering i s not accounted for i n t h i s model, so p r e d i c t i o n s near the end of the limb must be regarded with more suspicion than predictions c l o s e r to the center. The width of the c u f f seems to have no e f f e c t on the l o c a t i o n of the center of the region of negative a x i a l s t r a i n r e l a t i v e to the edge of the c u f f . 4.3.3 Thickness of Limb In the case of rectangular loading, the thickness of the limb strongly a f f e c t s the magnitude of the maximum negative a x i a l s t r a i n . A thick limb, i n general has a lower value of t h i s measure (30%-65% lower for a narrow and wide cuff r e s p e c t i v e l y ) than a t h i n limb. This pattern i s not observed i n the case of s i n u s o i d a l loading, however; i n t h i s case a limb of average thickness i s subject to a greater maximum negative a x i a l s t r a i n than a thinner or t h i c k e r limb. This unexpected behaviour can be understood by considering the f a c t that the p o s i t i o n of the region of negative a x i a l s t r a i n r e l a t i v e to the edge of the cuff i s e s s e n t i a l l y independent of a l l parameters other than the 101. shape of the loading d i s t r i b u t i o n and that the s i z e and shape of the region seems to depend only on the limb thickness. This suggests that t h i s r e g i o n w i l l be p a r t i c u l a r l y s e n s i t i v e to those features of the loading s i t u a t i o n which are p e c u l i a r to the v i c i n i t y of the cuff edge. Such features ought, then, to be non-dimensionlized using a reference length which has meaning i n the v i c i n i t y of the edge; since rapid changes In the s t r e s s f i e l d s occur near the c u f f edge, the i n f l u e n c e of more d i s t a n t e f f e c t s Is probably minor and the limb thickness i s probably a more appropriate reference dimension than the limb length. The loading d i s t r i b u t i o n s , however, have been defined at a x i a l p o s i t i o n s which have been non-dimensionalized by the limb length. A l l a x i a l d e r i v a t i v e s of the loading d i s t r i b u t i o n s then, are also non-dimensionalized by the limb length and are thereby rendered independent of the limb thickness. It may be supposed that the f i r s t d e r i v a t i v e of the loading d i s t i b u t i o n with respect to z i s a s i g n i f i a n t determinant of the magnitude of the maximum negative a x i a l s t r a i n . If t h i s d e r i v a t i v e were non-dimensionalized by the more appropriate limb thickness, then the same loading referred to the limb length w i l l give r i s e to lower-valued d e r i v a t i v e s i n thinner limbs than i n t h i c k e r ones. The magnitude of the maximum negative a x i a l s t r a i n for a thinner limb would be lower than expected i f t h i s f a c t o r were not understood. From another perspective, the puzzling observation comes about because an unstated assumption made when reading Table 2 i s i n f a c t wrong. When the d i v i s i o n into the two kinds of loading i s made, one assumes that the loading i n each of the nine subcases of the major d i v i s i o n s i s i d e n t i c a l . This i s only true i f the reference dimension Is taken to be the limb length. Once one understand that the only 102. meaningful dimension f o r edge-localized e f f e c t s i s the limb thickness, one r e a l i z e s that there are three separate loadings involved i n the nine subcases so that the subcases f o r d i f f e r e n t limb thicknesses cannot be compared d i r e c t l y . 4.4 S e n s i t i v i t y of S o l u t i o n to E l a s t i c Parameters 4.4.1 S e n s i t i v i t y to Er For t h i s t e s t , the t e s t RSAA was repeated with the e l a s t i c parmeter E set equal to 2 (te s t RSAA*2). Side-by-side comparisons of the r a d i a l s t r e s s e s , the a x i a l s t r e s s e s , the a x i a l s t r a i n s , and the h y d r o s t a t i c pressures are shown i n Figures 28 and 29. From these comparisons, i t can be seen that the corresponding contour maps i n each t e s t are very s i m i l a r to one another. In p a r t i c u l a r , the maximum value for the negative a x i a l s t r a i n i s the same i n both cases: roughly -0.22. 4.4.2 S e n s i t i v i t y to Poisson's r a t i o For t h i s t e s t , a t e s t s i m i l a r to SSAA was repeated with E^ set to 2 and v set to 0.25 (te s t SSAA*1). The same side-by-side comparisons described above are shown i n Figures 30 and 31. The r a d i a l s t r e s s and hydr o s t a t i c pressure plots are seen to be e s s e n t i a l l y s i m i l a r , although the r a d i a l s t r e s s contour l i n e s d i f f e r somewhat near the midpoint of the cuf f , and the hydrostatic pressure l e v e l s i n the SSAA case are higher than the l e v e l s at the corresponding point i n the SSAA*1 case. The case stress plots are very s i m i l a r i n shape, but the s t r e s s l e v e l s i n the SSAA case (highly Incompressible material) are approxi-mately twice the corresponding s t r e s s l e v e l s i n the SSAA*1 case. This 103. Radial S t ress R S A A • 1 « © o ^ //* {\ J I o o 9 1 D | 3 1 5 J 0.20 © © 1 \ f 1 | . , 1 i | ! 0.0 0.1 0.2 0.3 0.4 0 z Radial S t ress R S * A * 2 / • T 1 > . ' P1 •» \ \ » i P \ j, \ \ *V ° 1 ° \ 'o m——i 1 Axial S t ress R S A A 0.5 Figure 28. Comparison of radial and axial stress plots from tests RSAA and RSAA*2 Axial Strain RSAA \ \ - 2 I I II © O © 0.0 0.1 0.2 0.3 0.4 • • < z Axial Strain R S M , 2 "f o e | e V, y J v > — I 1 1 1 1 1 1— 0.0 0.1 0.2 0.3 0.4 z Hydrostatic Pressure ;ure 29. Comparison of a x i a l s t r a i n and hydrostatic pressure p l o t s from t e s t s RSAA and RSAA*2 1 0 5 . Radial Stress S S A A Axial Stress S S A A z Axial Stress S S A A * I Figure 30. Comparison of r a d i a l and a x i a l s t r e s s p l o t s from t e s t s SSAA and SSAA*1 106. Axial Strain Hydrostatic Pressure SSAA*I CI :s z Figure 31. Comparison of a x i a l s t r a i n and hydrostatic pressure p l o t s from t e s t s SSAA and SSAA*1 1 0 6 . r e s u l t s i n a x i a l s t r a i n s which are twice as great i n the SSAA case as i n the SSAA*1 case. The case SSAA*1 with E = 2 and v =0.25 w i l l serve as the model of r comparison f o r the other three e l a s t i c parameter s e n s i t i v i t y s t u d i e s . 4.4.3 S e n s i t i v i t y to v r Two t e s t s were c a r r i e d out, one with =0.5 ( t e s t SSAA*4) and the other with = 2 ( t e s t SSAA*5), to gain an i n t u i t i v e understanding of the s e n s i t i v i t y of the s t r e s s and s t r a i n f i e l d s to changes i n t h i s para-meter. In both of these t e s t s , E = 2 and v = .25. r The f i r s t t e s t , with v f = 0.5, i s shown i n a side-by-side compari-son with the reference t e s t i n Figures 32 and 33, and the second t e s t , with = 2, i s shown i n Figures 34 and 35. The corresponding plots i n each p a i r are almost i d e n t i c a l i n a l l cases except f o r the r a d i a l s t r e s s comparison with v = 2. In that case, the contour l i n e s for the r a d i a l s t r e s s (case = 2) are more v e r t i c a l near the midpoint of the c u f f . T h i s i s s i m i l a r to the case SSAA. In that case, Poisson's r a t i o i n the transverse plane r e s u l t i n g from a x i a l tension was 0.45, which i n d i c a t e s r e l a t i v e i n c o m p r e s s i b i l i t y ; i n t h i s case, the Poisson's r a t i o i n the transverse plane r e s u l t i n g from tension i n the transverse plane i s 0.5, which a l s o i n d i c a t e s r e l a t i v e i n c o m p r e s s i b i l i t y . It i s reasonable, therefore, to expect that the two plots w i l l be s i m i l a r . This reasoning does not apply to the a c t u a l values f o r the a x i a l s t r e s s and s t r a i n , however: with v =2, the maximum a x i a l s t r a i n i s the same as with v = ' r r 0.5 or v = 1 . r 107. Radial Stress S S A A * I Figure 32. Comparison of r a d i a l and a x i a l stress plots from t e s t s SSAA*1 and SSAA*4 108. Axial Strain S S A A * I z Axial Strain S S A A * 4 CM Figure 33. Comparison of a x i a l s t r a i n and h y d r o s t a t i c pressure p l o t s from t e s t s SSAA*1 and SSAA*4 Figure 35. Comparison of a x i a l s t r a i n and h y d r o s t a t i c pressure p l o t s from t e s t s SSAA*1 and SSAA*5 111. 4.4.4 Sensitivity to G _ r It was mentioned early i n this chaper that certain combination sof the elastic constants w i l l result In complex coefficients when the f i e l d equation i s separated. If G^is set equal to 2 together with E f, then such a situation obtains. The resulting "Solution" i s not at a l l physically r e a l i s t i c , but is shown in a side-by-side comparison with the reference case (SSAA*1 with = 2) for completeness (see Figures 36 and 37). If certain other values for the e l a s t i c constants were chosen, then an investigation into the sensitivity of the solution to changes in G^ could be performed. Time constraints unfortunately preclude such an investigation. 112. Figure 36. Comparison of a x i a l and r a d i a l s t r e s s plots from t e s t s SSAA*1 and SSAA*6 113. Figure 37. Comparison of a x i a l s t r a i n and hyd r o s t a t i c pressure p l o t s from t e s t s SSAA*1 and SSAA*6 114. CHAPTER 5 DISCUSSION AND DIRECTIONS FOR FUTURE WORK 5.1 V a l i d i t y of Damage Mechanism Estimates A c r i t i c a l evaluation of e a r l i e r work i n t h i s f i e l d has r e s u l t e d i a new proposed mechanism for the in v a g i n a t i o n of the myelin sheath near the nodes of Ranvier. This new mechanism, a x i a l compression, has some qu a n t i t a t i v e basis which sets i t apart from e a r l i e r proposed mechanisms C e r t a i n assumptions were made i n the d e r i v a t i o n of the q u a n t i t a t i v e estimates, however, which must be addressed before i t can be claimed with a high degree of c e r t a i n t y that the operative mechanism causing inv a g i n a t i o n i s indeed a x i a l compression. Two questions i n p a r t i c u l a r must be asked. F i r s t , does the myelin r e a l l y behave l i k e a s t r u c t u r a l m a t e r i a l , and i f so, how much energy i s required to cause the observed deformation? The answer to t h i s question must be based e i t h e r on experiment or a r a t i o n a l theory concerning the strength of myelin. As i t i s l i k e l y that an experimetnal answer w i l l be the only answer a v a i l -able for some time, t h i s would be a u s e f u l physiomechanical study. Second, the order-of-magnitude c a l c u l a t i o n treated the myelin sheath as e x i s t i n g i n a vacuum. In r e a l i t y , the sheath i s located i n a mass of t i s s u e which i s subject to both r a d i a l and c i r c u m f e r e n t i a l s t r e s s e s . It i s important to assess the e f f e c t of these stresses on th a n a l y s i s as presented. This importance can be seen by considering the a x i a l s t r e s s f i e l d i n the case of " s t i c k i n e s s " between the t i s s u e and the bone. In the p a r i t c u l a r case of s i n u s o i d a l loading with a cuff of average width applied to a limb of average thickness, the a x i a l s t r e s s 115. at the c u f f edge i s roughly -0.15, and the magnitude of t h i s s t r e s s increases as the center part of the cuff i s approached; i n the midregion, the a x i a l s t r e s s i s as high as -0.50. In s p i t e of the higher a x i a l stresses near the center of the c u f f , the a x i a l s t r a i n i s only negative near the c u f f edge; under the midpoint of the c u f f , the material i s a c t u a l l y i n a state of t e n s i l e s t r a i n while simultaneously being i n a s t a t e of a x i a l l y compressive s t r e s s . Since the order-of-magnitude analysis ignores the transverse stresses, i t s conclusions are open to question. 5.2 C r i t e r i a f o r Improved Model In an e a r l i e r chapter, the shortcomings of previously proposed models were described. Through consideration of those e a r l i e r models, a set of c r i t e r i a f o r an imporved model was developed. This set of c r i t e r i a included a number of observations which i t was hoped could be explained by an improved model. These observations were as follows: 1. The ti s s u e pressure i s r e l a t i v e l y constant (within 5% to 10%) from the s k i n l e v e l down to the bone. There i s some tendency f o r the tissue pressure to drop with i n c r e a s i n g depth, e s p e c i a l l y with l a r g e r limbs, but the r a d i a l nerve, the nerve c l o s e s t to the bone seems to be more susceptible to i n j u r y . 1 There i s some c o n t r a d i c t i o n i n the l i t e r a t u r e concerning t h i s tendency f o r the sub-cuff pressure to decrease with i n c r e a s i n g depth. Shaw and Murray (1982) measured tissue pressure i n the legs of several cadavers and reported pressure drops of 10% to 30% depending on the g i r t h of the limb (the larger the g i r t h , the larger the pressure drop). Thomson & Doupe (1948) and McLaren & Rorabeck (1985) on the other hand, report much smaller pressure drops i n an i n vivo s i t u a t i o n . In Thomson & Doupe's experiment, i n f a c t , there were regions of s l i g h t pressure increases with depth. 116. 2. A greater applied pressure causes an increased depth of i n v a g i n a t i o n . 3. A greater a p p l i e d presure causes more a f f e c t e d nodes i n each nerve f i b e r , as well as a greater area i n which damaged nodes are found. 4. A longer time of a p p l i c a t i o n of the tourniquet causes a marked increase i n the number of nodes a f f e c t e d , but no change i n the s e v e r i t y of the damage to any p a r t i c u l a r node. 5. The observed lesions occur almost s o l e l y at the edges of the c u f f and the d i r e c t i o n of i n v a g i n a t i o n i s from the sub-cuff region out. 6. While almost a l l nerves are damaged at the proximal edge of the c u f f , a s i g n i f i c a n t l y smaller p o r t i o n of them are damaged at the d i s t a l edge. 7. Smaller nerve f i b e r s are spared while l a r g e r nerves i n the same pressure f i e l d are damaged. 8. The r a d i a l nerve, which l i e s c l o s e to the bone i n the upper limb, i s r e l a t i v e l y susceptible to damage. This work has d e t a i l e d the development of a method by which the stress/strain/displacement f i e l d s i n an a n i s o t r o p i c a l l y e l a s t i c , axisymmetric object can be determined. This method has i n fa c t been used to generate contour maps of these f i e l d s . If a connection can be established between any of these f i e l d s and the proposed mechanism of damage, then the model's p r e d i c t i o n s can be checked against the observations l i s t e d above. If the predictions l i n e up reasonably well with the observations, then the model can be used with some confidence to suggest improvements i n the design of the tourniquet. 117. 5.2.1 Connection Between A x i a l S t r a i n F i e l d and Proposed Damage  Mechanism The proposed mechanism of damage i s a x i a l compressive s t r e s s . It has been demonstrated that t h i s mechanism can l i k e l y supply s u f f i c i e n t energy to the myelin sheath to cause the damage observed, i n contrast to the other proposed mechanism. In Section 5.1 the reasons why regions of compressive a x i a l s t r e s s i n a homogeneous e l a s t i c model do not neces-s a r i l y c o r r e l a t e d d i r e c t l y with regions of ac t u a l comrpession i n the a x i a l d i r e c t i o n were o u t l i n e d . Since experimental evidence c l e a r l y shows that damaged nerves exhibit compressive deformation, i t seems reasonable to attempt to c o r r e l a t e the regions of observed damage with the regions of a x i a l l y compressive s t r a i n . 5.2.2 Comparison of P r e d i c t i o n s with Observations Before adopting the above hypothesis that the regions of a x i a l l y compressive s t r a i n c o r r e l a t e with the regions of damage as a working hypothesis f o r the redesign of the tourniquet, i t i s necessary to check that the predictions of the model, i n t e r p r e t e d according to t h i s hypo-t h e s i s , are i n accord with the seven observations described above. 1. Observation: The sub-cuff pressures are e s s e n t i a l l y a f u n c t i o n of the a x i a l p o s i t i o n ; only small changes with depth are apparent. P r e d i c t i o n : As was described i n Section 4.2.1 i n the comparison of the model's r e s u l t s with the experimental work of Thomson & Doupe, t h i s feature i s p r e d i c t e d . In f a c t , the curves of the transverse pressure plots correspond even more c l o s e l y than the plots of the h y d r o s t a t i c pressure to experimental observations. The reasons f o r t h i s were outlined i n that same s e c t i o n . 118. 2. Observation: A greater applied pressure causes a greater degree of invag i n a t i o n . P r e d i c t i o n : Since the stress f i e l d s are independent of the value of E, Young's modulus, as i s the boundary cond i t i o n of zero r a d i a l displacement at the bone, the predicted values for the s t r a i n components depend only on the value of P f o r a given max loading s i t u a t i o n ; t h i s r e l a t i o n s h i p i s a l i n e a r one. The model p r e d i c t s , therefore, that the a x i a l s t r a i n i s l i n e a r l y r e l a t e d to the applied pressure, and consequently, so i s the extent of invag i n a t i o n . 3. Observation: A greater applied pressure causes a greater number of damaged nodes i n each nerve f i b e r , as w e l l as a greater area of the ti s s u e i n which damaged nerves are found. P r e d i c t i o n : If the assumption i s made that a c e r t a i n threshold l e v e l of a x i a l s t r a i n corresponds to the onset of invagination, then the following scenario can be imagined: as the tourniquet pressure i s r a i s e d from zero, the f i r s t node to be daamged w i l l be the node c l o s e s t to the point of maximum negative a x i a l s t r a i n . As the pressure i s ra i s e d , the contour corresponding to the threshold l e v e l w i l l expand away from that f i r s t point and w i l l grow to encompass more and more nodes (see Figure 38). Since the expansion of t h i s contour i s d i r e c t e d a x i a l l y as w e l l as r a d i a l l y , more nodes i n each f i b e r w i l l be affected as the pressure increases and the area i n which damaged nerves w i l l be found w i l l increase. 4. Observation: A longer time of a p p l i c a t i o n of the tourniquet causes an increase i n the number of nodes damaged but causes no change i n the damage to any p a r t i c u l a r node. P r e d i c t i o n : The model contains 119. P low P high £ f : threshold strain * : maximum strain Figure 38. Expansion of threshold s t r a i n l e v e l with i n c r e a s i n g applied pressure 120. no time-dependent phenomena, so i t i s i n t r i n s i c a l l y unable to address t h i s observation. Speculation may be engaged i n , however, based on the proposed mechanism f o r damage. The myelin sheaths are pictured as being two tubes s l i d i n g over one another. Since they are the same s i z e to begin with and do not normally f i t together i n a deformed manner, they may behave i n a metastable manner. That i s , they may require some " t r i g g e r " , or d i s t u r b i n g force, to i n i t i a t e the process of i n v a g i n a t i o n . Once t h i s process begins, the maximum degree of i n v a g i n a t i o n w i l l be the same as that present i n the surrounding nodes. Another p o s s i b i l i t y i s that the t i s s u e behaves i n a v i s c o e l a s t i c manner. Both these p o s s i b i l i t i e s have some d i f f i c u l t y explaining the observation that, while r e l a t i v e l y l i t t l e damage may be sustained i n the f i r s t two hours of a p p l i c a -t i o n , i n the t h i r d hour there i s a marked increase i n the amount of damage; both hypotheses would suggest a l a r g e r amount of damage immediately upon i n f l a t i o n of the tourniquet, with subsequent damage being r e l a t i v e l y minor. The t r i g g e r theory may be adjusted to account for t h i s observation by hypothesizing that a f t e r some hours of c u f f a p p l i c a t i o n , some other f a c t o r becomes operative which enhances the t r i g g e r . Of ocurse, such a theory would be i n c r e a s i n g l y complex and i t s mechanisms would be d i f f i c u l t to e l u c i d a t e . This time-dependent phenomenon may well be the most d i f f i c u l t observation to account f o r . 5. Observation: The l e s i o n s are located almost s o l e l y at the edge of the cuff and the d i r e c t i o n of i n v a g i n a t i o n i s always d i r e c t e d away from the c u f f center. P r e d i c t i o n : As was pointed out i n Sections 121. 4.2.2.3 and 4.3.1, the center of the region osf negative a x i a l s t r a i n was always very close to the cuff edge; i n the case of s i n u s o i d a l loading, the center was j u s t i n s i d e the c u f f edge, while i n the case of rectangular loading, the center was just outside the c u f f edge. If the true loading i s , as suspected, intermediate between s i n u s o i d a l and rectangular, then the region of negative a x i a l s t r a i n w i l l be centered under the edge of the c u f f . The d i r e c t i o n of invagination i s more d i f f i c u l t to explain i n the context of t h i s model; while i t i s true that the sub-cuff m a t e r i a l i s being exuded from under the c u f f , that i n i t s e l f i s no reason to expect that the i n v a g i n a t i o n w i l l always occur i n one d i r e c t i o n . The distance between the myelin sheaths on e i t h e r side of a node of Ranvier i s so small that the d i f f e r e n c e i n transverse s t r e s s e s w i l l be n e g l i g i b l e and so cannot be proposed as the reasons for the d i f f e r e n c e . It may be that some mechanism r e l a t e d to the expulsion of axoplasmic f l u i d from under the cuff i s responsible for t r i g g e r -ing the i n v a g i n a t i o n i n a p a r t i c u l a r way. It i s c l e a r that there i s no predisposing geometrical f a c t o r i n the shape of the myelin sheaths on e i t h e r s i d e of the nodes of Ranvier because at both edges of the cuff the d i r e c t i o n of invagination i s away from the sub-cuff region. Again, t h i s w i l l be a very d i f f i c u l t observation to explain. 6. Observation: A s i g n i f i c n a t l y greater proportion of the nerves are damaged at the proximal edge of the c u f f than at the d i s t a l edge. P r e d i c t i o n : The model presented i n t h i s paper has been formulated as being symmetric i n z; i t i s therefore unable to address t h i s 122. observation. This phenomenon may be r e l a t e d to the d i f f e r e n t shapes of the muscle masses at the proximal and d i s t a l edges of the c u f f ; a more d e t a i l e d look at t y p i c a l limb shapes e s p e c i a l l y close to the ends of the limb, would be needed to assess t h i s prob-a b i l i t y . It should be noted that the proximal edge of the c u f f i s les s l i k e l y to s u f f e r ischaemia since i t i s adjacent to the unoccluded part of the body; i f ischaemia were the operative mechanism here, i t would be expected that the d i s t a l end would s u f f e r the greatest amount of damage. 7. Observation: Smaller, unmyelinated nerve f i b e r s are spared, while l a r g e r , myelinated f i b e r s i n the same stress f i e l d s are damaged. P r e d i c t i o n : The order-of-magnitude c a l c u l a t i o n showed that nodal displacements were d i r e c t l y r e l a t e d to the radius. Larger neurons, therefore, are predicted to experience l a r g e r nodal displacements i n the same stress f i e l d s . 8. Observation: The r a d i a l nerve shows p a r t i c u l a r s u s c e p t i b i l i t y to damage by applied tourniquet pressue. P r e d i c t i o n : The region of negative a x i a l s t r a i n has i t s greatest magnitude at roughly h a l f the t i s s u e thickness above the bone; the model would not, therefore, predict any p a r t i c u l a r s u s c e p t i b i l i t y of the r a d i a l nerve unless some c h a r a c t e r i s t i c of that nerve made i t more susceptible to damage than another nerve (such as a greater percentage of motor neurons i n the r a d i a l nerve than i n other nerves. 123. In summary, the model developed i n t h i s paper p r e d i c t s s e v e r a l observations q u i t e w e l l , i n c l u d i n g the h y d r o s t a t i c pressure d i s t r i b u t i o n under the cu f f , the increase i n depth of invagination with increased pressure, the increase i n both the number of nodes a f f e c t e d and the t o t a l area i n which damaged nodes would be found with increased pressure, the sparing of smaller neurons, and the l o c a t i o n of the les i o n s at the edge of the c u f f . Several obervations were unable to be addressed by the model; these included the increase i n the number of damaged nodes with time, the d i r e c t i o n of invagination, and the d i f f e r e n t f r a c t i o n s of damaged nerves at the proximal and d i s t a l edges of the c u f f . None of these observations are incompatible with the proposed model, and suggestions were made which might explain these phenomena within the context of the model. F i n a l l y , the s u s c e p t i b i l i t y of the r a d i a l nerve reported by G r i f f i t h s and Heywood i s not supported. If t h e i r claim that the r a d i a l nerve i s more su s c e p t i b l e than other nerves i s cor r e c t , then the proposed model f a i l s to predit t h i s . On the whole, the proposed model seems more capable than Auerbach's of being used i n cuff redesign as the connection between the regions of negative a x i a l s t r a i n and nerve damage i s more p l a u s i b l e . Many important observations are explained, and several others are not incompatible with the model, so i t i s reasonable to assume that the model w i l l be of some use i n cuff redesign. 124. 5.3 Cuff Design Recommendations 5.3.1 Other Factors A f f e c t i n g Use as Design Tool 5.3.1.1 Neglect of Shear Stress Applied at Outer Surface In modelling the loading d i s t r i b u t i o n applied by the tourniquet, only r a d i a l pressure loads were considered. It i s extremely l i k e l y that i n a d d i t i o n to the r a d i a l pressure loads, shear stresses are applied by the tourniquet to the limb. Again, time c o n s t r a i n t s precluded an i n v e s t i g a t i o n into the e f f e c t s of such loadings. Some q u a l i t a t i v e comments can nevertheless be made concerning the e f f e c t s of shear stress applied to the top of the limb. In dis c u s s i n g the e f f e c t s of the assumption of a f r e e or s t i c k y i n t e r a c t i o n between the t i s s u e and bone at t h e i r i n t e r f a c e , i t was noted that a s t i c k y i n t e r a c t i o n would r e s u l t i n both higher a x i a l stresses and higher a x i a l s t r a i n s , presumably because the ti s s u e was prevented from simply moving l a t e r a l l y from under the c u f f and so compensating f o r the app l i e d load; instead, the material was forced to compress i n the a x i a l d i r e c t i o n to r e s i s t the applied load. Neglecting any p o s s i b l e shear st r e s s e s a p p l i e d at the top of the limb i s analogous to assuming free movement at the bone/tissue i n t e r -face. Since i t i s reasonable to assume that the f r i t i o n a l f o r c e between the tournqiuet and the limb w i l l prevent free movement, shear stresses r e s i s t i n g l a t e r a l movement of the t i s s u e can be expected. This would presumably serve to increase the magnitude of the a x i a l s t r a i n i n the same way that shear s t r e s s at the bone/tissue i n t e r f a c e would. The e f f e c t of th i s neglect of the shear stress on the usefulness of the model as a design t o o l i s d i f f i c u l t to estimate. In any case, 125. ignoring the presence of the shear s t r e s s i s not l i k e l y to lead to an overestimate of the magnitude of the a x i a l s t r a i n . 5.3.1.2 Presence of Large S t r a i n s In c e r t a i n t e s t s , very large s t r a i n s are observed. In RSNW, f o r example, a x i a l s t r a i n s of 0.3 are c a l c u l a t e d f o r a P /E r a t i o of 1. max This r a t i o i s t y p i c a l l y l a r g e r than that; values of 2 or 3 would not be uncommon. Since the s t r a i n s are d i r e c t l y p r o p o r t i o n a l to t h i s r a t i o , the c a l c u l a t e d s t r a i n s would be on the order of 0.5 to 1. These are most c e r t a i n l y not i n f i n i t e s i m a l deformations and, consequently, the small deformation assumptions do not apply. Without a greater f a m i l i a r i t y with large deformation problems, i t i s d i f f i c u l t to suggest the probable e f f e c t s on the s o l u t i o n of t h i s problem of the presence of l a r g e r deformations, so any conclusions based on t h i s model should be held only t e n t a t i v e l y i f the applied pressure i s r e l a t i v e l y high. 5.3.1.3 Limited Form of Anisotropy A v a i l a b l e f o r C a l c u l a t i o n s Because of the d i f f i c u l t i e s mentioned e a r l i e r i n connection with the s o l u t i o n of the separated equations with complex c o e f f i c i e n t s , only a l i m i t e d range of anisotropy could be considered i n the context of t h i s model. Unfortunately, the e l a s t i c c o e f f i c i e n t s which most accurately r e f l e c t the e l a s t i c propeties of t i s s u e define a problem which was not solvable by the method o r i g i n a l l y proposed. In order to carry out an a n a l y s i s , therefore, a q u a s i - i s o t r o p i c case was studied. The v a l i d i t y of the r e s u l t can be questioned i f i t i s f e l t that the s o l u t i o n i s s e n s i t i v e to the choice of e l a s t i c constants. 126. 5.3.1.3.1 S e n s i t i v i t y of S o l u t i o n to Changes i n Anisotropy Several tests were c a r r i e d out to determine the s e n s i t i v i t y of the s o l u t i o n to the d i f f e r e n t e l a s t i c parameters. It was found that doubling t h e r a t i o of t h e transverse e l a s t i c modulus to the a x i a l e l a s t i c modulus produced l i t t l e change i n the shape of any of the p l o t s , and no change i n the magnitude of the maximum negative a x i a l s t r a i n . The r a t i o of e l a s t i c moduli was therefore judged to be r e l a t i v e l y i n s i g n i f i c a n t . A second t e s t involved changing the value of Poisson's r a t i o from 0.45 to 0.25. Although t h i s change did not s i g n i f i c a n t l y a f f e c t the shapes of any of the contour maps, the enhanced c o m p r e s s i b i l i t y of the m a t erial resulted i n a 50% drop i n the magnitude of the a x i a l s t r e s s f i e l d and the a x i a l s t r a i n f i e l d . The value of Poisson's r a t i o was judged to be very s i g n i f i c a n t . Two more t e s t s i n v e s t i g a t e d the i n f l u e n c e of changes i n the value of the r a t i o of Poisson r a t i o s when the a x i a l Poisson r a t i o was set at 0.5. It was found that changes i n t h i s r a t i o had l i t t l e or no e f f e c t on the value of the maximum negative a x i a l s t r a i n and on the shapes of the contour maps. The only noticeable e f f e c t was i n the case of = 2 i n which the r a d i a l stress contour l i n e s resembled more the l i n e s i n the t e s t with v = 0.45 than the t e s t with v = 0.25. Because changes i n t h i s parameter caused l i t t l e change i n the a x i a l s t r a i n f i e l d s , i t was judged to be r e l a t i v e l y i n s i g i f i c a n t . One t e s t was attempted with a changed value f o r G^. This t e s t s did not give p h y s i c a l l y r e a l i s t i c r e s u l t s because i t was discovered that the p a r t i c u l a r choice of c o e f f i c i e n t s made r e s u l t e d i n complex separated equations, which were not solvable by the techniques developed i n the 127. e a r l i e r a n a l y s i s . No judgment as to the s i g n i f i c a n c e of t h i s parameter could therefore be formed. To summarize, the only parameter judged s i g n i f i c a n t was the value of Poisson's r a t i o as measured when a x i a l tension i s applied. This judgment lends credence to the d e c i s i o n to use the q u a s i - i s o t r o p i c case as the basis for a study of the e f f e c t s of the tourniquet parameters on the a x i a l s t r a i n f i e l d . It should be noted, though, that only doublings or halvings of the q u a s i - i s o t r o p i c values were considered; i f the aniso-t r o p i c values of r e a l t i s s u e are s i g n i f i c a n t l y d i f f e r e n t than those considered here, i t may be b e n e f i c i a l to repeat t h i s a n a l y s i s for a broader range of e l a s t i c constants. 5.3.1.3.2 D i f f i c u l t y i n A s c e r t a i n i n g 2-D M a t e r i a l Properties The question of the s e n s i t i v i t y of the s t r e s s / s t r a i n f i e l d s to changes i n the e l a s t i c constants i s important because very l i t t l e information i s a v a i l a b l e on the two-dimensional mechanical p r o p e r t i e s of t i s s u e , e s p e c i a l l y i n i n vivo s i t u a t i o n s . If i t can be assserted with a reasonable amount of confidence that most of the a n i s o t r o p i c p r o p e r t i e s do not s e r i o u s l y influence e i t h e r the shape or magnitude of the a x i a l s t r a i n f i e l d , then only rough estimates of the e l a s t i c parameters are necessary to give a reasonable estimate of the s t r a i n f i e l d . 5.3.1.4 D i f f i c u l t i e s i n S o l u t i o n of F i e l d Equation The f i e l d equation may be solved by the a n a l y t i c a l technique described i n t h i s paper i f the problem i s defined over a rectangular region. If i t i s desired to i n v e s t i g a t e the e f f e c t of a tourniquet on a tapered limb, for example, then another s o l u t i o n method must be used. 128. Numerical techniques would seem to be a u s e f u l set of methods to consider. They would not be l i m i t e d by the presence of a non-rectangular region. In a d d i t i o n , they would be able to handle a r b i t r a r y loadings d i r e c t l y , without being required to sum the solutions to sev e r a l problems as the a n a l y t i c a l technique does. The major d i f f i c u l t y with using numerical techniques on t h i s problem i s that the boundary conditions are given i n terms of high order d e r i v a t i v e s of the stress f u n c t i o n , and consequently, the r e s u l t i n g set of equations i s not diagonally dominant; simple i t e r a t i v e s o l u t i o n methods w i l l not converge. This s i t u a t i o n requires extreme care i n s o l u t i o n . 5.3.2 General Design Recommendations Since most of the e l a s t i c parameters were judged to be r e l a t i v e l y i n s i g n i f i c a n t i n t h e i r e f f e c t s upon the a x i a l s t r a i n f i e l d , i t was f e l t that cautious recommendations could be made concerning the redesign of the pneumatic tourniquet c u f f . Three general considerations o b t a i n here; the f i r s t r e l a t e s to the shape of the loading d i s t r i b u t i o n , the second to the width of the c u f f , and the t h i r d to the presence or absence of shear s t r e s s between the cuff and the s k i n . F i r s t , i t i s c l e a r from the d i s c u s s i o n i n Section 4.3.1 on the e f f e c t of the loading d i s t r i b u t i o n shape that i t i s better to have a s i n u s o i d a l loading pattern than a rectangular one. This conclusion i s strongest f o r thin-limbed people. For such limbs, the s i n u s o i d a l d i s t r i b u t i o n generated a maximum compressive a x i a l s t r a i n of only 50% 129. the magnitude of that generated by a rectangular d i s t r i b u t i o n . For l a r g e r limbed people, the reduction i n maximum s t r a i n was roughly 20%. Second, from the d i s c u s s i o n i n Section A.3.2 concerning the e f f e c t s of d i f f e r e n t cuff widths, i t can be concluded that, i n general, a wider c u f f reduces the maximum compressive a x i a l s t r a i n . This e f f e c t i s greatest i n large-limbed people, with possible reductions of 50% from the a x i a l s t r a i n s produced by a narrow c u f f . In thin-limbed people, the reductions are on the order of 25%. This recommendation should be considered to have the most weight i n regions where the limb can s t i l l be considered c y l i n d r i c a l ; i t i s not c l e a r that making the cuff so wide that i t reaches to the j o i n t s at e i t h e r end of the muscle mass i t i s applied to would be b e n e f i c i a l . F i n a l l y , the d i s c u s s i o n i n Section 5.3.1.1 on the p o s s i b l e e f f e c t s of ignoring the shear stress applied by the f r i c t i o n force operating between the c u f f and the s k i n suggests that such stresses increase the magnitude of the maximum negative a x i a l s t r a i n . If t h i s i s so, then one way to lower t h i s maximum s t r a i n would be to f i n d a way to e l i m i n a t e the transmission of shear stresses between the cuff and the s k i n . 5.4 Summary of D i r e c t i o n s f o r Future Work There are s e v e r a l d e t a i l s which would be u s e f u l to c l e a r up, none of which require techiques more s o p h i s t i c a t e d than those found i n t h i s work. These are summarized here: 1. The order-of-magnitude c a l c u l a t i o n f o r small nerve f i b e r s ought to be done to determine whether or not t h i s mechanism predicts that such nerves are more l i k e l y to be spared. 130.-2. The tendency of the s o l u t i o n to the rectangular loading problem when the number of terms i n the Fourier decomposition r i s e s above 13 or so to e x h i b i t l a r g e d e v i a t i o n s from the s o l u t i o n s to the lower order approximations ought to be i n v e s t i g a t e d . 3. The behaviour of the s o l u t i o n to the q u a s i - i s o t r o p i c problem as isotropy i s approached should be studied i n l i g h t of the s t r a i n c i r c u m f e r e n t i a l s t r e s s and s t r a i n and r a d i a l displacement and s t r a i n maps seen i n Figures 23 and 24. 4. A d i f f e r e n t t e s t should be run with a l t e r e d values f o r the e l a s t i c constant G to determine the s e n s i t i v i t y of the s o l u t i o n to changes r i n t h i s parameter. 5. The e f f e c t s of a shear s t r e s s a p p l i e d at the s k i n l e v e l should be i n v e s t i g a t e d . There are several other d i r e c t i o n s t h i s work could go i n which are s u b s t a n t i a l l y more involved e i t h e r t h e o r e t i c a l l y or experimentally. For example, i t would be u s e f u l to have an experimentally-based idea of what the loading d i s t r i b u t i o n imparted by a tournqiuet to a limb looks l i k e . The predictions of t h i s model should be tested i n an experimental i n v e s t i g a t i o n to determine i f the c u f f design recommendations are i n fa c t c o r r e c t . Such an i n v e s t i g a t i o n could also determine whether any reduction i n the s e v e r i t y of nerve damage due to an improved c u f f design was s i g n i f i c a n t enough to preclude the need for further work on t h i s problem. I f f u r t h e r work turns out to be warranted, then i t w i l l be d e s i r -able to generalize the s o l u t i o n method to non-rectangular domanis and perhaps to r e l a x the c o n d i t i o n of symmetry with respect to z. This l a s t 131. step w i l l open the way to understanding why d i f f e r e n t f r a c t i o n s of nerve f i b e r s are a f f e c t e d at the proximal and d i s t a l edges of the tourniquet. One of the major questions not addressed by t h i s model i s the time-dependence of the s o l u t i o n . I t may be u s e f u l to model the limb m a t e r i a l as a l i n e a r l y v i s c o e l a s t i c m a t e r i a l to get some idea of the i n f l u e n c e of such time-dependent parameters. Three other improvements i n modelling the t i s s u e p r o p e r t i e s would be, f i r s t , to consider the t i s s u e to be capable of large deformations, second, to onsider the t i s s u e to be n o n l i n e a r l y e l a s t i c , and t h i r d , to treat the t i s s u e as being compsed of two separate l a y e r s , one of f a t and one of muscle. I t was mentioned i n Chapter 3 i n the d i s c u s s i o n on modelling that muscle t i s s u e i s subject to nervous c o n t r o l ; a f i r s t step i n accounting f o r that e f f e c t i s to consider the muscle to be i n a s t a t e of i n i t i a l s t r e s s , as even a relaxed muscle has some tone. As a f i n a l suggestion f o r future work, i t should be noted that there i s a s c a r c i t y of experimetnal data a p p l i c a b l e to t h i s problem. The next major piece of work should be experimental i n nature and should seek to prove or disprove the major ideas posited i n t h i s paper. Only then should f u r t h e r t h e o r e t i c a l work be done, l e s t the gap between what we know and what we think become too l a r g e . 132. CHAPTER 6 CONCLUSIONS The major conclusions of t h i s work are summarized below: 1. I n t e r p r e t a t i o n of Ochoa et a l . ' s Work a) The observed l e s i o n s are probably due to mechanical deformation. b) The operative damage mechanism i s q u i t e p o s s i b l y an a x i a l compressive stress mechanism, i n c o n t r a d i c t i o n to Ochoa et a l . ' s suggestion that some f l u i d phenomenon i s responsible. This proposal i s supported by an order-of-magnitude a n a l y s i s . 2. C r i t i q u e of E a r l i e r Models a) G r i f f i t h s and Heywood's model was too l i m i t e d to be u s e f u l because of i t s one-dimensional formulation. b) Auerbach's model was reasonably adequate, but he did not make c l e a r the connection between the q u a n t i t i e s h i s model predicted and the mechanism of nerve damage, nor d i d he consider a n i s o t r o p i c m a t e r i a l p r o p e r t i e s . 3. Modelling of Problem An a n a l y t i c a l s o l u t i o n technique i s computationally f e a s i b l e for a problem defined over a rectangular region i n r and z. 4. Evaluation of Model P r e d i c t i o n s a) A comparison of the predicted s t r e s s f i e l d s with an e x p e r i -mental study by Thomson and Doupe provides some basis f o r concluding that r e l a t i v e motion at the bone/tissue i n t e r f a c e i s u n l i k e l y . 133. b) The loading d i s t r i b u t i o n imposed by a r e a l tourniquet i s l i k e l y a cross between a rectangular and a s i n u s o i d a l d i s t r i b u t i o n . c) Ignoring the shear s t r e s s at the s k i n due to the tournqiuet w i l l not lead to an overestimate of the negative a x i a l s t r a i n . d) The maximum negative a x i a l s t r a i n i s a reasonable i n d i c a t o r of regions of damage from both a c o r r e l a t i v e and causal point of view. e) The proposed model explains most of the important c l i n i c a l and experimental model. f ) The model does not d i r e c t l y p r e d i c t the r e l a t i v e s u s c e p t i -b i l i t y of the r a d i a l nerve. g) The magnitudes of points i n the a x i a l s t r a i n f i e l d are s e n s i -t i v e to changes i n v , Poisson's r a t i o , and E, Young's modulus, but are r e l a t i v e l y i n s e n s i t i v e to the other e l a s t i c constants. The s e n s i t i v i t y to cannot be reported. h) The model i s us e f u l i n redesigning the tourniquet c u f f . 5. Cuff Design a) A smoother loading d i s t r i b u t i o n reduces the magnitude of the maximum negative a x i a l s t r a i n ; t h i s e f f e c t i s greatest f o r th i n limbs. b) An increase i n cu f f width reduces the magnitude of the maximum negative a x i a l s t r a i n ; t h i s e f f e c t i s greatest f o r thick limbs. c) A decrease i n the shear s t r e s s between the limb and the tourniquet cuff reduces the magnitude of the maximum negative a x i a l s t r a i n ; t h i s e f f e c t was not q u a n t i f i e d . 134. Thicker limbs experience a lower magnitude of the maximum negative a x i a l s t r a i n than thinner limbs loaded i n the same manner. If a l l the design recommendations are followed, the magnitude of the maximum negative a x i a l s t r a i n can be reduced by 50-70%, with the greatest reduction being i n thinner limbs. 135. BIBLIOGRAPHY Auerbach, Steven M. (1984). Axisymmetrlc f i n i t e element a n a l y s i s of tourniquet a p p l i c a t i o n on limb. J . Biomechanics, 17, 861-866. Bentley, F.H. and Schlapp, W. (1943). The e f f e c t of pressure on conduction i n p e r i p h e r a l nerve. Journal of Physiology (London), 102, pp. 72-82. Bruner, J u l i a n M. (1970). Time, pressure and temperature f a c t o r s i n the safe use of the tourniquet. Hand, 2, 39-42. Chow, W.W. and Od e l l , E.I. (1978). Deformations and stresses i n s o f t body t i s s u e s of a s i t t i n g person. J . Biomech. Engineering, 100, 79-87. Denny-Brown, D. and Brenner, C. (1944a). P a r a l y s i s of nerve induced by d i r e c t pressure and by tourniquet. Archives of Neurology and  Physiology, 51, 1-26. Denny-Brown, D. and Brenner, C. (1944b). Lesion i n pe r i p h e r a l nerve r e s u l t i n g from compression by spring c l i p . Archives of Neurology  and Psychiatry (Chicago, 52, pp. 1-19. Fowler, T.J., Danta, G., G i l l l a t t , R.W. (1972). Recovery of nerve conduction a f t e r the pneumatic tourniquet. Journal of Neurology,  Neurosurgery and Psychiatry, 35, pp. 638-47. G r i f f i t h s , J.C. and Heywood, O.B. (1973). Bio-mechanical aspects of the tourniquet. Hand, 5, 113-118. Grundfest, H. and McKeen, C a t t e l l . (1935). Some e f f e c t s of h y d r o s t a t i c pressure on nerve a c t i o n p o t e n t i a l s . Americal J . of Physiology, 113, 56-67. Hurst, L.M., Weiglein, 0., Brown, W.F. and Campbell, G.J. The pneumatic tourniquet: a biomechanical and e l e c t r o p h y s i o l o g i c a l study. P l a s t i c  and Reconstructive Surgery, 67(5), pp. 648-52. Klenerman, L e s l i e . (1980). Tourniquet time — how long? Hand, 12, 321-234. Lewis, T., P i c k e r i n g , G.W. and Rothschild, P. (1931). C e n t r i p e t a l p a r a l y s i s a r i s i n g out of arrested blood flow to the limb. Heart, 16, pp. 1-32. McEwen, J.A. (1980). Complications of and improvements i n pneumatic tourniquets used i n surgery. Medical Instrumentation, 15, 253-257. McLaren, A.C. and Rorabeck, C H . (1985). The pressure d i s t r i b u t i o n under tourniquets. J . Bone and J o i n t Surgery, 67, 433-437. 136. Mase, George E. (1970). Theory and Problems of Continuum Mechanics. Schaumm's Outline S e r i e s . McGraw-Hill, New York. Ochoa, J . , Fowler, T.J. and G i l l i a t t , R.W. (1972). Anatomical changes i n p e r i p h e r a l nerves compressed by a pneumatic tourniquet. J .  Anat., 113, 433-455. Rasminsky, M. and Sears, T.A. (1972). Internodal conduction i n undissected demyelinated nerve f i b e r s . J . of Physiology (London), 227, pp. 323-350. Shaw, James A. and Murray, David G. (1982). The r e l a t i o n s h i p o between tourniquet pressure and underlying s o f t - t i s s u e pressure In the thigh. J . Bone and J o i n t Surgery, 64-A, 1148-1152. Timoshenko, S. (1956). Strength of M a t e r i a l s . Third E d i t i o n . Von Nostrand, New York. Timoshenko, S. and Goodier, J.N. (1970). Theory of E l a s t i c i t y . Third e d i t i o n . McGraw-Hill, New York. Thomson, A.E. and Doupe, J . (1949). Some f a c t o r s a f f e c t i n g the ausculatory measurement of a r t e r i a l blook pressures. Can. J . of  Research, 27E, 72-80. Vander, Sherman, Luciano. Human Physiology. Further d e t a i l s unrecorded. Yanada, H. (1970). Strength of B i o l o g i c a l M a t e r i a l s . Williams and Wilkins Co., Baltimore. Ziegart, Lewis (1978). referenced by Auerbach (1984). Further d e t a i l s unrecorded. APPENDIX A MECHANISMS OF IMPULSE TRANSMISSION APPENDIX A 138. MECHANISMS OF IMPULSE .TRANSMISSION P a r a l y s i s and lack of sensation occur because nerve impulses are blocked by some means. Anaesthetics such as those that d e n t i s t s use work by chemically i n h i b i t i n g the transmission of impulses from the sensory organs to the neurons, by i n h i b i t i n g the transmission from one neuron to another, or by i n h i b i t i n g the changes i n the structure of the c e l l membrane necessary f o r the transmission of impulses. Since the a p p l i c a t i o n of a tourniquet does not involve anaesthetics, i t i s neces-sary to understand how impulses can be blocked by non-chemical means. In a d d i t i o n , since tourniquets t y p i c a l l y stop transmission i n the region under the tourniquet (Hurst, 1980), where there are no synapses, i t w i l l be h e l p f u l to look at how transmission can be disrupted part way along the axon. Before t h i s i s done, i t must be understood how nerve impulses are transmitted i n the normal s t a t e . As might be expected, nerve transmission i s an e l e c t r i c a l pheno-menon. The neuron's i n t e r i o r chemical composition i s d i f f e r e n t from the composition In the e x t r a c e l l u l a r f l u i d and i t i s maintained that way by a semi-permeable c e l l membrane. The major d i f f e r e n c e i n composition that i s relevant here i s the d i f f e r e n c e i n potassium and sodium i o n concentration. The i n t e r i o r of the c e l l has a much higher concentration of potassium than the e x t r a c e l l u l a r space, while the reverse i s true f o r sodium. In addition, the membrane i s 50-75 times more permeable to potassium than to sodium. As a r e s u l t , p o s i i v e l y charged potassium ions from the i n s i d e of the c e l l d i f f u s e out of the c e l l , thereby rendering the outside of the c e l l p o s i t i v e with respect to the i n s i d e (see Figure Figure A l . Compositional d i f f e r e n c e s between c e l l i n t e r i o r and e x t e r i o r Note s e l e c t i v e permeability of membrane, p o t e n t i a l d i f f e r e n c e , and concentration d i f f e r e n c e . 140. A l ) . The net outflow of potassium ions stops when the e l e c t r o s t a t i c gradient i s s u f f i c i e n t to match the d r i v i n g force due to the d i f f u s i o n gradient (see Figure A 2 ) 1 . When t h i s occurs, the membrane p o t e n t i a l i s at a value of approximately -70 mV, with the i n s i d e of the c e l l being negative with respect to the e x t r a c e l l u l a r f l u i d . The above d e s c r i p t i o n sets the stage f o r considering the transmis-s i o n of nerve impulses. Information i s sent along an axon i n the form of membrane p o t e n t i a l s which d i f f e r from the r e s t i n g value. These changes p o t e n t i a l s f a l l i n t o two categories: 1) graded potentials f o r short distance s i g n a l l i n g , and 2) action potentials f o r long distance transmission. The mechanisms f o r these two types of p o t e n t i a l s are s i m i l a r , so i t w i l l be h e l p f u l to consider how the c e l l generates and transmits graded p o t e n t i a l s . A graded p o t e n t i a l begins when some event external to the neuron occurs. This external event causes a l o c a l change i n the membrane p o t e n t i a l , e i t h e r p o s i t i v e or negative from the r e s t i n g value, which i s us u a l l y i n proportion to the l e v e l of the stimulus. Assume f o r the purpose of di s c u s s i o n that the change i s +5 mV so that i n the v i c i n i t y of the stimulus, the membrane p o t e n t i a l i s -65 mV. At the s i t e of the In r e a l i t y , t h i s e q u i l i b r i u m i s never t r u l y attained because the membrane i s not completely impermeable to sodium and the d i f f u s i o n of the sodium ions sets up an opposing e l e c t r o s t a t i c gradient. In the absence of sodium ions and at p h y s i o l o g i c a l concentrations, the eq u i l i b r i u m p o t e n t i a l for potassium would be approximately -90 mV r e f e r r e d to the e x t r a c e l l u l a r p o t e n t i a l . In the presence of sodium, the membrane p o t e n t i a l i s a c t u a l l y c l o s e r to -70 mV. This r e s u l t s i n a s l i g h t net emigration of potassium ions which i s made up f o r by an ac t i v e transport mechanism which moves potassium ions back i n t o the c e l l against the concentration gradient. This c o n t r o l l i n g mechanism allows a dynamic eq u i l i b r i u m to be reached. e x t e r i o r i n t e r i o r OrnV 35 mV 7 0 mV Figure A2. Increasing p o t e n t i a l gradient opposes d i f f u s i o n of K. due to concentration gradient 142. stimulus, on the i n s i d e of the c e l l , the p o t e n t i a l i s r e l a t i v e l y higher than at the nearby regions. Assume furth e r that the cause of t h i s increased p o t e n t i a l i s an increase of p o s i t i v e ions on the i n s i d e of the c e l l . Accordingly, current w i l l begin to flow away from the stimulus s i t e on the i n s i d e of the c e l l i n an attempt to re s t o r e the r e s t i n g p o t e n t i a l . As the efects of t h i s migration are f e l t f u r t h e r a f i e l d , the membrane p o t e n t i a l of the adjacent regions i s changed a l s o . In t h i s way, the p o t e n t i a l i s propagated b i - d i r e c t i o n a l l y away from the s i t e of stimulus (see Figure A3). The i n f l u e n c e of such a p o t e n t i a l i s l i m i t e d to a few m i l l i m e t e r s , however, f or two reasons. The f i r s t i s simply the charge d i s p e r s a l ; the introduced charge spreads out along the i n t e r i o r of the c e l l and concequently the l o c a l p o t e n t i a l approaches the o r i g i n a l value. Furthermore, the membrane i s not completely impermeable to charge; i t i s , i n a sense, leaky. The potassium ions f i n d a lowered e l e c t r o s t a t i c p o t e n t i a l opposing t h e i r d i f f u s i o n i n t o the e x t r a c e l l u l a r f l u i d and, i n t h e i r d i f f u s i o n , they carry p o s i t i v e charge across the membrane which helps to restore the r e s t i n g p o t e n t i a l . An a c t i o n p o t e n t i a l i s q u a l i t a t i v e l y d i f f e r e n t than a graded p o t e n t i a l i n that the generated s i g n a l does not die out as i t t r a v e l s along the axon, nor does the amplitude of the s i g n a l carry information. Rather, the frequency of emission of the a c t i o n p o t e n t i a l s c a r r i e s the information. It i s s i m i l a r to a graded p o t e n t i a l , though, i n the way that i t begins. Again, an external event causes a l o c a l change i n the membrane p o t e n t i a l . I f , however, the magnitude of t h i s change i s greater than a c e r t a i n threshold value (u s u a l l y +5 to +15 mV) then 143. +5mV p o s i t i o n r e l a t i v e to s t i m u l u s Figure A3. Time course of graded p o t e n t i a l +IOOmV p o s i t i o n r e l a t i v e to s t i m u l u s Figure A4. Time course of a c t i o n p o t e n t i a l 144. another mechanism becomes operative; the permeability of the membrane to sodium begins to increase. This begins a p o s i t i v e feedback c y c l e i n which the increased perme-a b i l i t y to sodium r e s u l t s i n a net i n f l u x of sodium ions which i n turn makes the i n s i d e of the c e l l l e s s negative with respect to the extra-c e l l u l a r space. This increased p o t e n t i a l causes an increase i n the permeability to sodium and the c y c l e continues u n t i l , i n the space of a f r a c t i o n of a m i l l i s e c o n d , the permeability to sodium has incrased 500-f o l d and the membrane p o t e n t i a l has r i s e n from -70 mV to +30 mV! At t h i s point, another mechanism deactivates the sodium permeability for a c e r t a i n time (known as the r e f r a c t o r y period) and a c t i v a t e s a t e n - f o l d increase i n the permeability to potassium. This causes a rapid drop i n the membrane p o t e n t i a l back down to the r e s t i n g l e v e l . The r e a l l y important event, i n terms of impulse transmission, which occurs i n t h i s c y c l e i s the e x c i t a t i o n of the adjacent regions of the membrane. The membrane on both sides of t h i s stimulus experiences a l o c a l current flow s u f f i c i e n t to change the p o t e n t i a l by at l e a s t i t s threshold value, thus t r i g g e r i n g the same action-potential-generating c y c l e described above. Since the o r i g i n a l s i t e i s i n i t s r e f r a c t o r y period, the a c t i o n p o t e n t i a l a r i s i n g adjacent to i t cannot stimulate i t again, so the d i r e c t i o n of propagation i s away from the o r i g i n a l s i t e i n both d i r e c t i o n s (see Figure A4). Without t h i s r e f r a c t o r y period, a s i n g l e stimulus would give r i s e to a never-ending s e r i e s of a c t i o n p o t e n t i a l s t r a v e l l i n g i n both d i r e c t i o n s . I t i s important to note that because of the a l l - o r - n o t h i n g response of the membrane, an a c t i o n p o t e n t i a l i s transmitted undistorted; i t has the same strength at the end of i t s t r a v e l s as i t did at the beginning. In t h i s way, information 145. can be transmitted from the extremities of the body to the c e n t r a l nervous system. The l a s t point to be covered i n t h i s d i s c u s s i o n i s the f u n c t i o n of the myelin sheath. Two f a c t o r s a f f e c t the speed of propagation of an a c t i o n p o t e n t i a l : the axon diameter and whether or not i t i s myelinated. If the axon diameter i s large, then the r e s i s t a n c e to current flow i s lower and the regions adjacent to an a c t i o n p o t e n t i a l are brought to suprathreshold p o t e n t i a l s more r a p i d l y . If the axon i s myelinated, then the neuron behaves l i k e an insu l a t e d wire; i t i s no longer 'leaky'. Since le s s current flow out through the membrane, points f u r t h e r away can be brought to the threshold p o t e n t i a l i n the same amount of time as a nearer point i n a non-myelinated neuron. Action p o t e n t i a l s cannot occur along a p o r t i o n of the axon insulated by myelin; instead, they are generated at the points where the membrane i s exposed to the e x t r a c e l l u l a r f l u i d , the nodes of Ranvier. These are located at regular i n t e r v a l s along the axon and the a c t i o n p o t e n t i a l appears to leap from one node to the next as an a c t i o n p o t e n t i a l at one node t r i g g e r s the next one i n the sequence. The combination of a large diameter neuron and myelination i s potent; a c t i o n p o t e n t i a l s can be propagated at speeds greater than 400 km/h. APPENDIX B COMPARISON OF PROPOSED MECHANISMS 147. APPENDIX B  COMPARISON OF PROPOSED MECHANISMS The mechanisms which have been proposed to account f o r the nerve damage observed by Ochoa e_t a l . are the f o l l o w i n g : 1) The higher shear forces generated at the narrowing of the node of Ranvier are responsible for dragging the myelin sheath on one si d e of the node i n t o the other. 2) The pressure gradient a c t i n g on the annulus formed by the narrowing at the node of Ranvier would force the myelin sheath on one side of the node i n t o the other. 3) A x i a l compressive s t r e s s a c t i n g on the nerve i s high enough to force the myelin sheath on one side of the node in t o the other. These mechanisms w i l l be compared by a s i m p l i f i e d a n a l y s i s which w i l l hopefully y i e l d an order-of-magnitude estimate of t h e i r p o t e n t i a l for causing the damage claimed. A t y p i c a l neuron w i l l have a myelin sheath approximately 0.1 times i t s diameter. Its Young's modulus w i l l be r e l a t i v e l y independent of s i z e and we can take E = 15 kPa f o r a l l neuron diameters ( f o l l o w i n g Chow's work (1978) on soft t i s s u e s ) . We t r e a t the invaginating myelin sheathes as l i n e a r l y e l a s t i c tubes and assume n e g l i g i b l e changes i n the wall thickness on deformation. We a l s o neglect any e f f e c t s at the ends of the tubes. 148. Undeformed State: Deformed State: (one tube i n s i d e the other) O r i g i n a l circumference: 2irr Change i n circumference: 2Trr/10 S t r a i n : E - ^ r / 1 0 m Q 2irr The hoop s t r e s s , a, i s given by Ee a = Ee = 15,000 * 0.1 = 1500 N/m2 149. The bearing pressure can be c a l c u l a t e d p * diameter 1 = a * ( 2*wall thickness) d i a m e t e r of deformed sheathes The r a d i a l force per unit length i s given by: _ _ 4 i r „ , Fr = Zurp = — r a N/m Two mechanisms which would r e s i s t i n v a g i n a t i o n are the resistance to d i s t o r t i o n and the f r i c t i o n a l f o r c e between the o v e r r i d i n g myelin sheaths. The energy required to override these two mechanisms can be estimated as f o l l o w s . 150 Consider f i r s t the energy required to expand one myelin sheath a compress the other. The energy per unit volume i s given by 1 E e 2 15,000(0.I) 2 _ c J u - - f f e « _ i _ 75 — m3 The volume of compressed and expanded material i s given by V = 2-irdtJt = 0.8 -rrr2£ m3 The t o t a l energy of d i s t o r t i o n i s , therefore, U = uV = 0.8 -nr2l (75) a = 60 irr2£ J The energy required to overcome f r i c t i o n can be estimated i f the c o e f f i c i e n t of f r i c t i o n i s assumed. Taking p =0.3, we have the f r i c t i o n a l f o r c e given by uF r£. F = uF «. N = 4TT/9 uar£ N The t o t a l energy required to cause an invagination of length I i given by U f = / F f U ) d £ = f-par£2 J o 151. A ratio between the f r i c t i o n a l energy and distortion energy can be defined: A large motor neuron w i l l have a radius of roughly 5xl0~ 6 m, while a sensory neuron w i l l have a radius of one tenth this size, roughly 0. 5xl0~ 6 m. Empirical studies have shown that I can be on the order of 100 to 300 ym. For these values, the value of R.,, w i l l be between 30 f / d and 100 for larger neurons, and between 300 and 1000 for smaller neurons. We can conclude that the energy required to overcome f r i c t i o n is at least one to two orders of magnitude greater than the energy required for distortion. Although this i s less true for larger neurons, the f r i c t i o n a l force is s t i l l the dominant one, so for purposes of simplicity, the following estimates w i l l be calculated ignoring the energy required for distortion. We now consider the three mechanisms and estimate the forces generated: 1. Shear Force Due to Flow in Narrowed Section 2-n\iaTl2/9 = \iol 60*r 24 270 r 0 = Because of the increase in the local velocity in the narrowed region indicated by the double arrows, the shear stress on the walls increases. Ochoa proposes that the mechanism for invagination is that 152. t h i s increased shear s t r e s s drags the upstream myelin-sheathed segment i n t o the downstream one. Consider f i r s t a f l u i d element moving i n a tube (assume the f l u i d moves as a bolus). If we assume i n c o m p r e s s i b i l i t y , then the volume flow rate, vA, w i l l remain constant. The w a l l shear s t r e s s , T , i s taken to be proportional to the v e l o c i t y of the bolus. We consider the s i t u a t i o n i n which the f l u i d i s viscous enough to move i n a steady s t a t e ; the pressure gradient i s not a c c e l e r a t i n g the f l u i d . ~TJ V Force balance: 4— ( i r r 2 ) x = x(2nr )x dx v n n . . dP 2T r n dP C a n c e l l i n g x and IT gives — =» — ; x = T J — — n Consider what happens when the tube narrows. Since the volume flow rate i s a constant, v a -7- 0 — . A 0 r * n 1 dP x 1 Since x a v, x a — ; — a — a — . ' 9 dx r 3 r z n r d n n The pressure gradient shows a cubic dependence on the radius of the tube; the gradient can therefore change markedly through a narrowing. In our s i t u a t i o n , the axon narrows from a radius of 4 ym to a radius of 3.5 ym f o r a l a r g e r neuron.* The pressure gradient, therefore, increases by a f a c t o r of (4/3.5) 3, or approximately 1.5, above the gradient In the internode i n the v i c i n i t y of the narrowing. *The r a t i o i s the same f o r a l l neuron diameters. 153. The o v e r a l l pressure gradient can be estimated from e m p i r i c a l observations that the maximum applied pressure f a l l s . t o zero over a distance of approximately 30 mm near the edge of the c u f f (Thomson, 1949). The average pressure gradient i s therefore given by: dP 70,000 Pa o *i \TD / — = —*r- o 2.3 MPa/m dx 30 mm In the area of narrowing, the gradient increases to » 3.5 MPa/m. This l a s t step i s j u s t i f i e d because the length of narrowing i s r e l a t i v e l y small compared with the i n t e r n o d a l distance and the increase i s also f a i r l y minor. The assumption that the pressure gradient r i s e s i n the nodal region w i l l not make the t o t a l pressure drop s i g n i f i c a n t l y l a r g e r than the maximum applied pressure. In the narrowing, then, the shear s t r e s s i s given by: r n dP 0.7r dP 1 T = -r- = — s — -r— * 1.23r MPa 2 dx 2 dx If we assume that the force a v a i l a b l e f o r i n v a g i n a t i o n i s that shear force developed over the area of invagination, we have: Fs = 2irr T ~ 2.45 i r r 2 MN/m We can define a r a t i o of the force a v a i l a b l e from the shear mechanism to the force required to overcome f r i c t i o n by = J L _ = 2.45xlQ6 T r r 2 s/f u F r 4 T T U O r/9 « 1.2x10^ r For the neuron s i z e s we are considering, t h i s r a t i o w i l l be roughly 0.06 f o r la r g e r neurons and 0.006 for smaller neurons. C l e a r l y , t h i s 154. mechanism can provide only a small f r a c t i o n of the energy required to overcome f r i c t i o n . ; 2. Pressure Gradient Acting Across the Annulua Ochoa also suggested that the pressure gradient a c t i n g across the narrowing of the axoplasm-fIlled c a v i t y of the axon could cause i n v a g i n a t i o n . From the d i s c u s s i o n concerning the shear s t r e s s mechanism, we have the pressure gradient across the region of inv a g i n a t i o n equal to = 3.5 MPa/m. The bearing area i s the annulus of area: A = Tr(r 2 - r 2 ) ; r » 0.8r; r , * 0.7r a o I o i The force a v a i l a b l e for invagination i s given by: p a dx o i o i = 3.5x106* (1.5r)(0.1r) = 5.25xl0 5i r 2 N/m Another force r a t i o for t h i s mechanism can be defined by 155. V f F 5.25x105^ r 2 4 IT yo r/9 2.6xl0 3r For a l a r g e r neuron, t h i s r a t i o i s = 0.013 while f o r a smaller one, the r a t i o Is » 0.0013. Again, t h i s mechanism cannot provide enough energy to overcome the f r i t i o n a l r e s i s t a n c e . 3. A x i a l Compressive Stress Preliminary r e s u l t s from t h i s t h e s i s work i n d i c a t e d that compres-sive stresses on the order of one tenth of the maximum applied cuff pressure can be expected i n the a x i a l d i r e c t i o n . The applied pressure i s approximately 500 mmHg or roughly 70,000 Pa. If the a x i a l compressive s t r e s s i s one tenth of t h i s value, then we can develop a compressive force on the nerve according to the following equation: Fc = 0.1 P i r r 2 = 7000TT r 2 N . Again, a force r a t i o for t h i s mechanism can be defined by _ f c = 7000 r r r 2 c/f ~ F f 4iryar£/9 = 35 r / i If the compressive force i s to exactly balance the f r i c t i o n f o r c e , 1 must be given by I = 35r, o 180 ym for larger neurons, w 18 ym f o r smaller neurons 156. The p r e d i c t e d nodal displacement f o r l a r g e r neurons i s c e r t a i n l y on the order of the observed nodal displacements. This makes the hypothesis of a x i a l compression as the damage-causing mechanism more p l a u s i b l e than any of the other hypotheses. Summary The compressive mechanism can conceivably generate enough compres-sive force to cause a nodal displacement on the order of 200 um i n l a r g e r neurons, which i s the same order of magnitude as the observed displacements. The shear stress and pressure gradient'mechanisms generate forces i n proportion to the length of invagination, but t h e i r magnitudes are two orders lower than the force required to overcome f r i c t i o n a l r e s i s t a n c e . The compressive mechanism i s , therefore, the most p l a u s i b l e one. Smaller neurons r e s i s t i n v a g i n a t i o n more than l a r g e r ones; a smaller neuron might only become invaginated by 20 ym when a larger one undergoes an i n v a g i n a t i o n of 200 um. This mechanism, therefore, explains observations that smaller neurons are not damaged as exten-s i v e l y as l a r g e r neurons. APPENDIX C DERIVATION OF EQUATIONS 158. APPENDIX C DERIVATION OF EQUATIONS C . l Moment E q u i l i b r i u m The s t r e s s s t a t e at a point i n a s o l i d can be defined by a set of nine s t r e s s components (see Figure C I). These include the three d i f f e r -ent normal str e s s e s on the faces of the d i f f e r e n t i a l element and the s i x shear st r e s s components i n the planes of the faces. If the element i s not undergoing any angular a c c e l e r a t i o n , then the moments of the stresses acting on the faces of the d i f f e r e n t i a l element w i l l be zero. If we consider, therefore, the moments about the z a x i s , we get the r e l a t i o n M - x + Q(r+6r) 69 6z + x~ r9 2 r r 59 6z ~ - T 9r 6r 5z r69 2 - T = 0 6 9 + 0 - r(xt + x 7 )/2 9r 9r 6z -v 0 + (at - aT)/69 * r66 2/4 6r * 0 - r ( x r 9 - x^ ) + — r69 2/4 - r(x " T9r> " 0 We have, therefore, T „ = x„ . ' r9 9r 159. Figure CI. Stresses on elemental volume 160. This r e l a t i o n s h i p can be demonstrated f o r the other complementary shear stress components. In general, at e q u i l i b r i u m = T j i a n ^ orthogonal coordinate system and consequently, only s i x s t r e s s components are necessary to define the s t r e s s at a point. Under conditions of axisymmetric loading, the components x^^ and T^Q w i l l both be zero because the deformations which would be produced by such s t r e s s e s are not consistent with the assumption of axisymmetry. C.2 Equations of E q u i l i b r i u m If we consider the f o r c e balances f o r the three coordinate d i r e c t i o n s , we can determine the d i f f e r e n t i a l equations of e q u i l i b r i u m . In t h i s d e r i v a t i o n , the body forc e i s assumed to be zero. This i s a reasonable assumption because the pressure d i f f e r e n c e due to the e f f e c t of g r a v i t y i s on the order of 1 kPa, while the a p p l i e d c u f f pressure i s roughly 50 kPa 1; the error a t t r i b u t a b l e to t h i s cause w i l l be l e s s than 2%. If we take the limb diameter to be approximately 10 cm, then the change i n h y d r o s t a t i c pressure from one side of the limb to the other due to the e f f e c t of g r a v i t y w i l l be roughly the same as the change found i n a water column of 10 cm depth. This pressure i s approxi-mately 10,000 N/m3 * 0.1 m = 1 kPa. The cuff pressure, on the other hand, w i l l be roughly 300 - 500 mmHg, or * 50 kPa. 161. Radial e q u i l i b r i u m : Z F - a+ (r +566z) - a"(r"696z) r r r + ( T + - T~ )r6r69 rz rz ( T 6 r ~ T e r ) ( S r , S z c o s 2~ - ( a * + a~)6r6z s i n = 0 D i v i d i n g by - 6r 69 6z gives (note: r + = r ~ + Sr) 3T a 0 = l i m a + ( r " + 6r) - a " + r _ r r r 3z r 6r*0 3a a -a. 3x * + JZ-±+ " = o 3r r 3z C i r c u m f e r e n t i a l E q u i l i b r i u m : The assumption of axisymmetry implies that any d e r i v a t i v e s with respect to 9 are zero. If t h i s i s true, then the c i r c u m f e r e n t i a l force balance reduces to an i d e n t i t y : 0 = 0 . A x i a l E q u i l i b r i u m : (For s i m p l i c i t y , the l i m i t i n g process i s bypassed) 3a I F = (a + dz)rdrd9 - a rdrd9 z z 3z z 3T + (T + dr)(r+dr)d9dz rz 3r - x rd9dz = 0 rz 162. 3a 3T 3T .*. r + T + r + drldrdedz = 0 . 3z rz 3r 3r ' 3a T 3T • —5. + -EL + r z = o The two eq u i l i b r i u m equations are summarized below 3a a -a. 3T -i r , r 0 . rz _ r a d i a l : T — + + — - — = 0 . 3r r 3z 3a T 3x a x i a l : - — H 1 — = 0 . 3z r 3r C.3 Strain-Displacement Relations For small displacements, the s t r a i n i n elongation can be defined a the r a t i o of the increase i n length of an element i n a body upon defor-mation to the o r i g i n a l length of that element. The shear s t r a i n s are defined as the average angular change i n two elements o r i g i n a l l y at r i g h t angles i n the undeformed body. In the same way that we could define nine s t r e s s components to express the stress state at a point, so too can we define nine s t r a i n components. The s t r a i n s due to elongation of a body element are e , e r 9 and e , while the shearing s t r a i n s are e e „ , e and t h e i r z r9 0z zr complementary s t r a i n s . When axisymmetry i s assumed, £ rg, £g z and t h e i r complementary s t r a i n s are zero. When a body undergoes deformation, each point i n the body i s dis p l a c e d . The displacement components are u i n the r a d i a l d i r e c t i o n , i n the c i r c u m f e r e n t i a l d i r e c t i o n , and w i n the a x i a l d i r e c t i o n . The 163. coordinates of a-point i n the deformed body are given by the mapping (r,0,z) + (r+u, 9+v, z+w) . By using the d e f i n i t i o n of s t r a i n as Ai/lQ, we can define the s t r a i n s i n terms of the displacements. Consider f i r s t an element undergoing a r a d i a l displacement u, which i s a function of the o r i g i n a l coordinates. . . — > , r r+6r r+u r+u+5r+~- 6r 9r A£ ^ 3u 6r _ 3u . 3u £ 3r 6r 3r * " e r 3r * o Consider the e f f e c t of t h i s r a d i a l displacement on the s t r a i n i n the c i r c u m f e r e n t i a l d i r e c t i o n : I = r69 £ = (r+u)69 I = r69 ; lc = (r+u)S9 ; .*. A£ = u69 . o f M = u 39 . = u Z r 39 5 " £9 r ' o 164. Since the s i t u a t i o n under consideration i s axisymmetric, there w i l l be no c i r c u m f e r e n t i a l displacement, and consequently, e need not be w r i t t e n as a fu n c t i o n of v. I f we have an a x i a l displacement, we have the simple r e l a t i o n s h i p 9ui 3z * Since angular s t r a i n i s defined as the average angular change between two elements o r i g i n a l l y at r i g h t angles, we have the following r e l a t i o n s h i p : r a = 3u 3z 5 8(0 3r 1 f . Q \ 1 ,3u , 3GK £r z = 2 ( a + B ) = 2 (37 + 37} ' The strain-displacement r e l a t i o n s f o r the axisymmetric case are summarized below 3u u _ _3u>_ _ _ _3_u _3co £ r " 3r ' e9 " r ; 6 z " 3z ; rz 3z 3r * Remembering that the nine stress components were reduced to s i x by a consideration of moment equ i l i b r i u m , we might ask whether the nine s t r a i n components can be s i m i l a r l y reduced to s i x . If we look at the strain-displacement r e l a t i o n given above f o r the shearing s t r a i n s , we 165. see that i t i s symmetric i n the i n d i c e s ; the complementary s t r a i n s are therefore equal by d e f i n i t i o n . C.4 E l a s t i c Relationships C.4.1 I s o t r o p i c E l a s t i c Relationships E l a s t i c parameters are the values which connect the equations of s t r e s s to the equations of s t r a i n . When we extend an e l a s t i c object, we notice that the extension i s a f u n c t i o n of the applied f o r c e . If the m a t e r i a l i s l i n e a r l y e l a s t i c , then the extension i s given by a simple formula, extension = applied force/constant, or, i n terms of stresses and s t r a i n s , s t r a i n = stress/Young's modulus. We f i n d that i f we extend a three-dimensional o b j e c t i v e , the object experiences a l a t e r a l c o n t r a c t i o n . If the s t r a i n of extension i s given by e^, the l a t e r a l c o n t r a c t i o n i s given by ~ V £ K where v i s c a l l e d Poisson's r a t i o . If the material i s l i n e a r l y e l a s t i c , such contractions and extensions can be superimposed; i f the m a t e r i a l i s also i s o t r o p i c (the e l a s t i c constants E and v are constant i n a l l d i r e c t i o n s ) , then f o r a general s t a t e of s t r e s s , the s t r a i n s are given by e = 1/E (a - v (a + a ) ) , x x y z 1 6 6 . e = 1/E (o - v (a + 0 ) ) , y y x z e = 1/E ( 0 - v (a + 0 ) ) , z z x y This r e s u l t i s quite general and applies to any orthogonal system of coordinate axes. Therefore, i n c y l i n d r i c a l coordinates, e r = 1/E ( o r - v ( o e + a z ) ) , e 9 = 1 / E ( a 9 " V (°r ^ z ^ ' e = 1/E ( 0 - v (a + 0 ) ) . z z r o C.4.2 A n i s o t r o p i c E l a s t i c R e l a t i o n s h i p s C.4.2.1 Anis o t r o p i c Hooke's Relations The l i n e a r r e l a t i o n s h i p between stresses and s t r a i n s given above f o r the i s o t r o p i c case i s known as Hooke's Law. This law can be gener-a l i z e d to include l i n e a r l y e l a s t i c materials with a n i s o t r o p i c e l a s t i c constants. In i t s generalized form, each p o s s i b l e s t r e s s component i s given by a l i n e a r combination of a l l possible s t r a i n components m u l t i -p l i e d by e l a s t i c q u a n t i t i e s analogous to the Young's modulus of Hooke's Law. If we use tensor notation, t h i s generalized law can be expressed as a. . = C. ., ,e, , , where C. ., , has 34 or 81 components, i j i j k l k l i j k l T h i s e l a s t i c constant tensor i s too general f o r our purposes, however. We r e c a l l that both the stress components and the s t r a i n components are symmetrical i n t h e i r i n d i c e s , so C. ., , = C. .,. = C... , = r i j k l l j l k j i k l ^ j i l k " ^ n*- s s e t °f e q u a l i t i e s r e s u l t s i n only 36 independent e l a s t i c 167. constants. This can perhaps be seen more r e a d i l y i f we r e c a l l that there are only s i x independent stress components and s i x independent s t r a i n components: i f each s t r e s s component can be expressed as a l i n e a r combination of each s t r a i n component m u l t i p l i e d by an e l a s t i c constant, there can be at most 36 independent e l a s t i c constants. We can s i m p l i f y the index system by r e p l a c i n g the double i n d i c e s with a s i n g l e index as follows (since xy and yx are equivalent, they can be considered a s i n g l e index; the same i s true for the other two complementary index p a i r s ) : a X °11 X a y = °22 e y = £22 a z = °33 e z = £33 T xy = °12 e xy = £12 T xz = °13 e xz = £13 T yz = 023 e yz = E23 The s t r e s s can now be expressed as o, = C, , e . , where C, , represents an k k l 1 k l array of 36 e l a s t i c constants. This number can be reduced f u r t h e r by a consideration of the s t r a i n energy of a d i f f e r e n t i a l element. The exact d i f f e r e n t i a l of the volumetric s t r a i n energy i s given by the sum of the products of each str e s s component with the exact d i f f e r e n t i a l of the corresponding s t r a i n component, so that dU = a, 6e, (repeated indices i n d i c a t e summation). 168. Since the s t r a i n energy i s independent on each s t r a i n component, we can write 3U dU = - — de, = a, de, . 3e. k k k k Therefore ° k = c k i e i = lr • k If we take the quadratic equation, U = 1/2 e i e k > t * i e n t n e energy balance equation i s s a t i s f i e d . This expression for the volumetric s t r a i n energy implies symmetry i n the i n d i c e s k and 1, which reduces the number of independent e l a s t i c constants from 36 to 21. The i m p l i c a t i o n of symmetry can be seen from the f o l l o w i n g argument which assumes, without loss of g e n e r a l i t y , that the only s t r a i n s are e^ and e 2 . As sume U = 1/2 [ C l l £ l 2 + ( C 1 2 + C 2 1 ) e i e 2 + C 2 2 e 2 2 ] , C l k £ k = C 1 1 E 1 + C 1 2 E 2 * For t h i s e q u a l i t y to hold, 2 C 1 2 = C 1 2 + C 2 1 ; or C 1 2 = C 2 1 . Since t h i s argument could be applied to any two v a l i d i n d i c e s , we can e s t a b l i s h t h i s symmetry f o r a l l p o s s i b l e p a i r s of e l a s t i c constants. As a consequence, only 21 independent e l a s t i c constants remain. The e l a s t i c constant matrix can therefore be represented as 3U _ C 1 2 + ° 2 1 3e, " C l l £ l + 2 E 2 " 169. c l l C 1 2 C 1 3 C U C 1 5 C 1 6 C 1 2 C 2 2 C 2 3 C 2 , C 2 5 C 2 6 C 1 3 C 2 3 C 3 3 C 3 . C 3 5 C 3 6 C l , C 2 . C 3 - C 4 - C , 5 <%6 C 1 5 C 2 5 C 3 5 C« C 5 5 C 5 6 C 1 6 C 2 6 C 3 6 C 4 6 C 5 6 C 6 6 The number of constants can be reduced s t i l l f urther i f we consider that, due to the symmetry i n the a n i s o t r o p i c character of the t i s s u e , several d i f f e r e n t coordinate systems can be considered to be equivalent. For example, i f we think of the t i s s u e s as being s i m i l a r to a u n i a x i a l f i b r e - r e i n f o r c e d r e s i n , and i f we define the x-axis to be p a r a l l e l to the d i r e c t i o n of the f i b r e s , then we can see that i t does not matter which d i r e c t i o n i s defined as being p o s i t i v e or negative; the e l a s t i c constant matrix w i l l be unaffected by t h i s choice. In the same way, the matrix w i l l be unaffected by a r o t a t i o n of the other two axes about the x-axis (see Figure C2). We can define a l i n e a r mapping from one coordinate system to another by means of a transformation matrix A, such that the coordinate vector i n the transformed system i s given by x y z The transformation matrix can then be used to transform the s t r e s s matrix i n the o r i g i n a l coordinate system i n t o the new coordinate system; i . e . , a' = A a A1". We can use t h i s transformation matrix to deduce fu r t h e r c o n s t r a i n t s on the choice of the e l a s t i c constants by using the p r i n c i p l e that c e r t a i n transformations w i l l not a f f e c t the e l a s t i c constant matrix. 170. Figure C2. E l a s t i c a l l y equivalent coordinate systems 171. We begin with the equivalence of two coordinate systems d i f f e r i n g only i n the d i r e c t i o n of the x-axis. For t h i s case, the coordinate components i n the second system are given by x' = -x, y' = y, and z' = z. The corresponding transformation matrix, A, i s given by -1 0 0 A = 0 1 0 0 0 1 The st r e s s matrix i s transformed according to -1 0 0 [a 1] = 0 1 0 0 0 1 1 . - 0 xz -0 0 xy y o a xy xz o o xy y -o -o xy xz yz o 0 yz z yz o o o xz yz z -1 0 0 0 1 0 0 0 1 The s t r a i n matrix undergoes an analogous transformation. Since the e l a s t i c constant matrix remains the same under t h i s transformation, we have an equivalence between the s t r e s s / s t r a i n r e l a t i o n s h i p s i n the two coordinate systems. M -0 X C l l C 1 2 C 1 3 C 1 5 C 0 y C 1 2 C 2 2 C 2 3 C 2 . C 2 5 C o z C 1 3 C 2 3 C 3 3 C 3 - C 3 5 c 0 xy C l , C 2 - C 3 - S. Ss c a xz C 1 5 C 2 5 C 3 5 Se C 5 5 c 0 yz C 1 6 C 2 6 C 3 6 S 7 C 5 6 c xy xz yz 172. x xy -a xz -a yz 11 C12 C14 C15 C16 12 C22 C23 C 2 , C25 C26 13 C23 C33 C34 C35 C36 14 C24 C34 C44 °45 C46 15 C25 C35 C46 C55 C56 16 C26 C36 C47 C56 C66 -E xy -e xz -e yz We compare the expressions f o r from each of the two systems above; xy 14 x 24 y + C 0. e + C. , e + C, ce 34 z H xy 45 xz 46 yz -a =C.,e +C 0.e +C.,C + C, , e + C, ce + C, c e xy 14 x 24 y 34 z 44 xy 45 xz 46 yz Since each s t r a i n component i s independent, the c o e f f i c i e n t s of each s t r a i n component i n each of the two expressions must be equal f o r the e q u a l i t y to hold. We have, therefore, that C^^ = " C ^ . which can only be true i f C., = 0 . This r e l a t i o n i s a l s o true f o r C_,, C_,, and C, r . Ik 2 4 ' 34' i+6 If we compare the c o e f f i c i e n t s of and we simply get two i d e n t i t i e s : = C ^ , and = C ^ . By repeating t h i s comparison f o r the expressions f o r a , we f i n d xz that C 1 5 , C 2 5 » C 3 5 , and C 5 g a l l equal zero. The e l a s t i c constant matrix r e s u l t i n g from t h i s p a i r of comparisons i s given by c l l C12 C13 0 0 C16 C12 C22 C23 0 0 C26 C13 C23 C33 0 0 C36 0 0 0 C4 4 C45 0 0 0 0 C45 C55 0 C16 C26 C36 0 0 C66 173. We can repeat t h i s a n a l y s i s f o r an i n v e r s i o n of e i t h e r of the other two axes. If we perform the transformation given by x' = x, y' = -y, and z' = z, we f i n d that C l g , C 2 g , C 3 g , and C^ 5 a l l equal zero. The two transformations considered thus f a r do not span the f u l l set of p o s s i b l e ones. We w i l l consider two more transformations: 1) a r o t a t i o n of 90° about the x-axls and, 2) an a r b i t r a r y r o t a t i o n about t h i s a x i s . This sequence of transformations was chosen to s i m p l i f y the algebra involved at each stage. The 90° transformation r e s u l t s i n a transformed s t r e s s matrix which can be compared to the s t r e s s matrix i n the o r i g i n a l coordinate system. a ~-a X C l l C12 C13 0 0 0 e X a z C12 C22 C23 0 0 0 e z a y a xz C13 0 C23 0 C33 0 0 0 0 0 0 £ y e xz •a xy •a yz 0 0 0 0 0 0 0 0 C55 0 0 C66 -e xy - e _ yz_ a X " c l l C12 C13 0 0 0 ~ £ X a y a z C12 C13 C22 C23 C23 C33 0 0 0 0 0 0 £ y e z a xy a xz 0 0 0 0 0 0 0 0 C55 0 0 e xy £ XZ a yz 0 0 0 0 0 C66 £ yz We compare the expressions for from each of the two systems above; o = C 1 T e + C 1 0 e + C,,£ = C,,E + C. ,e + C.-£ . X 11 X 12 z 13 y 11 X 12 y 13 z 174. Again, because the s t r a i n components are independent, the c o e f f i c i e n t s of the corresponding s t r a i n components i n each equation must be equal. By considering the c o e f f i c i e n t s of e , then, we can conclude that C 1 2 = C13* Th e expressions f o r a y i e l d the e q u a l i t y C 2 2 = C 3 3 , and those from o y i e l d C. , = C c c . xy 44 55 The e l a s t i c constant matrix r e s u l t i n g from these comparisons i s given by rckiJ • c l l C12 C12 0 0 0 C12 C22 C23 0 0 0 C12 C23 C22 0 0 0 0 0 0 0 0 0 0 0 0 C55 0 0 0 0 0 0 C66 The transformation matrix A for an a r b i t r a r y r o t a t i o n a about the x-axis i s more complicated than the ones we have seen so f a r . The transformed coordinates are given by x' = x, y' = y cosa + z since, z' = -y sina + z cosa. The corresponding transformation matrix A i s given by 1 0 0 [A] = 0 cosa sina 0 sina cosa • transformation of the stress and s t r a i n components gives 175. x -a -a xy i xy r yz c o s 2 a + a s i n 2 a + 2 a sina cosa z yz c o s 2 a + a s i n 2 a - 2 a sina cosa z yz cosa + a sina xz sina + o cosa xz ( c o s 2 a - s i n 2 a ) + (o - a ) s i n a cosa z y C l l C 1 2 C 1 2 0 0 0 c 1 2 c 2 2 c 2 3 0 0 0 C 1 2 C 2 3 C 2 2 0 0 0 0 0 0 C. . 0 0 0 0 0 0 c,. kk 0 0 0 0 0 c 66 x e c o s 2 a + e s i n 2 a y z + 2 e s i n a cosa yz e c o s 2 a + £ s i n 2 a y z - 2 e sina cosa yz e cosa + e s i n a xy xz -e sina + e cosa xy xz e ( c o s 2 a - s i n 2 a ) yz ' + (e - e ) s i n a cosa z y _ x xy xz yz 11 C 1 2 C 1 2 0 0 0 12 C 2 2 C 2 3 0 0 0 12 C 2 3 C 2 2 0 0 0 0 0 0 S. 0 0 0 0 0 0 C55 0 0 0 0 0 0 C 6 6 — e x e y e z e xy e xz e _ y z _ The l a s t element i n the transformed s t r e s s vector i s a ( c o s 2 a - s i n 2 a ) + (a - a ) sina cosa yz z y = Crr (e ( c o s 2 a - s i n 2 a ) + (e - e ) sina cosa) 6 6 yz z y If we replace the stresses on the l e f t hand side of t h i s equation with t h e i r representations i n terms of s t r a i n s from the untransformed r e l a t i o n s , we get 176. a (cos2oc - s i n 2 a ) + (a - a ) sina cosa yz z y = Crr e (c o s 2 a - s i n 2 a ) + sina cosa * 6 6 yz 12 x 23 y 22 z 12 x 22 y 23 z = Ccc e (c o s 2 a - s i n 2 a ) + ( C 0 0 - C„,)(e - e ) s i n a cosa 66 yz 22 23 z y When we compare t h i s expression with the r i g h t hand side of the f i r s t equation, we see that f o r the e q u a l i t y to hold, we must have C g 6 = C 2 2 C 2 3 - Any other comparison leads only to an i d e n t i t y , so we have exhausted the co n s t r a i n t s on the number of independent e l a s t i c constants; with the form of anisotropy we have assumed, f i v e independent e l a s t i c constants are required to completely s p e c i f y the re l a t i o n s h i p s between the stresses and s t r a i n s . The f i n a l form of the e l a s t i c constant matrix i s therefore given by c l l C 12 C13 0 0 0 C 12 C 22 C 22~ C 66 0 0 0 C 12 C 22~ C 66 C 22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C66 If we rename the indices f o r s i m p l i c i t y , we get the form [C] = c l C 5 c l 0 0 0 C 5 C 2 C 2 - C 4 0 0 0 C 5 c 2 - c , C 2 0 0 0 0 0 0 C3 0 0 0 0 0 0 C 3 0 0 0 0 0 0 c 4 177. The e l a s t i c constant matrix C i s used to give the stresses i n terms of the s t r a i n s : a = Ce . We w i l l f i n d that i t i s a l s o u s e f u l to have the reverse r e l a t i o n s h i p , s t r a i n s i n terms of stre s s e s : e = C _ 1 a. We can symbolically i n v e r t C to give a matrix of the form [ C - l ] -V V 0 0 0 r ' V 0 0 0 r ' ^6 °2 0 0 0 0 0 0 V 0 0 0 0 0 0 c 1 0 0 0 0 0 0 The two constants, C 3 ' and C ^ ' are simply the inverses of C 3 and C ^ . The other inverse constants can be determined by i n v e r t i n g the 3x3 submatrix i n the upper l e f t hand corner of C . C g ' i s not independent of the other f i v e constants. The determinant of t h i s submatrix w i l l be defined as IC I which i s 1 s 1 given by |CJ = Ck (Cl(2C2- - 2 C 5 2 ) . The other c o e f f i c i e n t s are given by the following expressions: V V V V C ' r ' - V 2 C 2 - V / |C s | , - < C 1 C 2 - Cs2^K^ - 1 / C 3 , • -wi csi' - ( C 5 2 " C 1 ( C 2 " V^IS 178. C.4.2.2 Anis o t r o p i c Parameters In the d i s c u s s i o n on i s o t r o p i c m a t e r i a l , we concerned ourselves with only two e l a s t i c constants: Young's modulus and Poisson's r a t i o . There i s a t h i r d constant, G, which r e l a t e s the shearing s t r e s s and shearing s t r a i n through the following r e l a t i o n : T = 2Ge. This constant i s not independent of Young's modulus and Poisson's r a t i o , but i s given E by: G = 2(i+ v) * ^ *- s o t r oP*- c m a t e r i a l , therefore, requires only two e l a s t i c constants to s p e c i f y i t s e l a s t i c p r o p e r t i e s . These two constants are p h y s i c a l l y meaningful i n that they c o r r e s -pond d i r e c t l y to a measurable m a t e r i a l property, namely the r a t i o of s t r e s s to s t r a i n i n a u n i a x i a l stress test f or E, and the r a t i o of l a t e r a l c o n t r a c t i o n to a x i a l extension i n a s t r e t c h t e s t f o r v. We w i l l f i n d i t h e l p f u l conceptually to s p e c i f y f i v e s i m i l a r e l a s t i c constants f o r the i s o t r o p i c case. We begin by d e f i n i n g E and v for the a n i s o t r o p i c case i n an analo-gous manner; we extend the above d e f i n i t i o n f o r E by s p e c i f y i n g that i t i s to be measured along the p r i n c p a l a x i s , and for v by s p e c i f y i n g that i t i s based on the l a t e r a l c o n t r a c t i o n i n the plane perpendicular to the p r i n c i p a l a x i s . We now define s e v e r a l other p o s s i b l e measured. If we impose an extension i n the transverse plane, we can determine a property analogous to E and c a l l i t E 2 . We w i l l l a t e r want t h i s i n dimensionless form, so we take the r a t i o of t h i s quantity to E and c a l l i t E ^ . We define two variants on v for extensions i n the transverse plane: contraction i n transverse plane/extension cont r a c t i o n along p r i n c i p a l axis/extension 179. We can define two shear stress moduli corresponding to G i n the isotropic case. The f i r s t is the modulus resulting from a shear defor-mation i n any plane containing the principal axis; this modulus we c a l l G^. The second is the modulus resulting from a shear deformation in the transverse plane; this we c a l l G ^ . These definitions specify seven possible measures; only five of these can be selected independently, however. We must consider the set of relations contained i n e = C _ 1 a to determine the connections between the seven measures. Consider f i r s t a uniaxial stress applied along the principal axis. The resulting strains are given by e = C * o ; e = C ' cr ; e = C ' a . x 1 x y 5 X z 5 X From our d e f i n i t i o n s of the seven e l a s t i c constants, we can conclude that C. 1 = 1/E. Also, e = C ' a = e C'/C,', so we have C ' = C,' 1 y o x X 5 1 5 e /e = -v/E. y x We now consider a u n i a x i a l s t r e s s a p p l i e d along the y - a x i s . The s t r a i n s are e = C ' a ; e = C ' a ; e = C ' a . x b y y z y z 6 y We conclude that C„' = 1/E. = 1/E E , as well as that C ' = -v„/E„. £. c. T b <L Z Also, since e = C ' e /C 0' = -v. e , we have that v, = -vE . Because x 5 y 2 3 y ' 3 r of t h i s r e l a t i o n , v g i s redundant; v and are s u f f i c i e n t . We therefore define a new constant, v , defined as v = v_/v. 180. The connections between the shear moduli and the elements i n the e l a s t i c constant matrix i s straightforward; C 3' = V2Glie ; CJ - l/2G t. The values G... and G can be non-dimensionalized by analogy with the 1* t i s o t r o p i c case. We r e c a l l that G = E/2(l+v). If we define an aniso-t r o p i c equivalent G i n the same way, then we can define two new values a as follows, G r l * = G l * / G a 5 G = G /G . r t t a With a l l these r e l a t i o n s , we can choose which e l a s t i c constants we wish to use and i n s e r t them i n t o C _ 1 . If we pick E, v, E , v and G , j. f ' ' r ' r r l * as our f i v e independent e l a s t i c constants, then we can write out C - 1 (note that even though G i s included, i t i s not independent of the other f i v e constants; i t s d e f i n i t i o n i n terms of the outer f i v e i s q u i t e complicated and not p a r t i c u l a r l y relevant to the work at hand) [CM - I-1 -v -v 0 0 0 -v 1/E r -vv /E r r 0 0 0 -v "vv /E r r 1/E r 0 0 0 0 0 0 1+v r l * 0 0 0 0 0 0 1+v r l * 0 0 0 0 0 0 1+v G r t 181. C.5 Co m p a t i b i l i t y Equations We have w r i t t e n four s t r a i n components i n terms of two d i s p l a c e -ments, u and w. Each displacement, i n the axisymmetric case, i s independent of 0 and dependent only on r and z. Since the four s t r a i n components are functions of only two independent v a r i a b l e s (the space coordinates), only two of the s t r a i n s can be s p e c i f i e d a r b i t r a r i l y . Once these two s t r a i n s are s p e c i f i e d , the displacement f i e l d i s determined and the other two s t r a i n s components must be derived from t h i s displacement f i e l d according to the s t r a i n / displacement r e l a t i o n s , which are summarized here; 3u _ u _ _3o)_ . _3u)_ 9u e r = 3r 5 £0 " r ; £ z " 3z 5 rz = 3r 3z * It w i l l be u s e f u l to deal s o l e l y with the s t r a i n measures rather than the displacements because the s t r a i n s are d i r e c t l y r e l a t e d to the s t r e s s e s . A u s e f u l approach to many problems i s to determine the s t r e s s f i e l d s , and consequently the s t r a i n f i e l d s , p r i o r to determining the displacements at each point i n the body. The above equations can then be used to c a l c u l a t e d the.displacements once the s t r a i n s are known. There are four equations a v a i l a b l e , however, to s p e c i f y only two unknowns. An a r b i t r a r y set of s t r a i n components w i l l not, i n general, lead to a s o l u t i o n for the displacement f i e l d unless c e r t a i n c o n s t r a i n t s are imposed on the s t r a i n components. These co n s t r a i n t s are c a l l e d the c o m p a t i b i l i t y equations and express those i n t e r r e l a t i o n s h i p s between the s t r a i n components which are required to ensure a s o l u t i o n f o r the displacement f i e l d . 182. The strain/displacement r e l a t i o n s can be rearranged to eliminate u and w. If t h i s i s done, two r e l a t i o n s h i p s among the s t r a i n components r e s u l t 3u 3 r e e 2 e 9 u = r e 9 ; 37 = E r = — = £9 + — * 3 £ 9 " £ r = e9 + r S r " ' 3 2 e *3 3 2 e *3 2 3 2 e *3 *3 r _ 3 du , z _ 3Jua , r z _ 3Juo 3 3u 3z 2 3r3z 2 ' a r 2 3z3r 2 ' 3 r 9 z 3r 23z 3z 23r 3 2e 3 2e 23 2 r z rz 3z 2 3 r 2 8 r 3 z These are the equations of c o m p a t i b i l i t y expressed i n terms of s t r a i n s . These equations can be expressed i n terms of stre s s e s by applying the e l a s t i c constant matrix. The f i r s t equation gives 1 + V V r 3 , ' V V r + V ~E ( V V = R 37 ( " V C T z — a r + E-> r r r The second equation gives (adopting the common notation for d e r i v a t i v e s : 3Y/3x = Y,x) a - v u ( " V °z + r °9 ),zz + ( a z ' V ( ° r + 0 e > \ r r r r ' ' 2(1+v) t  Gr l * r z » r z 183. This formulation permits the s o l u t i o n of e l a s t o s t a t i c problems s o l e l y i n terms of stresses because i f the stress components s a t i s f y the c o m p a t i b i l i t y equations, then the other q u a n t i t i e s of i n t e r e s t , the s t r a i n s and the displacements, can be derived d i r e c t l y from the s t r e s s e s . In p r i n c i p l e , then, e l a s t o s t a t i c problems can be solved by f i n d i n g a s t r e s s f i e l d over a region which s a t i s f i e s the two e q u i l i b r i u m equa-t i o n s and the two c o m p a t i b i l i t y equations, subject to whatever boundary conditions might be imposed. These boundary conditions may be given i n terms of s t r e s s e s , s t r a i n s , displacements, or some combination of these q u a n t i t i e s . This does not pose an insurmountable problem i n p r i n c i p l e ; any boundary conditions not expressed i n terms of s t r e s s e s can be changed to a stress formulation through the strain/displacement r e l a -t i o n s and the s t r e s s / s t r a i n r e l a t i o n s . C.6 The Use of a Stress Function The problem of f i n d i n g a set of s t r e s s f i e l d s which s a t i s f i e s the two e q u i l i b r i u m equations and the two c o m p a t i b i l i t y equations, together with the boundary conditions, i s a d i f f i c u l t one. This problem can be s i m p l i f i e d by d e f i n i n g a new f u n c t i o n , c a l l e d the stress function, from which a l l the s t r e s s components can be determined by d i f f e r e n t i a t i o n . The stress function must be defined such that the four equations w i l l be s a t i s f i e d concomitantly with the s o l u t i o n of the s t r e s s f u n c t i o n . If t h i s can be accomplished, the e l a s t o s t a t i c problem reduces to f i n d i n g the s o l u t i o n to a s i n g l e f o u r t h order d i f f e r e n t i a l equation with bound-ary conditions i n v o l v i n g up to fourth order d e r i v a t i v e s . 184. C.6.1 I s o t r o p i c Stress Function i n Rectangular Coordinates The s e l e c t i o n of an appropriate d e f i n i t i o n for the s t r e s s f u n c t i o n i s r e l a t i v e l y straightforward when the problem i s formulated f o r plane s t r a i n i n rectangular coordinates, and assumes i s o t r o p i c material p r o p e r t i e s . This d e r i v a t i o n w i l l be presented so that the p r i n c i p l e can be e a s i l y understood. (Note: Only one c o m p a t i b i l i t y equation i s required because there i s no s t r a i n component i n the d i r e c t i o n perpendicular to the plane of s t r a i n ; only three s t r a i n s remain.) Equi l i b r i u m equations: a + T = 0, x,x xy,y a + T = 0. y,y xy,x Compatibility equation: 3 2 3 2 ( — + — ) < a + a ) = 0 3x 2 3y 2 X 7 The stress components can be defined i n terms of the stress function, f , as a = <b ; a = <j> ; x = -<p x r , y y ' y r,xx xy r,xy The two e q u i l i b r i u m equations are automatically s a t i s f i e d by t h i s d e f i n i t i o n . 185. F i r s t e q u i l i b r i u m equation: a + T =<J> - A = 0 . x,x xy.y ,xyy ,xyy Second e q u i l i b r i u m equation: a + T = d> - d> = 0 . y,y xy,x >xxy r ,xyx The c o m p a t i b i l i t y equation leads to the f i e l d equation f or the s t r e s s f u n c t i o n which remains to be solved. 32 a2 %2 %2 ax 2 3 y2 x y 3 x2 a y2 >yy > x x + 2<b + d> = 0, ,xxxx ,xxyy »yyyy or where V 4^ = V2V2(() = 0, V2 - ( £ _ + l L . ) . 3x2 3 y2 This r e l a t i v e l y simple example demonstrates the stress f u n c t i o n p r i n c i p l e . Notice that the e l a s t i c constants do not enter the formulation because they are not present i n the c o m p a t i b i l i t y equation. 186. C.6.2 I s o t r o p i c Stress Function i n C y l i n d r i c a l Coordinates When the plane s t r a i n problem i s formulated i n c y l i n d r i c a l coordin-ates, the d e f i n i t i o n of the s t r e s s f u n c t i o n becomes more complicated. One p o s s i b l e set of str e s s d e f i n i t i o n s i s a = ( ( v - l H + - o) + v<t> ) , r v v , r r r ,r r , z z ,z' aa = (v<b + — - d> + vd> ) , 9 , r r r ,r ,zz ,z' a = ((2-v)<t> + — <t> + (l-v)4> ) , z Y , r r r T , r Y , z z ,z' T = ((l-v)d) + d> - v<)> ) . rz , r r r ,r ,zz ,r There are two important d i f f e r e n c e s between t h i s formulation and the one i n rectangular coordinates; f i r s t , the s t r e s s component d e f i n i t i o n s involve the e l a s t i c constants because they appear i n the c o m p a t i b i l i t y equations, and second, these d e f i n i t i o n s i d e n t i c a l l y s a t i s f y a l l but the second e q u i l i b r i u m equation, which forms the basis f o r the f i e l d equation f o r the s t r e s s f u n c t i o n . When the problem was formulated i n rectangular coordinates, the c o m p a t i b i l i t y equation was the one which l e d to the f i e l d equation. In c y l i n d r i c a l coordinates, the second e q u i l i b r i u m equation leads to the biharmonic equation V 1^ = 0, where 187. The t h i r d term drops out because of axisymmetry. Note that i n t h i s case, even though the e l a s t i c constants are used to define the s t r e s s e s , they do not enter the equation i n v o l v i n g the s t r e s s f u n c t i o n . C.6.3 A n i s o t r o p i c Stress Function i n C y l i n d r i c a l Coordinates The formulation i n c y l i n d r i c a l coordinates can be extended to take account of a n i s o t r o p i c e l a s t i c p r o p e r t i e s . If the assumption i s made that the s t r u c t u r e of the d e f i n i t i o n s w i l l be unchanged by the i n t r o d u c t i o n of anisotropy, then a generalized set of d e f i n i t i o n s can be w r i t t e n . Their a b i l i t y to s a t i s f y the four necessary s t r e s s equations ( e q u i l i b r i u m and c o m p a t i b i l i t y ) can then be v e r i f i e d . The generalized d e f i n i t i o n s are a = (a <b + — <j> + c <t> ) , r , r r r ,r T , z z ,z a Q = (d <b + - <b + f <j> ) , 0 T , r r r ,r T ,zz 7 , z' a = (g <b + - <b + i <b ) , z 6 , r r r Y , r y , z z ,z' T =(j<t> +-<b +1<J> ) . rz , r r r ,r ,zz ,r If t h i s formulation i s t r u l y analogous to the i s o t r o p i c case, then an appropriate choice of the c o e f f i c i e n t s a-1 should i d e n t i c a l l y s a t i s f y the two c o m p a t i b i l i t y equations and one of the e q u i l i b r i u m equations. The e l a s t i c constant matrix r e s u l t i n g from t h i s p a i r of comparisons i s given by 188. a + r , r o r-o + T =0 rz,z The d e f i n i t i o n s f o r the s t r e s s components i n terms of the s t r e s s f u n c t i o n can be s u b s t i t u t e d i n t o t h i s e q u i l i b r i u m equation. This r e s u l t s i n the following equation, U+J)<t> r r „ + (b + k + a - d) - ( e + k) -*I5. , r r r z r ^ 2 + (c - f ) _t£££ + ( C + l ) o> = 0 . r ,rzzz This process can be applied to each of the two c o m p a t i b i l i t y equations expressed i n terms of the s t r e s s components. The f o l l o w i n g two equations r e s u l t , 1+vv vv ( _ J L ( d _ a ) - v h _ _ J L b + | _ ) * r r r 1+VV VV <j> + ( - g-i (e - b) + vh + - £ b - f - ) r r r 1+vv + (f - c X - ^ ) * r + r [ (-vg - ^ 1 a + f ) * + (-iv - ^ c + f ) * ] - 0, r r ' r r vv , , , a r , 2(l+v) , , . , , ' C N N and (-vg + — d ^ — - 1 + i - v(c + f ) ) E E G ,rrzzz r r r VV (j) + (-vh + £ — e) _2_ E E _ r 189. + (-vi + | f ) <b E E ,zzzzz r r + (g - v(a + d) - 2 < 1 + v ) j ) <j> G , r r r r z r ' O A O A .^ i x 2 ( l + v ) k w , ,rrz , , r +(h - v(b + e) ^——)(<!> „ „ „ 8 + — l -r ' r 2 r 3 Since each p a r t i a l d e r i v a t i v e term i n these three equations i s independent of the other p a r t i a l d e r i v a t i v e terms i n the same equation, the c o e f f i c i e n t s of each term must be i d e n t i c a l l y equal to zero i f the three equations are to be automatically s a t i s f i e d . Each equation y i e l d s f i v e c o e f f i c i e n t equations, so there w i l l be a t o t a l of f i f t e e n such equations i n twelve unknowns. The r a d i a l e q u i l i b r i u m equation y i e l d s the f o l l o w i n g equations, (1) a + j = 0 ; a = - j (2) b + k + a - d = 0 (3) e + k =' 0 ; e = -k (4) c - f = 0 ; c = f (5) c + 1 = 0 ; c = -1 These equations (excluding (2)) can be used to express a,c,e and f i n terms of j,k and 1. Eight unknowns remain, as do eleven equations. The f i r s t c o m p a t i b i l i t y equation gives the following f i v e equations (a,c,e and f have been replaced by t h e i r equivalents i n terms of j,k and 1), 190. 1+vv vv (6) ( - g — - ) ( d + j) - vh - ^ b - | - = 0, r r r 1+vv vv (7) - ( - _ L ) ( b + k) + vh + b + |- = 0, r r r 1+vv (8) (1 - D C - g — - ) - 0 ; 0 - 0, r vv (9) -vg + 1 - i j +|-= 0, r r vv « vv -1 (10) - v i + ^ l - | - - 0 ; ( - g £ — ) 1 = v i . r r r Equation (8) reduces to an i d e n t i t y upon s u b s t i t u t i o n . The l a s t equation (10), expresses i i n terms of 1. Equations (6) and (7) can be added to give the expression 1+vv (-g—-) (d + j - b - k) = 0 r The e l a s t i c constant term can then be factored out, leaving an equation which i s simply the negative of equation ( 2 ) . One of equations (6) and (7) i s therefore redundant. This leaves eight equations i n seven unknowns. The second c o m p a t i b i l i t y equation gives the fo l l o w i n g set of equations (again, the var i a b l e s a,c,e,f and i have been expressed i n terms of j,k and 1), vv 1-vv , s < u ) - „ 8 - J _ - F i d + ( 2 v - _ ^ - H i ± » ) ) 1 . „ , r r r r , vv (12) -vh + | - + j - ^ k = 0, r r 191. 1-vv vv (13) (-g-JL + _ L - 1-) i - o ; 0 r r r - o, (14) g + v j - vd - 2 ^ 1 + V ) j - 0 , r (15) h - vb + vk - 2 ( ^ + v ) k = 0 . G r Equation (13) i s simply an i d e n t i t y once the s u b s t i t u t i o n s are made. Further reductions become more d i f f i c u l t . Consider the su b t r a c t i o n of (12) from ( 9 ) ; (16) -v ( g - h) + ^ + v v r ( J ^ ) - 0. r r Rewrite equation (2) as (17) b - d = j - k Substitute (17) into (16) vv -1 (18) -v(g - h) + ( - g 1 — ) ( j - k) = 0 r Substract (15) from (14) to give (19) g - h + v(b-d) + v(j-k) - 2 ( ^ + v ) (j-k) = 0. r M u l t i p l y (19) by v, sub s t i t u t e for (b-d) using ( 1 7 ) , and sub s t i t u t e f o r v(g-h) using (18) to y i e l d 192. (20) [(^ £2) + 2v 2 - M l + v l ] ( j _ k ) = o. r r Since the term i n v o l v i n g the e l a s t i c constants i s not i d e n t i c a l l y zero, j=k. This being so, equation (17) implies that b=d. From (18), i t i s e a s i l y shown that g=h. This set of operations means that a l l the c o e f f i c i e n t s can be expressed i n terms of d,h,k, and 1. The equations (constraints) which remain to be a p p l i e d are equations (9),(11) and (14). Note that when the c o e f f i c i e n t s are expressed i n terms of d,h,k and 1, equation (6) equals equations (9) and (12) and equation (14) equals equation (15). The three equations, (9),(11) and (14), are summarized as (21) -vE h + vv k + d = 0, r r ' vv -1 2E (1+v) (22) -vE h - k - vv d + (2vE + — — ) 1 = 0, r r r v G r (23) h + vk - vd - 2 ( ^ + v ) k = 0. G r Add (21) to v E r (23). 2vE (1+v) (24) ( v 2 E ^ + vv ) k + (1 - v 2E ) d = 0. r G r r r This equation can be used to give d i n terms of k. The c o e f f i c i e n t , h i s e a s i l y found from (23). 193. (25) h = ( 2 ( ^ + V ) - v) k + vd r _ r2(l+v) .. . v 2 , 2 E r < 1 + V ) v + — (—=- vE - v Ik l - v 2 E r r 2(l+v) - vG (1+v ) r r G ( l - v 2 E ) r r k . To f i n d 1 i n terms of k, f i r s t subtract (22) from (21). vv -1 2E (1+v) (26) (vv r+l)(k+d) - (2vE r + —S L_ ) 1 = 0 . Express d i n terms of k and s u b s t i t u t e i n (26). -v 2E + 2vE (l+v)/G - vv (vv +1)(1 + E E k r l - v 2 E r vv -1 2E (1+v) = (2vE + — £ ) 1. r v G r Rewrite t h i s expression for 1 as 1-vv - 2v 2E + 2vE (l+v)/G v(l+vv )k r r r r r vv -1 + 2v 2E - 2vE (l+v)/G l - v 2 E r r r r r -v(l+vv )k l - v 2 E r A l l the c o e f f i c i e n t s are now expressed i n terms of k, and no co n s t r a i n t s remain, k can therefore be sel e c t e d a r b i t r a r i l y . If k i s taken as 194. l-v*E 1+vv r then when the a n i s o t r o p i c constants approach the i s o t r o p i c values, k w i l l approach the value of the i s o t r o p i c c o e f f i c i e n t described e a r l i e r . Using t h i s value f o r k, then, the other c o e f f i c i e n t s can be given as a = e = - j = -k, 2E (1+v) b = d = r — ^ — ( — v E - v ) 1+vv G r r ' r r c = f = -1 = v , - u - 2(1+v)  8 " h " G (1+vv ) " V ' r 1-v v r With the c o e f f i c i e n t s defined i n the above manner, the r a d i a l e q u i l i b r i u m equation and the two c o m p a t i b i l i t y equations are automatically s a t i s f i e d . The a x i a l e q u i l i b r i u m equation leads to the f i e l d equation f o r the s t r e s s f u n c t i o n . The a x i a l e q u i l i b r i u m equation i s T O + T + — = 0 . z,z z r , r r Replace the stress components with the a n i s o t r o p i c stress f u n c t i o n formulations to give 195. <i k <j> k <j> , r r r r ^ r f 2 r3 + (g+1) <b + (h+1) * T Z Z + i <f> = 0 . 6 ' ,rrzz v ' r ,zzzz Replace each of the c o e f f i c i e n t s with i t s equivalent i n terms of k and the e l a s t i c constants and then d i v i d e through by k; 2 <j) <b <b, <b + iIIL t£I + —E + 0(d> + > r Z Z ) + 8 d» - 0, , r r r r r ^ 2 r3 ,rrzz r ' ,zzzz ' where a ~ v ( l + vv ) ) , l - v 2 E G r r l - v 2 v 2 r 3 = E ( l - v * E )' r r This then i s the f i e l d equation for an a n i s o t r o p i c m a t e r i a l . In the i s o t r o p i c case, a = 2 and 3 = 1 . This f i e l d equation i s to be solved subject to boundary conditions i n v o l v i n g both stress components and displacements. It i s u s e f u l , therefore, to e x p l i c i t l y derive the equations f o r the displacements i n terms of the stress f u n c t i o n . The r a d i a l displacement, u, i s e a s i l y obtained from the r e l a t i o n u = re . r u = ±=— [(d-vE g - vv a) <b EE r° r T , r r z r + (e-vE h - vv b) — ^ + ( f - vE i - vv C) <b ] r r r r r ,zzz 196. When a l l the necessary s u b s t i t u t i o n s are made t h i s equation reduces to u = H . * w h e r e , = |y. ( v ( G r - i ) - i ) + ^ § - i . ' r r . The a x i a l displacement i s somewhat more d i f f i c u l t to obtain. Consider the f o l l o w i n g two strain-displacement r e l a t i o n s 3co . 3co , 3u e = - — ; 2e = r — + - — . z 3z ' rz 3r 3z Solving for co gives co = / e z dz + R(r) - J ( 2 e r z - | i ) d r + Z(z) where R and Z are functions of r and z r e s p e c t i v e l y . The f i r s t expression f o r co, when expressed i n terms of the s t r e s s f u n c t i o n , i s co = / e dz + R(r) 1 z (g-v(a+d)) * + (h-v(W-e)) , r r r + ( i - v(c+f)) d> + R(r) . , zz r r 197. The second expression f o r co i s 2(l+v)x » " /< 2 erz " i l ) d r + Z < z ) = /< G E " f > d r + Z<Z> E l » r , r r r ,zz E ,rzz ^ G , r r + ^ + ( ^ G t> \zz + Z ( z ) * r r ' ( 2 8) u = ZSggJL ( + + _j£) + 1 ( _ L - 2 v 2 ) * + Z ( z ) r ' r ' If the two expressions f o r co contained i n equations (27) and (28) are compared, i t i s apparent that they d i f f e r only i n the f i n a l term. Since the two expressions must be equal, R(r) = Z(z); t h i s i s only p o s s i b l e i f R(r) = Z(z) = K, a constant. If K i s taken to be zero, then the expression for the a x i a l displacement co i s co = l.(d> + + £ <b E , r r r ' E , zz where _ 2(l+v) k N G r and 1-vv r 198. With the d e f i n i t i o n s developed i n t h i s s e c t i o n , then, an aniso-t r o p i c e l a s t o s t a t i c problem can be posed as the problem of f i n d i n g a s o l u t i o n to the s t r e s s f u n c t i o n f i e l d equation subject to s t r e s s and displacement boundary conditions expressed i n terms of the stress f u n c t i o n . C.7 Non-Dimensionalization Through the process of non-dimensionalization, the number of va r i a b l e s involved i n t h i s problem can be reduced and the r e l a t i v e importance of d i f f e r e n t parameters can be made more apparent. Before the var i a b l e s can be non-dimensionalized, though, i t i s important to l i s t a l l the v a r i a b l e s and c o e f f i c i e n t s of i n t e r e s t . These can be organized into several groups. The f i r s t group are those parameters which s p e c i f y the e l a s t i c properties of the t i s s u e . There are f i v e of these parameters, E,E , v , v and G ^ , only one of which, E, has u n i t s . The next group are the str e s s e s , s t r a i n s and displacements. The s t r a i n s are dimensionless, but the stresses have units of Pa and the displacements have u n i t s of length. F i n a l l y , there are the geometric parameters which describe the shape of the limb and the loading imposed by the tourniquet. The limb i s described by three parameters, i t s length, I, bone diameter, a, and tis s u e thickness, t, and the loading by the maximum applied pressure, P , the width of the c u f f , c, and the shape of the pressure d i s t r i b u -max c t i o n , r, (the c u f f i s assumed to be mounted symmetrically about the midpoint of the limb). To describe the true s i t u a t i o n with t h i s small number of parameters i s n e c e s s a r i l y to l i m i t the a b i l i t y of the model to 199. properly describe any p a r t i c u l a r limb, but the l i m i t e d number of para-meters should be s u f f i c i e n t to understand the influence of such major parameters as the maximum app l i e d pressure, the width of the c u f f , and the shape of the pressure d i s t r i b u t i o n . A l l the dimensional parameters have u n i t s of e i t h e r pressure (Pa) or length (m). It should be po s s i b l e , therefore, to choose a reference pressure and length. The only p o s s i b l e candidates f o r the reference pressure are E and the maximum applied pressure, P . The value E i s a max property of the material and i s uncorrelated with the str e s s l e v e l s i n the limb, while P should c o r r e l a t e very highly with the induced max ' e J s t r e s s e s . It i s therefore reasonable to s e l e c t P as the reference max pressure. The new, non-dimensionalized s t r e s s parameters and v a r i a b l e s are the followi n g , o" cf. a T r 9 z rz E a : T = •= ; E = rp P ' 9p P ' zp P ' rzp P ' p P max r max max max max There i s more f l e x i b i l i t y i n the choice of a reference length. Several parameters could be used to cha r a c t e r i z e the limb, i n c l u d i n g the thickness of the tiss u e and the length of the limb. Since t h i s study w i l l be of i n t e r e s t to c l i n i c i a n s , i t i s important that the parametric study be easy to r e l a t e to the c l i n i c a l s i t u a t i o n . If the limb length i s chosen as the reference dimension, then the non-dimensionalized thickness parameter w i l l d i r e c t l y r e f l e c t the thickness or he f t i n e s s of the limb. In add i t i o n , the c u f f width r a t i o w i l l d i r e c t l y r e f l e c t the wideness or narrowness of the c u f f . The new, non-dimensionalized length parameters are the following, 200 . J: _ a_ _ _c (Note: the space v a r i a b l e s can also be nondimensionalized with respect to the thickness: = z l l ; r^ = r / i . In subsequent uses of these space v a r i a b l e s , they w i l l be r e f e r r e d to simply as z and r, with t h e i r dimensionless character understood.) The s t r e s s f u n c t i o n i t s e l f has u n i t s of Pa«m 3. The s t r e s s f u n c t i o n can therefore be non-dimensionalized by d i v i d i n g i t by P £ 3 . This max r e s u l t s i n a dimensionless stress f u nction „ . - i . P £ 3 max The f i e l d equation i s to be solved subject to boundary conditions of the form: k 2 a = (k. i|> + — i|> + k, ip ) , p 1 , r r r T , r 3 r , z z ,z ' k 5 T p - ( k i * > r r + r-+ ir + k 6 * f , z > , r ' APPENDIX D REDUCTION OF COMPLEX SOLUTIONS TO REAL SOLUTIONS 2 0 2 . APPENDIX D REDUCTION OF COMPLEX SOLUTIONS TO REAL SOLUTIONS The s o l u t i o n method used i n t h i s development involved separating a f i e l d equation with purely r e a l c o e f f i c i e n t s i n t o two equations with po s s i b l y complex c o e f f i c i e n t s , y and x* These values were r e l a t e d to the c o e f f i c i e n t s a and 8 i n the f i e l d equation by a = y + X a n d 3 = TX If y and x have an imaginary component, then they must be complex conjugates because a and 8 are r e a l . The s o l u t i o n s obtained from the separated equations were found to be products of a r e a l f u nction i n z and a (possibly) complex function i n r. T h e . -Fu\.nc-t'ion< tf> r were Bessel's functions of zero order of the f i r s t and second kind, and could be written as J (i/TT~ r) , Y (i/TT~ r ) , o Y o Y J ( i v T " r) , Y ( i / K ~ r) , o X ° X where K and K are found from the s o l u t i o n to the separated equations. Y x As i s shown i n Section 3 . 3 . 9 . 5 , K and K are equal to the product of a Y X r e a l number and y and x r e s p e c t i v e l y . If then, Y and x are complex conjugates, and K must also be. This l a s t point i s not important f o r the argument which follows immediately, but w i l l be u s e f u l s h o r t l y . Consider now one of the separated equations, 203. * r i|> + — ^ + y * = 0 • (Dl) , r r r ,zz This equation has two possible s o l u t i o n s Z J r ) and ZY^l/^ r ) , where Z i s a r e a l function of z. I f y i s complex, then the other separated equation has the form \> + — + Y * * = 0 . (D2) , r r r 1 r , z z The question a r i s e s as to whether or not the complex conjugate of a s o l u t i o n to the f i r s t equation w i l l solve t h i s equation. It i s r e l a -t i v e l y simple to e s t a b l i s h that t h i s i s i n f a c t so. Taking <? - Z(R r + i R ± ) and i>* = Z(R r - ±R±) , where R and R^ are the r e a l and imaginary components of the s o l u t i o n f u n c t i o n i n r, and s u b s t i t u t i n g i n t o (Dl) and (D2) r e s p e c t i v e l y y i e l d s the following two equations, R' Z(R; + 7^) + Z"<YR . " R' + 1Z(R£ + 7 1) + z " < Y i R r + ^ r R i } = ° ' ( D 3 ) and R' Z(R" + — ) + Z"(Y R - Y,R,) r r r r i i R' - i Z ( R l + — ) - IZ"(Y,R + Y R.) = 0 . (D4) 1 r i r r i 204. (D3) i s s a t i s f i e d by d e f i n i t i o n , which implies that both the r e a l and imaginary parts are zero. Since the l e f t hand side of (D4) has the same r e a l and imaginary terms (with a s i g n r e v e r s a l on the imaginary term), they must also be zero and (D4) i s therefore s a t i s f i e d . This r e s u l t proves that i f y i s complex and i f ib i s a s o l u t i o n to the f i e l d equation, ty* must also be a s o l u t i o n . The two solutions to (Dl) which have been discussed are * - J (i /IC~ r)Z and - Y (i /TT" r ) Z . o y o Y The analogous solutions to the conjugate equation are i|> = Jji/IT' r)Z and i|> - Yjt/IT r ) Z , where = K*. The r e s u l t proved above implies that ip = J * ( i / T ~ r)Z and <|> = Y * ( i / T ~ r)Z, o Y o Y are solutions to the f i e l d equation. It i s possible, though not necessary here, to show that J (i /IT~ r) = J ( i / E * " r) = J * ( i / ! T " r) o x o Y o Y and Y ( i / F " r) = Y * ( i / K ~ r) = Y * ( i / l C — r) o x o x o x The important point here i s that i f both J Z and J*Z are so l u t i o n s r o o to the f i e l d equation, so i s any l i n e a r combination of the two. Two new so l u t i o n s can be generated by d e f i n i n g two functions of r as follows , 205. J = -kj +J*) = Re(J ), r 2 o o o ' and J - - y ( J -J*) = Im(J ) . i Z o o o This d e r i v a t i o n can be repeated f o r Y and Y*. The four new s o l u t i o n s r o o generated by t h i s method are a l l r e a l functions of r; consequently, the a r b i t r a r y constants used to s a t i s f y the boundary conditions do not have to be complex. zobj 3cn/ 208: APPENDIX E COMPENDIUM OF AXIAL STRAIN PLOTS FOR ALL TEST RUNS 209. Axial Strain Figure E l . RSNN, RSNA, RSNW 210. Axial Strain z Axial Strain CM km - -C^-»otrr^vv / — o . 10 — _ r ^ ~ ' j r I ' M * I I I ' 0.0 0.1 0.2 0.3 0.4 z Axial Strain CM 1 1 1 1 1 1 ' 1 0.0 0.1 0.2 0.3 0.4 z Figure E2. RSAN, RSAA, RSAW 2 1 1 . Axial Strain Axial Strain Axial Strain z Figure E3. RSWN, RSWA, RSWW 2 1 2 . Axial Strain z Axial Strain z Axial Strain .5 Figure E4. SSMM, SSMA, SSMW 213. Axial Strain Axial Strain Axial Strain Figure E5. SSAN, SSAA, SSAW 2 1 4 . Axial Strain Axial Strain Axial Strain .a Figure E6. SSWN, SSWA, SSWW Axial Strain Axial Strain • 1 b f / A 1 | 0.4 1 i If \ 0 0 0.1 0.2 0.3 0.4 0 Z Axial Strain o m o 1 t 1 1 1 ^ i d 'P 1 7I?1 I i j i i i I I 0 0 0.1 0.2' 0.3 0.4 0. z Figure E7. RFNN, RFNA, RFNW Axiai Strain Figure E8. RFAN, RFAA, RFAW 2 1 7 . Axial Strain Figure E9. RFWN, RFWA, RFWW Axial Strain Axial Strain Axial Strain Figure E10. SFNN, SFNA, SFNW Axial Strain '.S z Axial Strain o ( / / / / ; \ O \ \ § P \ \ \° 11 IT 40 0.40 — ;o o.so— o fv e 0.10 / ^ ^ - 0 . 0 0 .—Tj.OO-' " 1 & Axial Strain Figure E l l . SFAN, SFAA, SFAW 220. Axial Strain Axial Strain Figure E12. SFWN, SFWA, SFWW Axial Strain Figure E13. SSAA*1, RSAA*2, SFAA*3 222. Axial Strain z Axial Strain Axial Strain Z Figure E14. SSAA*4, SSAA*5, SSAA*6 

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