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Comparison and application of rheological constitutive functions for whole human blood Easthope, Peter Lyall 1979

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COMPARISON AND APPLICATION OF RHEOLOGICAL CONSTITUTIVE FUNCTIONS FOR WHOLE HUMAN BLOOD by PETER LYALL EASTHOPE B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Pathology) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard The U n i v e r s i t y o f B r i t i s h Columbia Oct 1979 (c) Peter L y a l l Easthope, 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ( ^ 7 * ? Q c i *T :-6 A b s t r a c t This work develops an e m p i r i c a l method f o r i n v e s t i g a t i o n of the flow p r o p e r t i e s of blood and a p p l i e s i t to a c l i n i c a l l y o r i e n t e d problem; The development focuses on the c h a r a c t e r i z a t i o n of the flow p r o p e r t i e s of a blood sample. According t o the theory of continuum mechanics the steady s t a t e flow p r o p e r t i e s of a m a t e r i a l are c h a r a c t e r i z e d completely by i t s c o n s t i t u t i v e ( B u r c h f i e l d , 1972) f u n c t i o n which r e l a t e s the shear s t r e s s measured i n a rheometer to the shear r a t e and hematocrit of the sample; Eleven f u n c t i o n s d e r i v e d from v a r i o u s sources were examined f o r t h e i r a b i l i t y to f i t flow data from t h i r t y — o n e normal i n d i v i d u a l s , eleven of whom were using o r a l c o n t r a c e p t i v e s . (The remainder were not using any drugs.) A shear r a t e range of 0.0312 to 124 s _ 1 was used at hematocrits from 0.29 to 0;55. A n o n - l i n e a r curve f i t t i n g procedure allowed an o r d e r i n g o f the f u n c t i o n s to be e s t a b l i s h e d with r e s p e c t to t h e i r goodness of f i t ; The f u n c t i o n f i r s t employed by Walburn and Schneck (1976), T = Xi exp (X2 H+XL> /H2) D 1 - * 3 where T = shear s t r e s s , D = shear r a t e , H = hematocrit and Xx to X4 are a d j u s t a b l e parameters, was found to be the most s u c c e s s f u l . This c o n s t i t u t i v e f u n c t i o n was then used to examine data obtained from a po p u l a t i o n of normal women a t va r i o u s times during the menstrual c y c l e * as a hemorheological c y c l e had been r e p o r t e d t o occur over t h i s p e r i o d . The c o n c e n t r a t i o n s of s e v e r a l plasma p r o t e i n s were a l s o determined and p l o t t e d over time* No evident c y c l e of hemorheological p r o p e r t i e s or p r o t e i n c o n c e n t r a t i o n s was found. i i i CONTENTS Abs t r a c t ................................................ .i-i Contents ..,.......*.........*. I ..*....,........*,•...-.... i i i T a b l es . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ F i g u r e s . ., ,,. , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i Acknowledgements ........i................................xi N o t a t i o n ....... , i..s. x i i 1. i n t r o d u c t i o n ..,,.,*....*.................... i **.*..*, 1 1.1 M o t i v a t i o n For Study Of Hemorneology ,,,,,,.,..,,..,.,,1 1i2 References Of General I n t e r e s t .•,*............. 3 1,i3 Q u a l i t a t i v e Features Of Hemorheology ................ 3 1.4 Development Of Current Knowledge .....*.. .. 5 1*.5 C l i n i c a l I m p l i c a t i o n s ...............................3 1.6 Theory ........... I.... i , , * * , . - • 11 1.7 Cu r r e n t Status Of Hemorheology 16 2. Experimental Methods ................................... 20 2,1 Laboratory Methods . . „ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.1 A c q u i s i t i o n Of Blood Samples *.....................20 2.1.2 Viscometry ....20 2.1.3 M a n i p u l a t i o n s .................................. 21 2.1.4 I n t e r p r e t a t i o n Of Observations ....................22 2.1.5 Plasma P r o t e i n Determination ......................22 2*2 The Subject P o p u l a t i o n * .,* ........................... 22 3. T h e o r e t i c a l Methods *...*..... ...24 3*1 I n t e r p r e t a t i o n Of o b s e r v a t i o n s ......................24 3i2 Theory Of F i t t i n g Of The CF To Observations 24 3,3 S t a t i s t i c a l T e s t i n g And Comparison Of CFs .......*...24 3,«4 N o r m a l i z a t i o n Of Parameters ..*.,....,..........*....27 i v 3.5 Graphing ............................................28 3.6 Programs Used For Data A n a l y s i s ..................... 28 4, R e s u l t s And D i s c u s s i o n ................... 29 4 . 1 T e s t i n g Of C o n s t i t u t i v e F u n c t i o n s .29 4.2 A p p l i c a t i o n To Examination Of Menstrual C y c l e ; . . . . . . . 34 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . , . 88 B i b l i o g r a p h y ...................... 89 V Tables Table 1: Features Of The F u n c t i o n s Considered ............. 36 Table 2A: Standard D e v i a t i o n s Of S t r e s s (s) For Each CF Ap p l i e d To Each Sample; Steady S t a t e ; S2 Shear Rate Range; ( u n i t s : mPa ) ...................................37 Table 2B: P r o b a b i l i t i e s That Successive P a i r s Of Orderings Of sm In Table 2A Are C o r r e c t .......................... 38 Table 3A: Standard D e v i a t i o n s Of S t r e s s (s) For Each CF Applied To Each Sample; Steady S t a t e ; S1 Shear Rate Range; ( u n i t s : mPa ) ...................................39 Table 3B: P r o b a b i l i t i e s That Successive P a i r s Of Orderings Of sm In Table 3A Are C o r r e c t 40 Table 4A: Standard D e v i a t i o n s Of S t r e s s (s) For Each CF Ap p l i e d To Each Sample; Peak; S2 Shear Rate Range; ( u n i t s : mPa ) . i 41 Table 4B: P r o b a b i l i t i e s That S u c c e s s i v e P a i r s Of Orderings Of sm In Table 4A Are C o r r e c t ,.,.42 Table 5A: Standard D e v i a t i o n s Of S t r e s s (s) For Each ' CF Applied To Each Sample; Peak; S1 Shear Rate Range; ( u n i t s : mPa ) .......................................... 43 Table 5B: P r o b a b i l i t i e s That S u c c e s s i v e P a i r s Of Orderings Of sm In Table 5A Are C o r r e c t .......................... Table 6: P o p u l a t i o n Averages And Standard D e v i a t i o n s Of Parameters Of Best F i t For Steady State Data; S1 Shear Rate Range ......................................... .... 45 Table 7: P o p u l a t i o n Averages And Standard' D e v i a t i o n s Of Parameters Of Best F i t For Steady State Data; S2 Shear v i Bate Range 4 6 Table 8: P o p u l a t i o n Averages And Standard D e v i a t i o n s Of Parameters Of Best T i t For Peak Data; S1 Shear Rate Range .. , , , . . . . . . . . . . . . . . . i 4 7 Table 9: P o p u l a t i o n Averages And Standard D e v i a t i o n s Of Parameters Of Best F i t For Peak Data; S2 Shear Rate Range 48 Tabl e 10: R a t i o s Of (Xi f o r S 2 ) / ( X i f o r S1) For Steady State Data ..... .U ...... i . . . . . . . .,,,49 Table 11: R a t i o s Of ( X i f o r S 2 ) / ( X i f o r S1) For Peak Data .50 Table 12: Mean Values Of Hematocrit And CF Parameters For I n d i v i d u a l s In Menstrual C y c l e Survey ..................51 Table 13: Plasma P r o t e i n s 52 Table 14: Mean Values Of Plasma P r o t e i n C o n c e n t r a t i o n s For I n d i v i d u a l s In Menstrual C y c l e Survey ,...53 Table 15: P l o t t i n g Symbols ............... . . . . . i . i . . . . . j i 5 4 V l l F i g u r e s F i g u r e 1: The D i s t r i b u t i o n Of s Values For Function 7, Table 3A, Steady State, 0.0312 s-» With Mean And Standard D e v i a t i o n I n d i c a t e d ........................... 26 F i g u r e 2: Sample 29, Function 7, Steady State Data F i t Over S2 £Xi) = {2.70,3.66,.389,-.00495) 55 F i g u r e 3: Sample 29, Function 7, Steady State Data F i t Over 51 £Xi} = {4.32,3.58,.798,-10407} . U . ; , . U . . . , , ...55 F i g u r e 4: Sample 29, Function 7, Peak Data F i t Over S2 {Xi} = {1.772,4.67,.451 ,.0145} 56 F i g u r e 5: Sample 29, Fu n c t i o n 7, Peak Data F i t Over S1 {Xij = {1.2 2,6.27,. 761,. 0227] . 56 F i g u r e 6: Sample 29, Function 7, Peak Data F i t Over S2 {Xi} = {1.772,4.67,;451,:0145j 57 F i g u r e 7: Sample 29, Function 7, Peak Data F i t Over S1 {Xi) = {1.22,6.27,.761,.0227} r ...57 F i g u r e 8: Sample 18, Fu n c t i o n 7, Steady State Data F i t Over 52 {Xi} = {.845,6. 08,, 441,i 0487} . . . „ » , ; , . . . . ... . . 58 F i g u r e 9: Sample 29, Function Of Huang, Steady State Data F i t Over S2 {Xij = {. 126,.83, . 209, 1. 5} ................. 58 F i g u r e 10: Sample 29, Function 2, Steady S t a t e Data F i t Over S2 {Xi} = {1. 96,76. 1,. 0758, 2. 19} .............. 59 F i g u r e 11: Sample 29, Function 3, Steady State Data F i t Over S2 {Xi} - {2. 37, 49. 4,j. 118,1 .79j A........... 59 F i g u r e 12: Sample 29, Function 4, Steady S t a t e Data F i t Over S2 {Xi} = {31,9,. 0794, 1.75, 2.40} . . . . . A .. , ....60 Fi g u r e 13: Sample 29, Function 5, Steady State Data F i t V l l l Over S2 {Xi} = {15*3,. 156,2.02,2. 12} .... 60 Figure 14: Sample. 29, Function 6, Steady State Data Fit Over S2 {Xi} = {-4. 74,,* 00982, 2. 67, 2. 37] ................61 Figure 15: Sample 29, Function 8, Steady State Data Fit Over S2 {Xij = {. 756, . 547, 3. 70, 2. 77} 61 Figure 16: Sample 29, Function 9, Steady State Data Fit Over S2 {Xij = {8. 25,. 00920, . 0309, 1. 97} ,..* , 62 Figure 17: Sample 29, Function 10, Steady State Data' Fit Over S2 {Xi} = {5, 93, . 00892,. 0223,2. Q3j I i ........... I 6 2 Figure 18: Sample 29, Function 11, Steady State Data Fit Over S2 {Xij - {6,50,.357,4440,6.54} . . . . . . . , * . 6 3 Figure 19: Sample 29, Function 2, Steady State Data Fit Over S2 {Xi} = {1, 96,76. 1,> 0758, 2, 19} 63 Figure 20: Sample 29, Function 3, Steady State Data Fit Over S2 {Xij = {2. 37, 49, 4,. 118, il ,79j ...,.*..64 Figure 21: Sample 29, Function 4, Steady State Data Fit Over S2 {Xi} = {31.9,,0794,1.75,2.40} 64 Figure 22: Sample 29, Function 5, Steady State Data Fit Over S2 {Xi} = {15*3,. 156,2102,2. 12j . i * . . i 65 Figure 23: Sample 29, Function 6, Steady State Data Fit Over S2 {Xij = {-4.74,.00982*2.67,2.37} 65 Figure 24: Sample 29, Function 7, Steady State Data Fit Over S1 {Xi} = {2*70,3.66,. 389,-.00495} ........ *...*,.,66 Figure 25: Sample 29, Function 8, Steady State Data Fit Over S2 {Xij = {. 756, , 547,3. 70, 2. 77} ....66 Figure 26: Sample 29, Function 9, Steady State Data Fit Over S2 {Xi} = {8.25,.00920,.0309,1,97} ................ 67 Figure 27: Sample 29, Function 10, Steady State Data' Fit Over S2 {Xi} = {5*93,.00892,*0223,2;03} ................ 67 Fi g u r e 28: Sample 29, F u n c t i o n 11, Steady S t a t e Data F i t Over S2 {Xij = {6.50,.357,.440,6.54j ................... 68 F i g u r e 29: Normalized Hematocrit Versus Normalized Time For One da l e 69 Figur e 30: Normalized X1 Versus Normalized Time For One Male x . . . . . . . . . . . u,..: .*;.,;,.. «....,. 70 F i g u r e 31: Normalized X2 Versus Normalized Time For One Male ......... , ...71 Fi g u r e 32: Normalized X3 Versus Normalized Time For One Male ............... , ; . . . A.72 F i g u r e 33: Normalized X4 Versus Normalized Time For One Ma l e 73 Fi g u r e 34: Normalized Hematocrit Versus Normalized Time For Women ;. .................................................74 Fi g u r e 35: Normalized X1' Versus 1 Normalized Time F o r Women .75 Fi g u r e 36: Normalized X2 Versus Normalized Time For Women .76 Fi g u r e 37: Normalized X3 Versus Normalized Time For Women .11 F i g u r e 38: Normalized X4 Versus Normalized Time F o r Women .78 Fi g u r e 39: Normalized F i b r i n o g e n C o n c e n t r a t i o n Versus Normalized Time For Women ..............*............... 7 9 F i g u r e 40: Normalized Anti-thrombin I I I Co n c e n t r a t i o n Versus Normalized Time For Women ....................... 80 Fi g u r e 41: Normalized Albumin C o n c e n t r a t i o n Versus Normalized Time For Women . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 81 Fi g u r e 4 2: Normalized IgM C o n c e n t r a t i o n Versus Normalized Time For Women . , . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 F i g u r e 43: Normalized oc2-Macroglobulin C o n c e n t r a t i o n Versus X Normalized Time For Women i 8 3 F i g u r e 44: C o r r e l a t i o n P l o t Of Standard D e v i a t i o n Of F i t , s, Versus Normalized XI 84 F i g u r e 45: C o r r e l a t i o n P l o t Of Standard D e v i a t i o n Of F i t , s, Versus Normalized X2 ................ , , i . . . . . . . . . 85 F i g u r e 46: C o r r e l a t i o n P l o t Of Standard D e v i a t i o n Of F i t , s. Versus Normalized X3 ...86 F i g u r e 4 7: C o r r e l a t i o n P l o t Of Standard D e v i a t i o n Of F i t , s. Versus Normalized X4 ,..,,87 i 1 F i g u r e 48: Measuring System ............................I..105 x i Acknowledgements I would l i k e to thank Dr. Donald Brooks f o r s u p e r v i s i n g my work* Mrs. Mandy Hoskins f o r t e c h n i c a l a s s i s t a n c e and s e v e r a l members of the UBC Computing Center s t a f f f o r programming ad v i c e . F i n a n c i a l support was provided by grant MT 5759 from the Medical Research C o u n c i l to Dr. D. E. Brooks; Notation S c a l a r s ( a r r a y s of rank 0 ) D shear r a t e or v e l o c i t y g r a d i e n t or d e t r u s i o n r a t e e plasma v i s c o s i t y f s c a l e f a c t o r of rheometer output s i g n a l H volume f r a c t i o n of e r y t h r o c y t e s (hematocrit) L (width of flow channel)/(diameter of e r y t h r o c y t e ) R rheometer output s i g n a l T shear s t r e s s x normalized value of x X i = X parameter which a d j u s t s the CF i V e c t ors and A r r a y s of rank 1 denoted by one underscore M a t r i c e s and Arrays of rank 2 denoted by a double underscore Operations A«B = £ A i B i dot product; i Q « B ) j = £ A i B i j i (B»A) i = £ B i jA j — i (A«A)ij = £ AikAkj matrix product as 33 k V g r a d i e n t operator 1 1. I n t r o d u c t i o n 1 . 1 M o t i v a t i o n f o r study of hemorheology Eheology [greek rheo, t o flow + l p g i a , d i s c o u r s e ] i s concerned with those p r o p e r t i e s of m a t e r i a l s which c h a r a c t e r i z e t h e i r deformation i n response t o a p p l i e d f o r c e s . The s u b j e c t i s a d i s t i n c t p a r t of continuum mechanics. However, complete r h e o l o g i c a l i n f o r m a t i o n , while necessary, i s not s u f f i c i e n t t o s o l v e a flow problem s i n c e other i n f o r m a t i o n of continuum mechanics i s r e g u i r e d . The r h s o l o g i c a l property of a m a t e r i a l on one s c a l e of r e s o l u t i o n a r i s e s as an average over f l u c t u a t i n g q u a n t i t i e s on a f i n e r s c a l e . A t h e o r e t i c a l f o r m u l a t i o n of t h i s f e a t u r e w i l l be c a l l e d an i n f r a s t r u c t u r a l problem. For most m a t e r i a l s i n f r a s t r u c t u r a l problems have not been s o l v e d although the s o l u t i o n s would be of g r e a t value, T h i s problem w i l l be d i s c u s s e d f u r t h e r i n the t h e o r e t i c a l s e c t i o n ( 1 , 5 ) . Hemorheology i s t h e r e f o r e the rheology of blood and i t s c o n s t i t u e n t s . At present blood h i s t o l o g y and b i o c h e m i s t r y have been more h i g h l y developed and a p p l i e d c l i n i c a l l y than hemorheology. However t h i s s i t u a t i o n i s , at l e a s t i n p a r t , a consequence of the r e l a t i v e d i f f i c u l t y r a t h e r than the r e l a t i v e importance of the s u b j e c t s . Biochemistry deals with molecular s t r u c t u r e and dynamics, h i s t o l o g y with c e l l u l a r s t r u c t u r e and p h y s i o l o g y with f u n c t i o n of organs and the organism. R e l a t i n g the two former s u b j e c t s to the l a t t e r i s u l t i m a t e l y an a p p l i c a t i o n o f continuum mechanics f o r which hemorheological i n f o r m a t i o n i s e s s e n t i a l . In d e s c r i p t i v e terms 'proper 1 f u n c t i o n i n g of the 2 c i r c u l a t i o n i s necessary f o r t r a n s p o r t of blood gases, m e t a b o l i t e s , hormones and water. The system i s complex and a l l p a r t s and f e a t u r e s must remain i n some degree of balance. Thus, f o r example, when a bi o c h e m i c a l d i s t u r b a n c e causes e r y t h r o c y t e s to aggregate more s t r o n g l y than normal the pressure drop a c r o s s a p e r f u s i o n pathway i s e l e v a t e d thereby r e q u i r i n g an i n c r e a s e d power output of the heart.. With regard to a more l o c a l i z e d e f f e c t , i n c r e a s e d aggregation of e r y t h r o c y t e s i s a form of a l t e r a t i o n of the blood which may provide one o f Virchow's t r i a d of requirements f o r thrombosis (Goldman, 1973). In e i t h e r case the a l t e r e d blood rheology would be a necessary l i n k i n the sequence l e a d i n g from d i s t u r b e d c o n s t i t u t i o n of the e r y t h r o c y t e membrane or plasma to a p a t h o p h y s i o l o g i c a l c o n d i t i o n . Hemorheology i s al s o i m p l i c a t e d i n the continuum mechanical problems which a r i s e when attempting t o i n t e r p r e t some ex y i y o o b s e r v a t i o n s on macroscopic samples of blood o r c e l l suspensions - which i s the primary concern of t h i s t h e s i s - and when designing d e v i c e s f o r e x t r a c o r p o r e a l c i r c u l a t i o n of blood. For i n s t a n c e P o i s e u i l l e ' s i n t a r e s t i n blood flow i n v e s s e l s i n s t i g a t e d the s o l u t i o n of the problem of flow of a Newtonian f l u i d i n a c i r c u l a r tube ( P o i s e u i l l e , 1840; Wiedemann, 1856). However, s i n c e blood i s non-Newtonian t h i s model i s an inadequate approximation f o r most purposes. Subsequently with use.of the Casson (1959) c o n s t i t u t i v e f u n c t i o n (see below) Watanabe* Oka and Yamamoto (1963) obtained a more r e a l i s t i c r e s u l t . 3 1.2 References of general i n t e r e s t I f a r e f e r e n c e d e a l i n g with the ge n e r a l t h e o r y of continuum mechanics i s sought the t r e a t i s e e d i t e d by Eringen (1971) i s c u r r e n t l y the best a v a i l a b l e . . B i b l i o g r a p h i c i n f o r m a t i o n i n i t i s l i m i t e d but i n d i r e c t l y i t g i v e s access to the e n t i r e s u b j e c t . For the ge n e r a l theory and techniques of r h e o l o g i c a l i n v e s t i g a t i o n the monograph of Coleman, Markovitz and N o l l (1966) i s preferable;. However f o r some recent developments i t may be necessary t o r e s o r t to Walters (1975). On the s u b j e c t of hemorheology the monographs of Charm and Kurland (1974) and Whitmore (1968) should be r e f e r r e d t o f i r s t and those o f Di n t e n f a s s (1971, 1976) r e s o r t e d t o only f o r s u b j e c t s not covered i n the former* The review of hemorheology by M e r r i l l (1969) i s we l l w r i t t e n and c o n c i s e , but i f a c u r r e n t and comprehensive d i s c u s s i o n i s p r e f e r r e d , t h a t o f Chien (1975) i s recommended, 1.3 Q u a l i t a t i v e f e a t u r e s of hemorheology P r i o r to s e c t i o n 116 r e f e r e n c e w i l l be made t o a ' f o r c e ' a c t i n g on an element of f l u i d and the a s s o c i a t e d 'deformation' of the f l u i d element* In s e c t i o n 1*6 the f o r c e and deformation w i l l be given more p r e c i s e meanings, M e r r i l l ' s (1969) example of honey as a Newtonian l i q u i d w i l l be invoked here f o r i l l u s t r a t i o n . When a f o r c e i s a p p l i e d t o a sample of honey a deformation r e s u l t s with the l o c a l r a t e of deformation being p r o p o r t i o n a l t o the f o r c e . , A l i q u i d f o r which t h i s p r o p o r t i o n a l i t y holds t r u e i s c a l l e d a Newtonian l i q u i d . Water, motor o i l , and many other commonplace l i q u i d s , i n c l u d i n g plasma, are a l s o Newtonian. There are other m a t e r i a l s f o r which the r a t e 4 of deformation i s not p r o p o r t i o n a l to a p p l i e d f o r c e but which n e v e r t h e l e s s can flow. M e r r i l l (1969) c i t e s the example of mayonnaise which, u n l i k e honey* w i l l not d r i p e n t i r e l y from a spoon* although acted upon c o n t i n u o u s l y by the f o r c e of g r a v i t y . . For any m a t e r i a l the r e l a t i o n s h i p between the f o r c e s a p p l i e d to a s m a l l element and the deformation of t h a t element i s an . i n t r i n s i c m a t e r i a l property. Vfrien the,, force and deformation effects are given precise, quantitative meanings (shear stress and shear strain rate res-pectively), they canabe relatecLby a^imthemaitical function."'This function i s cal-lcdl .the constitutive ffunction..{&!)ji®f .thebmaterial. The ratio of shear stress-,to shear strain rate :is definedjas- -the .dynamic:viscosity. For 1 a New-tonian liq u i d the'.,viscosity i s independent of r a t e : of deformation whereas f o r a non-Newtonian l i q u i d i t i s not.. i s o u t l i n e d i n s e c t i o n 1.1 the r h e o l o g i c a l property of a m a t e r i a l , represented mathematically by the CF, along with other i n f o r m a t i o n of continuum mechanics al l o w s a complete flow problem to be s o l v e d - provided the mathematical d i f f i c u l t i e s can be overcome,. The approach o f experimental rheology i s to choose a simple flow system and then t o deduce the CF from o b s e r v a t i o n s o f the system. Although p r i m a r i l y the CF must r e l a t e the f o r c e s to the r a t e s of deformation* the i n f r a s t r u c t u r a l f e a t u r e of s e c t i o n 1.1 must a l s o be c o n s i d e r e d . In q u a l i t a t i v e terms the components i n v o l v e d are: 1. Plasma v i s c o s i t y : Blood v i s c o s i t y i s approximately l i n e a r i n plasma v i s c o s i t y , 2. Hematocrit: Blood v i s c o s i t y i n c r e a s e s n o n - l i n e a r l y with hematocrit (Chien et a l . , 1966); 5 3. I n t e r c e l l u l a r a ggregation: As e r y t h r o c y t e s flow past one another i n t e r c e l l u l a r bonds are formed and broken* An i n c r e a s e i n s t r e n g t h o f aggregation causes an i n c r e a s e i n blood v i s c o s i t y * Macromolecules i n c l u d i n g f i b r i n o g e n and dextran are known t o p a r t i c i p a t e i n the aggregation of e r y t h r o c y t e s (Brooks, Goodwin and Seaman, 1974). 4, E r y t h r o c y t e mechanics: In the.flow of blood, the motion of any e r y t h r o c y t e i s impeded by i t s neighbours.. T h i s e f f e c t i s f reduced as e r y t h r o c y t e s become more f l e x i b l e . Thus, i n c r e a s e d f l e x i b i l i t y o f e r y t h r o c y t e s i s a s s o c i a t e d with decreased v i s c o s i t y of blood. The f l e x i b i l i t y of e r y t h r o c y t e s i s not e a s i l y characterized*. I t i n v o l v e s two main components: a) I n t e r i o r : The e r y t h r o c y t e contents are Newtonian and are c h a r a c t e r i z e d by a known v i s c o s i t y (Cokelet and Meiselman, 1968) . b) Membrane: The kinematics of the membrane i s extremely complex i n blood flow,. I t appears e s t a b l i s h e d t h a t the e r y t h r o c y t e membrane prese n t s a s m a l l bending s t i f f n e s s i n comparison to l a r g e r s t i f f n e s s f o r other deformations. Both v i s c o u s and e l a s t i c p r o p e r t i e s have been r e p o r t e d . Because of these s e v e r a l d i f f i c u l t f e a t u r e s complete d e s c r i p t i o n of the mechanics of an e r y t h r o c y t e i s a formidable problem. Evans and Skalak (1979) have presented a monograph on t h i s s u b j e c t . 1.4 Development of c u r r e n t knowledge The s t r o n g dependence of blood v i s c o s i t y on hematocrit from 0.00 to 0.95 was demonstrated by Chien et a l . (1966). Stone, Thompson and Schmidt-Nielsen (1968) presented the hematocrit dependence of v i s c o s i t y f o r blood from s e v e r a l s p e c i e s with 6 d i f f e r i n g e r y t h r o c y t e shapes. Wells et - a l . (1964) reported t h a t the v i s c o s i t y of a suspension of e r y t h r o c y t e s i n a f i b r i n o g e n s o l u t i o n i n c r e a s e d with the c o n c e n t r a t i o n of the l a t t e r ; . Putnam, Kevy and Repogle (1967) and M e r r i l l et a l ; _ (1966) found t h a t a d d i t i o n of f i b r i n o g e n t o blood ex yiyp was a s s o c i a t e d with i n c r e a s e d v i s c o s i t y ; M e r r i l l et a l . (1965b) found t h a t blood v i s c o s i t y i n c r e a s e d with n a t i v e f i b r i n o g e n c o n c e n t r a t i o n (2.1-4;6 mg/ml) i n normal donors. However, Begg and Hearns (1966) found no c o r r e l a t i o n over a wider range of f i b r i n o g e n c o n c e n t r a t i o n s (2.76-10;70 mg/ml) but used only shear r a t e s (see below) from 50 s~» to 260 s~». Weaver, Evans and Walder (1969) found a r e l a t i o n s h i p between blood v i s c o s i t y and f i b r i n o g e n c o n c e n t r a t i o n but concluded t h a t i n normal i n d i v i d u a l s t h i s would not have a s i g n i f i c a n t e f f e c t on c a r d i o v a s c u l a r dynamics. M e r r i l l * Cheng and P e l l e t i e r (1969) found the Casson (1959) y i e l d s t r e s s (see below) to be p r o p o r t i o n a l to the square of the plasma f i b r i n o g e n c o n c e n t r a t i o n i n normal blood. Chien et a l . . (1970) added p u r i f i e d f i b r i n o g e n to e r y t h r o c y t e suspensions and found t h e i r v i s c o s i t i e s to i n c r e a s e with f i b r i n o g e n c o n c e n t r a t i o n s r a n g i n g from zero through normal. Thus i n s p i t e of some e a r l y c o n t r a d i c t o r y r e s u l t s i t i s now w e l l e s t a b l i s h e d t h a t f i b r i n o g e n i n f l u e n c e s the flow p r o p e r t i e s of blood; Meiselman et a l ; (1967) s t u d i e d the r e l a t i o n s h i p between the c o n c e n t r a t i o n s of dextrans of v a r i o u s molecular weights and the v i s c o s i t y o f whole blood i n c o n c e n t r i c c y l i n d e r (Couette, 1890) and c a p i l l a r y v iscometers. Dextrans of molecular weight 40,000 or gr e a t e r were found t o i n c r e a s e i n t e r c e l l u l a r adhesion 7 as shown by the Casson (1959) y i e l d s t r e s s (see below) and v i s c o s i t y at low shear r a t e * Brooks et alv (1974) showed t h a t e r y t h r o c y t e aggregation and hence suspension v i s c o s i t y depends upon both dextran and i o n i c c o n c e n t r a t i o n s , Normal e r y t h r o c y t e s are h i g h l y deformable.. However t h i s deformation o f the e r y t h r o c y t e s depends upon both membrane and i n t e r n a l p r o p e r t i e s . F i s c h e r , S t o h r - L i e s e n and Schmid-Schonbein (1978b) have presented photomicrographs showing the d e f o r m a b i l i t y and motion of an e r y t h r o c y t e i n a v e l o c i t y g r a d i e n t . E a r l i e r work by F i s c h e r et a l . (1978a) has shown t h a t the d e f o r m a b i l i t y o f the e r y t h r o c y t e depends s t r o n g l y upon the number o f d i s u l p h i d e bonds i n the intramembranous s p e c t r i n network. Based on data obtained with a c o n e - p l a t e viscometer, Charache et a l . (1967) gave the v i s c o s i t y of e r y t h r o c y t e hemolysates as a f u n c t i o n of hemoglobin c o n c e n t r a t i o n and s t a t e d t h a t the r e s u l t s were non-Newtonian. Schmidt-Nielsen and T a y l o r (1968) used a c a p i l l a r y tube viscometer and obtained r e s u l t s s i m i l a r t o those of Charache et a l l , (1967). C o k e l e t e t a l . (1968), using a c o n c e n t r i c c y l i n d e r viscometer (Couette, 1890), confirmed p r e v i o u s r e s u l t s r e g a r d i n g the e f f e c t of hematocrit and a l s o showed d e f i n i t e Newtonian behaviour when hemoglobin c o n c e n t r a t i o n d i d not exceed t h a t normally o c c u r r i n g i n e r y t h r o c y t e s . Ham e t - a l . (1968) gave r e s u l t s s i m i l a r t o those of Schmidt-Nielsen and T a y l o r (1968). Thus the: major p a r t of the e r y t h r o c y t e content appears t o be a Newtonian l i q u i d . The mechanical behaviour of the e r y t h r o c y t e membrane 8 i n v o l v e s s e v e r a l d i f f i c u l t f e a t u r e s . Because of the complexity and extent of the s u b j e c t i t cannot be d i s c u s s e d f u r t h e r here* The i n t e r e s t e d reader should c o n s u l t Evans and Skalak (1979). 1.5 C l i n i c a l i m p l i c a t i o n s A v a s t amount of work has been done i n attempting to f i n d simple c o r r e l a t i o n s between d i s e a s e s and abnormal hemorheology. In many i n s t a n c e s unequivocal r e s u l t s have not been e s t a b l i s h e d . The review by Chien (1975) covers the s u b j e c t w e l l and only a b r i e f survey i s giv e n here, A study r e l e v a n t to any c l i n i c a l hemorheological assessment of women was c a r r i e d out by D i n t e n f a s s (1971, p, 21) who presented data showing a time dependence o f blood v i s c o s i t y with the same p e r i o d as the menstrual c y c l e i n normal women. Sin c e the blood was t e s t e d without a n t i c o a g u l a n t i t i s p o s s i b l e t h a t i t was c l o t t i n g d u r i n g the t e s t . One aim of t h i s t h e s i s was to t e s t t h i s h y p o t h e s i s using a n t i c o a g u l a t e d blood. S e l l a b l e and accurate r e s u l t s are needed s i n c e i n some p a t h o l o g i c a l s t a t e s the hemorheological a b e r r a t i o n s may be comparable i n magnitude t o normal p h y s i o l o g i c a l f l u c t u a t i o n s . Aronson, Magora, and Schenker (1971) found an i n c r e a s e i n blood v i s c o s i t y p a r a l l e l i n g the i n s t i t u t i o n of o r a l c o n t r a c e p t i v e (OC) therapy i n some i n d i v i d u a l s and noted t h a t the woman having the most r a p i d and l a r g e s t i n c r e a s e i n blood v i s c o s i t y proved to be the only one i n the t e s t group to develop t h r o m b o p h l e b i t i s * They suggested t h a t r h e o l o g i c a l measurements might be used i n s c r e e n i n g f o r t h i s problem. E l e v a t e d blood v i s c o s i t y has been reported i n coronary heart d i s e a s e , a t h e r o s c l e r o s i s , c e r e b r o v a s c u l a r d i s e a s e s and 9 p e r i p h e r a l v a s c u l a r d i s e a s e s i n c l u d i n g Raynaud's syndrome. In some cases e l e v a t e d v i s c o s i t y has been a t t r i b u t e d t o one or more of e l e v a t e d hematocrit (Mayer, 1 9 6 4 ; S t a b l e s et a l . , 1 9 6 7 ) , a l t e r e d c o n c e n t r a t i o n of some plasma p r o t e i n s ( D i n t e n f a s s , J u l i a n and M i l l e r , 1 9 6 6 ; Langsjoen, and Inman* 1 9 6 8 ; Bygdeman and Wells, 1 9 6 9 ) or e l e v a t e d c o n c e n t r a t i o n of f i b r i n o g e n s p l i t products (Copley, L u c h i n i and Whelen, 1 9 6 6 ; V e r s t r a e t e , Vermylen and Donati, 1 9 7 1 ) . In diabetes m e l l i t u s , abnormal hemorheology has been c o r r e l a t e d with e l e v a t e d plasma v i s c o s i t y . S p e c i f i c f e a t u r e s have been found such as c o r r e l a t i o n of e r y t h r o c y t e aggregation with r e t i n o p a t h y and neuropathy ( D i t z e l , 1 9 7 1 ) 1 . S u r g i c a l o p e r a t i o n s may r e s u l t i n a reduced hematocrit, e l e v a t e d plasma f i b r i n o g e n or e l e v a t e d e r y t h r o c y t e aggregation with r h e o l o g i c a l m a n i f e s t a t i o n s ; Dormandy and Edelman ( 1 9 7 8 ) found t h a t 'an e l e v a t e d blood v i s c o s i t y before o p e r a t i o n [ s i c ] i s s i g n i f i c a n t l y a s s o c i a t e d with the development o f p o s t o p e r a t i v e D, V. T. {deep vein thrombosis Q'.. However, t h e i r data showed a wide o v e r l a p between the normal and D.V.T. groups. The most severe e f f e c t s occur when blood i s passed through an e x t r a c o r p o r e a l c i r c u i t ; T h i s causes d e n a t u r a t i o n of plasma p r o t e i n s , hemolysis and hematocrit r e d u c t i o n due t o priming of the e x t r a c o r p o r e a l c i r c u i t with a b l o o d - f r e e l i q u i d . A n e s t h e t i c s do not appear to have any s i g n i f i c a n t d i r e c t e f f e c t upon blood rheology (Behar and Alexander, (1966; Aronson e t - a L , 1 9 6 8 ) . A r e s p i r a t o r y d i s o r d e r may cause secondary polycythemia, thus i n c r e a s i n g blood v i s c o s i t y , ; Decreased p e r f u s i o n consequent t o i n c r e a s e d v i s c o s i t y aggravates the hypoxia which the 10 polycythemia would otherwise c o u n t e r a c t (Segel and Bishop, 1966). A c i d o s i s w i l l cause decreased f l e x i b i l i t y of the e r y t h r o c y t e which w i l l c o n t r i b u t e to the e l e v a t i o n of the blood v i s c o s i t y ; A l l the p r e v i o u s l y mentioned problems are p a r t i c u l a r l y severe when o c c u r i n g i n i n f a n t s because of i n h e r e n t high h e m a t o c r i t (Wintrobe, 1967) and s t i f f e r y t h r o c y t e s (Gross and Hathaway, 1972) r e l a t i v e to the a d u l t ; In r e c e n t years hemodilution has become an accepted t h e r a p e u t i c procedure (Messraer and Schmid-Schonbein, 1975). The o b j e c t i v e i s to maximize the t r a n s p o r t c a p a c i t y of the c i r c u l a t i o n . Increase of hematocrit causes i n c r e a s e of oxygen c a p a c i t y per volume of whole blood however i t a l s o causes i n c r e a s e o f v i s c o s i t y which reduces flow r a t e . Thus under normal p h y s i o l o g i c a l c o n d i t i o n s , a h e m a t o c r i t around .40 would appear to be o p t i m a l . However, i n c l i n i c a l s i t u a t i o n s , decreased c a r d i a c s t r e n g t h along with i n c r e a s e d aggregation of e r y t h r o c y t e s or i n c r e a s e d p e r f u s i o n r e s i s t a n c e : o f t i s s u e s may r e s u l t i n a lower optimum hematocrit (Chien, 1971). The r e d u c t i o n of hematocrit i s u s u a l l y achieved by i n f u s i o n of a l i g u i d without e r y t h r o c y t e s . T h i s procedure has been a p p l i e d i n cardiopulmonary bypass, myocardial i n f a r c t i o n , h y p e r v i s c o s i t y of the neonate and c i r c u l a t o r y shock. The g e n e r a l p i c t u r e seems t o be t h a t , w h i l s t r h e o l o g i c a l a b n o r m a l i t i e s accompany a number of abnormal s t a t e s , s e v e r a l d i f f i c u l t i e s have prevented a complete c h a r a c t e r i z a t i o n of such e f f e c t s ; namely: 1; Many i n v e s t i g a t o r s have made r h e o l o g i c a l measurements at 11 d i f f e r e n t h e m a t o c r i t s and shear r a t e s ; comparison on a uniform b a s i s has r a r e l y been made* 2. 'Normal* blood v i s c o s i t i e s and plasma p r o t e i n c o n c e n t r a t i o n s o f t e n o v e r l a p 'abnormal' values. T h i s d i f f i c u l t y i s compounded by 1*.above. 3. Viscometers with v a r i a b l e shear r a t e , shear r a t e s below 10-i s - i and s m a l l sample volume have only r e c e n t l y become a v a i l a b l e (Contraves &G). The instrument c u r r e n t l y i n use i n our l a b o r a t o r y (Contraves LS-2) p r o v i d e s shear r a t e s much lower than p r e v i o u s l y used i n c l i n i c a l work i n a range where changes i n i n t e r c e l l u l a r aggregation cause s i g n i f i c a n t changes i n v i s c o s i t y ; . 4. Because the. i n f r a s t r u c t u r a l problem has not been s o l v e d , the observed r h e o l o g i c a l p r o p e r t i e s of blood cannot be r e l a t e d t h e o r e t i c a l l y t o the m i c r o s c o p i c and chemical p r o p e r t i e s of the e r y t h r o c y t e s and the chemical p r o p e r t i e s of plasma. 1.6 Theory The r a t i o , (width of flow channel)/(diameter of e r y t h r o c y t e ) , which w i l l be denoted L , i s s i g n i f i c a n t i n e s t a b l i s h i n g the g e n e r a l nature of the flow of blood. Thus i n a c a p i l l a r y where L i s s l i g h t l y s m a l l e r than 1 e r y t h r o c y t e s move i n s i n g l e f i l e * The shape of each e r y t h r o c y t e i s almost c o n s t a n t and i t s a x i s o f symmetry remains c o i n c i d e n t with the c a p i l l a r y a x i s . The i n f r a s t r u c t u r a l problem i n t h i s case i s to express the flow r a t e through a c a p i l l a r y as a f u n c t i o n of diameter, l e n g t h , p r e s s u r e drop* plasma v i s c o s i t y , number of e r y t h r o c y t e s and p r o p e r t i e s of the e r y t h r o c y t e ; T h i s has been accomplished by Skalak (1978).. When L>1 - t h i s i n c l u d e s a l l v e s s e l s other than c a p i l l a r i e s 12 and a l l f a m i l i a r viscometers - l a t e r a l motion, r o t a t i o n , deformation, and aggregation of e r y t h r o c y t e s are each s i g n i f i c a n t and the s i t u a t i o n becomes f a r more complicated. The l a r g e number of p o s s i b l e c o n f i g u r a t i o n s makes d i f f i c u l t the c a l c u l a t i o n of averages of observable q u a n t i t i e s and l a r g e f l u c t u a t i o n s i n observable q u a n t i t i e s can be expected t o be a s s o c i a t e d with the occurrence of l a r g e aggregates, Thesee f l u c t u a t i o n s would hamper experimental determination of average values; Spontaneous occurrence of hematocrit g r a d i e n t s r e s u l t i n g from both flow and sedimentation may f u r t h e r complicate the i n t e r p r e t a t i o n of the experimental s i t u a t i o n . The i n f r a s t r u c t u r a l problem has been so l v e d f o r c e r t a i n systems much si m p l e r than b l o o d . In these cases there was minimal need f o r approximation, the averaging was elementary and the r e s u l t s were c l o s e to exact. An example i s the case i n which L » 1 with non-i n t e r a c t i n g r i g i d spheres a t low volume f r a c t i o n i n a Newtonian f l u i d with both spheres and f l u i d having the same d e n s i t y ( E i n s t e i n , 1906); As more f e a t u r e s are added t o the models e.g.: a l l o w i n g a l a r g e volume f r a c t i o n of p a r t i c l e s - the d i f f i c u l t i e s of s o l u t i o n i n c r e a s e , and more e x t e n s i v e approximations must be made (e.g. Casson, 1959; B a t c h e l o r , 1950), Thus i t might appear hopeless f o r these e s t a b l i s h e d t h e o r i e s t o be extended t o one which c o u l d be u s e f u l l y a p p l i e d t o blood; However, a sagacious paper by Lew (1969) may p r o v i d e the groundwork f o r a s u c c e s s f u l development. U n t i l t h i s i s achieved observed r e l a t i o n s h i p s between v a r i a b l e s must be expressed by the most expedient a v a i l a b l e mathematical formalism, For the viscometer used i n t h i s i n v e s t i g a t i o n L=63. 13 In the a n a l y s i s of a flow with t h i s l a r g e L-value, the s i m p l e s t approach i s to assume that the average values of o b servable v a r i a b l e s can be represented by continuous f u n c t i o n s of p o s i t i o n and time; T h i s i m p l i e s t h a t the r e l a t i o n s h i p s between these v a r i a b l e s are a l s o continuous f u n c t i o n s . T h i s i s c a l l e d the continuum h y p o t h e s i s and the imaginary' m a t e r i a l which conforms p r e c i s e l y t o i t without any i n f r a s t r u c t u r a l f e a t u r e s i s c a l l e d the continuum. a p r e c i s e c h a r a c t e r i z a t i o n of the flow p r o p e r t i e s of blood must i n v o l v e q u a n t i t a t i v e measures of i t s rate of deformation (or more g e n e r a l l y , h i s t o r y of deformation) and of the f o r c e e f f e c t which i s a s s o c i a t e d with the deformation* The subsequent o b j e c t i v e i s t o e s t a b l i s h as p r e c i s e l y as p o s s i b l e the r e l a t i o n s h i p which e x i s t s between ' f o r c e s ' o c c u r r i n g i n blood and the a s s o c i a t e d deformations. I t i s a f a m i l i a r f a c t t h a t v e l o c i t y i s a v e c t o r having a d i r e c t i o n . Objects analogous to v e c t o r s but having two independent d i r e c t i o n s a s s o c i a t e d with them e x i s t and are c a l l e d t e n s o r s of rank 2, The deformation of i a l i q u i d continuum i s represented by such'a t e n s o r , c a l l e d the s t r a i n r a t e * S i m i l a r l y the ' f o r c e e f f e c t ' a c t i n g upon an element of the continuum must be represented by a rank 2 tensor c a l l e d the s t r e s s r a t h e r than by a f o r c e v e c t o r . The natures of these t e n s o r s are d i s c u s s e d f u r t h e r i n appendix 1; Although i n general the s t r a i n r a t e and s t r e s s may be f u n c t i o n s of p o s i t i o n and time the concern of rheology i s o n l y to e s t a b l i s h the r e l a t i o n s h i p between these two t e n s o r s at the same l o c a t i o n and i n s t a n t . T h i s r e l a t i o n s h i p i s an i n t r i n s i c m a t e r i a l property represented by the c o n s t i t u t i v e f u n c t i o n (CF). 14 c o n s t i t u t i v e f u n c t i o n : D,e,H, {Xij — > T (1) SYMBOL NAME DIMENSION UNITS T s t r e s s t e n s o r MT-2L" 1 mPa * D s t r a i n r a t e tensor T - i e plasma v i s c o s i t y H T - I L - I mPa»s * H hematocrit {Xij s e t of n o n - p h y s i c a l dimensions deduced u n i t s deduced parameters which from context from context a d j u s t the f u n c t i o n * note: 1 mPa = 10" 3 N«m- 2 = 1 mPa«s = 1 cp 10-2 dyn»cm-2 The n o t a t i o n above says t h a t the values of D ,e,H and {Xi} uniquely determine the value of T under reasonable assumptions* In the work of t h i s t h e s i s e m p i r i c a l f u n c t i o n s of t h i s form are examined.. On the other hand, the o b j e c t i v e of the i n f r a s t r u c t u r a l problem i s to e s t a b l i s h such a f u n c t i o n t h e o r e t i c a l l y from a r e a l i s t i c p h y s i c a l model..The {Xij would then be r e p l a c e d by d e f i n i t e p h y s i c a l parameters which must i n c l u d e q u a n t i t a t i v e measures of e r y t h r o c y t e s t i f f n e s s , shape and aggregation* As e x p l a i n e d i n appendix 1 each of T_ - and D have 9 components i n matrix representation. However, i n 2-dimensional flows with a p p r o p r i a t e c o o r d i n a t e s a l l components except 2 i n each of T and D are zero, i , e * : T]Z = TZi # 0 * DIZ = D2I (2) T;j = 0 = Dy U , j) i ( (1,2) , (2,1) } The nonzero component of s t r e s s i s c a l l e d the shear s t r e s s and twice the nonzero component of s t r a i n r a t e , 2 D , 2 , i s c a l l e d the 15 shear r a t e , v e l o c i t y g r a d i e n t or d e t r u s i o n r a t e . S t r i c t l y speaking the f i r s t term should not be used with t h i s meaning s i n c e i t can a l s o be a p p l i e d t o the t e n s o r or tensor component. However, i t i s f r e q u e n t l y t r e a t e d as synonymous with the other two terms. The symmetry of the t e n s o r s a r i s e s from a c o n s i d e r a t i o n of r o t a t i o n a l motion i n the case of s t r e s s and from c o n s t r u c t i v e d e f i n i t i o n i n the case of s t r a i n r a t e . The CF may then be denoted CF : D,e,H, (Xi) — > T (3) where T=T,2 and D= 2D[2, . There are s e v e r a l CFs of t h i s form which have been developed from simple phenomenological models or which are j u s t reasonable mathematical c h o i c e s . Probably the best known of these i s t h a t of Casson (1959). I n t e r p r e t e d i n the present context i t i s T = (k0+k, D»/ 2) 2 (4) k 0 = c, (<1-H)-C* -1) k, = c, ((k 0/eW2) -1) where c, and cz are constants f o r a p a r t i c u l a r m a t e r i a l * T h i s CF has a y i e l d s t r e s s , k§, which i s present with zero shear r a t e . . M e r r i l l (1969) reviewed the a p p l i c a t i o n of t h i s CF to experimental data* The aspect of most i n t e r e s t here i s the treatment of the Casson (1959) y i e l d s t r e s s . With a r o t a t i o n a l viscometer, the y i e l d s t r e s s i s not a c t u a l l y measured; but determined ±>y e x t r a p o l a t i o n t o zero shear r a t e on a p l o t of (shear s t r e s s ) * / 2 vs. (shear r a t e ) * / 2 . However, with contemporary viscometers* the Casson (1959) p l o t c o n f l i c t s with the observed s t r e s s which drops towards the o r i g i n i n the low range of shear r a t e s p r e v i o u s l y i n a c c e s s i b l e . _ T h i s phenomenon has been a t t r i b u t e d t o the formation of an e r y t h r o c y t e - f r e e 16 boundary l a y e r , although t h i s has not been r i g o r o u s l y proven* Quemada (1975a,b,c) developed a f u n c t i o n which he showed to be an improvement over Casson's(1959) f u n c t i o n , p r i m a r i l y by the absence of a y i e l d s t r e s s . P h i l l i p s and Deutsch (1975) hypothesized a general CF i n tensor form, l i n e a r i n s t r e s s , s t r a i n and t h e i r f i r s t d e r i v a t i v e s with r e s p e c t t o time, thereby i m p l y i n g v i s c o e l a s t i c p r o p e r t i e s * Walburn and Schneck (1975) used a computer r o u t i n e which began with one independent v a r i a b l e and p r o g r e s s i v e l y added more u n t i l a f u n c t i o n depending upon hematocrit, c o n c e n t r a t i o n of t o t a l p r o t e i n l e s s albumin and shear r a t e was c o n s t r u c t e d . Huang (1972) has presented s e v e r a l e x p r e s s i o n s based upon the 1 theory of n o n - e q u i l i b r i u m thermodynamics. 1.7 C u r r e n t s t a t u s of hemorheology Since t h e r e i s not yet an i n f r a s t r u c t u r a l theory f o r blood, hemorheology cannot be used t o determine a b s o l u t e values of e r y t h r o c y t e p r o p e r t i e s such as aggregation energy or i n t e r n a l v i s c o s i t y . However, an e m p i r i c a l l y determined CF has important a p p l i c a t i o n s . In the s o l u t i o n of flow problems r e l e v a n t to p h y s i o l o g y and to the design of c a r d i o v a s c u l a r p r o s t h e t i c d e v i c e s , and i n hemorheological r e s e a r c h , a macroscopic CF i s e s s e n t i a l . . T h a t o f Casson (1959) has probably been most used. However, i t w i l l be shown that much b e t t e r forms are a v a i l a b l e . Of g r e a t e r i n t e r e s t here i s the c h a r a c t e r i z a t i o n of normal and p a t h o l o g i c a l s t a t e s of the blood, s i n c e the c e l l u l a r parameters upon which the CF of blood depend d i f f e r s i g n i f i c a n t l y i n many disease s t a t e s ( D i n t e n f a s s , 1971). F u l l e x p l o i t a t i o n of c l i n i c a l hemorheological data might w e l l p r o v i d e 17 i n f o r m a t i o n r e l a t i n g , f o r i n s t a n c e , b l o o d f l o w p r o p e r t i e s t o h y p e r c o a g u l a b i l i t y . In most hematology l a b o r a t o r i e s , the e r y t h r o c y t e sedimentation r a t e (ESS) i s the only r h e o l o g i c a l method used. Because the r a t e of descent of an aggregate g e n e r a l l y i n c r e a s e s with the number of e r y t h r o c y t e s i t c o n t a i n s , the ESS g i v e s q u a l i t a t i v e i n f o r m a t i o n r e g a r d i n g aggregation.. However the process has not yet been analysed a c c u r a t e l y (Burton et a l . , 1969; Shangkuan, Huang and Copely, 1977).. At p r e s e n t , major problems a r i s e i n the comparison and c o r r e l a t i o n of c l i n i c a l hemorheological data p r i m a r i l y because blood i s a non-Newtonian f l u i d whose p r o p e r t i e s depend s t r o n g l y upon the hematocrit, a parameter which d i f f e r s s i g n i f i c a n t l y between people and can change s i g n i f i c a n t l y i n an i n d i v i d u a l . To compare data from d i f f e r e n t i n d i v i d u a l s i t i s necessary e i t h e r t o prepare blood samples at a c o n s i s t e n t hematocrit when making v i s c o s i t y measurements, a proceedure which i s d i f f i c u l t e x p e r i m e n t a l l y , or to use some e s t a b l i s h e d mathematical e x p r e s s i o n to • c o r r e c t ' v i s c o s i t y values determined at the a c t u a l h e m a t o c r i t t o a standard one. To compare i n d i v i d u a l s , v i s c o s i t y v a l u e s must a l s o be r e f e r r e d t o a r e f e r e n c e shear r a t e . In a r h e o l o g i c a l examination of a blood sample, the shear r a t e s are not a c c u r a t e l y predetermined, p a r t i c u l a r l y i n the low shear r a t e range (D<35s~ 4), s i n c e they depend upon the c o n s t i t u t i v e c h a r a c t e r of the i n d i v i d u a l sample as well as on the boundary c o n d i t i o n s o f the experimental flow f i e l d . Therefore i t i s d i f f i c u l t t o o b t a i n data r e f e r r e d to s t a n d a r d c o n d i t i o n s of hematocrit and shear r a t e . Yet, i f a c t u a l data 18 p o i n t s a r e t o be u t i l i z e d , comparison of hemorheological i n f o r m a t i o n i n any other way i s i n v a l i d * These problems may be overcome by f i t t i n g a CF to shear s t r e s s data d e r i v e d over a range of shear r a t e s and h e m a t o c r i t s and using the parameters of the f u n c t i o n as the b a s i s f o r comparison among i n d i v i d u a l s . The s e t of parameters {Xi} takes on s p e c i f i c v a l u e s t h a t c h a r a c t e r i z e a p a r t i c u l a r sample of blood. T h e o r e t i c a l l y an abnormal c o n d i t i o n such as s t r o n g aggregation of e r y t h r o c y t e s would be a s s o c i a t e d with abnormal values of one or more of the parameters. T h i s procedure has the a d d i t i o n a l advantage t h a t f i t t i n g of the curve p r o v i d e s a s t a t i s t i c a l averaging of the data which reduces random experimental e r r o r . Use of the CF approach should f a c i l i t a t e comparisons of c l i n i c a l hemorheological data. T h i s form of a n a l y s i s has r e c e n t l y been advocated by Huang (1977). The CF which i s chosen to r e p r e s e n t the data should be t h a t which, averaged over the p o p u l a t i o n , conforms most c l o s e l y to o b s e r v a t i o n s * The t h e o r e t i c a l f o u n d a t i o n of the f u n c t i o n i s of secondary concern, although i t would be f o r t u n a t e i f the best f i t t i n g f u n c t i o n had a sound t h e o r e t i c a l b a s i s , The o b j e c t i v e of the present work was to t e s t the a b i l i t y of CFs, e i t h e r s e l e c t e d from the l i t e r a t u r e or c o n s t r u c t e d to e x h i b i t the a p p r o p r i a t e g e n e r a l c h a r a c t e r , t o f i t o b s e r v a t i o n s from a p o p u l a t i o n of normal i n d i v i d u a l s . The optimal values of the numerical parameters i n each case were those which gave the minimum r o o t mean square of d e v i a t i o n s ; The f u n c t i o n determined t o f i t best i n p o p u l a t i o n average was then used to examine whether the hemorheological c h a r a c t e r i s t i c s of normal women 19 v a r i e d through the menstrual c y c l e . Future work should be a b l e to e s t a b l i s h whether or not the use of o r a l c o n t r a c e p t i v e s causes hemorheological p r o p e r t i e s t o vary beyond normal ranges. Such i n f o r m a t i o n would c o n t r i b u t e t o an understanding of the c i r c u l a t o r y problems sometimes a s s o c i a t e d with o r a l c o n t r a c e p t i o n . 20 2* Experimental methods v 2. 1 Laboratory methods 2.1*1 A c q u i s i t i o n of blood samples For -the purpose of t e s t i n g the CFs one sample was obtained from each of 30 women ranging i n age from 19 t o 44 years and thre e samples were obtained from one male aged 23 years* None of these donors were r e c e i v i n g any medication with the exce p t i o n of the 11 women using o r a l c o n t r a c e p t i v e s . To e s t a b l i s h the p o s s i b l e changes through the menstrual c y c l e , 4 t o 8 samples were taken from each of 12 females at i n t e r v a l s d u r i n g the menstrual c y c l e . Three samples were taken from one male over a comparable span o f time t o provide data for comparison*. For. preliminary i n v e s t i g a t i o n 8 samples were a l s o o b tained from a woman using o r a l c o n t r a c e p t i v e s . With the donor i n a seated p o s i t i o n , blood was drawn from an a n t e c u b i t a l v e i n i n t o a p l a s t i c s y r i n g e using a 21-gauge needle and d i s c h a r g e d i n t o a commercial EDTA tube (Sherwood Medical I n d u s t r i e s , HRI 8881-010543). Blood was r o u t i n e l y withdrawn between 0730 and 1000 h; l e s s than 10% of the samples had v i s i b l y cloudy plasma. 2.1*2 Viscometry The rheometer used was a Contraves LS-2 with a c o n c e n t r i c c y l i n d e r measuring system; in n e r c y l i n d e r s t a t i o n a r y , outer c y l i n d e r r o t a t i n g , The dimensions of the annular sample space were: 11 mm i n s i d e diameter, 12 mm out s i d e diameter and 8 mm height,* A guard r i n g , which prevents t r a n s m i s s i o n of s t r e s s near the l i q u i d s u r f a c e was made from methyl methacrylate. T h i s 21 device provided no measurable e f f e c t when s t a n d a r d i z i n g o i l was t e s t e d but was found t o be necessary t o o b t a i n Newtonian behaviour f o r plasma. A water c i r c u l a t o r maintained the sample temperature a t 25.0+0.1 C. The output s i g n a l , p r o p o r t i o n a l to shear s t r e s s , was recorded c o n t i n u o u s l y on a c h a r t r e c o r d e r . The p r o p o r t i o n a l i t y constant was determined with o i l s of known v i s c o s i t y . 2;1i3 Manipulations Measurements were made at 10 shear r a t e s ranging i n geometric p r o g r e s s i o n from approximately 0.0312 s - 1 to 124 s-» at v a r i o u s h e m a t o c r i t s . In some cases c a l c u l a t i o n s were done with a subset of only s i x shear r a t e s * These two ranges of shear r a t e s w i l l be d i s t i n g u i s h e d by: 51 = ( shear r a t e s | .031 s - 1 < shear r a t e < 3.1 s _ 1 jj ' (5) 52 = { shear r a t e s | .031 s~* < shear r a t e < 124 s-» j The: volume f r a c t i o n of e r y t h r o c y t e s , H, was determined with an IEC microhematocrit c e n t r i f u g e . The c o r r e c t i o n f o r i n t e r c e l l u l a r space ( 1 % ) was i n s i g n i f i c a n t r e l a t i v e to other e r r o r s and was negl e c t e d * The values of H were ranged from 0i 29 t o 0.55. The measurements a t p h y s i o l o g i c a l hematocrits were run with the blood -taken d i r e c t l y from the sampling tube. The blood was then separated i n t o plasma and e r y t h r o c y t e s by c e n t r i f u g a t i o n at 1000 g f o r 180 t o 300 s so t h a t the m a j o r i t y of p l a t e l e t s remained i n the plasma. P l a t e l e t s were removed from the plasma by c e n t r i f u g a t i o n at 24,000 g f o r 120 t o 180 s and the chosen a r t i f i c i a l h e m a t o c r i t s were obtained by remixing plasma and c e l l s , The o c c a s i o n a l cloudy plasma was f i l t e r e d through a 22 0.45 i^xm c e l l u l o s e a c etate f i l t e r ( W i l l i p o r e Corp.) . The sample volume used i n the viscometer was 0.7 mis P r i o r t o measurement at each shear r a t e the sample was mixed by r a i s i n g and l o w e r i n g the i n n e r c y l i n d e r through a displacement of about 5 mm. No more than 35 s e l a p s e d between completion of mixing and i n i t i a t i o n o f shearing* Measurements on any sample over the range of shear r a t e s were completed w i t h i n 30 minutes of sample p r e p a r a t i o n . The plasma v i s c o s i t y was determined a t 25s0±0*1 C using a c a p i l l a r y tube viscometer (Cannon Instruments 100 A980) with a flow time f o r water of 72 s. 2.1.4 I n t e r p r e t a t i o n o f o b s e r v a t i o n s Two data were taken from the viscometer c h a r t r e c o r d s . The peak value c o n s i s t e d of the maximum of the s t r e s s - t i m e curve at any shear r a t e , a value which, i f present, was a t t a i n e d w i t h i n the f i r s t 30 s of each run. The second datum was the steady s t a t e value taken when the shear s t r e s s became constant* 2.1.5 Plasma p r o t e i n determination The plasma p r o t e i n c o n c e n t r a t i o n s were determined using r a d i a l immunodiffusion p l a t e s produced by the Behring d i v i s i o n of Hoechst Pharmaceuticals. The p r o t e i n s , t h e i r a b b r e v i a t i o n s , the catalogue numbers of the p l a t e s and the r a t i o s of p o p u l a t i o n standard d e v i a t i o n t o p o p u l a t i o n mean (published by Becker et a i i , 1968) are g i v e n i n Table 13. 2.2 The s u b j e c t p o p u l a t i o n Four o f the s u b j e c t s responded t o p o s t e r s which were i n i t i a l l y used to e l i c i t p a r t i c i p a t i o n . However, t h i s approach proved to be unproductive. Personal requests t o i n d i v i d u a l s were found t o be more s a t i s f a c t o r y * The o n l y prominent common f a c t o r s 23 of the s u b j e c t ' s backgrounds were t h a t most were e i t h e r s tudents of n u r s i n g or members of the s t a f f of the Health Sciences Center Extended Care H o s p i t a l . Approval of the p r o t o c o l employed was obtained from the OBC Human Experimentation Committee; 24 3. T h e o r e t i c a l methods 3.1 I n t e r p r e t a t i o n of o b s e r v a t i o n s The shear s t r e s s and shear s t r a i n r a t e vary s l i g h t l y with p o s i t i o n i n the rheometer. However, i t i s only necessary t o r e f e r t o one p o i n t i n the sample i n order to determine the CF. T h i s r e q u i r e d p o i n t i s taken as any p o i n t on the c y l i n d r i c a l s u r f a c e . o f the bob (the inner s t a t i o n a r y measuring element),. E v a l u a t i o n of the shear s t r e s s there i s b a s i c a l l y a problem of s c a l i n g but e v a l u a t i o n of s t r a i n r a t e at t h a t s u r f a c e i s more i n v o l v e d and r e q u i r e s the a l g o r i t h m of K r i e g e r and E l r o d d e s c r i b e d i n appendix 3. The e r r o r i n the s t r e s s c a l c u l a t i o n due t o the end e f f e c t at the lower end of the bob i s d i s c u s s e d i n appendix 4. 3.2 Theory of f i t t i n g of the CF to o b s e r v a t i o n s The success of any CF i n f i t t i n g the data must be e v a l u a t e d by some o b j e c t i v e c r i t e r i o n . The c r i t e r i o n used i s t h a t of i maximum l i k e l i h o o d . A g e n e r a l 1 development i s given by bard (1974). The development s p e c i f i c t o t h i s s i t u a t i o n i s o u t l i n e d i n Appendix 5,. . 3.3 S t a t i s t i c a l t e s t i n g and comparison of CFs The f u n c t i o n s compared, l i s t e d i n Table 1, were obtained from a v a r i e t y of sources. In a number of cases the hematocrit dependence i s given by a f a c t o r of exp(parameter*H) s i n c e approximately t h i s dependence has been observed e x p e r i m e n t a l l y f o r whole blood by a number of authors (e;g». Weaver et a l . . 1969; Dormandy, 1974), In none of the equations i s a s p e c i f i c parameter a s s o c i a t e d with e r y t h r o c y t e aggregation i n c l u d e d . 25 Function 1 i s a s i m p l i f i e d form of the Walburn-Schneck f u n c t i o n , (see below), having only 2 parameters. F u n c t i o n s 2 through 6 were formed by combining a l i n e a r shear r a t e dependence with one which goes e x p o n e n t i a l l y t o zero a t s m a l l D and adding a h e m a t o c r i t dependence. When H — > 0 they reduce t o the Newtonian r e p r e s e n t a t i o n of plasma; otherwise they have no t h e o r e t i c a l background. F u n c t i o n 7 was f i r s t presented by Walburn and Schneck (1976) who used a computer r o u t i n e which determined the f u n c t i o n of b e s t f i t i n the shear r a t e range of 24 s - 1 to 230 s _ 1 * As o r i g i n a l l y presented the parameter was m u l t i p l i e d by ( t o t a l p r o t e i n c o n c e n t r a t i o n minus albumin c o n c e n t r a t i o n ) i n an attempt t o remove i n d i v i d u a l v a r i a b i l i t y from the value of X^. T h i s f a c t o r has been e l i m i n a t e d a l b e i t with some l o s s o f c o n s i s t e n c y amongst samples from d i f f e r e n t people. Function 8 was d e r i v e d by Quemada (1975a,b,c) as an improved v e r s i o n of Casson's f u n c t i o n with zero y i e l d s t r e s s * F u n c t i o n s 9 and 10 are based on the eguation of P h i l l i p s and Deutsch (1975). These authors d i d not i n c l u d e an H dependence so t h i s v a r i a b l e was i n t r o d u c e d i n two ways as shown i n Table 1, I t i s e x p e r i m e n t a l l y observed f o r some samples that when ln(T) i s p l o t t e d versus In(D) two q u a s i - l i n e a r r e g i o n s appear. Function 11 r e p r e s e n t s t h i s behaviour with X^ being a measure of the shear r a t e at which the slope changes* Convergence of the o p t i m i z a t i o n r o u t i n e was never obtained with any o f the f o u r forms of the f u n c t i o n presented by Huang and F a b i s i a k (1977), Consequently i t i s not i n c l u d e d i n Table 1. These authors presented t h e i r f u n c t i o n i n the f o l l o w i n g form and s p e c i f i e d numerical v a l u e s f o r c,A>n and-.tg ''#lich"are constant for a particular sample of blood,' 26 T = eD ± c&D nexp (-cD nt 0) ( 6 ) the + s i g n i s taken f o r a d i l a t a n t f l u i d and the - sign f o r a t h i x o t r o p i c o r s h e a r - t h i n n i n g f l u i d . With t h e hematocrit i n t r o d u c e d the f u n c t i o n was examined i n the f o l l o w i n g forms: T = eD ± XjD* 2exp (X^ H-X3D"^2) (7) T = exp(X 4H)»(eD ± Xj D X*exp (-X3 D xz)) (8) The c h o i c e ' o f • or - s i g n and two forms of hematocrit dependence gave four d i f f e r e n t forms, a l l o f which were t e s t e d . The f i t of the v a r i o u s f u n c t i o n s t o blood taken from one person cannot be c o n s i d e r e d a g e n e r a l t e s t of these f u n c t i o n s . Since blood flow behaviour i s known to change from person to person* some f u n c t i o n s c o u l d be biased towards p a r t i c u l a r i n d i v i d u a l s . As i n d i c a t e d s c h e m a t i c a l l y i n F i g u r e 1 the magnitude of s (s=the standard d e v i a t i o n of f i t , see Appendix 5) v a l u e s d i s p l a y e d a p o p u l a t i o n d i s t r i b u t i o n * The mean of the d i s t r i b u t i o n was denoted sm and i t s standard d e v i a t i o n sd. sd=5,01 12 H : L~ — io -| e Jsnt=6.99 number of s values 5 •» 1 2 0 4 6 8 s value 1 o 12 FIGURE 1 The d i s t r i b u t i o n of s values f o r f u n c t i o n 7, Table 2A, steady s t a t e , S1 shear r a t e range with mean and standard d e v i a t i o n i n d i c a t e d . The; o r d e r i n g of sm values - s m a l l e s t t o l a r g e s t - was recorded. Based on the d i s t r i b u t i o n s of s values over the pop u l a t i o n i t was p o s s i b l e t o estimate the p r o b a b i l i t i e s t h a t s u c c e s s i v e p a i r s 27 of o r d e r i n g s of sm values were c o r r e c t (see appendix 7 f o r d e t a i l s ) . A f t e r determining parameter values by minimizing s a s t a t i s t i c a l a n a l y s i s of these values was performed. The o p t i m a l parameter values were averaged over the. p o p u l a t i o n and po p u l a t i o n standard d e v i a t i o n s were calculated,. 3*4 N o r m a l i z a t i o n of parameters In many cases the value of a p h y s i o l o g i c a l parameter d i f f e r s among i n d i v i d u a l s and i n any p a r t i c u l a r i n d i v i d u a l i t may change with time. Thus although the period of the menstrual c y c l e averaged about 28 days d i f f e r e n c e s of s e v e r a l days appeared i n the data. S i m i l a r l y the hematocrit and the parameters o f best f i t of a CF f o r an i n d i v i d u a l may d e v i a t e a p p r e c i a b l y from the p o p u l a t i o n mean. To d e a l with these e f f e c t s data were normalized. Let a s e t of values of a v a r i a b l e be {yj | 1<i<n}. The normalized v a r i a b l e i s d e f i n e d by: (yi normalized with r e s p e c t t o y) = ft = Yi /Y (9) In examining the v a r i a t i o n of a q u a n t i t y through the menstrual c y c l e the elapsed time from the preceding menstruation was normalized with r e s p e c t t o the i n d i v i d u a l ' s p e r i o d . t i = t j / T (10) t j = time between menstruation and drawing of the i t h blood sample; T = time between s u c c e s s i v e menstruations of i n d i v i d u a l The values of hematocrit or the o p t i m a l CF parameters f o r an i n d i v i d u a l were normalized with r e s p e c t to the mean value f o r t h a t i n d i v i d u a l . When the data was then p l o t t e d the r e s u l t s from the whole p o p u l a t i o n could be superimposed and examined f o r a 28 c y c l i c f e a t u r e without c o n f u s i o n caused by d i f f e r e n c e s between i n d i v i d u a l s of c y c l e p e r i o d s or normal values of v a r i a b l e s . 3*5 Graphing A program was w r i t t e n to graph the data p o i n t s of a sample and t o p l o t the a s s o c i a t e d CFs. The f i g u r e s were produced i n the UBC Computing Center by t h i s program. The graphs of CF parameters, h e m a t o c r i t and plasma p r o t e i n s versus time were p l o t t e d by a HP 9872A p l o t t e r * . 3,6 Programs used f o r data a n a l y s i s Appendix 9 c o n t a i n s the computer programs used i n t h i s work i n the f o l l o w i n g order: O p t i m i z a t i o n P o p u l a t i o n a n a l y s i s of o p t i m i z a t i o n r e s u l t s Graphing These programs are documented i n t e r n a l l y by comment statements. 29 4. Results and d i s c u s s i o n 4.1 T e s t i n g of c o n s t i t u t i v e f u n c t i o n s The f u n c t i o n s t e s t e d . and some of t h e i r l i m i t i n g c h a r a c t e r i s t i c s which are of i n t e r e s t f o r t h e o r e t i c a l i n t e r p r e t a t i o n appear i n Table (I. A l l f u n c t i o n s have zero y i e l d s t r e s s . A l l f u n c t i o n s have p o s i t i v e v i s c o s i t y a t zero shear r a t e although i n some cases t h i s v i s c o s i t y i s i n f i n i t e . When D becomes l a r g e the v i s c o s i t y should remain l a r g e r than the continuous phase v i s c o s i t y but f i n i t e * U n f o r t u n a t e l y , with the Walburn-Schneck f u n c t i o n t h i s l i m i t i s zero*. For blood T(D,e,H,{Xij) i s an even f u n c t i o n of D; i . e . T(-D,e,H, (Xij) = -T (D,e,H, {Xij ) (11) Of a l l f u n c t i o n s c o n s i d e r e d only those of P h i l l i p s and Deutsch ( f u n c t i o n s 9 and 10) have t h i s symmetry. The standard d e v i a t i o n s , s, f o r each f u n c t i o n when a p p l i e d to samples of v a r y i n g hematocrit from each i n d i v i d u a l are l i s t e d i n T a b les 2A and 3A f o r steady s t a t e and Tables 4A and 5A f o r peak value data. T a b l e s 2A and 4A gi v e the r e s u l t s obtained when the f u n c t i o n s are f i t to data on the S2 shear r a t e range whereas Tables 3A and 5A a r e . f o r data on the S1 range. The f i t s i n the cases of low shear r a t e subsets were t e s t e d s i n c e t h a t r e g i o n i s p a r t i c u l a r l y important i n d e t e c t i o n of c e l l - c e l l i n t e r a c t i o n s (Brooks e t a l . , 1974). The standard d e v i a t i o n v a l u e s i n these t a b l e s have been averaged over a l l i n d i v i d u a l s t o o b t a i n sm c h a r a c t e r i z i n g each f u n c t i o n and the standard d e v i a t i o n about t h i s mean has been denoted as sd. These two parameters are g i v e n i n Ta b l e s 2A 30 through 5A and are the r e s u l t s upon which the goodness of f i t e s t i m a t i o n was based f o r the v a r i o u s f u n c t i o n s ; The most n o t a b l e f e a t u r e i s t h a t the Walburn-Schneck f u n c t i o n y i e l d s the s m a l l e s t value of s (Tables 2A to 4A) and a l s o has r e l a t i v e l y s m a l l values of sd,. In Table 4A where f u n c t i o n 4 f i t best the s value f o r the Walburn-Schneck f u n c t i o n was not e x c e p t i o n a l l y l a r g e r ( l e s s than 30%); I t i s noteworthy t h a t f o r Quemada's f u n c t i o n some of the s v a l u e s are comparable t o those f o r the Walburn-Schneck f u n c t i o n while o t h e r s are l a r g e r than 100 mPa (e.g. T a b l e s 3A, sample 22), Small values of s i n d i c a t e curves f i t t i n g w e l l while l a r g e values of s are inay be associated with a s i n g u l a r i t y occuring when ((1-X ZH) • (1-X EX 3H) X- JD*/z = 0 (12) C o n d i t i o n s l e a d i n g to the s i n g u l a r i t y might have been avoided by c o n s t r a i n i n g the parameters but t h i s was not attempted. Another p o s s i b l e cause of bad f i t (i»e. l o c a l minima) i s d i s c u s s e d i n Appendix 5. ' Most of the p r o b a b i l i t i e s of c o r r e c t o r d e r i n g s of s v a l u e s , given i n T a b l e s 2B-5B, are c o n s i d e r a b l y l a r g e r than 0> 5* Judging from these p r o b a b i l i t i e s the o r d e r i n g s can be c o n s i d e r e d to be reasonably w e l l e s t a b l i s h e d . Walburn and Schneck used the r 2 parameter as a measure of goodness of f i t , p e r f e c t f i t being a s s o c i a t e d with r 2 = 1.000 (see Appendix 8). For comparison with r e s u l t s presented i n t h i s way the values of r 2 averaged over the sample p o p u l a t i o n are a l s o given i n Tables 2A through 5A. With itwo e x c e p t i o n s the o r d e r i n g of r 2 matches that of sm values. One exception occurs i n the steady s t a t e data i n the S1 shear r a t e range and one occurs i n the peak value data i n the same shear 3 1 rate range. I t can be shown from the d e f i n i t i o n of r 2 (appendix 8 ) that the ordering of r 2 and s need not be the same. These r e s u l t s demonstrate that the r 2 parameter i s a less sensitive indicator of goodness of f i t than s. Since the l a t t e r can be t h e o r e t i c a l l y j u s t i f i e d (Appendix 5) i t was applied in t h i s work. Tables 6, 7, 8, and 9 give average and standard deviation values of the parameters for each type of data (steady state, peak) over the two ranges of shear rates f o r a l l functions considered. The averages are taken over the sample population in these cases and the standard deviations are calculated with respect to the corresponding averages. 'Average1 CFs for the population r e s u l t i f the parameter averages from Tables 6, 7, 8 and 9 are substituted into t h e i r respective functions. Figures 2 to 8 show various samples of data with f i t curves for the Walburn-Schneck function. Figures 2 to 7 are for data from person 29, This case was chosen because.it gave the best f i t amongst a l l steady state data f i t over the S2 shear rate range. Figures 2 and 3 show the r e s u l t s of f i t t i n g over S1 and S2 shear rate ranges respectively for the steady state data. Figures 4 and 5 give analogous information for the peak data. Figure 6 refers to the same data and range of shear rates of f i t as figure 4, but only the S1 range i s plotted*. This magnifies the region of figure 4 near the o r i g i n so that the discrepancies of f i t at low shear rates are v i s i b l e * Figure 7 refers to the same.data as Figure 5. Here parameters derived by f i t t i n g over the lowest six shear rates are used to plot function 7 over the S2 shear rate range so that the e f f e c t of extrapolation i s 32 d i s p l a y e d . F i g u r e 8 shows the r e s u l t s f o r sample 18 i n the steady s t a t e data f i t over the S2 shear r a t e range where i t i s the second from worst f i t f o r t h i s f u n c t i o n . Even i n t h i s case the curves are seen t o rep r e s e n t the data reasonably w e l l * F i g u r e 9 i s a graph of Huang's f u n c t i o n (equation (6) i n t e x t ) . The parameter values recommended by Huanq and F a b i s i a k (1977) were used with the e x c e p t i o n t h a t e = 1.5 was chosen s i n c e the p u b l i s h e d value of *11 was u n r e a l i s t i c f o r the plasma v i s c o s i t y * The graph l a c k s the form which a l l o w s the other f u n c t i o n s to f i t the data. F i g u r e s 10 through 18 are' f o r f u n c t i o n s 2 through 11 r e s p e c t i v e l y , a p p l i e d t o sample 29, steady s t a t e data f i t over the S2 range. F i g u r e s 19 through 28 are f o r the same c o n d i t i o n s as F i g u r e s 10 through 18 r e s p e c t i v e l y but i n 19 through 28 apparent v i s c o s i t y r a t h e r than s t r e s s i s p l o t t e d on the o r d i n a t e . F i g u r e s 10 through 28 allow a v i s u a l comparison of the f u n c t i o n s . In almost a l l cases the s values f o r the steady s t a t e data are s m a l l e r than the corresponding values d e r i v e d from peaks of the shear s t r e s s - time output. T h i s probably r e f l e c t s the g r e a t e r v a r i a b i l i t y i n the peak values r a t h e r than the i n a b i l i t y of the f u n c t i o n s used to f o l l o w the shear r a t e dependence. The reason f o r the appearance of peaks i n the t o r q u e - time r e c o r d s i s not completely c l e a r . I t has been suggested t h a t i t i s due to a w a l l e f f e c t i n the rheometer, r e p r e s e n t i n g a migration of c e l l s away from the v e s s e l boundaries (Cokelet e t a l . , 1963). A l t e r n a t i v e l y , s i n c e i t i s only present i n aggregated suspensions i t seems p o s s i b l e t h a t i t could r e p r e s e n t a shear dependent b u i l d up of aggregation. P h i l l i p s and Deutsch (1975) 33 have shown that a simple v i s c o e l a s t i c f l u i d could also exhibit such behaviour,. In any event, since the peak stresses are often used as an i n d i c a t i o n of the degree of aggregation present in a sample, t h e i r values were included i n the analysis. It i s noteworthy that no single set of parameters seems to adequately represent the shear stress over the whole range of D examined. I t i s apparent from Figures 6 and 7 that stresses at high shear rates cannot be accurately predicted from low shear rate values, nor can the low shear behaviour be derived safely from curves f i t to data taken over the f u l l range accessible (where the high shear values dominate the f i t t i n g procedure) . This would suggest that none of the equations examined have much fundamental si g n i f i c a n c e with respect to the basic f l u i d mechanics of these systems, On the other hand, function 7 i s the best form of function in thi s regard* I t i s c l e a r from the res u l t s presented i n Tables 10 and 11 that the parameters of best f i t for the S1 and S2 shear rate ranges are very s i m i l a r for this equation, i n contrast to a l l but the simple power law expression,* This i s true both for the steady state and peak value data. It remains to be seen whether or not there i s any underlying physical reason for these observations.. Among the functions, then, that of Walburn and Schneck, function 7, would appear to be the most sati s f a c t o r y and should probaily be used f o r representation of data where f e a s i b l e . However, the unsatisfactory l i m i t of vi s c o s i t y when D —^ > 0 for t h i s eguation should be kept i n mind. Future work might be directed towards combining the Walburn and Schneck hematocrit dependence with other shear rate dependencies that eliminate the 34 above objection and allow more rapid changes i n slope at low shear rates. Such a function might well provide a consistent f i t over a l l accessible rates of shear. 4.2 Application to examination of menstrual cycle 1 Table 12 gives the means of hematocrit and CF parameters and table. 14 gives the means of the plasma protein concentrations for each i n d i v i d u a l . Figures 29 through 43 show the variation of hematocrit, CF parameter values and fibrinogen, anti-thrombin I I I , albumin, IgM and oc2-macroglobulin concentrations ( a l l normalized as explained i n 3.4) through the menstrual cycle. In these, and subsequent plots i n d i v i d u a l s are distinguished by unique p l o t t i n g symbols as spe c i f i e d i n Table 15. These graphs have no evident c y c l i c feature* However, i t i s of intere s t to note the narrowness of the bounds on the range of fluctuation with time of an ind i v i d u a l ' s normalized plasma protein concentration. The r a t i o s of population standard deviations to population mean i n Table 14 are comparable to the published values i n Table 13. The population standard deviations for the optimal parameters are much larger than the population standard deviations for the hematocrit or proteins. This raises the following question. Do values which are f a r from the corresponding mean re s u l t from l o c a l rather than global minima of the optimization problem or do they represent r e a l physiological variations? In figures 44 through 47 standard deviations of f i t (s) are plotted against normalized parameter values. The presence of l o c a l minima would be manifest i n these graphs as a co r r e l a t i o n between s and X i ; i . e , aberrant parameter values would be associated with 3 5 r e l a t i v e l y l a r g e standard d e v i a t i o n s . I t i s apparent t h a t t h i s does not occur. Therefore i t can be concluded t h a t an a b e r r a n t parameter r e p r e s e n t s an a b e r r a t i o n of the flow p r o p e r t i e s o f the blood sample* T A B L E 1 Features of the f u n c t i o n s considered A t t r i b u t e IFunctiSS-Hdlburo & Schneck Queaada P h i l l i p s & Deutsch 10 P h i l l i p s T(e,H.O, II ,) H a J D->0 D li» I D-X» 0 li» I H->0 D T i s odd Function of 0 e m p i r i c a l l y deteroined ( 1 - 1 11) e eip(X H)D 2 1 X | B i p | I H)D»X i l ( 1 - e i p ( - I 0))) 4 1 2 3 I |eip{t B)D«I ( 1 - e i p ( - I UD) | | 4 1 2 3 I (D»I H(1-eip<-I D ) ) | e i p ( I HI « 1 2 3 I (OH ( l - e i p ( - I HD|||ezp(X H| « 1 2 3 un c e r t a i n X (exp(X 11) »X X H) 1 1 2 3 X (exp(I U)«X I H) 4 1 2 3 (1*1 X H)e i p ( I HI t 1 2 3 I ( U I X H)exp(X H) « 1 2 3 e<( | <oo I exp(X U| I axp(X U) 4 1 I axp(I H) 14 3 I exp(I Ii) « 3 I D(UX (1-eip(-X UD|))eip(I U) I ('*« X H)exp(I U) I (1»I )exp(I U) 1 2 I 2 (1-1 H) X ezp(I H»I /U|D 3 I 2 4 -1 -1/2 U I D 1 -1 -1/2 (1-1 H)» (1-1 I U|I D 2 2 3 1 UX D 2 X eip(X H| 1 4 a D exp UX 0 3 I H HD 2 1-X I ii 2 3 X 3 i p ( l U| 1 4 1-X H 2 1»X HO 3 X (V-I I 3 14 X -X V -1 2 (» \30 1 X I 1 2 exp(X HI I 4 3 I X 1 2 exp(X U ) X 4 3 oo(i.<oi 1>X D 2 U I D 3 V = In (D) «5 P E R S O N S E X 1 2 3 4 5 6 7 8 9 1 0 I 1 12 I 3 14 1 5 1 6 1 7 1 8 19 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 31 31 F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F M M . M H E M A T O C R I T S AT W H I C H M E A S U R E M E N T S W E R E M A D E . 5 1 , . 4 1 , . 3 0 . 5 0 , .41 . 5 2 , . 4 4 , .40 . 4 7 , .42 . 4 9 , .42 .5 0 , .41 . 5 0 , . 4 5 , . 4 9 , . 3 8 . 5 1 , .45 . 5 0 , . 5 1 , .5 I , . 4 9 , . 5 0 , . 5 0 , .48 , . 5 0 , . 5 0 , . 4 9 , . 5 0 , . 5 0 , . 4 0 .47 .45 . 5 1 , .41 . 5 0 , .40 . 5 5 , . 4 3 .49 , . 4 6 .45 . 4 0 .43 . 3 9 . 4 3 . 5 3 , . 4 0 . 5 0 , .40 . 4 2 .40 .44 . 4 1 . 4 8 , .45 . 4 9 , .42 .50 , .40 . 4 9 , .45 . 4 9 , . 4 5 . 5 2 , .44 sd . 4 0 .40 . 3 9 . 4 0 . 3 0 . 3 8 , . 3 9 , .40, .38 . 3 0 .30 .30 . 3 0 . 3 1 . 3 0 .31 . 3 0 .40 . 3 0 . 3 0 .40 . 4 0 , .4 0 .34 . 3 0 .29 ,30 O R D E R I N G O F sm 2 4 . 16 20 .32 20.3 2 1 2 . 8 9 2 8 . 9 4 1 1 . 0 2 16.77 1 8 . 5 8 1 6 . 4 3 8.67 2 4 . 0 4 1 3 . 8 9 11.81 2 5 . 0 4 33.93 30. 73 2 3 . 5 5 1 1 . 5 1 1 7 . 6 8 2 6 . 9 2 0.0 12. 68 19 . 5 1 2 1 . 79 2 5 . 3 8 1 7 . 3 7 2 0 . 0 3 1 7 . 0 7 1 7 . 0 1 1 7 . 5 7 1 2 . 5 2 2 9 . 5 5 1 1 . 8 8 1 9 . 3 8 6.45 . 9 9 2 4 4 9 7. 10 6.41 9. 43 5. 12 4. 50 5. 50 12.48 5. 59 10. 82 7.78 8. 54 11. 0 6 1 0 . 17 8.97 3 1 . 8 2 9. 15 8. 10 1 1 . 0 6 18. 10 7.69 7.40 7. 68 8.46 6. 75 6. 35 7.37 8. 76 5. 27 5. 52 8. 84 6.94 9. 49 5.97 8. 91 4. 88 .99823 3 8.73 7 .58 0.0 4.81 12.29 6.51 12 .67 b .66 1 1 . 1 6 9 .35 7.46 11 .44 12 .03 9.84 3 3 . 1 4 10.06 0.0 13 .03 1 8 . 2 5 9.10 8.97 9.61 9.54 8 .46 7.61 9.20 9.97 5. 53 6.35 1 0 . 6 8 7.36 10.40 6.57 1 0 . 1 4 5.03 9 9 7 7 9 5 5.54 7.45 9 . U 5.36 4.50 4.87 1 2 . 1 1 4.90 1 0 . 2 7 5.75 9. 14 1 0 . 6 0 7.63 7.88 3 0 . 2 9 8.67 0.0 9.02 1 6 . 4 0 0.0 0.0 5.58 6.17 5.22 4.91 0.0 7.13 5.14 4.39 1 0.03 6.78 8.63 5.65 8.25 5.03 .99842 2 1 1 . 1 3 5.95 1 3 . 2 5 8.89 4 .26 5. 06 12. 16 5.12 10. 35 6 .43 8.12 10. 71 8 .72 8. 11 3 1 . 14 1 1 . 0 8 9.81 1 0 . 4 1 1 6 . 7 2 7.32 6.30 6. 14 9. 17 10 .03 5. 39 9.84 7.38 0.0 4 .64 0.0 6.62 1 4 . 1 9 5.27 9 .35 5 .04 . 9 9 8 0 6 4 12. 17 15.66 14.22 1 0 . 9 7 2 1.60 1 4 . 0 5 1 7 . 8 9 1 2 . 5 5 17. 12 15.8 0 1 5 . 3 9 26. 06 1 8 . 3 6 1 5 . 3 9 35. 22 1 2 . 2 5 10. 55 1 7 . 9 4 2 0 . 1 7 10. 38 1 2 . 9 6 1 7 . 2 6 10. 17 1 1 . 2 9 1 1 . 9 6 1 1 . 9 7 1 5 . 3 3 1 0 . 3 6 8.52 1 2 . 4 4 8. 44 1 5 . 2 1 1 6 . 2 9 1 5 . 0 3 5.29 .99563 6 0.0 5.64 4 . 9 3 4.06 4 . 8 5 4 . 4 0 1 0 . 1 0 4.43 7.41 6.75 8.32 6. 3 5 7.86 4 . 9 5 32. 1 3 7 . 1 9 1 1 . 1 5 1 1 . 6 1 6.54 4. 8 0 4.61 6.43 4.94 4.44 4 . 0 6 4 . 9 0 5.06 5. 36 3 . 2 3 5 . 3 3 7.72 7 . 8 0 6.43 10.20 84. 85 1 7 0 . 2 5 9. 38 9 1 . 6 6 6 6 . 3 1 1 6 . 6 7 5 5 . 8 4 6 7 . 4 7 2.75 1 1 1 . 1 4 9.00 7 5 . 8 3 6 7 . 7 7 1 2 5 . 4 8 7 3 . 4 9 2 3.39 7.39 8 3 . 8 0 6 8. 59 74. 56 1 0 9 . 5 7 6 7. 51 1 1 0 . 3 6 6 8 . 6 9 5 1 . 57 14.79 1 1 8 . 1 8 6.02 6 7 . 5 2 7.05 1 2 0 . 9 6 1 2 2 . 8 0 6.99 6 5 . 4 8 5.01 4 4 . 0 6 9 9 8 7 6 . 8 9 2 2 8 1 11 1 3 . 7 0 10. 8 3 2 4 . 3 0 8.45 7.28 2 8 . 6 3 2 9 . 3 3 1 6 . 9 7 2 5 . 8 9 1 0 . 8 4 2 8 . 1 3 2 1 . 3 1 2 0 . 2 1 2 6 . 3 6 31 .41 2 0 . 4 3 1 8 . 5 4 14. 7 3 2 6 . 4 1 1 5 . 8 8 1 5 . 9 1 1 5 . 5 2 1 1 . 1 8 1 7 . 6 5 9.91 17. 74 2 2 . 4 0 14. 1 5 1 4 . 3 5 1 8 . 7 1 10. 3 6 2 5 . 3 9 1 4 . 0 8 1 8 . 4 1 6. 6 9 . 9 9 3 3 2 8 10 2 1 . 0 4 2 6 . 69 1 0 . 8 4 6.73 14.47 1 0 . 5 9 0.0 1 4 . 9 8 1 2 . 7 2 2 3 . 2 5 1 6 . 3 5 1 5 . 8 9 1 7.87 1 4 . 9 1 3 2 . 0 6 0.0 1 7 . 1 5 1 4 . 6 4 2 2 . 0 4 11.41 2 0 . 7 4 1 2 . 9 5 1 3 ; 11 1 2 . 13 19 .09 1 1 . 9 0 1 3 . 7 4 8.65 1 2 . 2 4 1 5 . 6 7 1 0 . 6 3 19. 85 1 1 . 8 4 1 5 . 6 8 5.43 . 9 9 5 1 8 7 11 2 3 . 9 9 19. 82 20.1 7 1 2 . 7 0 2 9 . 03 1 0 . 2 1 1 6 . 3 6 1 8 . 2 9 1 5 . 5 9 7. 03 2 3 . 8 1 1 3 . 1 4 10.28 2 4 . 8 1 3 2 . 19 3 0 . 8 0 2 3 . 4 8 7.52 1 7 . 0 7 2 6 . 9 2 1 6 . 5 7 1 4 . 5 7 1 9 . 6 6 2 1 . 5 9 2 5 . 3 3 1 8 . 0 7 2 8 . 3 4 1 6 . 9 8 16.92 1 7 . 3 7 1 3 . 1 8 4 7 . 0 0 1 0 . 6 9 1 9 . 6 8 8.20 . 9 9 1 8 0 1 0 (D P> n tr «+ Ui 9> B p> 13 tt 1— P-. (D •4 • o> m U) < r+ (D P> P) (+ Pi H-•<! o et tn tn (+ P> o (+ Hi cn rt t-3 w tt 3* to (0 03 cn ir> cn cn W —• t\j P> cn tt — * tt Hi P> o tt <t> fl> 1-1 P> P) o 0 Cr (D O ht) PJ e T3 s —) H-rt-W fl> • • O i : s (+ O PI 3 8 T A B L E 2B P r o b a b i l i t i e s t h a t s u c c e s s i v e p a i r s of o r d e r i n g s of sm i n Table 2A are c o r r e c t O R D E R S : U 21 P R D B A B I I I T Y T H A T ( S D A V ( 7 ) < S D A V t 41 I * 0 . 8 3 4 D R D E R S : 21 3 : P R O B A B I L I T Y T H A T ( S D A V ( 4 ) < S D A V l 2 ) 1 = 0 . 7 0 2 O R D E R S : 31 4 ; P R D B A B I L I T Y T H A T ( S O A V I 2 ) S D A V l 5 ) ) = 0 . 5 3 6 O R D E R S : 4E, 5 : P R O B A B I L I T Y T H A T ( S D A V l 5 1 < S D A V l 3 1 1 = 0 . 7 3 3 O R D E R S : 5 S 6 : P R O B A B I L I T Y T H A T ( S O A V ( 3 ) < S D A V l S ) ) = l . D 3 3 O R D E R S : 6 & 7 ; P R D B A B I L I T Y T H A T < S D A V ( 6 ) S D A V l 1 0 ) t =0 . S 3 7 D R D E R S : 7& 8 ; P R D B A B I L I T Y T H A T ( S D A v ( LO ) < S D A V l 9 1 1 = 0 . 9 6 4 O R D E R S : 8 & 9 : P R D B A B I L I T Y T H A T ( S D A V l 9 ) < S D A V ( 1 ) ) = 0 . 7 2 4 O R D E R S : 9 £ 1 0 : P R D B A B I L I T Y T H A T ( S D A V ( L ) < S D A V t I 3) ) = 0 . 5 6 6 D R D E R S : l o & i l ; P R O B A B I L I T Y T H A T ( S O A v ( 1 3 ) < S D A V l 8 ) ) = 1 . 0 0 0 PEKSON SEX HLMATUCRITS AT WHICH MEASUREMENTS MERE HADE 1 F .51, .41, .30 1.37 0.0 1.77 1.30 1.80 0.0 2 F .50, .41 3.38 4.41 3.66 4.27 0.0 0.0 3 F .52, • 44, .40 1.35 3.37 1.46 1. 07 1.04 2.63 4 f • 47, .42 1.15 0.0 1.43 1.41 0.95 1. 11 5 F .49, .42 1.68 0.0 2.31 0.52 0.54 2.63 6 F .50, .41 1.30 1.48 1.34 0. 99 1.01 0.0 7 F .50, .45, .40 2.21 3.75 2.43 0.0 2. 86 0.0 d F .49, .38 1.43 3. 53 2.26 1.28 1.43 2.72 9 F .51, .45, .40 1.59 3.49 1.62 0.0 1. 54 0.0 10 F .50, .40 2.41 5.69 6.14 3.15 3.17 2.82 11 F .51, .47, .39 2.20 3. 05 2.20 2. 01 2.01 0.0 12 F • 51, .45 , .40 1.23 3.88 1.73 1.47 2.46 1.60 13 F .51, .41 1.80 0. 0 7.01 2.84 2.86 3.48 14 F .50, .40, .30 1. 73 1.92 1.73 0. 79 1.21 0.0 15 F • 55, • 43, .38, .34 1.27 4.46 5.6o 0.0 0.0 0.0 16 F .49, .46, .39, .30 1.75 0.0 1.99 0. 99 1.02 2.22 17 F -49, .45, .40, .29 1.54 2. 59 1.60 1.32 1.33 2.09 18 F . 50, .40, .38 0.0 3.88 2.57 1.73 1.85 0.0 19 F .50, .43, .30 3.84 7.65 4.35 0. 0 4. 05 3.06 20 F .48, .39, .30 0.70 0.0 1.29 0.41 1.45 0.0 21 F . 50, .43, .30 1.24 4. 19 2.30 1.24 2.07 2.47 22 F .53, .40 0.93 1.00 4.43 1.09 1.20 0.0 23 F . 50, .40, .30 0.78 3.94 1.94 1.58 0.0 1.63 24 F .50, .42, .31 1.95 0. 0 2.73 0. 0 2.53 2.6b 25 F .49, .40, .30 0.96 0.0 1.62 0. 81 0.93 1.36 26 F .50, .44, .31 0.61 0.0 1.74 0.75 1.49 1.52 27 F .50, .41, .30 1.30 1. 19 1.76 1. 12 1.72 0.0 28 F .48, .45 , .40 1.35 2. 15 2.11 0.0 1.08 2.43 29 F ' .49, .42, .30 1.08 2.67 3.46 0.61 0.63 2.11 30 F .50, .40, .30 1.30 0.0 2.23 0.fl5 1. 79 2.07 31 M .49, .45, .40 1 .96 2.10 2.46 2. 10 2.11 2.13 31 M • 49, .45, .40, .30 1.39 2. 88 1.32 0.0 1.44 0.0 31 M .52, .44, .40 1.15 3. 52 1.76 1.21 1.64 0.0 sm l.5o 3.34 2.56 1.42 1.71 2.26 sd 0.69 1. 48 1.43 0.b7 0.80 0.60 r 2 .98889 .95050 .96934 .99044 .98673 .97610 ORDERING OF sm 3 9 ti 2 6 7 1.04 3.2/ 0.9* 0.9u l . O i 0.79 2.0o 1 .2o 1.5J 2.0o 1.7* 1.1/ 0.9* 0.87 1.2a 0.0 1.22 2 . 4 i 2.1* 0.5J 1 . I d 0.92 0.5* 1.91 0.8^ 0.55 1.0* 1.36 0.9* l.Oo 1.9* I .03 0.92 1 .3u O.bi .99232 1 8 9 10 1.24 1.43 1.45 20.75 2.97 4. 17 21.42 1.46 1.47 1.15 1.15 1.26 3.19 1.15 1.17 0.95 1.22 1.20 2.26 2.47 2. 56 0.0 1.86 1.85 21.35 1.79 1.79 0.0 2.45 0. 0 21.22 2.02 2.04 22.26 1.83 1.85 23.61 1.74 1.73 1.21 1.08 1.21 20.04 1.76 1. 64 0.0 1.29 1.36 12.35 1.60 1.62 22.64 2. 16 2.31 19.77 3.26 0.0 17.27 0.80 0.90 17.86 1.49 1.58 20.70 0.94 0.0 16.02 1.32 1.37 15.47 0.0 0.0 0.99 1.14 1.24 0.98 1.16 1.23 20.32 1.15 1.88 15.89 1.50 1.80 1.24 1.05 0.0 1.55 0.95 0.98 0.0 0.0 2. 18 17.96 1.33 1.34 19.68 1.48 1.54 13.16 1.58 1.68 8.94 0.59 O.c.3 07892 .96919 .98799 10 4 5 fD P> O W sr rt S» cn a p> P i & c u H a* (D • • p i f D Ul <: f + H -fD 0) (U <+ P > P-O W W r+ P> o c+ M i f D * • Ul c + w M f D Ul Ui ui cr fD H> Ul l-l H H i P> O f + 1") fl> f D H P> P> O t ) tr f D Ci p i C » T3 r+ P* Ul f D • • P i B f + >X3 O U> i-3 00 bd co co 4 0 TABLE 3B P r o b a b i l i t i e s t h a t s u c c e s s i v e p a i r s of o r d e r i n g s of sm i n Table 3 A are c o r r e c t O R D E R S : 14 2 J P R O B A B I L I T Y T H A T I S D A V l 71 < S J A V ( 4) t =0.717 C f t D E R S : 2& 3; P R O B A B I L I T Y T H A T ( S D A V l 4) < S D A V I 1) ) =0.749 O R D E R S : 34 4; P R O B A B I L I T Y T H A T ( S D A V l 1) < S U A V ( 9) i =0.550 O R D E R S : 44 5 ; P R O B A B I L I T Y T H A T ( S D A V l 9J < S J A V ( 10) ) =0.725 O R D E R S : 54 6 ; P R O B A B I L I T Y T H A T ( S D A V I I U ) < S D A V I 5) ) =0.566 O R D E R S : 64 7; P R O B A B I L I T Y T H A T ( S D A V ( 5) < Si>AV( 61 I =0.997 O R D E R S : 74 8 ; P R O B A B I L I T Y T H A T I S D A V l 6) < S J A V I 3 ) ) =0.856 O R D E R S : 84 9; P R O B A B I L I T Y T H A T I S D A V I a < S D A V ( 2) ) =0.974 O R D E R S : 9410; P R O B A B I L I T Y T H A T ( S D A V C 2i < S u A V ( 8) ) =1.000 P E R S O N S E X 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 31 31 F F F F F F F F F F F F F F F F F F F F F F F F F F F F M M M H E M A T O C R I T S A T W H I C H M E A S U R E M E N T S W E R E M A D E .51, .41, .30 .50, .41 .52, .44, .40 .47, .42 .49, .50, .42 .41 .50, .45, .40 .49, .38 .51, .45, .40, .50, .40 .51, .47, .39 .51, .45, .40 .51, .41 .50, .40, .30 .49, .46, .39 .30 .49, .45, .40 .29 .50, .48. • 4 3 , .39, .50, .43, .53, .40 .50, .40, .50, .49, .50, .50, .48, .49, .50, • 49, .49, .52, sm 3d 7 2 O R D E R I N G O F sm .42 , .30 .30 .30 .30 .31 .40, .30 .44, .31 .41, .30 .45, .40 .42, .30 .40, .30 .45, .40 .45, .40, .44, .40 , 30 24.69 22.26 22.81 14.41 29.70 13.85 18.32 20.3 1 18.48 14.22 25.02 23. 80 16.48 26.02 32.21 24.71 22.14 27.60 19.53 17.17 20.44 23.26 26.44 19.57 21 .82 18.58 18.05 19.45 17.95 30.84 14.49 21.41 4.90 .99117 9 7. 50 6. 78 9.42 7.31 5. 00 5.93 14. 45 6.51 1 I. 99 7. 15 8.97 27.27 9. 70 10.44 9.90 9.27 23 7 13 94 8. 92 10.85 10. 28 7. 64 8. 12 8. 42 9.95 6. 30 6. 46 8.83 6. 95 10. 10 7. 64 9. 65 4. 62 .99801 2 10.36 8.68 10.30 7.24 5.77 7 .30 15.17 8.14 13 .32 9.76 8.14 28.17 12.64 12.88 12 .19 11 .34 23.54 10.46 11.85 13.14 12. 88 10.77 10.90 11.85 12.66 6.95 8.67 12 .50 8.08 12 .12 9.17 11.51 4.49 .99731 5 5.49 5. 31 8.85 7.42 4.97 5.32 13.63 5.77 10.79 4. 71 9. 58 26.54 6.23 8.72 9.12 8.06 21.25 6. 30 6. 96 9.33 7.46 5.48 5.95 5.73 7.46 6.11 4. 85 6.16 6.55 9.08 6.41 8.24 4.68 .99845 1 7.11 6. 00 8. 97 7. 17 17.21 5.52 0.0 6. 19 11.52 5. 85 8. 52 25.91 8. 09 9.79 10. 35 9.16 21.69 0.0 8.11 9.83 9.39 7.12 0.0 7.79 8.54 6.45 0. 0 8.28 6.59 10.20 0.0 9. 67 4. 82 , 99797 3 16.50 23.60 22.34 13.97 19.93 20.30 22.47 18.40 26. 14 24.43 20.83 26.25 33. 51 21.47 19.71 17.19 23. 21 22.09 18.40 2 2.87 15.08 1 7. 93 18.54 18.53 14. 40 16.65 14.25 19. 19 26.25 20. 72 10.78 20.19 4. 56 .99255 8 7.58 10.95 11.46 7.52 10.24 9.13 0.0 9.23 12.58 12.95 11 .26 21 .53 13.73 8.26 11.61 9.88 10.20 7.75 9.36 12.07 8.23 9. 14 9.13 9.37 10.01 8.42 6.99 9.79 13.91 11.35 10. 88 4.40 5.89 86.96 5.89 0.0 5. 08 20.95 5 8.61 76.57 7.73 70.99 90.62 74.23 73. 19 81. 50 12.66 22.34 26.32 39.57 115.41 0.0 76. 55 73.38 0.0 7.69 111.31 8.12 7. 70 66.01 114.01 120.60 10.49 52.30 2.78 40.19 .99795 .92458 4 l l 11.48 11.63 15.86 11. 14 27.25 25. 29 20.58 19. 08 21 .46 10.32 19.7 3 36.89 22.48 18. 56 24. 16 24.04 23. 77 11.03 0.0 17. 13 13.06 10.68 27.47 10.83 14. 32 14.45 9. 38 17. 14 14.75 17.52 14.84 17.88 6.51 .99363 7 10 12.37 23.41 14.42 19.72 11.74 12.83 17.68 12.94 19.20 10.64 13.39 31.46 14.29 18.80 23.15 12.75 24.6 3 0.0 15.15 15.69 13.81 19.04 16.97 0.0 13.36 12.59 19.16 12. 52 25.44 14.12 11 .34 16.82 5. 37 .99457 6 l l 23.85 21 .20 23.03 13.04 28.15 10.90 17.24 18.58 15.80 9.60 24.47 0.0 11.38 24.97 31.22 23.66 20. 50 26.98 16.75 53.53 19.35 21 .99 25. 13 17.73 41 .06 17.24 16.93 18.36 32.07 35.10 10.72 22.35 9. 46 ,98949 10 w r i -m p i P> 3 o p> p r p i M U l P > P> B O i T3 ro H < (D • • P> ( + • a P" ro O PI ** W . , o U l H i NJ l / l U l <+ H9 tr H !> <D (D 03 P) cn tr* H c n W H •P P> cn i+ (D H i O p i H 3 i Q ro CD P> n t r 0 O a ct- p i Ul T3 13 H s H * *a ro p ) P i —• d-o T A B L E HB P r o b a b i l i t i e s t h a t s u c c e s s i v e p a i r s of o r d e r i n g s of sm i n Table 4A are c o r r e c t ORDERS: I f , 2 : PROBABILITY T H A T ( S O A V l 4 ) < S O A V l 2 1 1 = 0 . 8 8 3 ORDERS: 2 £ 3 ; P R O B A B I L I T Y T H A T ( S D A V l 2 ) < S D A V l 5 ) ) = 0 . 5 0 5 O R D E R S : 3 G 4 ; P R O R A B I L I T Y T H A T ( S D A V l 5 ) < S D A V t 7 1 ) = 0 . 7 7 7 O R D E R S : 4 C 5: P R O B A B I L I T Y T H A T I S D A V l 1) < S D A V l 3 ) 1 = 0 . 8 6 0 O R D E R S : 5€ 6: P R O B A B I L I T Y T H A T ( S D A V ( 3 ) < S D A V { 1 0 ) I = 1 . 0 0 0 O R D E R S : 6 f . 7 ; P R O B A B I L I T Y T H A T ( S D A V l 1 0 ) < S D A V l 9 ) ) = 0 . 7 5 3 O R D E R S : 7T. 8: P R O B A B I L I T Y T H A T I S O A V I 9 1 < S D A V l 6 ) ) = 0 . 9 4 6 O R D E R S : 88 9: P R O B A B I L I T Y T H A T <SO A V I 6 ) < S D A V l I ) ) = 0 . 3 4 4 O R D E R S : 9 &io; P R O B A B I L I T Y T H A T ( S D A V l I ) < S D A V l 1 3 1 > = 0 . 6 8 7 O R O E R S : 1 0 & U ; P R O B A B I L I T Y T H A T ( S D A V 1 1 3 ) < S O A V l 8 ) ) = 1 . 0 0 0 R S O N S E X H E M A T O C R I T S AT W H I C H MEASUREMENTS W E R E MADE 1 F . 5 1 , . 4 1 , . 3 0 2 F . 5 0 , .41 3 F . 5 2 . . 4 4 , . 4 0 4 F . 4 7 , .42 5 F . 4 9 , . 4 2 6 F . 5 0 , . 4 1 7 F . 5 0 , . 4 5 . .40 8 F . 4 9 , . 38 9 F . 5 1 , . 4 5 , . 4 0 10 F . 5 0 , . 4 0 11 F . 5 1 , . 4 7 , . 3 9 12 F . 5 1 , . 4 5 , . 4 0 13 F . 5 1 , .41 14 F . 5 0 , . 4 0 , .30 16 F . 4 9 , . 4 6 , . 3 9 , . 3 0 17 F . 4 9 , . 4 5 , . 4 0 , .29 19 F . 5 0 , . 4 3 . . 3 0 2 0 F .48 , . 3 9 , .30 21 F . 5 0 , . 4 3 , . 3 0 2 2 F . 5 3 , . 4 0 23 F . 5 0 , . 4 0 , . 3 0 24 F . 5 0 , . 4 2 . . 3 1 2 5 F . 4 9 , . 4 0 , .30 2 6 F . 5 0 , . 4 4 , .31 2 7 F . 5 0 , . 4 1 , . 3 0 2 8 F . 4 8 , . 4 5 , . 4 0 2 9 F . 4 9 , . 4 2 , . 3 0 30 F . 5 0 , . 4 0 , . 3 0 3 1 M . 4 9 , .45 , .40 32 M . 4 9 , . 4 5 , . 4 0 . . 3 0 31 M . 5 2 , . 4 4 , . 4 0 sm sd' 72 ORDERING OF sm 1 2 1.89 2.33 2.67 6.16 2.08 3.47 1.16 1.72 2 . 7 8 2.88 2.00 0.0 3.00 3.51 2.04 1.25 2.08 1.56 2.46 1.23 0.0 3.82 2.36 3.30 1.10 1.68 3.58 3 . 4 9 2. 58 2.61 1.57 2.33 3.20 0.0 1.87 2.66 1.71 3.01 1.72 1.45 1.33 2.88 1.43 2.19 2. 34 2.96 2.29 3.00 2.13 2.48 1.52 2. 21 1.00 3.70 1.93 2.18 2.46 2.58 2. 63 2.34 1.94 2.58 2.09 2.67 0.62 0.98 . 9 8 8 2 0 . 9 7 9 1 6 4 9 3 4 0.0 1.32 6.16 0.0 1.92 0.0 1.13 0.0 2.53 1.43 1.72 1.02 0.0 0.0 1.15 0.0 1.71 1.50 1.73 0 . 9 0 0.0 1.92 2.51 0.0 0.0 2.76 3.48 2.12 2.83 0.0 2.22 2 . 4 0 3.99 3.12 2.78 1.43 3.27 1.48 1.43 1.30 0.0 2.10 2.83 0.0 3.07 0.0 3.14 1.80 0.0 2 . 5 5 1.64 1.21 1.71 0.82 2.44 2 . 1 5 1.36 2.04 2.07 1.86 2.33 1.47 2.45 1.76 1.08 0.61 . 9 8 1 9 8 . 9 9 1 4 1 7 2 5 6 1.33 0.0 3 . 6 2 2.89 3.6 1 1.47 1.93 2.88 1.79 0.0 1.02 0.0 3.64 0.0 2 . 8 6 1.05 0.0 0.0 1.00 1.69 3.07 0.0 0.0 0.0 1.08 1.40 2.86 6.21 2.23 0.0 2 . 3 4 2.85 3.40 2.27 2.02 2. 27 2 . 5 3 3.41 0 . 9 6 0.0 1.84 0.0 2 . 3 7 2.78 2.67 0.0 2 . 3 6 0.0 0.0 0.0 2 . 1 1 2.61 1.98 0.0 2.14 0.0 0.0 0.0 2 . 4 0 0.0 1.53 3.12 2. 2 5 2.64 0.80 1.25 . 9 8 5 2 0 . 9 7 7 8 2 6 8 7 8 0.99 0.0 2.47 0.0 2.01 2 6 . 7 0 1.06 1.51 1.11 0.0 0 . 8 9 0.0 2.87 3.01 1.75 0.0 1.61 2 7 . 0 5 1.41 0.0 2 . 4 0 0.0 1.39 2 8 . 4 3 1.00 2 7 . 7 8 1.94 0.0 0.0 0.0 1.13 0.0 0.0 2 2 . 4 2 1.19 1.11 1.35 2 2 . 3 0 0.82 2 5 . 9 1 1.18 2 0 . 2 3 0.87 2 0 . 2 0 1.72 1 7 . 2 7 1.32 2 0 . 9 7 1.70 2 5 . 2 1 0.91 2 1 . 3 9 0.79 1.13 0.82 2 3 . 0 4 1.73 2 8 . 6 5 1.49 2 6 . 6 7 1.19 2 3 . 7 3 1.42 1 9 . 7 5 0.53 9.47 . 9 9 4 4 9 - . 1 0 2 9 1 1 10 9 10 1.80 1.84 0.0 3.49 1.15 1.23 1.41 2.01 1.91 2.66 1.50 1.52 3. 3 6 3.39 1.22 1.14 1.88 1.84 1.43 1.57 1.96 0.0 2 . 3 1 2.48 1.74 2.29 2.71 2.98 2 . 3 9 2.46 2 . 0 4 2.07 2.42 2.42 2.02 2.18 2 . 0 9 2.28* 1.16 1.21 2.14 2 . 3 3 2 . 0 1 2.17 2 . 6 9 2.68 2 . 4 0 2.56 2 . 2 3 2 . 2 9 1.63 1.65 1.32 1.37 1.74 1.88 1.67 1.61 1.94 1.92 1.95 2.00 1.94 2.12 0 . 5 0 0.59 . 9 8 9 5 2 . 9 8 7 6 2 3 5 TABLE 5B Probabilities that successive pairs of orderings of sm in Table 5A are correct ORDERS: 16 2: PRDBABI LITY THAT ( SDAVl 71 < S0AV( 4)) =0.932 ORDERS: 26 3; PROBABILITY THAT (SDAVl 41 < SOAVl = 0.8 74 ORDERS: 36 4; PROBABI LITY THAT (SDAVl 91 < SDAV( i n =0.856 ORDERS: 46 5; PROBABILITY THAT (SDAV( 1) < SDAVl 10) ) = 0.561 ORDERS: 56 6; PROBABILITY THAT ISDAVUO) < SDAVl 5)1 =0.753 ORDERS: 66 7; PROBABILITY THAT (SDAV( 5) < SDAVl 3 ) ) =0.772 ORDERS : 76 8; PROBABI LI TY THAT ISDAVl 31 < SDAVl 61) = 0.58 3 ORDERS: 86 9; PROBABILITY THAT (SDAVC 6 ) < SOAV ( 2 ) I = 0 .541 ORDERS: 9610; P R O B A BILITY THAT (SDAVl 2) < SDAVl 3)) =1,000 '1 45 TABLE 6 P o p u l a t i o n averages and standard d e v i a t i o n s of parameters of best f i t f o r steady s t a t e data; S1 shear r a t e range FUNCTION NUMBER X l S T D DEV *2 S T D DEV * 3 S T D D E V S T D D E V 23 5 . 5 8 1 0 . 9 7 5 2 E - 0 1 0 . 7 2 9 0 0 . 5 8 7 2 E - 0 1 13 - 6 1 . 3 5 6 5 . 6 6 3 9 9 . 1 1 4 0 . 5 0 . 4 9 5 7 0 . 7 2 2 2 E - 0 1 0 . 3 2 8 0 0 . 9 1 8 4 E - 0 1 18 6 . 4 8 2 0 . 8 2 0 6 2 9 . 5 5 2 2 . 0 0 5 . 2 8 8 3 . 8 5 4 0 . 7 0 0 5 0 . 4 1 3 3 13 1 . 6 9 1 0 . 1 9 3 8 5 . 2 0 1 i . 7 4 4 3 . 7 4 7 0 . 4 3 9 2 2 . 1 8 7 0 . 4 5 9 0 19 7 . 0 8 1 5 . 2 4 4 4 . 3 9 7 4 . 6 6 4 4 . 0 4 0 0 . 3 4 1 2 1 . 0 9 9 0 . 5 4 1 7 15 - 0 . 5 8 5 1 1 . 3 3 6 1 . 0 7 2 1 . 4 1 3 ' 5 . 9 4 1 0 . 9 3 9 2 2 . 1 1 5 0 . 9 5 2 3 15 1 . 0 1 6 • 0 . 8 7 2 4 E - 0 1 6 . 2 0 7 0 . 7 0 8 7 E - 0 1 0 . 7 2 4 0 0 . 7 0 1 2 E - 0 1 0 . 4 4 0 0 E - T - 0 1 0 . 1 7 4 8 E - 0 1 ' 12 0 . 8 0 3 1 0 . 9 7 7 8 E - 0 1 1 . 0 0 8 0 . 5 1 1 5 E - 0 1 2 . 4 9 4 0 . 9 2 7 4 E - 0 2 0 . 9 2 8 8 E - 0 1 0 . 3 2 6 6 E - 0 1 21 4 . 8 2 1 1 . 0 9 5 0 . 7 9 5 4 0 . 2 1 3 8 2 . 2 2 5 0 . 7 4 3 4 4 . 5 6 1 0 . 4 6 0 3 1 0 24 4 . 2 4 5 1 . 2 9 9 1 . 6 0 6 0 . 7 5 7 4 4 . 6 3 9 2 . 5 3 5 4 . 7 8 9 0 . 5 6 1 6 TABLE 7 ' ' P o p u l a t i o n averages and standard d e v i a t i o n s of parameters of best f i t f o r steady s t a t e data; S2 shear rate range FUNCTION NUMBER n 1 30 2 29 3 25 * 23 5 16 6 26 7 14 8 21 9 19 10 18 13 • 27 X l STD DEV X 2 STD DEV STD DEV STD DEV 5.057 0.1361 1.991 0.4136 2.330 0.4014 31.76 2.351 14.94 0.6676 -2.045 1.591 0.7721 0.7110E-01 1.345 0.2407 7.772 2.340 5.848 0.9442 6.493 0.3460 0.4302 0.2652E-01 79. I 5 15.06 44.96 8.567 0.1210 0.2489E-01 0.2698 O.3062E-01 0.5317E-02 0.3889E-02 6.004 0.6398E-01 1.070 0. 1370 0.3776E-02 0.2657E-02 0.4763E-02 0.2200E-02 0.3550 0.3509E-01 0. 1148 0. 2208E-01 0.2152 0. 4996E-01 1 .750 0.3954 2.207 0.3082 2.813 0.4138 0.4478 0. 1444E-01 2 .584 0.4144 0.1274E-01 0.1092E-01 0. 120 6E-01 0.6323E-02 0.3835 0.7044E-01 2.378 0.4369 2.024 0.3704 2.641 0 .4610 2. 158 0.3025 2. 539 0.4640 0 .7667E-01 0.1793E-01 1.291 0.4217 1.939 0.5565 2.144 0.3515 7.465 0.4507 TABLE 8 Population averages and standard d e v i a t i o n s of parameters of best f i t f o r peak data; S1 shear r a t e range FUNCTION X. X 2 Xj X 4 NUMBER n STD DEV STD DEV STD OEV STD DEV 26 6.174 0.1006 0.7826 0.5888E-01 20 7. 418 1.1CT8 249.6 223.2 1 .691 0.8918 0 .4251 0.2609 19 6.72 0 C.8606 29.47 22.00 7 .436 3.229 0.7057 0.3323 17 9.030 12.00 2.780 1.668 3.459 0.2829 2. 387 1.010 16 14.75 4.059 1.439 0.1549 3.801 0.4289 0.6948 0.2633 11 0.1793 0.7886 0.9942 2.032 6.049 0.6746 2.930 1.639 16 I .343 0.1092 6.216 0.5245E-01 0.7868 0. 6505E-01 0.4434E-01 0.1544E-01 10 0.8937 0.1157 0.9944 0.3523E-01 2.498 0.8927E-02 0.1753 0.4339E-0 1 22 5.910 1.462 0.5860 0.1646 1.706 0.4749 4.682 0.5034 10 22 5.070 1.214 1.191 0.4046 3 .390 1.231 4.950 0.4556 TABLE 9 Population averages and standard d e v i a t i o n s of parameters of best f i t f o r peak data; S 2 shear r a t e range NOTION UMBER n X l STD DEV . X 2 STD DEV x 3 STD DEV x 4 STD DEV 1 23 5 . 3 3 9 0.903 I E - 01 0 . 4 9 1 0 C . 2 1 5 1 E - 01 2 25 1.91 1 0 . 3 1 9 8 7 2 . 19 1 4 . 2 3 0 . 2 3 0 3 0 . 2 2 4 7 E - 01 2 . 4 6 5 0 . 3 8 2 4 3 24 2 . 2 4 3 0 . 3 1 8 0 3 7 . 3 3 7.6B5 0.4412 0 . 5 8 2 3 E - 01 2 . 1 2 3 0 . 3 2 7 2 4 24 3 0 . 0 5 2 . 1 5 8 C . 2 3 9 8 0 . 2 6 3 7 E - 0 1 1 .735 0 . 3 4 4 9 2 . 6 7 1 0 . 4 2 0 8 5 22 1 4 . 0 5 1.531 0.4948 0 . 5 7 9 3 E - 01 2 . 0 4 9 0. 3 6 0 1 2. 3 3 3 0 . 3 7 4 4 6 17 - 1 . 8 1 6 1.145 0 . 5 5 1 1 E -0 . 3 6 9 5 E -0 2 •02 2 . 9 2 6 0 . 2 9 7 8 2. 527 0 . 2 8 8 4 7 22 1 .239 0 . 4 8 8 3 5.641 0 . 6 9 3 1 0 . 5 1 0 6 0 . 3 8 3 9 E - 01 0 . 5 2 6 5 E 0 . 2 8 5 1 E 8 27 1. 142 0 . 5 8 7 5 0.8572 0 . 3 5 0 3 3.2 54 1.144 1.926 0 . 9 3 7 1 9 19 6 . 2 5 7 1 .607 0. 1 2 6 3 E -C . 7 0 7 4 E -0 2 0 3 0 . 3 2 0 7 E -0 . 2 3 4 1 E -02 02 1.833 0 . 6 7 2 5 1 0 16 5.799 C . 7 8 8 9 0 . 2 8 0 3 E -0 . 1 5 4 0 E -02 0 2 0 . 7 0 9 9 E -0 . 4 6 4 3 E -0 2 02 2 . 0 0 2 0 . 3 7 9 8 13 11 6 . 1 1 2 0 . 1 1 0 5 E - 01 0 . 3 1 6 3 0 . 4 6 4 3 E - 0 2 0 . 4 6 3 7 0 . 2 5 2 8 E - 01 8 . 4 2 9 0 . 2 6 6 1 TABLE 10 R a t i o s of (Xi f o r S 2 ) / ( X i f o r S1) f o r steady s t a t e data 1,S2 2,S2 3,52 4,S2 Function — number X X maan+std* dev. 1,S1 2,S1 3,S1 4,S1 1 0.906 0.590 0.748+0,223 2 -0.0325 0. 198 0.232 7.25 1.912+3.561 3 0.359 1. 521 0.0407 2.889 1. 20 2+1,29 2 4 18,78 0.023 0.467 1,208 5.120+9.120 5 2. 110 0.061 0.546 1,964 1. 170+1.022 6 3.495 0.0050 0.473 1,201 1,294+1.548 7 0.760 0.967 0.619 1.743 1. 022+0*501 8 1.675 1-062 1;0636 13.90 4*418+6.328 9 1.612 0.0047 0,i0057 0.425 0.512+0.760 10 1. 378 0.0030 0*0026 0,448 0.548+0.648 50 TABLE 11 Ra t i o s of ( X i f o r S 2 ) / ( X i f o r S1) f o r PEAK data X X X X 1,S2 2,S2 3,S2 4,S2 Function mean+std; dev. number X X X X 1,S2 2,S2 3,S2 4,S2 1 0*904 0*932 0,* 918+0*020 2 -8.27 1*599 Oi 293 Oi 772 -1.402+4.611 3 0.965 1.003 0.711 0.993 0.918+0,139 4 Oi 187 1.871 1. 083 0.916 1* 014+0.691 5 0.480 3*056 1.063 1.582 1; 545+Ti 103 6 -3.293 1*078 | 0. 932 0.722 -0.120+2*101 7 0*824 0.999 0. 920 0. 992 01934+0.081 8 0.899 1.014 0*998 0. 530 0.860+0.226 9 0.816 1.357 1.304 0.972 1. 113*0.260 10 0.837 1.348 1l 368 0:967 1. 130+0.269 51 TABLE 12 Mean values of hematocrit and CF parameters for individuals in menstrual cycle survey ;rson H X1 X2 X3 X4 1 ,40 2.03 4.62 .447 .017 2 .43 .74 6.05 U59 .070 7 *44 2.81 4. 10 .442 .002 8 .38 1. 94 4. 71 .408 .022 9 144 2i04 4.54 .452 .014 10 .41 .95 6, 10 .449 .0 46 11 145 4.03 3.46 .436 -1018 12 .42 1,94 4. 67 .451 .013 13 .40 1.00 6*04 .367 .020 15 .40 2.06 4. 68 .432 .005 16 .45 3.10 lid 9 .477 1006 18 .38 1.49 5. 44 .471 .032 30 .45 3.02 3. 72 .422 -.004 31 .45 .76 6. 02 .456 .078 5 2 TABLE 13 Plasma p r o t e i n s p r o t e i n a b b r e v i a t i o n f i b r i n o g e n F i b * anti-thrombin I I I A n t i - t . IgM IgM. albumin Albi. oc2-macroglobulin CX2-M. Behring population s t d . dev. R.I-D. p l a t e — — catalogue no* p o p u l a t i o n mean OTBN 03 OTBI 03 0TD0 03 OTBG 03 OTBU 03 . 1 6 . 4 2 . 1 3 . 2 6 TABLE 14 lan values of plasma protein concentrat ions for individuals in i menstrual cycle survey Person Fib: Anti-t. IgM Alb. a 2 - H , mg/dl mg/dl mg/dl g/dl mg/d 1 296 33 240 4.1 242 2 330 — 90 4.4 374 7 349 40 159 4.7 215 8 285 37 138 4:7 329 9 304 31 450 4.2 310 10 372 39 136 4:7 218 11 293 33 113 4. 8 360 12 273 33 257 5. 2 294 13 301 41 128 4. 8 307 15 414 — 16 — — 18 503 —* — — 30 413 35 170 4. 7 380 31 — — mean 344 36 188 4. 6 303 std, dev . 69 3.6 106 .32 61 std. dev./mean .20 . 10 .56 ,07 .20 TABLE 15 P l o t t i n g symbols Person Symbol 1 1 2 2 7 7 8 8 9 9 10 A 11 B 12 C 13 D 15 F 16 G 18 I 30 U 31 V 55 5i# SflMPLE 1 0.0 46.667 93.333 140.0 RCDW/DR) (SXX - 1 ) FIGURE 2 Sample 29, function 7, steady state data f i t over S2 {Xi} = {2.70,3:66,.389,-.00495} R(DW/DR) (SKK - 1 ) FIGURE 3 Sample 29, function 7, steady state data f i t over S1 {Xii = {4.32,3.58,.798,-.0407} 56 .48 0 - 0 2 . 1 4 . 2 RIDW/DRJ I.SXH-1J FIGOSE 4 Sample 29, f u n c t i o n 7, peak data f i t over S2 {Xij = {1.772,4.67,., 451, .0145} a _ .48 R ( D W / D R ) ( S X X - 1 ) FIGURE 5 Sample 29, f u n c t i o n 7, peak data f i t over S1 {Xi} = {1.22,6.27,. 761,.0227] 5 7 . o • a . CD LO CO Q _ SI# SAMPLE PERKS FCN 7 PIT ]0 0 . 0 4 6 . 6 6 7 RCDW/DR) 93.333 (SXK-1J 1 4 0 . 0 FIGURE 6 Sample 29, f u n c t i o n 7 , peak data f i t over S2 {Xi} = {1.772,4.67,.451,.0145} o LO LO ' • i n C L L O Q _ to IS SI# SAMPLE PEAKS PCN 7 PIT 6 4 6 . 6 6 7 9 3 . 3 3 3 RfDW/DR) (SXX - l ) CD A 1 4 0 . 0 FIGURE 7 Sample 29, f u n c t i o n 7, peak data f i t over {Xij = {1.22,6.27,.761,.0227* S1 58 a 0 . 0 4 6 . 6 6 7 9 3 . 3 3 3 1 4 0 . 0 R(DW/DR) (SXH -1) FISORE 8 Sample 18 r f u n c t i o n 7, steady s t a t e data f i t over S2 {Xij = {,845, 6.08,. 441,. 0487} a a — a o l 0 . 0 4 6 . 6 6 7 93.333 1 4 0 . 0 R(DW/DR) (SXX - l ) FIGUSE 9 Sample 29, f u n c t i o n of Huang, steady s t a t e data f i t over S2 {Xii = {. 126,.83,* 209, 1, 5j 59 o • o . to ' " i n CLVD Q_ S A M P L E STEADY PCN 2 F I T 10 1 S T A T E 4 6 . 6 6 7 R (DW/DR) 9 3 . 3 3 3 ( S X X -1) 1 4 0 . 0 FIGURE 10 Sample 29, f u n c t i o n 2, steady s t a t e data f i t over S2 {Xi} = {1.96, 76. 1, .0758, 2. 19) •—-a. t o 0 . 0 SI# S A M P L E 1 STEADY S T A T E FCN 3 F I T 10 4 6 . 6 6 7 9 3 . 3 3 3 R(DW/DR) ( S X X - l ) 1 4 0 . 0 • FIGURE 1 1 Sample 29, f u n c t i o n 3, steady s t a t e data f i t over S2 {Xij = {2. 37,49.4,, 118, 1.79} 60 o • a _ .—* L O L O E g -C O R S I # S A M P L E S T E A D Y P C N 4 P I T 1 0 1 S T A T E 4 6 . 6 6 7 R ( D W / D R ) 9 3 . 3 3 3 140.0 FIGUEE 1 2 Sample 29, f u n c t i o n 4 , steady s t a t e data f i t over S2 {Xi} = {3119, .0794, 1.75,2.40i a • a. a LO C O P S I # S A M P L E S T E A D Y P C N 5 P I T 1 0 1 S T A T E 4 6 . 6 6 7 9 3 . 3 3 3 R I D V / D R ) ( S H H - 1 ) 140.0 FIGUEE 1 3 Sample 29, f u n c t i o n 5, steady s t a t e data f i t over S2 {Xi} = {15.3,. 156,2.02,2, 12i 61 a -a. co C X L O 0.0 S I # 5 R M P L E 5 T E R D Y FCN 6 P I T 10 1 STATE 46.667 RCDW/DR) 9 3 . 3 3 3 140.0 FIGURE 14 Sample 29, function 6, steady state data f i t over S2 {Xi} = {-4s. 74,. 0098 2,2.67,2.37} o to ( X L O Q _ SI# SAMPLE STEADY FCN 8 F I T 10 1 STATE 46.667 R(DV/DR) 9 3 . 3 3 3 140.0 FIGURE 15 Sample 29, function 8, steady state data f i t over S2 {Xij = {,756,.547,3.70,2.77} 62 CD Q_ SI# SAMPLE STEADY FCN 9 FIT 10 1 STATE 4 6 . 6 6 7 R(DW/DR) 93.333 (SXX - l ) 140.0 FIGURE 16 Sample 29, f u n c t i o n 9, steady s t a t e data f i t over S2 {Xij = {8.25, . 00920,.0309,1.97j a O : E LO E g -Q_ •E: 0 . 0 SI# SAMPLE 1 STEADY STATE FCN 10 FIT 10 4 6 . 6 6 7 R(DW/DR) 93.333 (SXX-1) 1 4 0 . 0 FIGURE 17 Sample 29, f u n c t i o n 10, steady s t a t e data f i t over S2 {Xi} = {5.93,,00892,,0223,2*03} 63 5i# SAMPLE 1 0.0 45.667 93.333 140.0 R(DW/DR) ( S X H - 1 ) FIGURE 18 Sample 29, f u n c t i o n 11, steady s t a t e data f i t over S2 {Xi} = {6.50,-357,.440,6. 54) SI# SAMPLE 1 STEADY STATE 0.0 45.667 93.333 140.0 R(DW/DR) ( S X X - l ) FIGURE 19 Sample 29, f u n c t i o n 2, steady s t a t e data f i t over S2 {Xij = {1.96,76. 1,-0758,2.19} 64 cn IN SRMPLE 1 STEADY STATE FCN 3 F I T 10 46.667 93.333 8 : i H O . 30 140.0 FIGURE 20 Sample 29, f u n c t i o n 3, steady s t a t e data f i t over S2 {Xij = {2. 37, 49.4, . 118, 1. 79} fN SI# SAMPLE 1 STEADY STATE FCN 4 F I T 10 46.667 93.333 R(DW/DR) ( S X K - 1 ) FIGURE 21 Sample 29, f u n c t i o n 4, steady s t a t e data f i t over S2 {Xi} = {31.9, i 0794, 1. 75,2,40} 65 CN V OJ •CM 8rt> SI# SAMPLE STEADY FCN 5 FIT 10 1 STATE 0.0 46.667 93.333 R(DV/DR) (SXH-1) 140.0 FIGURE 22 Sample 29, f u n c t i o n 5, steady s t a t e data f i t over S2 {Xi} = {15.3, . 156, 2.02, 2. 12} OJ CN fN ;B.QD C O c a a SI# SAMPLE 1 STEADY STATE FCN 6 FIT 10 to 0.0 46.667 93.333 R(DW/DRI (SXX-1) ~1 140.0 FIGURE 23 Sample 29, f u n c t i o n 6, steady s t a t e data f i t over S2 {Xij = {-4.74, . 00982,2. 67,2.37} 66 CD CM SI# SAMPLE STEADY STATE FCN 7 F I T 10 4 6 . 6 * 7 9 3 . 3 3 3 prnu/npi r^¥¥-ii * 8 : $ 0 . 3 0 1 4 0 . 0 FIGURE 24 Sample 29, f u n c t i o n 7, steady s t a t e data f i t over S1 {Xij = {2.70,3.66,.389,-.00495} f M 5 I# SAMPLE 1 STEADY STATE FCN 8 10 - H O . 30 0 . 0 4 6 . 6 6 7 9 3 . 3 3 3 R(DW/DR) C S X X - l ) 1 4 0 . 0 FIGURE 25 Sample 2.9, f u n c t i o n 8, steady s t a t e data f i t over S2 {Xi) = {.756,.547,3.70,2.77} 67 cn SI* SAMPLE STEADY FCN 9 FIT 10 STATE & 8 : $ 0.30 4 6 . 6 6 7 R(DW/DR) 9 3 . 3 3 3 (SXX-1) 1 4 0 . 0 FIGURE 26 Sample 29, f u n c t i o n 9, steady s t a t e data f i t over S2 {Xi} = {8,25, .00920, .0309, 1.97} cn fN S I # SAMPLE 1 STEADY STATE FCN 10 FIT 10 0 . 0 4 6 . 6 6 7 9 3 . 3 3 3 R(DW/DR) (SHK-1) 0 .30 1 4 0 . 0 FIGURE 27 Sample 29, f u n c t i o n 10, steady s t a t e data f i t over S2 {Xij = {5.93,.00892,.0223,2.03} 68 CD 9 • C M SI# SAMPLE STEADY FCN 13 FIT 10 STATE 0.0 45.667 93.333 R (DW/DR) (SXX-1) 140.0 FIGURE 28 Sample 29, function 11, steady state data f i t over S2 {Xij = {6.50,; 357,.440,6.54j o CO ce: L U Q _ ca o LU ZD _ l < > 1. 1 0 + 1. 08+ 1. 08+ 1. 04+ 1. 02 1 . 0 0 0. 8 8 + 0. 98+ 0 . 94 0 . 92+ 0 . 9 0 +1. 10 1. 08 +1. 08 +1 .04 +1. 02 + 1 . 0 0 +0. 98 + 0 . 9 8 +0. 94 0 . 9 2 0 . 90 TIME 4- 2 8 DAYS FIGURE 29 cn 1 0 X x v M E A N V A L U E O F X1 F O R P E R S O N p p p p p r r B ro *. eg CD 8 M B s B (a 8 8 8 0.001 I I I 1 I I I - I I I < l — h -0 . 1 0 0 . 2 0 0 . 30+ 0 . 4 0 0 . 50+ 0 . 8 0 j 0 . 70+ 0 . 80+ 0 . 9 0 1. 0 0 I I I I I l - H — I — I -B ro i\> N CO B *. 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O CD TI o -< O 0 . 6 0 0 + 0 . 7 0 0 + 0 . 8 0 0 + 0 . 8 0 0 + 1 . 0 0 0 1—I—I—I—I—I—I—I—I—|—I—I—I—I—I—I—I—I—I—I—I—hr0 .00B 0 9 0 9 01 P H—I—I—I—I—I—I—I—h 0 9 o D a m cn H—I—I—I—I—I—I—I—I—I—I—h 08 A L B U M I N C O N C E N T R A T I O N + M E A N F O R P E R S O N P P P P P 0 . 0 0 0 r - • > • - a co : r n a T T - n 3 : »—i m CD C • i -TO m r -m TZ. \-> CD : —\ zn. • o n o -< o r -0 . 1 0 0 0 . 2 0 0 + 0 . 3 0 0 + 0 . 4 0 0 + 0 . 5 0 0 + 0 . 8 0 0 + 0 . 7 0 0 0 . 8 0 0 + 0 . 8 0 0 1 . 0 0 0 ' 2 H — I 1—I 1—I—I 1 — h o o c o > > to o p H — I 1 1—I 1 1 1 — h IV) H — I — I — I — I — I — I — I — H 1 - 0 . 0 0 0 CO > o c o m > r~ m CO H — I — I — I — I — I — I — I — H 18 I g M C O N C E N T R A T I O N - M E A N F O R P E R S O N < o r~ m P P 0 . 0 0 0 0 . 1 0 0 + 0 . 2 0 0 + rn 0 . 3 0 0 + > co m a —I 0 . 4 0 0 CD CZ '-I-PO m r- 0 . 5 0 0 m NJ CD 0 . 8 0 0 + 0 . 7 0 0 0 . 8 0 0 + 0 . 8 0 0 + 1 . 0 0 0 P P P CD a p r § § H — I — I — I — I — I — I — I — h o OD O ID C O CD OD N OD O H — I — I — I — I — I — I — I — h a a H — i — i — i — i — i — i — i — i — i — i — h - 0 . 0 0 0 c OD a m (0 > r~ m —I 1 1 1 1 1 1 1 1 1 1 h 28 a 2 - M A C R O G L O B U L I N C O N C E N T R A T I O N T M E A N F O R P E R S O N p p p p r I . I I . i . I 8 I I 0 . 0 0 0 4 1 1 1 1 1 h - H 1 1 1 1 1 1 1 1 1 1 j h ID 0 . 1 0 0 + 0 . 2 0 0 + m 0 . 3 0 0 + co : m a — i 0 . 4 0 B + " m o CZ •!• TO m l - 0 . 5 0 0 ' rn CO CD O -< o 0 . 6 0 0 + 0 . 7 0 0 + 0 . 8 0 0 0 . 8 0 0 + 1 . 0 0 0 P ID O O > c > 10 H—I—I—I 1 h o > CO 03 m a • (0 ID ro m > r~ m co i—I—I—I—I—I—I—I—I—I—I—I—h 0 . 0 0 0 £8 4 0 . 0 H H H 1 I !• I •H 1 1 1- H 1 1 1 1 1 1-3 8 . 0 + 3 2 . 0 + 2 8 . 0 + > Q 2 4 . 0 + Q 2 0 . 0 + co 1 8 . 0 + 1 2 . 0 + 8 . 0 + 4 . 0 + G G G 8 F A 1 .» 1 " B I U B U G 7 1 8 0 . 0 d (M a* H 1 ± 1 ± H- H + 1 + 1 ± 1 ± H 1-0.0 ea a* a* a* a* F I G U R E 44 CO 4 0 . 0 d H 1 1 1 I H H 1- H 1 1 1 1 H H H 3 6 . 0 + 3 2 . 0 + 2 8 . 0 + > LU Q 2 4 . 0 + Q co 2 0 . 0 + co 1 6 . 0 + D 2 1 2 . 0 + 8.8+ 4 . 0 + D I B 0 . 0 d — + — t -d H 1 I 1- H + t~ 1 - 0 . 0 d 8 d d d ti 8 ol X2 FIGURE 45 co c n 9* 4 0 . 0 3 6 . 0* * 3 2 . 0 - -2 8 . 0 - -Lu 2 4 . 0- • Q Q 2 0 . 0 - -CO H 1 I I I 1 I I 1 1 I 1 I -| 1 1 I 1 1 I 1-c o 1 6 . 0 + 1 2 . 0 + 8 . 0 + 4 . 0 + 0 . 0 ; 7 8 8 -I H H 1 I H + H I I 1 H G G D D 1 A 1 N a* o a* a s 8 8 a* X3 F I G U R E 4 6 00 a 48.0H H 1 1 1 1 1 I 1 1 1 1 1 1 1 I 1- H 1 1 1 1 1 1 1 1 t 32.0+ 28.0+ 24.0+ Q 20.0+ co oo 18.8+ 12.0+ 8.0+ 4.0+ 7 I F A 8 8 1 C G G 4- 7 0 . 0 4 H + 1- a ' a ' a ' a ' a ' a ' a ' a ' a ' A H +0.0 a rf i i i I tA ti J i i i a a5 rf a rf H G U R E 47 88 C o n c l u s i o n Of the CFs examined t h a t of Walburn and Schneck was found to f i t best f o r steady s t a t e data over both shear r a t e ranges c o n s i d e r e d and f o r peak data over the s m a l l e r range of shear r a t e s * The peak data over the S2 shear r a t e range was f i t best by f u n c t i o n 4, However, even i n t h i s case the Walburn-Schneck f u n c t i o n was f o u r t h best* When the Walburn-Schneck f u n c t i o n was f i t t o samples comprising a normal p o p u l a t i o n and the o p t i m a l parameters were p l o t t e d , no c y c l i c v a r i a t i o n of these parameters i n the menstrual c y c l e was e v i d e n t . T h i s r e s u l t does not agree with t h a t of D i n t e n f a s s (1971, p* 21). Future work should have the f o l l o w i n g o b j e c t i v e s : 1, The so c a l l e d i n f r a s t r u c t u r a l problem f o r L>>1 should be s o l v e d . 2* The non-uniqueness of the s o l u t i o n of the o p t i m i z a t i o n problem should be examined so t h a t anomalies due t o s o l u t i o n s a s s o c i a t e d with l o c a l minima can be avoided. 3. The r e l a t i o n s h i p s between the values of CF parameters, e r y t h r o c y t e s t i f f n e s s and e r y t h r o c y t e aggregation should be examined. 4. The r o l e of the a n t i c o a g u l a n t should be examined s i n c e the measurements w i l l not be c l i n i c a l l y u s e f u l unless they can be r e l a t e d t o c o n d i t i o n s i n v i v o i n a r e p r o d u c i b l e manner. 89 B i b l i o g r a p h y A b b r e v i a t i o n s of c u r r e n t j o u r n a l t i t l e s conform t o ISO 833-1974 or ANSI Z39.5-1969 (these two standards are e g u i v a l e n t except f o r minor d i f f e r e n c e s ) . A b b r e v i a t i o n s of o b s o l e t e t i t l e s are taken from Index Hedicus f o r the year of p u b l i c a t i o n ; Aronson, H* B-Z., Levesque, P. 8., Charm, S- and Esten, B. .1968. I n f l u e n c e of a n a e s t h e t i c s on the rheology of human blood. Can. Anaesth. Soc. J , 15:2 44-257. Aronson, H. B-Z., Hagora, F. and Schenker, J. G. 1971. E f f e c t of o r a l c o n t r a c e p t i v e s on blood v i s c o s i t y . Am.' Obstet, Gynecol. 110:999-1001. . Bard> Y. 1974. Nonlinear.parageter_.es t i m a t i o n * Academic Press, New York. B a t c h e l o r , G. K* and Green, J..T* 1950. The d e t e r m i n a t i o n of the bulk s t r e s s i n a suspension of s p h e r i c a l p a r t i c l e s to order c 2 . J . F l u i d Mech. 56:401. Becker, W., Rapp, W: , Schwick, H. G. and S t o r i k o , K. _ 1968* Methoden zur q u a n t i t a t i v e n bestimmung von plasma proteinen durch immunprazipitation* Z. K l i n . Chem. 6:113-224. Begg, T, . B. and Hearns, J* B. 1966. Components i n blood v i s c o s i t y . C l i n * S c i . 31: 87-93. Behar, M . G, and Alexander, S. C. 1966. I n _ v i t r o e f f e c t s of i n h a l a t i o n a l a n a e s t h e t i c s on v i s c o s i t y of human blood. Anesthesiology 27:567-573* B i r d , C. and Moore Lee, C* 1975* 0BC_NLP. U n i v e r s i t y of B r i t i s h Columbia Computing Center, Vancouver. Brooks, 0. E., Goodwin, J . W. and Seaman, G. V. F. 1974. Rheology of e r y t h r o c y t e suspensions: E l e c t r o s t a t i c f a c t o r s i n the dextran-mediated aggregation of e r y t h r o c y t e s . B i o r h e o l o g y , 11:69-77, i B u r c h f i e l d , R. W. . 1972* [Ed. J Supplement to-the - Oxf ord. ;English d i c t i o n a r y , Oxford U n i v e r s i t y P r e s s , London! Vol* 1. Heaning 6, Burton, R. R., Sluka, S* J * , Krone, R* B. and Smith, A. H*. 1969. The p h y s i c a l c h a r a c t e r i s t i c s of e r y t h r o c y t e s e t t l i n g i n a l i g u i d medium* J . Biomech*.2:389-396. Bygdeman, S. and Wells R. J . 1969. J . A t h e r o s c l e r * Res*.10:33-39. C a l l e n , Hi B., 1963: Thermodynamics. John Wiley and Sons, Inc* New York. . Campbell, R, G. 1973. Foundations of f l u i d flowtheory.. Addison-90 Wesley P u b l i s h i n g Co., London. Casson^ N. 1959. A flow eguation f o r p i g m e n t - o i l suspensions o f the p r i n t i n g i nk type, I n : C. C. M i l l t Ed.3 Bheolpqy o f . d i s p e r s e systems. Pergamon Pr e s s , New York. Charache, s , , Conley, C. L., Waugh D. F . , Ugoretz, R. J , and S p u r r e l l , J . R. 1967, Pathogenesis of hem o l y t i c anemia i n homozygous hemoglobin C di s e a s e , J . . C l i n . Invest..46:1795-1811. Charm S. E., Kurland G, S* 1967. S t a t i c method f o r determining blood y i e l d s t r e s s . . Nature (Lond) , 216:1121-1123.. Charm, . S, E. and Kurland • Gl . S. 1974,; Blood_flow_and m i c r o c i r c u l a t i o n . Wiley, New York; Charm, S. . E. , Kurland G* S i and Schwartz h. 1969; Absence of t r a n s i t i o n i n v i s c o s i t y o f human blood between shear r a t e s o f 20 and 100 s e c — * . J . Appl, P h y s i o l , 26:389. Chien, S,, Usami, S., Dellenback, R, J , and Gregerson, M. I . 1970* Shear-dependent i n t e r a c t i o n of plasma p r o t e i n s with e r y t h r o c y t e s i n blood rheology; Am* J , P h y s i o l , 219: 143-153. Chien, S., Dsami, S,, T a y l o r , EL . M., Lundberg, J . L:. and Gregersen, M. I, 1966,'Effects of hema t o c r i t and plasma p r o t e i n s on human blood rheology at low shear r a t e s . J . Appl. P h y s i o l . 21:81-87, Chien, S. 1975. B i o p h y s i c a l behavior of red c e l l s i n suspensions; pp.1031 t o 1133 In: D; MacN. Surgenor {.Ed. ij The -red blood c e l l . Academic Press, New York. C o k e l e t , G, H., Meiselman, H. J . 1968. R h e o l o g i c a l comparison of hemoglobin s o l u t i o n s and e r y t h r o c y t e suspensions,; Science, 162:275-277. Cokelet, G. R., M e r r i l l , E,; W. , G i l l i l a n d , E; R;, Shin, H., B r i t t e n ^ A,, and Wells, R. E; 1963. The rheology of human blood -measurement near and at zero shear r a t e , Trans. Soc* Rheol. 7:303-317, Coleman, B. D., Markovitz, H. and N o l l , W; 1966. V i s c o m e t r i c flows__of non-Newtonian f l u i d s , , S p r i n g e r - V e r l a g , New York. Colquhoun, D. 1971. L e c t u r e s .on b i o s t a t i s t i c s , Oxford U n i v e r s i t y Press, Oxford. Copley, A,. . L i , L u c h i n i , B. W. and Whelan, E, W. . 1966;. On the r o l e o f f i b r i n o g e n - f i b r i n complexes i n flow p r o p e r t i e s and s u s p e n s i o n - s t a b i l i t y of blood systems, pp. 375 to 388. In: Copley A. L. I Ed,J Hemorheolqyl Pergamon Press, Oxford. 7 Couette, ft. M i 1890, E t u d i e s sur l e frottement des l i q u i d e s . Ann. Chemi Phys. 21:433-510. 91 Davis, H. F, 1967. I n t r o d u c t i o n to ve c t o r a n a l y s i s ; A l l y n and Bacon Inc,, Boston. D i n t e n f a s s , L., J u l i a n , D. G* and M i l l e r ^ G. . E. 1966. V i s c o s i t y of blood i n normal s u b j e c t s and i n p a t i e n t s s u f f e r i n g from coronary o c c l u s i o n and a r t e r i a l thrombosis. Amer. Heart J . 71:587-600. D i n t e n f a s s L . 19711 Blood mjcrorheology y i s e o s i t y . f a c t o r s i n blood flow, ischemia-and_thrpabosis. Butterworth and Co. L t d . , London. D i n t e n f a s s , L, 1976* Rheology ..of blood ^ n ^ d i a q n o s t i c ^ a n d p r e v e n t a t i v e medicine-. Butterworth and Co* Ltd* , London. D i t z e l , J . 1971. Changes i n rheology and oxygen t r a n s p o r t f u n c t i o n o f the e r y t h r o c y t e s i n d i a b e t e s . In: D i t z e l , J . and Lewis, D. H. [ E d s ; ] M i c r o c i r c u l a t o r y ^ a p p r o a c h s to c u r r e n t t h e r a p e u t i c problems! S, Karger, Basel; pp. 123-131, D i t z e l , J . , Bang, H* 0* and Thorsen, N. 1968, Myocar d i a l i n f a r c t i o n and whole-blood v i s c o s i t y . Acta Med. Scand. 183:577-579, Dormandy, J . A. and Edelman J . . B* 1968, High blood v i s c o s i t y : an a e t e o l o g i c a l f a c t o r i n venous thrombosis. B r i t . J . Surg, 60:187-193, 1 t Dormandy, J ; A, 1974*.Medical and en g i n e e r i n g problems i n blood v i s c o s i t y . Biomed; Eng. 9:284-303. E i n s t e i n , A. 1906. A new dete r m i n a t i o n of molecular dimensions; pp; 36 to 621. In: R, F u r t h J. Ed, 3 I n v e s t i g a t i o n s on the theory of the Brownian movement. Dover P u b l i c a t i o n s , New York. Erin g e n , C. [Ed , ] Continuum.physics. Academic Press* New York, 4 v o l . (Vol. I Mathematics, V o l , I I Continuum mechanics of s i n g l e substance m a t e r i a l s . ) Evans, E. A. and Skalak, Ri 1979, Mechanics and.-_thermodynamics of .biomembranes.. CRC Press, C l e v e l a n d , F i s c h e r , T. M., Haest, C, 8. M., S t o h r - L i e s e n , M. , Kamp, D. and Deuticke, B. 1978a. S e l e c t i v e a l t e r a t i o n of e r y t h r o c y t e d e f o r m a b i l i t y by -SH reagents* Evidence f o r an involvement of s p e c t r i n i n membrane shear e l a s t i c i t y . Biochim. Biophys. Acta 510:270-282. F i s c h e r , T. M., S t o h r - L i e s e n , M. and Schmid-Schonbein* H. 1978b. The red c e l l as a f l u i d d r o p l e t : tank t r e a d - l i k e motion of the human e r y t h r o c y t e membrane i n shear flow. Science, 202:894-896. Fung, Y, C>; 1965. F o u n d a t i o n s : o f 1 s o l i d _ m e c h a n i c s * P r e n t i c e - H a l l Inc., New York. 92 Fung, Y. C* 1969. &„£irst_gourse_in continuum,,mechanics* P r e n t i c e - H a l l Inc., Englewood C l i f f s . Goldman* G. 1973. P r i n c i p l e s ^of^ medical_-science. McGraw-Hill Book Co., New York. p. 318. Gross, G, P. and Hathaway, W. E. 1972,.. F e t a l , e r y t h r o c y t e ^ d e f o r m a b i l i t y . P e d i a t . Ees. 6:593-599. Ham, T. H., Dunn, B. F., Sayre, fi. R. and Murphy, J*.R. .1968. P h y s i c a l p r o p e r t i e s of red c e l l s as r e l a t e d to e f f e c t s i n v i v o . I . Increased r i g i d i t y of e r y t h r o c y t e s as measured by v i s c o s i t y of c e l l s a l t e r e d by chemical f i x a t i o n , s i c k l i n g and h y p e r t o n i c i t y * Blood, 32:847-861. Huang, C. R., F a b i s i a k , W. 1977, A r h e o l o g i c a l equation c h a r a c t e r i z i n g both the time-dependent and the. steady s t a t e v i s c o s i t y of whole human blood. In:American I n s t i t u t e o f Chemical Engineers 70th Annual Meeting, f i c h e #49. American I n s t i t u t e of Chemical Engineers, New York. Huang, C. R. 1972. A thermodynamic approach t o g e n e r a l i z e d r h e o l o g i c a l equations of s t a t e f o r time-dependent and time-independent non-Newtonian f l u i d s . Chem* Eng. J . ,3: 100-104. K r i e g e r , I. M. and E l r o d , H. 1953. D i r e c t d e t e r m i n a t i o n of the flow curves of a non-Newtonian f l u i d . I I * Shearing r a t e i n the c o n c e n t r i c c y l i n d e r viscometer: J* Appl. Phys. 24:134-136. Landau, L,. D. and L i f s c h i t z , E. M. 1959, Theory-of e l a s t i c i t y ; Pergamon P r e s s , Oxford. Langsjoen, P. H. and Inman, T. W. 1968, Hemorrheologic o b s e r v a t i o n s i n acute myocardial i n f a r c t i o n . Angiology 19:247-256. i Langsjoen, P. H, 1966. Blood v i s c o s i t y changes i n acute myocardial i n f a r c t i o n ; Postgrad. Med* J . 39 (5):A42-A56. Lew, H.. S. 1969. Formulation of s t a t i s t i c a l equation of motion of blood; Biophys. J . 9:2 35-24 5. Marsh, B. D; and Pearson, J . R. A. 1972. The measurement of no r m a l - s t r e s s d i f f e r e n c e s u sing a cone-and-plate t o t a l t h r u s t apparatus; Rheol. Acta. 10:557. Mayer, G* A. 1964, Blood v i s c o s i t y i n he a l t h y s u b j e c t s with coronary heart disease* Can* Med. Assoc. J . , 91:951-954,. Meiselman, H. J . , M e r r i l l , E* W., Salzman, E, W., G i l l i l a n d , E. R. and P e l l e t i e r , G. A. 1967. E f f e c t of dextran on rheology of human blood: low shear viscometry. J . Appl. P h y s i o l : 22:480-486: M e r r i l l , E. W., Cheng, C* S. and P e l l e t i e r , G. A. 1969. Y i e l d s t r e s s of normal human blood as a f u n c t i o n of endogenous f i b r i n o g e n * J . Appl. P h y s i o l * 26:1-3. 93 M e r r i l l , E. W, , G i l l i l a n d , E. B., Lee, T. S. and Salzman, E. W. 1966* Blood rheology: effect of fibrinogen deduced by addition. Circii Ees. it8: 437-446. M e r r i l l , E. W., Margetts, W. G., Cokelet, G. R., B r i t t e n , A., Salzman, E. fl*, Pennell, E. B, and Melin, M. 1965a. Influence of plasma proteins on rheology of human blood; Proceedings of the Fourth International Congress on Rheology, Part 4. Symposium of biorheology. Interscience pubs. New York. M e r r i l l , E. H. , Margetts, W. G;, Cokelet, G, R, and G i l l i l a n d , E. . R. 1965b. The Casson equation and rheology of blood near zero shear,; Proceedings of the Fourth International Congress on Rheology, Part 4. Symposium of biorheology. Interscience Pubs. New York. M e r r i l l , E. W. 1969. Rheology of blood* Physiol; Revs. 49:863-888. Messmer, K. and Schmid-Schonbein, H; 1975* Bibliotheca_Hemat. No. 4 Unintentional :Hemodilution. S. Karger, B a s i l * Messmer, K. and Schmid-Schonbein,H.1972. Hemodilution* S. Karger, B a s i l . Moore Lee, C* . 1978, UBC CURVE* University of B r i t i s h Columbia Computing Center. Vancouver. P h i l l i p s , W. M. and Deutsch, S* 1975. Toward a co n s t i t u t i v e equation for blood; Biorheology, 12:383-389; P o i s e u i l l e , J . L. M. 1840._ Recherches experimental sur le mouvement des l i q u i d s dans les tubes de tres p e t i t s diametres. C. RlHebd. Seances Acad* Sci* 11:961-967,1041-1048. Powel, M. J, D* 1965. An e f f i c i e n t method for finding the minimum of a function of several variables without c a l c u l a t i n g derivatives* ComputiJ. 7: 155-162. Putnam, T, C., Kevy, S*. V* and Repogle, R.,L. 1965* Factors affe c t i n g the v i s c o s i t y of the blood* Surg* Forum, 16:126-129. Putnam* T. C , Kevy, S. V. and Repogle, R. L. 1967., Factors affecting the v i s c o s i t y of blood. Surg* Gyneco* Obstet* 124:547-552. Quemada, D. 1975. Rheologie des suspensions concentrees et du sang* Discussion d'une l o i viscosite-concentration, deduite d'un principe d'energie dissipee extremale,* C. E. Hebd. Seances Acad. Sci* Ser> B S c i . Phys: . 280: 793-795. Quemada, D. 1975. Rheologie des suspensions concentrees et du-sang. Thixotropie des suspensions de globules rouges et l o i v i s c o s i t e - v i t e s s e de cisaillement. C. R.. Hebd, Seances Acad. S c i , Serie B S c i . Phys. 281:69-72. I 94 Quemada, D, 1975. . Hemorheologie. Aggregation des hematies et p r o p r i e t e s non-Newtonian du sang: n o u v e l l e l o i r h e o l o g i e et v a l i d i t e de l a l o i de Casson pour l e sang. C. R. Hebd. Seances Acad. S c i . , S e r i e D S c i . Nat., 281:747-750. Schmidt-Nielsen, K; and T a y l o r , C. R. 1968* Red blood c e l l s : why or why not? Science, 162:274-275. Segel, N, . and Bishop, J . M. J . .1968. The c i r c u l a t i o n i n p a t i e n t s with c h r o n i c b r o n c h i t i s and emphysema at r e s t and d u r i n g e x e r c i s e , with s p e c i a l r e f e r e n c e t o the i n f l u e n c e of changes i n blood v i s c o s i t y and blood volume on the pulmonary c i r c u l a t i o n . J . C l i n . I n v e s t . 45:1555-1568. Shangkuan, Y. L., Huang, C. R; and Copely, A; L..1977. A n a l y s i s of s e d i m e n t a t i o n i n human whole blood from i t s c o n c e n t r a t i o n p r o f i l e of e r y t h r o c y t e s ; Biorheology, 14:69-73. Skalak, R, .1978. P h y s i c a l and mathematical models of blood flow: t h e o r e t i c a l a n a l y s i s In: Kroc f o u n d a t i o n workshop on: red blood c e l l mechanics and blood flow, Santa Ynez V a l l e y , C a l i f o r n i a . S t a b l e s , D, P* , Rubenstein, A. H., Metz, J . .and L e v i n , N. W. 1967. The P o s s i b l e Role Of Hemoconcentration In The E t i o l o g y Of M y o c a r d i a l I n f a r c t i o n ; Amer* Heart J . 73:155-159, Stone, H. 0., Thompson, H. K., J r . , and Schmidt-Nielsen, K. 1968* I n f l u e n c e o f e r y t h r o c y t e s on blood v i s c o s i t y * Am.. J . P h y s i o l * 214:913* V e r s t r a e t e , M., Vermylen, J . and Donati, M. B, 1971, .Plasmic degradation of f i b r i n o g e n . Scan* J . Haematol. Suppl. 13:3-14. Walburn, F. J . and Schneck, D* J ; 1976,; A c o n s t i t u t i v e equation f o r whole human blood* B i o r h e o l o g y , 13:201-210. Walters, K. 1975. Rheometry. Chapman and H a l l , ' London. Watanabe, T., Oka, S, and Yamamoto, M. 1963* A phenomenological theory of the sigma e f f e c t . B i o r h e o l o g y , 1:193-199. Weaver, J . P. A., Evans, A..and Walder, D. N..1969. The e f f e c t o f i n c r e a s e d f i b r i n o g e n content on the v i s c o s i t y of b l o o d . . C l i n ; S c i , 36:1-10, Wells, R, E., Gawronski, T. H., Cox, P. J , and Perera, R. D. 1964. I n f l u e n c e of f i b r i n o g e n on flow p r o p e r t i e s of e r y t h r o c y t e suspensions. Am. J..Physiol;.207:1035-1040. Whitmore, R. L. 1968. Rheology of the c i r c u l a t i o n . Pergamon Pre s s , Oxford. Wiedemann, G. 1856. Oeber die bewegung der f l u s s i g k e i t e n im k r e i s e der geschlossenen g a l v a n i s c h e n s a u l e und i h r e beziehungen zur e l e c t r o l y s e . Pogg. Ann* 99:177-274, 95 Wintrobe, Mi M. 1967; Clinical^bematologY.,- Lea and Febiger, Philadelphia. 96 APPENDIX 1 Tensor a n a l y s i s Tensor a n a l y s i s cannot be reviewed here but the d e f i n i t i o n of a t e n s o r and the i n t e r p r e t a t i o n of the s t r e s s tensor w i l l be d i s c u s s e d * Davis {1967, p. 240) d e f i n e s a rank r tensor as 'a s c a l a r valued f u n c t i o n of r v e c t o r v a r i a b l e s that i s l i n e a r i n each v a r i a b l e ' * Consider any two n-dimensional c o o r d i n a t e systems ( a i , . . . , a n j and £b.i,*. . ,bn} * . Many authors (e.g. Eung, 1969, p. 93) d e f i n e a rank r tensor as a s e t of n components {tpj.i,... 11<pi<n, 1<i<rj with r e s p e c t to the a c o o r d i n a t e system which transforms under c o o r d i n a t e t r a n s f o r m a t i o n i n t o { u q i . . , . q r | 1<qj<n,1<j,<r} with r e s p e c t to the b system a c c o r d i n g t o : 3aPl do? r + where summation over the pi i s implied* Davis' d e f i n i t i o n can be used to generate components and he shows ( i b i d , pi 272) t h a t they transform as s p e c i f i e d above. Two s p e c i f i c remarks are r e l e v a n t here. A rank 1 tensor i s j u s t a f u n c t i o n which generates the components of a v e c t o r i n Davis* d e f i n i t i o n and i s i d e n t i c a l l y a v e c t o r i n the component d e f i n i t i o n . Second, i n t h r e e dimensions a rank 2 t e n s o r w i l l have nine components which can be arranged i n a 3 x 3 array (or m a t r i x ) ; t h i s was the t r a d i t i o n a l r e p r e s e n t a t i o n of a tensor. The 'rate of deformation' of a m a t e r i a l i s c h a r a c e r i z e d by a rank 2 t e n s o r c a l l e d the s t r a i n r a t e . Numerous good d i s c u s s i o n s of the s t r a i n or r a t e of s t r a i n t e n s o r s are 9 7 a v a i l a b l e (Campbell, 1973, p. 93; Fung, 1969, p, 93; Landau, 1959, p. 1), However, t h a t of C a l l e n (1965, p.,213) w i l l be o u t l i n e d here. 0' Consider the f l u i d i n a f i d u c i a l s t a t e and l a b e l two p o i n t s : P having l o c a t i o n v e c t o r r and Q having l o c a t i o n v e c t o r r+dr. A f t e r a time dt the f l u i d deforms and P and Q move to P' and Q* r e s p e c t i v e l y * The displacement v e c t o r s s ( r ) and s (r+dr) are d e f i n e d by the diagram. Vector a d d i t i o n i m p l i e s d r • - d r = s ( r + d r ) - s ( r ) (1) s ( r ) i s expanded i n a T a y l o r s e r i e s as : s(r+dr) = s(r>+dr«Vs (r)+0 (dr»dr) (2) (1)*(2) ==> dr«-dr = dr»Vs(r) d e f i n e S(r) = V s ( r ) . (S i s a rank 2 matrix but not a tensor.) Thus 5 (r) p r o v i d e s a way of f i n d i n g the change i n r e l a t i v e p o s i t i o n of 2 p o i n t s i n the m a t e r i a l . However^ t h i s i n v o l v e s two superimposed e f f e c t s . One i s a r i g i d body r o t a t i o n and the o t h e r , which i s of prime i n t e r e s t here, i s deformation. S can be decomposed i n t o 2 p a r t s which correspond p r e c i s e l y t o r o t a t i o n and deformation,* i * e . . S = 1/2 (S+S T) +1/2 (S-S T) (4) deformation+rotation the reader i s r e f e r r e d t o C a l l e n f o r a f u l l e r d i s c u s s i o n of t h i s decomposition. Define 98 I = V 2 (|+|T) (5) I t can be v e r i f i e d t h a t E i s a rank 2 t e n s o r ; i t i s u s u a l l y c a l l e d the s t r a i n t e n s o r . For f l u i d flow the r a t e of s t r a i n i s the important q u a n t i t y . Thus (6 ) s - T a f " e;g.: a s p e c i f i c element of the matrix of D i n c a r t e s i a n D = ~ E = 1/2 c o o r d i n a t e s i s : = 1/2 2>i 1 L • i ! ^ The s t r a i n r a t e i s c a s t i n the Davis form as: D (m rn)=m»D«n Where m and n are u n i t v e c t o r s moving with the f l u i d . ( 7 ) Example : I t may be v e r i f i e d t h a t D (e,e) i s the r a t e of decrease I 2. of 1/2 of the small angle between convected curves tangent to the u n i t v e c t o r s e and e^ T I Transformation of the s t r a i n t e n s o r i n t o c y l i n d r i c a l c o o r d i n a t e s i s c a r r i e d out i n appendix 2 . The ' f o r c e e f f e c t * i n f l u i d flow i s a l s o a rank 2 t e n s o r which i s c a l l e d the s t r e s s and i s denoted here by T. I t s dimensions are F L ~ 2 and i n c e r t a i n i n s t a n c e s i t i s a p r e s s u r e . In g e n e r a l s t r e s s i s d e f i n e d as f o l l o w s . 9 9 A s m a l l s u r f a c e element with area A l o c a t e d i n s i d e the m a t e r i a l i s considered* Let the u n i t normal of the s u r f a c e be m. Let another u n i t v e c t o r n be s p e c i f i e d and l e t F be the t o t a l f o r c e a c t i n g on the s u r f a c e due t o t r a c t i o n of a d j o i n i n g m a t e r i a l . The s t r e s s a s s o c i a t e d with m, n i s T (m,n) = F»n = m-T^n (8) = component of F i n the n d i r e c t i o n here T i s a rank 2 matrix such t h a t m«T.=F ; . (9) example : T (e,e) = e*T«e=T l 2 1 2 i — 2 1 0 0 A P P E N D I X 2 Transformation of the s t r a i n t ensor from c a r t e s i a n c o o r d i n a t e s i n t o p o l a r c y l i n d r i c a l c o o r d i n a t e s Z location displacement c a r t e s i a n c y l i n d r i c a l p o l a r •x,uzsmxz. - 1 - X l U j.co5X " ^ - U L , C O S X g . - — a , — — c, = u, C0SX2.- X-iu^sinxz. C z. = u, e 1 n Xz. X, u 2_ cos Xz.. 9 c y c o \ 3 ^ 4 cow da r row. col 3c t 9 c 3 du, 3u, ^ 3 c 3 90.2, 3c, 3 c 3 3us 3 U 3 \3Ua 3 a , ax, 3X3 da*. 9 a « . 9a2. ax, 3x.t 9x 5 9 a ^ 9 a 3 da*, 3 X , a x z dx3 Let a , = x ,cos x a Qz = x, s m X 2 . a 3 = X 5 cosx^ s i n X z O -X,5inX2, a , o 0 0 I c o s Xz. X i S i n Kz 0 S i n X t , K | C 0 5 X g . O O O I x, u - a a a, G 0 O I a, X, a . o o 0 be the s t r a i n t ensor component i n c a r t e s i a n c o o r d i n a t e s . Manipulate the t r a n s f o r m a t i o n so t h a t i t has the a p p r o p r i a t e d e r i v a t i v e : 101 0ij = dCr | d c s \ 1 Z _ j _ z _ J_ 2 dXL dXj P o u r 3 a s d p d x d a r 3a s/dc d X i d x ; d u t d;at ^ X w dxw/3cv s da-t \ I dXg$ 3 a s d e s [ d a - y . \dxi axj d u t 9 C L * 1 3 X J dXi dxj 3uy-l.dx^ i C d a s 5 c 5 d u f ddr r dCr d X i °»J d u t da-t dx 3ut |_|_ d a s d c s 3 x j d u t l d X j / dxj dikf\dxx d c ; " d u 3 x i c h a n g e t h e o r d e r o f f a c t o r s t o c o n f o r m w i t h u s u a l m a t r i x m u l t i p l i c a t i o n 0,j- 1 IdixtJi d C r c d d a l row _j_ dlkti<3Cr dai -d u t r o w ^ X i c o l d X i J t J d e r col d a y row d u t yx>w d X i c o | 0 X i 0 • cu . a , 0 X i 0 0 l _ a , X» x , 0 0 1 a* 0 o M = U M U V * V ^ o 0 i " 1 0 0 = 0 X ? 0 0 0 1 1 0 0 U 3 , l • 0 0 0 0 l _ X , U ^ 3 102 0 i l = i ( M + M T K z a u S t , 2_ U 3 , « U,,2 z. > _ U I , 3 X|"U2-,3 _ U3,> u 3,1+^1,3 2. 2 u 3,3 c o n s i d e r 1 —>r f 2 —>0 r u = p h y s i c a l displacement i n 9 d i r e c t i o n p h y s i c a l 0 , e = ^ V T ' 0 r . f 1 ^ + §g=) =4( r " ^ J -d ( ru e ) r z 3Ur , d ( rua) _ ( r u 8 ) \ r a e In Couette flow; p l n y s i c a l <0R _ r c)U0 're 2 dr - K r i e g e r & E l r o d ^ ~ P h i l l i p s & Deutsch S u b s t i t u t e U Q = C O = ^ -^[physical <jl>re\=rdu& z=r'bco dt 103 APPENDIX 3 K r i e g e r and E l r o d ^ s a l g o r i t h m f o r determining shear r a t e The shear r a t e v a r i e s with r i n the annular sample space of the rheometer* For a Newtonian f l u i d the shear r a t e at the bob su r f a c e can be c a l c u l a t e d d i r e c t l y by e v a l u a t i n g an i n t e g r a l taken from the cup to the bob. However when the CF i s non-Newtonian and i s not yet determined e v a l u a t i o n of the s t r a i n r a t e r e q u i r e s use of a s e r i e s * The theory was developed by Kr i e g e r and E l r o d (1953) and the r e s u l t i s : D(T) =(w/ln(s)) l l + m ln(s)+1/3(ra In (s) ) * +1/3 (In (s) ) 2 1 'dUn(T)] d) s = r 2 / r , d(ln(f)} r, = r a d i u s of bob rz = r a d i u s of cup a smal l m o d i f i c a t i o n i s made f o r a p p l i c a t i o n ; f = s c a l e f a c t o r f o r rheometer=T/B R = output s i g n a l o f rheometer n o w d _ d _ d _ d d(lnCT)) d(ln(T))~d(ln(f)) d ( l n ( T ) - l n ( f ) ) d(ln(R)) (1) becomes D(fR) = w/ln(s) [l+m ln(s ) + 1 / 3(m In (s) ) 2+1/3 ( l n ( s ) ) 2 djf- ) 'adnlw)) d ( , n ( R l" The f i r s t and second d e r i v a t i v e s r e g u i r e d to e v a l u a t e the s e r i e s are obtained from a c u b i c s p l i n e which i s f i t t o the s e t of data p o i n t s . The c u b i c s p l i n e f i t t o n data p o i n t s { (R{ , wj ) j 1<i<n] i s the curve LJP'i (•) where Pi (•) i s a 3rd degree polynomial such i = i t h a t 1 0 4 Pi (Hi ) = W I -PE <Ri+i) = w i and d R d R The s p l i n e i s c o n s t r u c t e d and d e r i v a t i v e s obtained by the r o u t i n e SPLNFT of the DBC Computing Center (Moore Lee, 1 9 7 8 , pp. 61-66). SPLNFT w i l l not work unl e s s the a b s c i s s a e are s t r i c t l y i n c r e a s i n g so i t i s necessary t o s o r t the data p o i n t s with t h i s c r i t e r i o n a p p l i e d to {Hi}* A f t e r the s p l i n e and d e r i v a t i v e s are obtained the data p o i n t s along with the d e r i v a t i v e s are r e s t o r e d t o o r i g i n a l order* 1 0 5 APPENDIX 4 Observed s t r e s s and i t s e r r o r due t o end e f f e c t f o r blood i n a Couette viscometer These c a l c u l a t i o n s r e f e r to a Couette measuring system as shown i n f i g u r e 48* ,-— FIGURE 48 Measuring system The n o t a t i o n which w i l l be used i s : f = c a l i b r a t i o n constant of instrument 1 = l e n g t h of bob M = t o t a l torque of f o r c e s of l i q u i d on bob M, = torque of l i q u i d on s u r f a c e 1 of bob M3 = torque of l i q u i d on s u r f a c e 3 of bob R = output s i g n a l of instrument r = r a d i u s of bob T = shear s t r e s s on s u r f a c e 1 of bob w = angular v e l o c i t y of cup Torque e q u i l i b r i u m and v a r i o u s i d e n t i t e s are expressed by 106 £ 8=11=111! +M3 = M (14-Ms/H,) = (27Tr, 1)T, r, (l+Ma/M,) (1) The instrument i s c a l i b r a t e d with o i l to determitte the c o e f f i c i e n t T , - ; - T 7 7 - - T T - 7 r r r which i s then assumed to be Z i r r f 1 0 + M s / M , j constant i n a p p l y i n g equation (1) to determine an experimental r s t r e s s T . although L--- i s s t r i c t l y c o n s t a n t the f a c t o r 2 7 T i f 1 (1+t'i3/Ml ) v a r i e s s l i g h t l y with w, thereby i n t r o d u c i n g an e r r o r i n T . The o b j e c t i v e i s t o determine the bounds on d e v i a t i o n of the f a c t o r (1 + M3/M| ) - 1 under experimental c o n d i t i o n s with blood from i t s value i n the c a l i b r a t i o n . I t i s necessary t o express the CFs i n tensor form (note t h a t the Walburn and Schneck CF has been s t u d i e d i n a s c a l a r form). T h i s has not y e t been c a r r i e d out so the r e s u l t cannot be e v a l u a t e d . 1 0 7 APPENDIX 5 Theory of f i t t i n g of CF to o b s e r v a t i o n s The c o n s t i t u t i v e f u n c t i o n must be f i t t o the data a c c o r d i n g t o some o b j e c t i v e c r i t e r i o n . The c r i t e r i o n used i s t h a t o f maximum l i k e l i h o o d . A gen e r a l theory i s giv e n by Bard ( 1 9 7 4 ) . The development s p e c i f i c t o t h i s s i t u a t i o n w i l l be o u t l i n e d * R R = o b s e r v e d vexlue R r = = theoretical m e a n of d is t r ibut ion scale I scale Z scale 3 Assume t h a t the p r o b a b i l i t y d i s t r i b u t i o n of R i s normal with mean R* and standard d e v i a t i o n s; R* and s are f u n c t i o n s of the angular speed, w, and the s c a l e s e l e c t e d . Thus the d i s t r i b u t i o n f o r s c a l e i i s P i ( R ) V 2 7 T S; For s c a l e j P i ( R = 4 Si Let the width of the d i s t r i b u t i o n with r e s p e c t t o be the q u a n t i t y k such t h a t P ( k + R ' ) P ( R ' ) = a Thus the width of the i and j d i s t r i b u t i o n s can be compared by: 1 0 8 R(ki+R f) = a Pj(R') e • e S; Therefore the width of the d i s t r i b u t i o n i s l i n e a r i n s. In the case under c o n s i d e r a t i o n r e f e r r i n g to the d i s t r i b u t i o n of G , assume t h a t kocl/f. Thus the l a r g e s t d i s t r i b u t i o n width occurs on the most s e n s i t i v e s c a l e ; Choosing c a p p r o p r i a t e l y a l l o w s s = c / f . The s u b s c r i p t i now s e r v e s to denumerate the instrument s c a l e s ; For the j t h o b s e r v a t i o n on a sample: To determine £Xi] f o r the f u n c t i o n R = 1 / f T(D,e,H,(Xi)) i t i s necessary to maximize dp or In (dP) with r e s p e c t to v a r i a t i o n of { X i j . The same r e s u l t w i l l occur from minimizing j J J s u b s t i t u t e s\=c/f\_ 109 Denoting the L.H,S. by F( { X i ) ) i t i s seen that the c r i t e r i o n f o r determining {Xi} i s to minimize I 12-F|{X-J) = E ( T r T j ' f where T.* = f; R.1. = t h e o r e t i c a l l y p r e d i c t e d s t r e s s Tj = f-t R>« =observed s t r e s s To o b t a i n c(=s- tf L) maximize dP with r e s p e c t to v a r i a t i o n of c. 0 I n d P = £ ' l n j dlndP = f d R j 1 — 1 V IVZTT Si | z h J 1 S J f Rij - Rlj 1 [Rtj-Rii ] 1 s i i 1. sr 1 (-») as-L d c O - . ^ - ^ - R y f It C ^ • c 3 M 1 J - ' J Standard d e v i a t i o n of s t r e s s = f ^ s - ^ c = (1/n£](Tj -Tj ) 2) */2 j Note that m i n i m i z a t i o n o f F ( { X i ) ) and minimization of c({Xi}) would r e s u l t i n the same solution for,{xi}. The minimization problem i s discussed i n appendix 6. 110 APPENDIX 6 O p t i m i z a t i o n Suppose that a f u n c t i o n a l g(T(D,e,H, {Xij)) i s to be minimized with r e s p e c t to {Xi). One might begin by forming the p a l g e b r a i c equations: = o 1<i<p (1) The problem i s now i d e n t i f i e d as being i n one of two c a t e g o r i e s . I t i s • a l g e b r a i c a l l y s o l v a b l e 1 i f the system of equations can be s o l v e d t o provide e x p l i c i t f u n c t i o n s mapping the data i n t o {Xi}* Otherwise i t i s not a l g e b r a i c a l l y s o l v a b l e . U n f o r t u n a t e l y the problem a t hand f a l l s i n t o the l a t t e r category; Thus i t may be d e a l t with i n one of two ways: 1. A t r a n s f o r m a t i o n may be i n t r o d u c e d which d i s t o r t s the problem so t h a t the new a l g e b r a i c system i s s o l v a b l e . For example c o n s i d e r the problem o f f i t t i n g T(D,k,m) to o b s e r v a t i o n s , where: T = k D m (X, =k,Xz=m) (2 ) S t a t i s t i c a l a n a l y s i s of the e r r o r s (theory of maximum l i k e l i h o o d (Bard, 1974) l e a d s to the c r i t e r i o n of f i t commonly c a l l e d the l e a s t squares c r i t e r i o n . In t h i s case i t r e q u i r e s minimization of g(T(D,k,m)) with r e s p e c t to k,m where 9 | T ) = £ ( k D ^ - T j ' j 2 , 3 , §1= E 2 (k D ; - T J ) D ? = o = * E k Dr = ETJ D ; J J J 1 = i-J- <«) E D / m 111 H- E 2 ( k D j - ^ ] k D j 1 l n ( D j ) = 0 = ^ > £ (k Dj 1—Tj ]Dj" In (DJ ) = o (») J The l a s t equation above i s not a l g e b r a i c a l l y s o l v a b l e f o r m. To transform the problem the n a t u r a l l o g a r i t h m i s a p p l i e d t o equation (1) above g i v i n g l n ( T ) = ln(k)+mln(D) (6) Assuming the s t a t i s t i c a l a n a l y s i s can be a p p l i e d t o equation (6) t o analyse the e r r o r s of In (T) versus ln(D) i t r e q u i r e s t h a t g ( l n ( T ) ) be minimized with r e s p e c t t o ln(k) and m where 9(10(7))= £( ln(k) + m l n ( D j ) - T j ) 2 (7) ^2Qn(k)+mln(Dj)-Tj)-0 =>nln(k)+(Sln(Dj))m = E In {.Tj) (8) = Zz ln(k)+mln(Dj) -Tj) ln(Dj )=0 ' ^ > ( E ln(Dj))ln(k) + ( E ( l n ( D j ) f ) m = T j (9) E x p l i c i t formulas f o r k rm are o b t a i n a b l e from equations (8) and (9). This t r a n s f o r m a t i o n of the problem i s r e f e r r e d to as l i n e a r i z a t i o n when the a l g e b r a i c equations o f the transformed problem a r e l i n e a r although i n g e n e r a l any s e t of s o l v a b l e equations i s s u f f i c i e n t . The r o u t i n e employed by Walburn and Schneck (1976) u t i l i z e d l i n e a r i z a t i o n . One might presume t h a t the e r r o r s o c c u r r i n g when the 112 problem i s l i n e a r i z e d would be s m a l l enough t o n e g l e c t , however t h i s cannot be r e l i e d upon* Colquhoun (1971, pp. 257-272) g i v e s an example of a two parameter f u n c t i o n f i t by both methods and has found the e r r o r s of the parameters to be 4.8% and 5.9% when the problem i s s o l v e d n u m e r i c a l l y versus 24;7% and 45.6% when l i n e a r i z a t i o n i s used; The d i f f i c u l t y with l i n e a r i z a t i o n i s t h a t when v a r i a b l e s are transformed so are d e v i a t i o n s from the p r e d i c t e d values,; Since the t r a n s f o r m a t i o n i s not j u s t a s c a l i n g i t changes the r e l a t i v e s i z e o f d i f f e r e n t terms i n the g(T) sum and thus changes the s o l u t i o n £Xij which minimizes g(T) . When the problem i s reduced to a system of l i n e a r a l g e b r a i c equations the q u e s t i o n of uniqueness of the s o l u t i o n can be answered completely* Any l i n e a r a l g e b r a i c system has e i t h e r 0, 1 or 0 0 s o l u t i o n s . Zero s o l u t i o n s would correspond t o absence of a minimum and an i n f i n i t e s e t of s o l u t i o n s would r e s u l t from mi n i m i z a t i o n of a f u n c t i o n which i s j u s t a constant value; n e i t h e r o f these cases are of p h y s i c a l importance. The s o l u t i o n of the l i n e a r i z e d problem w i l l t h e r e f o r e be unique. The s i t u a t i o n i s l e s s d e f i n i t e i n the more g e n e r a l i z e d approach, however. 2,; The second way of h a n d l i n g m i n i m i z a t i o n problems which are not a l g e b r a i c a l l y s o l v a b l e i s t o s o l v e the o r i g i n a l problem by numerical methods; T h i s i s more accurate than l i n e a r i z a t i o n s i n c e no a r b i t r a r y d i s t o r t i o n i s i n v o l v e d . T h i s approach was f o l l o w e d here, using an a l g o r i t h m based on the method of conjugate d i r e c t i o n s devised by Powel (1965), Other a l g o r i t h m s were t r i e d but of those a v a i l a b l e at the OBC Computing Center (B i r d and Moore Lee, 1975) t h a t of Powel was found to be most 113 r o b u s t . A FORTRAN program with s i n g l e p r e c i s i o n v a r i a b l e s was used. U n f o r t u n a t e l y the theory of n o n l i n e a r o p t i m i z a t i o n and e s t i m a t i o n i s not w e l l developed so i t was necessary t o use the minimum value of = ( p; £ (Tj- -T- ) 2) (10) as an estimate of the standard d e v i a t i o n of the o b s e r v a t i o n s from the p r e d i c t i o n s * The e r r o r i n t r o d u c e d by t h i s approximation i n the l i n e a r case can be eval u a t e d as f o l l o w s . For a l i n e a r f u n c t i o n T(D,e,H,{Xij),' 1<i<p, the unbiased e s t i m a t o r of standard d e v i a t i o n i s SITJ-T/ : l/z n - p y For n=28 r p=4 the f r a c t i o n a l e r r o r i n our e s t i m a t o r vs. The c o r r e c t e s t i m a t o r , f o r the l i n e a r case would be. JL y/z -1—i v v n L, I n - p 2-i I e _ '_!_ o y/z n-p L, j n - P * * - | = .07 n I t t h e r e f o r e seems l i k e l y t h a t the e r r o r i n the n o n l i n e a r case would a l s o be s m a l l . In any event the parameter, s, used i s a measure o f goodness of f i t . The p o s s i b i l i t y of a n o n l i n e a r problem having s e v e r a l s o l u t i o n s c e r t a i n l y e x i s t s ; No g e n e r a l theorem s p e c i f y i n g the c r i t e r i a f o r e x i s t e n c e or non-existence of a unique s o l u t i o n or determining the number of s o l u t i o n s of a n o n - l i n e a r o p t i m i z a t i o n problem i s known. T h e r e f o r e , i f one minimum i s found i n a p a r t i c u l a r problem i t may be a l o c a l minimum, while the g l o b a l minimum remains t o be found. Which minimum i s found when an o p t i m i z a t i o n r o u t i n e i s a p p l i e d depends on the l o c a t i o n of the 114 i n i t i a l point used. Roughly speaking, a minimum c l o s e t o the i n i t i a l p o i n t w i l l be found* Thus i t i s p o s s i b l e t h a t some l a r g e values of s are a s s o c i a t e d with l o c a l minima whereas i n onl y some cases the g l o b a l minimum was obtained. 115 APPENDIX 7 The p r o b a b i l i t y t h a t the o r d e r i n g of the. averages of 2 sample s e t s i s the o r d e r i n g of the means of the d i s t r i b u t i o n s from which they are.sampled Probabi l i ty d e n s i t y y To begin l e t the allowed values of y be a d i s c r e t e sequence,*Y, 0'f» aeons taritcrnerement,. Y= {b, e, | b, i s a whole number} Let the two d i s t r i b u t i o n s and t h e i r corresponding s e t s of o b s e r v a t i o n s be i d e n t i f i e d and d i s t i n g u i s h e d by s u b s c r i p t s 1 and 2 . Thus f o r d i s t r i b u t i o n 1: mean= a, and s t d . dev. = s, p r o b a b i l i t y density=cr* exp (-(y, -a, ) 2 ( 2 s 2 ) - * ) =N (y, ) where the s e t of observations= {yi\ 11<i<n } c, = c o n s t a n t Replacing 1 with 2 g i v e s the s p e c i f i c a t i o n s f o r d i s t r i b u t i o n number 2 . The p r o b a b i l i t y t h a t the i t h o b s e r v a t i o n on s e t 1 has the value y (i =N (y,i ) e , where N ( • ) i s a normal d i s t r i b u t i o n The p r o b a b i l i t y a s s o c i a t e d with the s e t (y,i j n i n , = IT N(y,i )e,= (TT N(y,; ) ) e, ni i=i i=i The p r o b a b i l i t y of o b t a i n i n g the two s e t s of o b s e r v a t i o n s from t h e i r r e s p e c t i v e d i s t r i b u t i o n s = (TT N(y,i ) ) e V ' ( IT N ( y 2 j ))Qrt 1=1 I S J 1=1 J-l 116 = (Y T N ( Y | l )N(y2j)) (e, < n , - i > e z . c n a - i > ) e , ez P a | i 0 l l = p r o b a b i l i t y t h a t a, <a<>. = p r o b a b i l i t y d i s t r i b u t i o n of the combined s e t of o b s e r v a t i o n s = T (TTTT N (y, i )H(j i j .))e l <n .-i)e i (n r i>e l et — i i J =e, <n,-i > e s L < n - i J £ e, e 2 ( TT T J N (y , i ) N (y f c j )) a,<-a.2. 1 J Now e, , ej, are made s m a l l e r while l a r g e r values of b, and hz are s e l e c t e d so t h a t Y\'\  an^  Yz] become s u f f i c i e n t l y dense t h a t the sum can be r e p l a c e d by i n t e g r a t i o n : e, — > d a i e 2 — > d a z n < ^ r3a1<a2=(cla,) (daj J J da,da 2 C,1'e - c£*-e •»-» • a,<ci2. v c T ' I, Ii= J d a . -co exp = i d a - o o \ 1 1 1 rO.2. ~ ) d a , exp 2 1 2a;2 f y f py> 21 2 1 ) da v e> -co 2ar exp(-^ f(yf-y,i] P* diary,) e x p f - ^ l a . - y j " ' 117 1 -00 * J '-00 1 . V . : ' = c 3 e , j d a , e x p ~ 2 5 ? S ( ( a , - y a ) 2 + ( y ^ - y / J ) 00 . . = c 3 e,e 2 U (a r y J e x p ( - z 5 7 K -yz) ] ( ) l e t a i - y i = > Q i T - y , = w+-( y z - y , ) -00 ~t OO / . / V V --00 n, e Z -oo z=-oo Botate z,w c o o r d i n a t e s through angle - 9 so t h a t the boundary l i n e : z=w+ ( y 2.-yi) becomes c o i n c i d e n t with one c o o r d i n a t e l i n e . For a p o s i t i v e a n g l e ^ the t r a n s f o r m a t i o n i s <V w P c o s s i n S u b s t i t u t e ^ = - 0 cos*/' z cos 6 -5i n 9 stnG cosO p t h e b o u n d a r y l i n e i s 2 L = \ A / + ( y z . - y > ) w=z bound c o s G p 4-s'mQc^ = - s i n 8 p -+- c o s 9 + { y z - y , ) ( c o s G + s i n G ) p = ( c o s 9 - s i n 9 ) ^ + ( y a - y , ) ^ = = = > ( c o s 6 + s i n 0 ) c o s 8 - s i n 9 (Require t h a t ^£=0 on the boundary) ==> c o s 0 = sin© ==> 9 = 1T//4 S p e c i f i c a t i o n of the boundary becomes: \ V F V z j p J/2. y» Transformation now g i v e s : w 2 tarp+dra) =z(-p+ar zt-i(P+<l)1 y2.~yi., 00 ^ 00 C 3 C ^ ) d p ) d c l e x p ~ ( ^ ( p i - ^ p c l + ^ ) + ^ ( p V ^ p q + c l - ) ) - O O -00 L y*-y, a b 00 AS. -00 -00 TJf- OO M 00 ^ f b b -2- b z' =c*c. $ d p exp f a (l - (5)2) p ^ exp M<\+k $ -00 ' ' -00 K n, J 120 To determine c r e q u i r e oo 0 ^ 0 0 - o o -00 i - -Vf ) d r e 4 ") d i r e •vz: ^ I \ i - I I V V-'oo o o APPENDIX 8 The r 2 parameter yi = observed value yt = p r e d i c t e d value yi = mean of observed v a l u e s s = standard d e v i a t i o n of f i t 122 APPENDIX 9 Programs used f o r data a n a l y s i s 1 $SIG DEBR PRIO=N T= 10S PAGES=6 RETURN=CNTR FOSM=11X 15 CARDS=000 2 $R P.OPT.G 1=PR (60.00,65.60) 2=*SINK* -3 3=*DUMMY* 4=PR (8.000,41 .000) SEBCOM=*SINK* SPRINT=*S INK* 4 $S.IG 5 ***THE FOLLOWING ARE PROGRAM CONSTANTS AND INITIAL VA LUES OF 6 •••OPTIMIZATION PARAMETERS AND COMPRISE UNIT 4 --SEE THE COMMENTS 7 •••AT THE BEGINNING OF THE OPTIMATION PROGRAM. . 8 0 1 9 2 4 4 4 4 4 4 4 4 4 4 4 5 10 0 0 1 0 0 0 0 0 0 0 0 0 0 11 800 12 .0238 .044 .06 .081 .15 .373 .945 2.38 6. 15. 37.3 94 .5 129. , 13 1 0 1 0 1 1 1 1 0 0 0 0 0 14 1 1 0 1 1 1 1 1 1 1 1 1 1 15 1 0 1 0 1 1 1 1 1 1 1 1 1 16 5. . 41 1. 1. 1. 17 1.8 168. . 1 2.2 1-18 2.3 59. .23 2. 1. 19 2. . 4 2. 7 1 . 7 5 1. 20 10. 1. 3. 1.6 1. 21 40. .. 16 5. . 1, 0 1. 22 .9 6.0 .49 .10 1. J .8 6.1 .46 .05 1. 23 .8 1. 2.5 2.9 1. 24 1 . 5 .05 .10 5.0 1. 25 1.5 .05^ . 10 5.0 1. , 26 .13 .83 .21 5.5 27 .13 .83 .21 5 . 5 28 5.05 .405 .695 2,0863309 30.. 29 5. .51 1. 1. 1. 30 1.5 66. .2 3.03 1. 31 1.8 36. .45 2. 6 1. 32 45. .19 1.3 3.14 1. 33 21. .39 1.6 2.6 1. 34 -.86 .0032 2.4 3.2 1. 35 1.0 5.7 .51 . 082 1. 36 .74 .47 10. 3.1 1, 37 20. .0 34 , 164 1.30 1. 38 20.= .034 .164 1.30 1. 39 .38 .40 8.4 6. 1 1. 40 10. 1 .77 1. 1. 1* 41 5.05 .405 .695 2.0863309 30.. 42 •••TEE FOLLOWING IS TEST D ATA?*1******** ******** ******* 123 43 DETBUSION BATE TEST 44 1 45 1. 5 0 0 46 .3 47 56. 83609 -2. .-2. -2, ,193.8991 -2. -2. *2. .-2. .900. . 48 1 49 1. 5 0 1 FUNCTION 1 TEST 50 .3.4.5 51 .4192 .9406 .0952 4.6451 10.468 23.466 52.653 117. 28 0 260.01 585.93 52 .7692 1.6603 3.5585 7.5930 16.456 35.487 76.591 164.1 58 350.28 759. 12 53 1.4115 2.9304 6.0436 12.412 25.869 53.664 111.410 223 .77 471. 88 983.50 54 1 55 1.5 0 2 FUNCTION 2 TEST 56 .3 .4.5 57 .6718 1.6913 4.2142 10.3931 25.7857 61. 7166 138. 1468 271. 3696 473.3610 888.8597 58 .8725 2.1965 5.4724 13.4931 33.4580 79.9641 178.3353 347. 1749 594.7152 1092.8436 59 1.0798 2.7184 6.7724 16.6972 41.3939 98.8759 220.1984 427.1668 726.4801 1323.2027 60 1 61 1.5 0 3 FUNCTION 3 TEST 62 .3 .4 .5 63 .5484 1. 3814 3.4463 8.5260 21.3207 52.0064 121. 5101 2 58.3059 489.8942 940.1985 64 .7176 1.8070 4.5049 1 1. 1244 27. 6909 66. 8095 152. 3806 310.9531 572. 8521 1 125.7101 65 .8988 2. 2627 5.6368 13.8958 34.4429 82. 2788 183.8288 364.4200 669.0424 1358.8599 65.1 1 65. 2 1. 5 0 3 FUNCTION 3 TEST 65.3 .3.4 .5 65.4 .5484 1.3814 3.4463 8.5260 21.3207 52.0064 121. 5101 2 58.3059 489.8942 940.1985 65.5 .7176 1.8070 4.5049 11.1244 27.6909 66.8095 152.3806 310.9531 65.6 .8988 2.2627 5.6368 13.8958 34.4429 82.2788 183.8288 364.4200 669.0424 1358.8599 66 1 67 1.5 0 7 FUNCTION 7 TEST 6 8 .3 .4 .5 69 1.3108 2.8766 6.2680 13.5958 29.962 65.6968 144.1724 314.1458 681.4064 1501.657 70 1. 7708 3. 7105 7. 723 16.006 33.6713 70.498 147. 7189 30 7.4596 637.2093 1340.4804 71 3. 106 6. 2143 12. 355 24.4657 49. 1303 98. 2219 196.5119 390.7004 773.6723 1553.6361 72 1 73 1.5 0 9 FUNCTION 9 TEST 74 .3.4.5 75 .2754 .6943 1.7349 4.2992 10.6659 24.4451 51,7541 118 .4603 288. 9397 729.5968 124 76 .4496 1.1334 2.8319 7.0177 17.4102 39.9022 84.4791 19 3.3647 471.6410 1190. 9327 77 .7339 1.8501 4.6225 11.455 28.4189 65. 1329 137. 8966 3 15.6324 769.8672 1943.9788 78 1 79 1.5 0 10 FUNCTION 10 TEST 80 a 3 • 4 * 5 81 .2755 .6944 1.7357 4.3116 10.8513 26.3275 58.0329 123 .9164 291. 531 730.6495 82 .4496 1.1335 2.833 7.035 17.6675 42.4243 92.0242 199. 2998 474.3779 1192.0385 83 .7339 1. 8502 4,-6241 11. 4786 28.7663 68.4243 146.8908 322.2361 772.8572 1945.1829 84 1 85 1.5 0 11 HUANG 1 FUNCTION 11 TEST 86 .3.4.5 87 .2899 .736 1.8515 4.631 11.80 29.91 75.89 190.64 474. 9 1203.5 88 .2891 .7342 1.8477 4.623 11.788 29.89 75.86 190.62 47 4.93 1203. 477 89 .2882 .7323 1.8438 4.6155 11.773 29.863 75.8256 190.5 9 474. 93 1203.5 90 1 91 1.5 0 12 HUANG2 FUNCTION 12 TEST 92 .3.4.5 93 .3508 .8894 2.2351 5.585 14.22 36.00 91.237 229.00 57 0.33 1445.15 94 .3728 .9454 2.3756 5.936 15.114 38.26 96.98 243.4 606 . 2 153 6.1 95 .3963 1. 0048 2.5251 6.3095 16.06 40. 67 103.08 258.72 644.33 1632.7 96 1 97 1.5 0 11 HUANG3 FUNCTION 11 TEST 98 .3 .4 .5 93 .3163 .7923 1.9691 4.869 12.27 30.?1 76.93 191. 41 475 .11 1203.48 100 .3171 .794 1.9728 4.877 12.2811 30.73 76.96 191.44 47 5. 12 1203. 5 101 .318 .7959 1.S767 4.885 12.2966 30.76 76.997 191.463 475. 12 1203.4 102 1 103 1.5 0 12 HUANG4 FUNCTION 12 TEST 104 .3 .4 .5 105 .3772 .9457 2.3527 5.8235 14.68 36. 79 92,.27 229.77 57 0. 50 1 445.2 106 .4009 1. 0052 2.5007 6. 19 15.607 39. 1 1 98.08 244.23 60 6.39 1536.05 107 .4261 1. 0684 2.658 6. 579 16.5885 41. 566 104.247 259.5 9 644. 53 1632.7 108 1 109 1.5 0 11 HUANG 1 FUNCTION 11 TEST 110 .3 .4 .5 111 .2458 .6421 1.6552 4.233 11.03 28.59 74.16 189.36 474 .65 1203.47 112 .2038 .5526 1.4681 3.8536 10.293 27.33 72.51 188. 14 4 74. 372 1203.5 125 113 .1311 .3975 1.1439 3.196 9.016 25.1524 69.66 186.02 4 73.89 1203.5 114 1 115 1.5 0 12 HUANG2 FUNCTION 12 TEST 116 .3 .4 .5 117 1.5210 3.8567 9.692 24.217 61.66 156. 1 1 395.62 993.01 2473. 1 6266,5 118 2.6362 6.6846 16.798 41. 97 106. 87 270. 57 685.72 1721. 1 4286.4 10861.5 119 4.5692 11.59 29. 12 72.75 185.24 469. .1 188.5 2983.2 74 29. 5 18825.6 120 1 121 1.5 0 11 HUANG3 FUNCTION 11 TEST 122 .3 .4 .5 123 . 3604 .8862 2. 1654 5.267 13.039 32.03 78.659 192. 7 47 5.4 1203.5 124 .4024 .9756 2.3525 5.647 13.78 33.29 80.31 193.92 475 .68 1203.5 125 . 4751 1. 1307 2.6767 6.304 15. 05 35.47 83.16 196.04 47 6. 2 1203.5 126 1 127 1.5 0 \2 HUANG4 FUNCTION 12 TEST 128 .3 .4 .5 129 1.6355 4=. 1008 10.20 25.25 63.67 159.54 400. 12 996.35 2473.8 6266.5 130 2.8347 7. 107 17.68 43. 77 1 10. 36 276. 5 693.5 1726.9 42 87. 7 10861.4 131 4.913 12.32 30,65 75.86 191,27 479.3 1202. 2993,2 743 1. 75 18825.6 ,. 132 C THIS PBOGEAM FITS VABIOUS CONSTITUTIVE FUNCTIONS T 0 133 C HEMORHEOLOGICAL DATA TAKEN WITH A CGNTBAVES LS-2 B HEOMETEB. 134 C THE FITTING IS ACCOMPLISHED BY THE OPTIMIZATION BO UTINE 135 C POWEL. 136 C INTEBNAL DOCUMENTATION IS PROVIDED BY COMMENT STAT EMENTS 137 C**VARIABLE DEFINTIONS******************************* + $ $ $ # If. $ if. $ 4 $ if. $ 138 C BND11= AXIS OF BNY11 FOR SPECIFIED VALUE OF OMC 139 C BND12= AXIS OF BNY12 FOR SPECIFIED VALUE OF OMC 140 C BND21= LOWER BOUND ON HEMATOCRIT INDEX 141 C BND22= UPPER BOUND ON HEMATOCRIT INDEX 142 C BNY11(OMC) = LOWER BOUND ON DETRUSION BATE INDEX 143 C BNY12 (OMC)= LOWER BOUND ON DETRUSION RATE INDEX 144 C BVISC= VISCOSITY OF WHOLE BLOOD 145 C DISC= DISCBEPANCY BETWEEN PBEDICTED AND OBSEVED S IGNAIS 146 C FOR STBESSES 147 C EPS= EBROB ALLOWED IN X (I) 148 C EX: FLAG INDICATING WHETHEB EXCHANGE HAS OCCUBED D URING LAST 149 C INSPECTION OF ORDERING OF HEMATOCRITS 150 C F= OPTIMUM VALUE OF SUM OF SQUARES 151 C FNFLAG: INDICATES WHICH FUNCTIONS ABE TO BE OPTIMI 126 . . ZED. 152 C GAMA= DETRUSION RATE (S**-1) 153 C ID=IDENTITY OF PERSON FROM WHICH SAMPLE WAS TAKEN; USUALLY 154 C SOCIAL INSURANCE NUMBER 155 C IERR: ERROR PARAMETER FOR POWEL 156 C NF= NUMBEfi OF FUNCTION EVALUATIONS SINCE ENTERING POWEL 157 C NDATA(I)= NUMBER OF NON-BLANK DATA POINTS FOR HEMA TOCRIT I 158 C NFM= MAXIMUM ALLOWED VALUE•OF NF 159 C NFW=NUMBER OF PARAMETERS IN EQUATION NO. FN 160 C NOH=NUMEER OF HEMATOCRITS 161 C NOM AXIS OF NOMM FOR SPECIFIED VALUE OF 0 MC f i 162 C NOMM(OMC) = NUMBER OF DETRUSION RATES USED IN OPT IMIZATION 162.5 C NR -= NUMBER 0F R VALUES 163 C OM= 50TAI0NAL SPEEDS Ifl•! (RPM) FOR A PARTICULAR • FIT 164 ; C OMA= SET OF ALL ROTATION SPEEDS AVAILABLE TO PRO GRAM 165 C OMC= INDEX FOR OMFLAG INDICATING WHICH SET OF RO ATATIONAL ' 166 C > SPEEDS ARE TO BE USED. 167 C OMFLX (I,OMC) = FLAG INDICATING WHETHER ITH SPEED IS TO BE USED 168 i C IN OPTIMIZATION; OMC INDEXES VARIOUS CASES. 169 C OMFLAG(I) =COPY OF AXIS OF OMFLX FOR CERTAIN VAL UE OF OMC 170 C OMX(I,OMC) —ROTATIONAL SPEEDS IN RP*vl FOR CASE OMC 171 C PASS: FLAG FOR INDICATING WHETHER PARAMETERS OF FUNCTION 1 i 172 C HAVE BEEN INTIALIZED (=0 BEFORE'INITIALIZI NG) r 173 C PVISC= PLASMA VISCOSITY 174 C R=INITIAILY OUTPUT VALUE FROM LS-2 (0..GT.R.LT.100 . ) ; 175 C SUBSEQUENTLY IS' MULTIPLIED BY FACTOR TO GIVE 03S ERVED •! 176 C SHEAR STEESS. ! ; 177 C SI= SUM OF R VALUES 178 C RI2= SUM OF SQUARES OF R VALUES 179 C SAM=SAMPLE NUMBER ;SEQUENTIAL FROM 1 FOR EACH PERS ON 180 C SC= (INSTUMENT OUTPUT ON SOME SCALE) / (OUTPUT WHICH WOULD APPEAE 181 C ON SCALE 1) 182 C SD= STANDARD DEVIATION 183 C SGAMA= SQET (GAMA) 184 C SUBSET: IDENTIFIES SUBSET OF DATA CURRENTLY IN USE 185 C TAU= PREDICTED STRESS 186 C TRAP1= VARIABLE USED BY PROGRAM INTERUPT TRAP 187 C TEAP2= • • •• 188 C WORK: WORK SPACE USED BY POWEL 189 C X= PARAMETER VALUE 127 190 C XPI= INITIAL VALUES FOB PABAWETEBS WITH PEAK VALUE INITIAL DATA 191 C XSSI= INITIAL VALUES FOB PARAMETERS WITH STEADY ST ATE DATA 192 C XS= LN(E) 193 C Y = LN(OM) 194 C** DNIT OSAGE*** ************************************* ************* 195 C UNIT 1 EXPEEIMENTAL DATA 196 C UNIT 2 PBINTED OUTPUT 197 C UNIT 3 OUTPUT FOB PUNCHED CARDS 198 C UNIT 4 CONDITION SELECTION FLAGS AND INITIAL VALUE S OF PABAMETEES 199 C**STRUCTURE OF UNIT 1 ****************************** ************* 200 C OMC 201 c PVISC,ID,SAM,CUBVE 202 c H(1)..,.(6) 203 c E(1,1) (13,1) 204 c 205 c R (1,N).,... (13,N) 206 c 207 c B (1,NOH) .... (13,NOH) 208 c REPEAT PBEVIOOS 5 LINES FOR EACH SAMPLE 209 C**STBUCTUEE OF UNIT 2 ****************************** ************* 210 C EVIDENT FROM OUTPUT 211 C**STBUCTUEE OF UNIT 3 ****************************** ************* 212 C ID, SAM,H (1) „ ,,H (NOH) 213 c ID,SAM,FN, X (1) ,. . . X (NFN) 214 c ID,SAM,FN,SD,RSQR 215 C**STBUCTUEE OF UNIT 4 ****************************** ************* 216 C DETEST,OPTEST 216.5 c FNFLAG (1) . . . . (13) 217 c NX (1) .... (13) 219 c NFM 219.5 c OMA(1,1) .... (13,1) 220 c OMFLX (1,1).... (13,1) 222 c • • • • 223 c OMFLX (1,3) .... (13,3) 225 c XSSI(1, 1).... (5,1) 226 c 227 c XSSI(1,FN) ... , (5 #FN) 228 c 229 c XSSI (1, 13) ... . (5,13) 230 c XPI (1,1),.,. (1,5) 231 c 232 c XPI (1,FN)..,. (5,FN) 233 c 234 c XPI (1,13).... (5,13) 235 c ************************************************* ************** 236 SUBROUTINE MAIN 237 IMPLICIT REAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 128 238 LOGICAL EQUC 239 LOGICAL*1 CURVE, 240. INTEGER*2 DATFLG,OMC,NR 241 IN TEGER*4 BNY11,BND11,BNY12,BND12 , 242 1BND21,BND22,SUBSET,PASS,EX,SC, 243 1IERR,NFN,NOH,ID,SAM,TRAP1,TRAP2,FNFLAG,FN, 244 20MFLX,OMFLAG,DETEST,OPTEST 245 REAL*4 H 246 DIMENSION WORK (40) ,EPS (5) ,TRAP2(26) 24 7 COMMON/A/OMX(13,3) ,OM(13) ,GAMA(13,6) ,SGAMA(13,6 ) ,PVISC,H(6) , 24 8 1B(13,6) , DISC (13,6) , B V I S C (13, 6) ,X S S I (5, 13) , X P I (5 ,13), 249 2X(5),F,SD,RSQR,RI2,RI,TAU, 250 3NOH,NX(13) , FN FLAG (13) , FN, BNY 11 (3) ,BND11,BNY12(3 ),BND12, 251 3BND21,BND22, 252 4SUBSET,ID,SAM,NDATA (6) , NF,NFM,N1, OMFLX (13,3) ,OM FLAG (13) , 253 4NOMM (3) ,NOM, 254 5DETEST,OPTEST,DATFLG(13,6),OMC,NR 255 6/A1/NFN 256 CALL BEGIN 257 10 CALL DATAA 258 EXTERNAL FUNC,EFLEBR 259 C CUINFO ATTENDS TO 'SOFTWARE* INTERRUPTS. 260 CALL CUINFO('EFLUEM *,EFLERR) 261 FN=1 262 20 CONTINUE 263 IF (FNFLAG (FN) . EQ. 0) GOTO 150 264 C TRAP ATTENDS TO 'HARDWARE INTERRUPTS' 265 CALL TBAP (TBAP1,TBAP2,& 130) 266 C CALCULATE SQUABE BOOT OF DETRUSION RATE FOR USE IN FCN 8 267 IF (FN.NE. 8) GOTO 40 268 DO 30 M=1,NOH 269 DO 30 NN=1,NOM 27C SGAMA (NN,M)=SQBT (GAMA (NN, M) ) 271 30 CONTINUE 27 2 40 CONTINUE 273 C INITIALIZE BVISC S DISC 274 NFN=NX (FN) 275 60 IF (EQUC (CURVE, • P')) GOTO 80 276 DO 70 1=1,NFN 277 70 X (I) -XSSI (I,FN) 278 GOTO. 100 27S 80 DO 90 1=1,NFN 280 90 X (I) =XPI ( I ,FN) 281 100 IF (FN.EQ. 13)CALL DON (&120 ,8130) 282 BO 110 1=1,NFN 283 1 10 EPS (I)=.001*X (I) 284 NF=-1 285 CALL POWEL (F,X,NFN, 1000.,EPS,1,40, 286 10,WORK,IERR,FUNC,&130) 287 CALL RESULT 288 120 CALL PRINT 129 289 GOTO 140 290 ENTRY EFLRTN 291 130 CALL INTEPI 29 2 ENTRY ENT1 293 140 CALL PUNCH 294 150 CONTINUE 295 FN=FN+1 296 IF (FN.LT. 14) GOTO 20 297 GOTO 10 298 END 299 FUNCTION FUNC(PAB,NFN) 300 C THIS ROUTINE CALCULATES FUNCTION VALUES FOR POWEL. 301 IMPLICIT REAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 302 INTEGEB*2 DATFLG,OMC,NB 303 INTEGEB*4 BNY11,BND11,BNY12,BND12, 304 1BND21,BND22,SUBSET, 305 1NFN,NOH,FNFLAG,FN, 30 6 20MFLX,OMFLAG, DETEST, OPT EST 307 REAL*4 H 308 DIMENSION PAR (5) 309 COMMON/A/OMX(13,3) ,OM (13) ,GAMA (13,6) ,SGAMA (13,6 ) ,PVISC,H-(6) , 310 1R (13,6) ,DISC (13,6) ,BVISC (13,6) ,XSSI(5, 13) ,XPI (5 ,13), 311 2X(5),F,SD,5SQE,EI2,BI,TAU, 312 3NOH,NX(13) ,FNFLAG (13) ,FN,BNY11 (3) ,BND11, BNY 12 (3 ),BND12, 313 3BND21,BND22, 314 4 SUBSET,ID, SAM, NDAT A (6) ,NF,NFM,N1 , OMFLX (13,3) ,OM FLAG (13) , 315 4NOMM (3) ,NOM, 316 5DETEST,OPTEST,DATFLG (13,6),OMC,NB 317 NF=NF+1 318 IF(NF.EQ.NFM)CALL INTEPT 319 IF (FN.EQ. 13) GOTO 10 320 BND11=1 321 BND12=N0M 322 BND21=1 323 BND22=NOH 324 10 FUNC=0 325 GOTO(20,50,100,150,200,250,300,350,380,410,440, 490, 540) ,FN 326 C CONSTITUTIVE FCN 1 327 20 DO 40 M=BND21,BND22 328 EPAB 1 H=PVISC*EXP (PAB (1) *H (M) ) 329 PAB2H=1.-PAE(2) *H(M) 330 DO 40 NN=BND11,BND12 331 IF(EATFLG(NN,M) i EQ.0)GOTO 40 332 30 TAU=EPAR1H*GAMA(NN,M)**PAB2H 333 FUNC=FUNC+ (B(NN,M)^TAU)**2 334 40 CONTINUE 335 RETUBN 336 C CONSTITUTIVE FCN 2 • 337 50 DO 90 M=BND21,BND22 338 EPAE1 H=EXP (PAB (1) *H (M) ) 339 PAR2H=PAR (2) *H (M) 130 340 DO 90 NN=BND11,BND12 341 •• IF (DATFLG (NN, M)-EQ. 0) GOTO 90 342 60 AEG=-PAB (3) *GAMA (NN,M) 343 IF(ABG.LT.-180.218)GOTO 70 344 EXPO=EXP (AEG) 345 GOTO 80 346 70 EXPO=0. 34 7 80 TAU=PAE (4) * (GAMA (NN, M) *EPAE1H+PAE2H* (1.-EXPO) ) 348 FUNC=FUNC+ (R(NN,M) -TAU) **2 349 90 CONTINUE 350 RETURN* 351 C CONSTITUTIVE FCN 3 352 100 DO 140 M=BND21,BND22 353 EPAE1H=EXP (PAE (1) *H (M) ) 354 PAB3H=PAE (3) *H (M) 355 DO 140 NN=BND11,BND12 356 IF (DATFLG (NN, M) .EQ.Q) GOTO 140 357 110 ABG=-PAE3H*GAMA (NN , M) 358 IF (ABG.LT.-180.218) GOTO 120 359 EXPO=EXP (AEG) 360 GOTO 130 361 120 EXPO=0. 362 130 TAU=P AR (4) * (GAMA (NN,M) *EPAE1H+ PAE (2) * (1. - EXPO) ) 363 FUNC=FUNC+(B(NN,M)-TAU) **2 364 140 CONTINUE 365 RETURN 366 c CONSTITUTIVE FCN 4 367 150 DO 190 M=BND21,BND22 368 PAR 1 H=PAJ3 (1) *H (M) 36S EPAE3H=EXP (PAE (3) *H (M) ) 370 DO 190 NN=BND11,BND12 371 IF (DATFLG (NN, M) .EQ. 0) GOTO 190 372 160 ARG=-PAR (2) *GAMA (NN, M) 373 IF (ARG.LT.-180.218)GOTO 170 374 EXPO=EXP (AHG) 375 GOTO 180 376 170 EXPO=0.. 377 180 TAU=P AR (4) * (GAMA (NN, M) +PAR1H*(1. -EXPO) ) *EPAR3H 378 FUNC=FUNC+ (R (NN,M) -TAU) **2 379 190 CONTINUE 380 RETURN 381 c CONSTITUTIVE FCN 5 382 200 DO 240 M=BND21,BND22 383 PAR2H=PAR (2) *H (M) 384 EPAR3H=EXP (PAR (3) *H (M) ) 385 DO 240 NN=BND11,BND12 386 IF (DATFLG (NN, M) . EQ. 0) GOTO 240 387 210 ARG=-PAE2H*GAM A (NN , M) 388 IF(ABG.1T.-180.218)GOTO 220 389 EXPC=EXP (AEG) 390 GOTO 230 391 220 EXPO=0. 392 230 TAU=PAR (4) * (GAMA (NN,M) +PAR(1) * (1. -EXPO) ) *EPAR3H 393 FUNC=FUNC+(B(NN,M)-TAU)**2 394 240 CONTINUE > 395 BETUEN 131 396 C CONSTITUTIVE FCN 6 397 250 DO 290 M=BND21,BND22 398 PAB2H=PAB (2) *H (M) 399 EPAR3H=EXP (PAR (3) *H (M) ) 400 DO 290 NN=BND11,BND12 401 IF (DATFLG(NN,M).EQ.O)GOTO 290 402 260 ARG=-PAR2H*GAMA(NN,M) 403 IF(AEG.IT.-180.218)GOTO 270 404 EXPO=EXP (AEG) 405 GOTO 280 406 270 EXPO=0. 407 280 TAU=PAR (4) *GAMA (NN,ti) * (WPAR(1) * ( 1.-EXPO) ) *EPAE 3H 408 FUNC=FUNC+ (R(NN,M) -TAU)**2 409 290 CONTINUE 410 RETURN 411 C CONSTITUTIVE FCN 7 (WALBURN 6 SCHNECK) 412 300 DO 340 M=BND21,BND22 413 ARG=PAB (2) *H(M) +PAB(4)/H(M) **2 414 IF(AEG.LT.r180.218)GOTO 310 415 EXPO=EXP (AEG) 416 GOTO 320 417 310 EXPO = 0. . 418 320 PAE3H=PAB (3) *H (M) 419 DO 340 NN=BND1 1,BND<12 420 IF (DATFLG(NN,M) .EQ.O)GOTO 340 421 330 TAU=PAR (1) *EXPO*GAMA (NN,M) ** (1.-PAE3H) 422 FUNC=FUNC+ (R (NN,M) -TAU) **2 423 340 CONTINUE 424 EETUEN 425 C CONSTITUTIVE FCN 8 (QUEMADA) 426 350 DO 370 M=BND21,BND22 427 PAR2H-PAR (2) *H (M) 428 PAR23H=PAR (2) *PAR (3) *H (M) 429 DO 370 NN=END11,BND12 430 IF (DATFLG (NN, H) .EQ.O) GOTO 370 431 360 TAU=PAR (4)*GAMA (NN, M) * ( (1 . + 1./(PAR (1) *SGAMA (NN, 432 1/( (1.-PA.R2H) + (1.-PAR23H) / (PAR ( 1) *SGAMA (NN , M) ) )) **2 433 FUNC=FUNC+ (R (NN, M) -TAU) **2 434 370 CONTINUE 435 EETUEN 436 C CONSTITUTIVE FCN 9 (PHILLIPS & DEUTSCH) 437 380 DO 400 M=BND21,BND22 438 EPAE4H=EXP (PAE (4) *H (M) ) 439 DO 400 NN=BND11,BND12 440 IF (DATFLG (NN, M) .EQ.O) GOTO 400 441 390 TAU=PAE(1) *EP AB4H* ( (1.+PAB (2) *GAM A (NN, M) 442 1**2) / (1. fPAE (3) *GAM A (NN , M) **2) )*GAMA (NN, M) 443 FUNC-FUNC+(R (NN,M)-TAU) **2 444 400 CONTINUE 445 RETURN 446 C CONSTITUTIVE FCN 10 (PHILLIPS & DEUTSCH) 447 410 DO 430 J3=BND21, BND22 448 EPAE4H=EXP(PAE(4)*H(M) ) 1 3 2 4 4 9 PAR2H=PAE (2) *H (M) 4 5 0 P A E 3 H = P A E (3) *H (M) 4 5 1 DO 4 3 0 NN=BND11 ,BND 12 4 5 2 I F (D A T F L G ( N N , M ) .EQ.0)GOTO 430 4 5 3 4 2 0 TAU=PAB (1) * E P A B 4 H * ( (1.+PAB2H*GAMA (NN,M) 4 5 4 1**2) /(1.+PAR3H*GAMA (NN , M ) * * 2 ) ) *GAMA(NN,M) 4 5 5 FUNC=FUNC+ (E (NN,M) -TAU) **2 4 5 6 4 3 0 CONTINUE 4 5 7 E E T U E N 4 5 8 C C O N S T I T U T I V E FCN 11 (HUANG3) 4 5 9 4 4 0 DO 4 8 0 M=BND21,BND22 4 6 0 P A E 4 H = P A E (4) *H (M) 4 6 1 DO 4 8 0 NN=BND11 ,BND12 4 6 2 I F ( D A T F L G ( N N , M ) . E Q . O ) G O T O 480 4 6 3 4 5 0 AEG=PAE4H-PAE (3) *GAMA (NN, M) * * P A E (2) 464 I F ( A B G . L T . r 1 8 0 . 2 1 8 ) GOTO 4 6 0 4 6 5 EXPO=EXP (ABG) 4 6 6 GOTO 4 7 0 4 6 7 4 6 0 EXPO=0 4 6 8 4 7 0 TAU=6. 7*GAMA ( N N , M) +PAE (1) *GAMA (NN , M) **PAR (2) 4 6 9 1*EXPO 4 7 0 FUNC=FUNC+ (E ( N N , M) -TAU) **2 4 7 1 4 8 0 CONTINUE 4 7 2 WRITE ( 2 , 4 8 5 ) FUNC 4 7 3 4 8 5 FOE MAT (* • ,<311.4) 4 7 4 STOP 4 7 5 E E T U E N 4 7 6 C C O N S T I T U T I V E F C N 12 (HUANG4) 4 7 7 4 9 0 DO 5 3 0 M=BND21,BND22 4 7 8 E P A R 4 H = E X P (PAE (4) *H (M) ) 4 7 9 DO 5 3 0 NN=BND11,BND12 4 8 0 I F ( D A T F L G (NN, M) .EQ. 0) GOTO 5 3 0 4 8 1 ARG=-PAR (3) *GAMA (NN, M) **PAR (2) 4 8 2 I F ( A R G . L T - - 1 8 0 . 2 1 8 ) GOTO 5 0 0 4 8 3 EXPO=EXP (AEG) 4 8 4 GOTO 510 4 8 5 5 0 0 EXPO=0 4 8 6 5 1 0 T A U = E P A E 4 H * ( 6 . 7*GAMA (NN,M) +PAB (1) *GAMA (NN,M) * * P A B ( 2 ) 4 8 7 1*EXPO) 4 8 8 F U N C = F U N C * ( B ( N N , M ) - T A U ) * * 2 4 8 S 5 3 0 CONTINUE 4 9 0 WRITE ( 2 , 5 3 5 ) FUNC 4 9 1 5 3 5 FORMAT (' «,G13.4) 4 9 2 STOP 4 9 3 RETURN 4 9 4 C C O N S T I T U T I V E F C N 13 4 9 5 5 4 0 DO 6 0 0 M=BND21,BND22 4 9 6 DO 6 0 0 NN=BND11,BND12 4 9 7 I F ( D A T F L G ( N N , M ) . E Q . 0 ) G O T O 6 0 0 4 9 8 5 5 0 V=ALOG (GAMA (NN, M) ) * 5 . 4 9 9 GOTO ( 5 6 0 , 5 6 0 , 5 7 0 , 5 8 0 ) / S U B S E T 5 0 0 5 6 0 ARG=fl (M) * (PAR ( 1 ) - P A E (2) *V) 5 0 1 GOTO 590 5 0 2 5 7 0 A R G = H ( M ) * (X ( 1 ) - X ( 2) *V-X (3) * ( V - X ( 4 ) ) / ( 1 . - ( V / X (4) ) * * P A R ( 1 ) ) ) 133 503 GOTO 590 504 580 ARG=H(M)*(X(1)-X(2)*V-X(3)*(V-X(4))/(1.-(V/X(4) )**X(5))) 505 590 TAU=PVISC*GAMA (NN, M) *EXP (AEG) 506 FUNC=FUNC+ (E (NN,M) -TAU) **2 507 600 CONTINUE 508 EETUEN 509 END 510 SUBROUTINE EESULT 511 C THIS ROUTINE CALCULATES THE AEBAY OF DISCREPANCIES AND OTHER 512 C VALUES FOR THE OPTIMUM SOLUTION. 513 C IF PREDICTED BULK VISCOSITIES OR SOME OTHER PARAME TEES BASED 514 C ON THE EESULTS OF THE OPTIMIZATION ABE REQUIEED TH EY SHOULD 515 C BE PROGEAMMED HEBE. 516 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 517 INTEGEB*2 DATFLG, OMC , NE 518 INTEGER*4 BNY11,BND11,BNY12,BND12, 519 1BND21,BND22,SUBSET, 520 1NFN,N0H,FNFLAG,FN, 521 20MFLX,OMFLAG,DETEST,OPTEST 522 COMMON/A/OMX(13,3),OM(13),GAMA(13,6),SGAMA(13,§ ) ,PVISC,H(6) , 52 3 1E(13,6) ,DISC (13,6) ,BVISC(13,6) ,XSSI(5,13) ,XPI (5 ,13), 524 2X (5) ,F,SD,ESQE,EI2,BI,TAU, 525 3NOH,NX(13) ,FNFLAG (13) ,FN,BNY11 (3) ,BND11,BNY 12(3 ),BND12, 526 3BND21,BND22, 527 4 SUB SET, ID , SAM , NDATA (6) , NF , NFM, N1 , OMFLX (13, 3) ,OM FLAG (13) , 528 4NOMM (3) , NOM, 529 5DETEST,OPTEST,DATFLG(13,6),OMC,NE 530 6/A1/NIN 531 SD=SQET (F/N1) 532 ESQB=1.-F/ (BI2-BI**2/N1) 533 IF (FN.EQ.13)GOTO 10 534 BND11=1 535 BND12=NOM 536 BND21=1 537 END22=NOH 538 10 GOTO (20,50,100, 150,200,250,300,350,380,410,440, 490,540) ,FN 53 9 C CONSTITUTIVE FCN 1 540 20 DO 40 M=BND21,BND22 541 EPAB1 H=PVISC*EXP (X (1) *H (M) ) 542 PAR2H=1.-X (2) *H (M) 543 DO 40 NN=BND11,BND12 544 IF(DATFLG(NN,M).EQ.O) GOTO 40 545 30 TAU=EPAB1H*GAMA (NN,M)**PAB2H 546 GOTO 560 547 40 CONTINUE 548 EETUEN 549 C CONSTITUTIVE FCN 2 134 550 50 DO 90 M=BND21,BND22 551 EPAE1H=EXP (X ( 1) *H (M) ) 552 PAB2fi=X (2) *H(M) 553 DO 90 NN=BND11,BND12 554 IF (DATFLG (NN, M) .EQ. 0) GOTO 90 555 60 ABG=-X (3) *GAMA (NN, M) 556 IF (ARG.LT.-180. 218) GOTO 70 557 EXPO=EXP (ABG) 558 GOTO 80 559 70 EXPO=0. 56 0 80 TAU=X (4) * (GAMA (NN, M) *EPAB 1H+PAE2H* (1.-EXPO) ) 561 GOTO 560 562 90 CONTINUE 563 EETUEN 564 C CONSTITUTIVE FCN 3 565 100 DO 140 M=BND21,BND22 566 EPAB1H=EXP (X (1) *H (M) ) 567 PAB3H=X (3) *H (M) 568 DO 140 NN=BND11,BND12 569 IF (DATFLG(NN,M).EQ.O) GOTO 140 57 0 110 AEG=-PAB3H*GAMA(NN,M) 1 571 IF (AEG. IT.-180. 218) GOTO 120 572 EXPO=EXP (ABG) 573 GOTO 130 574 120 EXPO=0. 575 130 TAU=X (4) * (GAMA (NN, M) *EPAE1H+X (2) * (1.-EXPO) ) 576 GOTO 560 577 140 CONTINUE 578 EETUEN 579 C CONSTITUTIVE FCN 4 580 150 DO 190 M=BND21,BND22 581 PAE1H=X (1) *H (M) 582 EPAE3H=EXP (X (3) *H (M) ) 583 DO 190 NN=BND11,BND12 584 IF (DATFLG (NN, M) .EQ.0)GOTO 190 585 160 AEG=-X (2) *GAMA (NN, M) 586 IF (ABG.LT--180.218)GOTO 170 587 EXPO=EXP(AEG) 588 GOTO 180 589 170 EXPO=0. 590 180 TAU=X (4) * (GAMA (NN, M) +PAE1H* (1.-EXPO) ) *EPAB3H 591 GOTO 560 592 190 CONTINUE 593 EETUEN 5.94 c CONSTITUTIVE FCN 5 595 200 DO 240 M=BND21,BND22 596 PAB2H=X (2) *H(M) 597 EPAB3H=EXP (X (3) *H (M) ) 598 DO 240 NN=BND11,BND12 599 IF (DATFLG (NN,M) .EQ.O) GOTO 240 600 210 ABG=-PAE2H*GAMA(NN,M) 601 IF (AEG.LT.-180.218)GOTO 220 602 EXPO=EXP (ABG) 603 GOTO 230 604 220 EXPO = 0. 605 230 TAU=X (4) * (GAMA (NN, M) +X (1) * (1..-EXPO) ) *EPAE3H 135 606 GOTO 560 607 240 CONTINUE 608 RETURN 603 C CONSTITUTIVE FCN 6 610 250 DO 290 M=BND21,BND22 611 PAR2H=X (2) *H(J3) 612 EPAR3H=?EXP (X(3) *H(M) ) 613 DO 290 NN=BND11,BND12 614 IF (DATFLG(NN,M).EQ.0)GOTO 290 615 260 ARG=-PAB2H*GAMA (NN , M) 616 IF (ARG. LT. --180. 216) GOTO 270 617 EXPO=EXP (ARG) 618 GOTO 280 619 270 EXPO=0. 620 280 TAU=X (4) *GAMA (NN,h) * (1 + X (1) *(1.-EXPO) ) *EPAR3H 621 GOTO 560 622 290 CONTINUE 623 RETURN 624 C CONSTITUTIVE FCN 7 (WALBURN S SCHNECK) 625 300 DO 340 M=BND21,BND22 626 ARG=X (2) *H (M) +X (4) /H (W) **2 627 IF (ABG.LT.-180. 218)GOTO 310 628 EXPO=EXP (ARG) 629 GOTO 320 630 310 EXPO=0. . : 631 320 PAR3H=X(3)*H(M) 63 2 DO 340 NN=BND11,BND12 633 IF(DATFLG(NN,W).EQ.O)GOTO 340 634 330 TAU=X (1) *EXPO*GAMA (NN, M) ** (1,-PAR3H) 635 GOTO 560 636 340 CONTINUE 637 RETURN 638 C CONSTITUTIVE FCN 8 (QUEMADA) 639 350 DO 370 M=BND21,BND22 640 PAR2H=X (2) *H(M) 641 PAR23H=X(2)*X (3) *H (M) 642 DO 370 NN=BND11,BND12 643 IF (DATFLG (NN,M).EQ.O) GOTO 370 644 360 TAU=X(4) *GAWA (NN, M) * ( (1. + 1./,(X (1) *SGAMA (NN, M) ; 645 1/( (1.-PAB2H) + (1.-PAR23H)/(X(1) *SGAMA (NN, M) ) ) ) 646 z GOTO 560 647 370 CONTINUE 648 BETURN 649 CONSTITUTIVE FCN 9 (PHILLIPS S DEUTSCH) 650 380 DO 4C0 M=BND21,BND22 651 EPAE4H=EXP (X (4) *H (M) ) 652 DO 400 NN=BND11,BND12 653 IF (DATFLG (NN,U) .EQ.O)GOTO 400 654 390 TAU=X (1) *EPAB4H* ( (1.+X (2) *GAHA (NN, M) 655 1**2)/ (1- * X (3) *GAMA (NN, M) **2) ) *GAMA (NN, M) 656 GOTO 560 657 400 CONTINUE 658 BETURN 659 c CONSTITUTIVE FCN 10" (PHILLIPS & DEUTSCH) 660 410 DO 430 M=BND21,BND22 136 661 EPAR4H=EXP (X (4) *H (M) ) 662 PAR2H=X (2) *H (M) 663 PAR3H=X (3) *H (M) ' 664 DO 430 NN=BN-®1 1 r BND 12 665 IF (DATFLG (NN^M) .EQ.O) GOTO 430 666 420 TAU=X (1) *EPAE4H*( (1.+PAR2H*GAMA (NN,M) 667 1**2) / (1. +PAR3H*GAMA (NN, M) **2) )*GAMA(NN,M) 668 GOTO 560 669 430 CONTINUE 67C RETURN 671 C CONSTITUTIVE FCN 11 (HUANG 1) 672 440 DO 480 M=BND21,BND22 673 PAE4H=PAE (4) *H (M) 674 DO 480 NN=BND11,BND12 675 IF (DATFLG(NN,M).EQ.O) GOTO 480 676 450 ABG=PAB4H-PAB (3) *GAMA (NN, M) **PAE (2) 677 IF (AEG.LT.-180.218)GOTO 460 ' 678s, EXPO=EXP (AEG) 679 GOTO 470 680 460 EXPO=0 681' 47 0 TAU=6.7*GAMA (NN,M) -PAR (1) *GAMA (NN , M) **PAR (2) 682 1+EXPO 683 FUNC=FUNC+ (R (NN, M) -TAU) **2 684 480 CONTINUE 685 RETURN 686 c CONSTITUTIVE FCN 12 (HUANG3) 687 490 DO 530 M=BND21,BND22 688 EPAR4H=EXP (PAR (4) *H (M) ) 689 DO 530 NN=BND11,BND12 690 IF (DATFLG (NN, M) .EQ.O) GOTO 530 691 ABG=-PAE (3) *GAMA (NN, M) **PAR (2) 692 IF (ABG.LT.r18G.218)GOTO 500 693 EXPO=EXP(ARG) 694 GOTO 510 695 500 EXPO=0 696 510 TAU=EPAR4H* (6. 7 *GAMA (NN , M) -PAR (1) *GAMA (NN,M)**P AR (2) 697 1*EXPO) 698 FUNC=FUNC+ (R (NN, M) -TAU) **2 699 530 CONTINUE 700 RETURN 701 C CONSTITUTIVE FCN 13 702 540 DO 550 M=BND21,BND22 703 DO 550 NN=BND11,BND12 704 IF (DATFLG (NN,M) .EQ.O) GOTO 550 705 V=ALOG(GAMA(NN,M) ) +5. 706 ARG=H (M) * (X (1) -X (2) *V-X (3) * (V-X (4) ) / ( 1 . - (V/X(4) )**X(5))) 707 TAU=PVISC*EXP(ARG) *GAMA (NN,M) 708 GOTO 560 709 550 CONTINUE 710 RETURN 711 560 CONTINUE 712 570 BVISC (NN,M)=TAU/GAMA (NN, M) 713 DISC(NN,M) =1.-TAU/R (NN,M) 714 GOTO (40,90, 140, 190,240,290,340,37 0,400,4 30,4 80, 137 530,550) ,FN 715 END 716 SUBROUTINE DON(*,*) 717 C THIS ROUTINE FITS DON'S FUNCTION.. IT IS DEALT WIT H DIFFESENTLY 718 C THAN THE OTHER RUNCTIONS BECAUSE' VARIOUS SUBSETS 0 F THE DATA 719 C ARE USED. 720 IMPLICIT REAL*4(A-H,0-Z) ,INTEGER*4 (I-N) 721 INTEGER*2 DATFLG,0MC,NR 722 INTEGER*^ BNY11,BND11,BNY12,BND12, 723 1BND2 1,BND22,SUBSET, 724 1IERR,NFN,NOH,FNFLAG,FN, 725 20MFLX,OMFLAG,DETEST,OPTEST 726 REAL*4 H 727 DIMENSION WORK (40) , EPS (2), PAS (2) 72 8 COMMON/A/OMX(13,3) ,OM(13) , GAMA (13,6) ,SGAMA(13,6 ) ,PVISC,H(6) , 729 1R(13,6) ,DISC(13,6) ,BVISC(13,6) ,XSSI(5,13) ,XPI (5 ,13) , 730 2X(5) ,F,SD,RSQR,RI2,RI,TAU, 731 3NOH,NX(13) ,FN FLAG (13) ,FN,BNY11 (3) ,BND11,BNY12 (3 ),BND12, 732 3BND21,BND22, 733 4SUBSET,ID,SAM,NDATA(6) , NF, NFM, Nil , OMFLX (1 3 , 3) ,OM FLAG (13) , 734 4NOMM (3) ,NOM, 735 5DETEST,OPTEST,DATFLG(13,6),OMC,NR 736 6/A1/NFN 737 .. EXTERNAL FUNC 738 SDBSET=1 739 BND11=9 740 BND12=10 741 BND21=1 742 BND22=NOH 743 PAR(1) = X(1) 744 PAR(2) = X(2) 745 DO 10 1=1,2 746 10 EPS (I) =.001*PAR (I) 747 CALL EOWEL (F,PAR,2,1000.,EPS,1,50,0,WORK,IERR,F UNC,S6'0) 748 X(1)=PAfi(1) 749 X(2)=PAR(2) 750 SUBSEI-2 751 BND11=1 752 BND12=6 753 PAR (1)=X (1) +X (3) *X (4) 754 PAR (2)=X (2)+X (3) 755 DO 20 1=1,2 756 20 EPS (I)=.001*PAR (I) 757 CALL POWEL (F,PAR,2, 1000.,EPS,1,50,0,WORK,IERR,F UNC,S60) 758 X (3)=PAR (2)-X (2) 759 X (4) = (PAR (1)-X (1) )/X (3) 760 SUBSET-3 761 BND11=1 138 762 BND12=10 763 PAB(1)=X(5) 764 GOTO 40 765 30 EPS (1)=.001*PAB (1) 766 CALL EOWEL(F,PAB,1,1000.,EES,1,50,0,WOBK,IEEE,F ONC,660) 767 40 X(5)=PAE(1) 768 SUBSEI-4 769 F=FUNC(X,NFN) 770 CALL EESULT 771 50 BETUBN1 772 60 BETUEN2 773 END 774 SUBEOUTINE EFLEBB(MSG) 775 INTEGEB*2 MSG (2) 776 CALL SPEINT (MSG (2) ,MSG (1) ,0,LNE) 777 CALL EFLETN 778 EETUEN 779 END 780 SUBEOUTINE PUNCH 781 C THIS BOUTINE EBODUCES OUTPUT WHICH WILL BE PUNCHED ON CABDS 782 C FOE FUTUBE MACHINE PBOCESSING. 783 . IMPLICIT EEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 784 INTEGEE*2 DATFLG,OMC,NE 785 INTEGEE*4 BNY1 1,BND 11,BNY 12,BND12, 786 1BND21,BND22,SUBSET, 787 11EBB , NFN,NOH,ID,SAM,FNFLAG,FN, 788 20MFLX ,OMFLAG,DETEST,OPTEST 789 EEAL*4 H 790 COMMO N/A/OMX (1 3 , 3) ,OM(13) ,GAMA (1 3, 6) , SGAM A (13 , 6 ) ,PVISC,H(6) , 791 1B (13,6) , DISC (13,6) ,BVISC (13,6) ,XSSI (5,13) ,XPI (5 ,13), 792 2X (5) ,F,SD,ESQB,BI2,BI,TAU, 793 3NOH,NX (13) ,FNFLA"G (13) ,FN,BNY 11 (3) , BND 11, BNY 12 (3 ),BND12, 794 3BND21,BND22, 795 4 SUBSET, ID,SAM, NDAT A (6) , NF, NFM, Nil , OMFLX (1 3, 3) ,OM FLAG (13), 796 4NOMM (3) ,NOM, 797 5DETEST,OPTEST,DATFLG(13,6),OMC,NE 798 6/A1/NFN 799 IF(FN.EQ.1)WEITE(3,10)ID,SAM, (H(I) ,1=1,NOH) 800 10 FOBMAT(I9,» • ,12, ' •,6(F5.4,« •) ) 801 WBITE(3,20)ID,SAM,FN, (X(I) ,1=1,NFN) 802 20 FOBMAT(I9,« ',I2,« •,•FCN#«,12,5 (• «,EJ0.4)) 803 WBITE (3,30)ID,SAM,FN,SD,BSQE 804 30 F0EMAT(I9,« ',12,• « , • FC N#« ,12 , • «,F8.2, 1 » ,F10 .7) 805 EETUEN 806 END 807 SUBEOUTINE PEINT 808 C THIS BOUTINE PEODUCES PBINTED OUTPUT FOE IMMEDIATE EVALUATION. 809 IMPLICIT BEAL*4(A-H,0-Z),INTEGEB*4(I-N) 139 ) 810 INTEGER*2 DATFLG,OMC,NE 811 INTEGEE*4 BNY11,BND11,BNY12,BND12, 812 1BND21,BND22,SUBSET, 813 1IEEE,NFN,NOH,ID,SAM,FNFLAG,FN, 814 20MFLX,OMFLAG,DETEST,OPTEST 815 EEAL*4 H 816 COMMON/A/OMX (13,3) , OM (13) , GAM A (13,6) ,SGAMA (13,6 ) ,PVISC,H(6) , 817 1B(13,6) ,DISC(13,6) ,BVISC(13,6) ,XSSI(5,13) ,XPI(5 ,13), 818 2X (5) ,F,SD,BSQE,BI2,BI,TAU, 819 3NOH,NX(13) ,FNFLAG(13) ,FN,BNY11 (3) ,BND11,BNY12 (3 ),BND12, 820 3BND21,BND22, 821 4 SUB SET, ID , SAM, ND AT A (6) , NF , NFM, N1 , OMFLX (13,3) ,OM FLAG (13), 822 4 NO MM (3) ,NOM, 823 5DETEST,OPTEST,DATFLG(13,6),OMC,NE 824 6/A1/NFN 825 10 WEITE (2 , 20) FN 826 20 FOEMAT (//• PABAMETEBS IN CONSTITUTIVE EQN ',12, • ABE: •) 627 WBITE(2,30) (X (I) ,1 = 1, NFN) 828 30 FOEMAT (5612.4) 828. 1 IF (NE-NOM) 56,59,57 828.2 56 WBITE(2,53) 828.3 53 FOBMAT (• NUMBEB OF DETBUSION BATES EXCEEDS * , 828.4 1'NUMBIB OF B VALUES') 828.5 GOTO 59 828.6 57 WRITE (2,58) 828. 7 58 FORMAT (' NUMBER OF R VALUES EXCEEDS NUMBEB OF * 828.8 ' 1'DETBUSION RATES') 829 WRITE (2,40) 830 40 FORMAT(•0((LS2 METER READING-PREDICTED VALUE)/L S2 METER', 831 1 ' READING) WAS: ') 832 59 DO 60 M=1,NOH 833 60 WRITE (2,70) (DISC(NN,M) ,NN=1,BND12) 834 70 FORMAT (• ' ,13F7.3) 835 WRITE(2,80)SD 836 80 FOEMAT('OSTD DEV OF STEESS=•,1F8.4,• MPA *) 637 WBITE (2,90)ESQB 838 90 FORMAT('OR-SQUARE = ',F10.7) 839 RETURN 840 END 841 SUBROUTINE INTRPT 842 C THIS ROUTINE ASSIGNS DEFAULT VALUES WHEN OPTIMIZAT ION FAILS. 843 IMPLICIT EEAL*4(A-H,O-Z) ,INTEGEB* 4 (I-N) 844 INTEGEB*2 DATFLG,OMC,NR 845 INTEGER+4 BNY11,BND11,BNY12,BND12, 846 1BND21,BND22,SUBSET, 847 1IEEE,NFN,NOH,ID,SAM,FNFLAG,FN, 848 20MFL X,OMFLAG,DETEST,OPTEST 140 84 9 REAL*4 H 850 COMMON/A/OMX(13,3) #OM (13) ,GAMA (13,6) ,SGAHA(13,6 ) ,PVISC,H(6) , 851 1E(13,6) ,DISC(13,6) ,BVISC(13,6) ,XSSI(5,13) ,XPI(5 ,13), 852 2X (5) ,F,SD,RSQR,RI2,RI,TAU, 853 3NOH,NX(13) ,FNFLAG (13) ,FN,BNY11 (3) ,BND11,BNY12 (3 ),BND12, 854 3BND21,BND22, 855 4SUBSET,ID,SAM,NDATA (6) , NF , NFM, N1 , OMFLX (13,3) ,OM FLAG (13) , 856 4 NO MM (3) ,NOM, 857 5DETEST,OPTEST,DATFLG (13,6) ,OMC,NE 858 6/A1/NFN £59 WRITE (2,10)FN 860 10 FOEMAT('0*,•INTEBEOPT OCCUBBED WITH FUNCTION* • ,12) 861 IF (NF.NE.NFM) GOTO 30 862 WEITE (2,20) NF j 863 20 FOBM AT (* LIMITING NUMBEB OF FUNCTION EVALUATION * S REACHED:•,14) 864 30 SD=0,. 865 RSQB=0. 866 DO 40 1=1,5 667 X (I) =0. 868 40 CONTINUE 869 CALL ENT1 870 EETUEN 871 END 872 SUBEOUTINE NODATA(*) 873 C THIS BODTINE ASSIGNS DEFAULT VALUES WHEN DATA IS N OT AVAILABLE. 874 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 875 INTEGEB*2 DATFLG,OMC,NB 876 INTEGEB*4 BNY11,BND11,BNY12,BND12, 877 1BND21,BND22,SUBSET, 878 11ERR,NFN,NOH,ID,SA M,FNFLAG,FN, 879 20MFLX,OMFLAG,DETEST,OPTEST 880 REAL*4 H 881 COMMON/A/OMX (13,3) ,OM (13) ,GAMA (13,6) ,SGAMA (13,6 ),PVISC,H(6), 882 1R(13,6) ,DISC(13,6) ,BVISC(13,6) ,XSSI(5,13) ,XPI(5 ,13) , 883 2X (5),F,SD,RSQR,RI2,RI,TAU, 884 3NOH,NX(13) ,FNFLAG (13) ,FN ,BNY11 (3) ,BND11,BNY12 (3 ),BND12, 885 3BND21,BND22, 886 4SUBSET,ID,SAM,NDATA (6) ,NF,NFM,N1,OMFLX(13,3) ,OM FLAG (1 3) , 887 4NOMM (3) ,NOM, 888 5DETEST,OPTEST,DATFLG(13,6),OMC,NR 889 6/A1/NFN 890 WRITE (2,10) 891 10 FOBMAT('0',* NO DATA') 892 SD=0. 893 RSQB=0. 141 894 DO 20 1=1,5 895 X(I)=0. 896 20 CONTINUE 897 DO 30 FN=1,13 898 IF (FNFLAG (FN) .EQ. 0) GOTO 30 899 CALL PUNCH 900 30 CONTINUE 901 EETUEN1 902 END 903 SUBBOUTINE BEGIN 904 C THIS SUBEOUTINE INITIALIZES THE *PBOGBAM CONSTANTS • BY 905 C BEADING THEM FBOM A FILE. 906 IMPLICIT HEAL*4(A-H,0-Z),INTEGEB*4(I-N) 907 LOGICAL EQUC 908 LOGICAL*! CUBVE 909 INTEGEB*2 DATFLG,OMC,NR 910 INTEGEfi*4 BNY11,BND11,BNY12,BND12, 911 1BND2 1,BND22,SUBSET,PASS,EX,SC, 912 11EEE ,NFN,NOH,ID,SAM,FNFLAG,FN, 913 20MFLX,OMFLAG,DETEST,OPTEST 514 BEAL*4 H,OMA(13) 915 COMMON/A/OMX(13,3) ,OM (13) ,GAMA (13,6) ,SGAMA (13,6 ) ,PVISC,H(6) , 916 1E(13,6) ,DISC (13,6) ,BVISC (13,6) ,XSSI(5,13) ,XPI(5 ,13), 917 2X(5),F,SD,BSQR,EI2,EI,TAU, 918 3NOH,NX(13) ,FNFLAG (13) ,FN ,BNY11 (3) ,BND11,BNY12 (3 ) ,BND12, 919 3BND21,BND22, 920 4 SUBSET, ID, SAM,NDATA (6) , NF , NFM, N1 , OMFLX (13,3) , OM FLAG (13) , 921 4NOMM (3) ,N0M, 922 5DETEST,OPTEST,DATFLG (13,6),OMC,NR 923 6/A1/NFN 924 CALL FEEAD (4,•21: »,DETEST,OPTEST) 925 I F ( (DETEST.EQ.O) .OB. (OPTEST.EQ.O) ) GOTO 20 926 WEITE (2,10) 927 10 FOBMAT (• CANNOT TEST DETEUSION RATE CALCULATION AND 928 1 OPTIMIZATION SIMULTANEOUSLY') 929 41 STOP 930 20 IF (DETEST.EQ. 1)WEITE (2,30) 931 30 FOBM AT(' THIS IS A TEST OF THE DETEUSION BATE', 932 1' CALCULATION. THE INPUT MUST BE SUITABLE TEST DATA.•/ 933 2 * THE TEST IS PASSED IF THE DETEUSION BATES MATC H THE ', 934 3'INTENDED VALUES;I.E. FOB T=C*GAMA**(1/3) THE B ESULTS •, 935 4 'SHOULD BE (. 0368, 1. 4600, 146,. Q036) , •) 936 IF (OPTEST. EQ. 1) WEITE (2,40) 937 40 FOBM AT(' THIS IS A TEST OF THE OPTIMIZATION; TH E INPUT MUST*, 938 1» BE SUITABLE TEST DATA':' AND APPLIED TO A SPECIF AC FUNCTION•) 1 4 2 9 3 9 C A L L F E E A D ( 4 , ' I V : • , N X ( 1 ) , 1 3 ) 9 4 0 C A L L F E E A D ( 4 , ' I V : • , F N F L A G ( 1 ) , 1 3 ) 9 4 1 C A L L I B E A D ( 4 , * 1 : 1 , N F M ) 9 4 2 C A L L F E E A D ( 4 , ' B V : • , O M A ( 1 ) , 1 3 ) 9 4 3 DO 9 5 OMC= 1 , 3 9 4 4 C A L L F R E A D ( 4 , « I V : ' , O M F L X ( 1 , O M C ) , 1 3 ) 9 4 5 C C O P Y O M A T O O M X A C C O E D I N G T O O M F L X . 9 4 6 1 = 0 9 4 7 D O 4 5 N N = 1 , 1 3 9 4 8 I F ( O M F L X ( N N , O M C ) ) 4 1 , 4 5 , 4 3 9 4 9 4 3 1 = 1 + 1 S 5 0 OMX ( I ,OMC) =OMA ( N N ) 9 5 1 4 5 C O N T I N U E 9 5 2 C B E C O E D N U M B E B O F S H E A B B A T E S 9 5 3 N O M M ( O M C ) = I 9 5 4 9 5 C O N T I N U E 9 5 5 D O 1 0 0 F N = 1 , 1 3 9 5 6 1 0 0 C A L L F E E A D ( 4 , • B V : • , X S S I ( 1 , F N ) , 5 ) 9 5 7 D O 1 1 0 F N = 1 , 1 3 9 5 8 1 1 0 C A L L F E E A D ( 4 , ' B V : * , X P I ( 1 , F N ) , 5 ) 9 5 9 W R I T E ( 2 , 1 2 0 ) ( F N F L A G ( I ) , 1 = 1 , 1 3 ) , ( N X ( I ) , 1 = 1 , 1 3 ) 9 6 0 1 2 0 F O R M A T ( * « , T 2 8 , ' 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 ' / 9 6 1 1 ' F U N C T I O N S E L E C T I O N F L A G S : ' , 1 3 1 3 / 9 6 2 1 ' N U M B E B O F P A B A M E T E B S : ' , 1 3 1 3 ) 9 6 3 W B I T E ( 2 , 1 2 2 ) N F M 9 6 4 1 2 2 F O E M A T ( • M A X I M U M N U M B E B O F F U N C T I O N E V A L U A T I O N S / ' , 9 6 5 1 ' O P T I M I Z A T I O N ^ , 1 4 ) 9 6 6 W E I T E ( 2 , 1 4 0 ) O M A 9 6 7 1 4 0 F O E M A T ( * L S - 2 B O T A T I O N A L S P E E D S ( B P M ) : ' / 1 3 F 7 . 3 9 6 8 V ' B O I A T I O N A L S P E E D I N D I C E S ' i 9 6 9 2 / ' 1 3 4 5 7 1 0 1 3 1 6 ' , 9 7 0 3 ' 1 9 2 2 2 5 2 8 2 9 ' ) 3 7 1 D O 1 3 5 O M C = 1 , 3 9 7 2 W E I T E ( 2 , 1 2 5 ) O M C , ( O M F L X ( K , O M C ) , K = 1 , 1 3 ) 9 7 3 1 2 5 F O E M A T ( • L S - 2 B O T A T I O N S P E E D S U B S E T F L A G S F O E O M C = • , 1 1 9 7 4 1 / 1 4 , 1 2 1 7 ) 9 7 5 W B I T E ( 2 , 1 3 0 ) N O M M ( O M C ) 9 7 6 1 3 0 F O B M A T ( ' N U M B E B O F S P E E D S I N T H I S S U B S E T I S ' , 9 7 7 1 * P E E F O E M E D = ' , 1 2 ) S 7 8 1 3 5 C O N T I N U E 9 7 9 W R I T E ( 2 , 1 6 0 ) 9 8 0 1 6 0 F O E M A T ( • I N I T I A L V A L U E S F O E S T E A D Y S T A T E D A T A A B E : • ) 9 8 1 W E I T E ( 2 , 1 7 0 ) ( ( X S S I ( J , F N ) , J = 1 , 5 ) , F N = 1 , 1 3 ) 9 8 2 1 7 0 F O E M A T ( • ' , 5 F 8 . 3 ) 9 8 3 W E I T E ( 2 , 1 8 0 ) 9 8 4 1 8 0 F O E M A T ( ' I N I T I A L V A L U E S F O E P E A K D A T A A B E : ' ) 9 8 5 W E I T E ( 2 , 1 7 0 ) ( ( X P I ( J , F N ) , J = 1 , 5 ) , F N = 1 , 1 3 ) S 8 6 E E T U E N 9 8 7 E N D 9 8 8 S U B E O U T I N E D A T A A 9 8 9 C T H I S S U B E O U T I N E E E A D S T H E E X P E E I M E N T A L D A T A A N D P E 143 RFORMS SOME 990 C SIMPLE MANIPULATIONS ON IT. 991 IMPLICIT EEAL*4 (A-H,0-Z) , INTEGER*4 (I-N) 992 LOGICAL EQUC 993 LOGICAL*1 CURVE 994 INTEGEfi*2 DATFLG,OMC,NE 995 INTEGER*4 BNY11,BND11,BNY12,BND12, 996 1BND21,BND22,SUBSET,PASS,EX,SC, 997 1IERR,NFN,NOH,ID,SAM,FNFLAG,FN, 998 20MFLX,OMFLAG,DETEST,OPTEST 999 EEAL*4 H,LNS,P,SI 1000 DIMENSION XS (1 3) , Y (1 3) , 1001 1Y1 (13) ,Y2(13) ,P (91) , SI (2) ,OMP(13) 100 2 COMMON/A/OMX(13,3) ,OM(13) , GAMA (13,6) , SGAMA (13,6 ) ,PVISC,H(6) , 1003 1E(13,6) , DISC (13,6) ,BVISC (13,6) ,XSSI (5,13) ,XPI (5 ,13), 1004 2X(5),F,SD,RSQR,EI2,RI,TAU, 1005 3NOH,NX(13) ,FNFLAG (13) ,FN,BNY!l1 (3) , BND11, BNY 12 (3 ),BND12, 1006 3BND21,BND22, 1007 4SUBSET,ID,SAM,NDATA (6),NF,NFM,N1,OMFLX(13,3) ,OM FLAG (13) , 1008 4NOMM (3) ,NOM, 1009 5DETEST,OPTEST,DATFLG (13,6),OMC,NR 1010 6/A1/NFN 1011 CALL FBEAD(-2,'ENDF*,2) 1012 10 CALL FBEAD (1,*I*2; *,OMC) 1013 CALL FEEAD(1,•B:*,PVISC,* 21: 1,ID,SAM, 'STBING:*, CUEVE,1, 1014 1S20,£480) 1015 20 CALL FBEAD(-2,'NOFILL*,«TBUE«) 1016 CALL FBEAD(1,'R V: •,H (1) ,6,&30) 1017 30 CALL FBEAD (-2, ' NUMBER* , NOH) 1018 40 IF (H (1).EQ.O. ) CALL NODATA(610) 1019 C COPY SHEAB BATES, FLAGS, OBDEB OF SHEAR BATES TO B E FIT 1020 C TO BEDUCED BANK ASSAYS. 1021 NOM=NCMJ3 (OMC) 1022 DO 45 NN=1,NOM 1023 OM (NN)=OMX (NN,OMC) 1024 OMFLAG (NN)-OMFLX (NN,OMC) 1025 45 CONTINUE 1026 C INITIALIZE AND BEAD E 1 027 DO 50 M=1,6 1028 DO 50 NN=1,13 1029 50 E(NN,M)=-2. 1030 DO 60 N=1,NOH 1031 CALL FBEAD (1, * B V: * , B (1, N) , 13, &5 5) 1032 55 CALL FEEAD (-2 , • NUMBEE*, NE) 1041 60 CONTINUE 1042 C SORT H AND B FOR H DECREASING 1043 70 EX=0 1044 DO 90 M=2,NOH 1045 IF (H (M-1) .GT.H (M) ) GOTO 90 1046 HCOPY=H(M-1) 144 1047 H (M-1) =H (M) 1048 H(M)=HCOFY 1049 DO 8 0 NN=1;NOM 1050 ECOPY=E(NN,M-1) 1051 E (NN,M-1)=B (NN,M) 1052 E (NN,M) =BCCPY 1 053 80 CONTINUE 1054 EX=EX+1 1055 90 CONTINUE 1056 IF (EX.GT.O) GOTO 70 1057 C SET DATA DELETION FLAGS (DATFLG) 1058 DO 95 M=1,NOH 1059 DO 95 NN=1,NOM 1060 IF (.NOT. (S (NN,M).LE.O.) )GOTO 99 1061 DATFLG (NN, M) =0 1062 GOTO 95 1063 99 DATFLG (NN,M) = 1 1064 95 CONTINUE 1065 DO 93 M=1,NOH 1066 1=2 1067 DO 9 2 NN=I,NOM 1068 I F ( (DATFLG (NN-1,M) . EQ.O),.OB. (DATFLG (NN , M) .EQ. 0) )GOTO 92 1069 IF (.NOT. ( (NN,. LT. 5) - AND. (B(NN,M). LT.30.).AND. (E ( NN-1 ,M) . GE. 1070 1R (NN,M) ) ) ) GOTO 92 1071 DATFLG (NN-1,M)=0 1072 DATFLG (NN,M)=0 1073 92 CONTINUE 1074 93 CONTINUE 1075 C SCALE B 1076 DO 100 M=1,NOH 1077 SC=1 1078 DO 91 NN=1,NOM 1079 I F (DATFLG (NN, M) .EQ. 1) GOTO 94 1080 91 CONTINUE 1081 94 RCOPY=B (NN,M) 1082 I=NN+1 1083 DO 100 NN=I,NOH 1084 IF (DATFLG (NN,M) .EQ.0)GOTO 100 1085 97 I F ( ( (BCOPY.GT. 30. ) .OB. (NN.GT. 4)) .AND. (BCOPY.GT. fi (NN,M))) 1086 1SC=5*SC 1C87 ECOPY=E (NN,M) 1088 B (NN, M) =SC*B (NN,M) 1089 100 CONTINUE 1090 IF (OPTEST.EQ-0) GOTO 120 1091 C SET GAMA TO PBEDETEBMINED VALUES SO THAT OPTIMIZAT ION CAN BE 1092 C TESTED 1093 DO 110 M=1,NOH 1094 DO 110 NN=1,NOM 1095 110 GAMA (NN ,.M) =OM (NN) 1096 GOTO 280 1097 120 LNS=ALOG.(12./11.) 1098 DO 270 M=1,NOH 145 1099 C OBTAIN LN (OM) AND LN (E) WITHOUT EMBEDDED MISSING D ATA VALUES, 1100 C AND DETEEMINE THE NUMBEB OF DATA VALUES. 1101 1=0 1102 DO 140 NN=1,NOM 1103 IF (DATFLG (NN,M) .EQ.O) GOTO 140 1104 1=1+1 1105 XS (I) =ALOG (B (NN , M) ) 1106 Y(I)=ALOG(OM (NN)) 1107 OMP (I) =OM (NN) 1108 140 CONTINUE 1109 c INITIALIZE VABIABLES OF THE SPLINE FITTING ROUTINE 1110 SI (1)=0. 1111 SI(2)=0. 1 112 DO 145 J=1,13 1113 145 P(J)=0. 1114 c FIT A SPLINE TO THE LN (OM) VS. LN (R) 1115 CALL SMOOTH(XS,Y,P,1,SI,7,S160) 1116 GOTO 180 1117 160 WRITE(2,170) 1118 170 FORM AT(• SMOOTH RESTRICTIONS VIOLATED') 1119 STOP 1120 c FIND DERIVATIVES OF SPLINE 1121 180 CALL SMTH(XS,Y,Y1,Y2,1,& 190) 1122 GOTO 210 1123 190 WRITE (2,200) 1124 200 FORMAT(• SMTH RESTRICTIONS VIOLATED') 1125 STOP 1 126 c CALCULATE SHEAB BATES 1 127 210 DO 220 J=1,I 1128 Y (J)= (1. +Y1 (J) *LNS+ (Y1 (J) *LNS) **2/3. 1129 1+LNS**2/3.*Y2 (J)) *OMP(J) *. 1047 19755/LNS 1130 220 CONTINUE 1131 c UNPACK (IE. INTEODUCE MISSING DATA POINTS) 1132 240 NN=0 1133 DO 260 J=1,I 1134 250 NN=NN+1 1135 I F (. NOT,. (DATFLG (NN, M) . EQ. 0) ) GOTO 255 1136 GOTO 250 1137 255 GAMA (NN,M) =Y (J) 1138 260 CONTINUE 1139 270 CONTINUE 1 140 280 CONTINUE 1141 c WBITE ORIGINAL DATA 1 142 I F (PVISCGT. 1. ) WBITE (2,290) PVISC,ID,SAM 1 143 290 FOEMAT('0',11('PLASMA')/'OMEDIUM VISCOSITY = «,F8.6, : (CP) 1 144 16(' ' ) ,«ID#=',19, • ' , 1 2 ) 1145 IF(PVISC.LT,1.)WBITE(2,300)PVISC,ID,SAM 1146 300 FOEMAT('0',11('SALINE')/'OMEDIUM VISCOSITY = ' ,F8, 6, (CP) 1147 16(« •),'J:D#=',19,' «,I2) 1 148 IF(EQUC (CURVE,'P') ) GOTO 320 1149 WRITE (2,310) 1150 310 FORMAT('+',T57,•STEADY STATE') 1 151 GOTO 340 146 1 152 320 WRITE (2,330) 1 1 5 3 330 FORMAT(« + «,T57 ,•PEAKS•) 1154 340 WRITE (2,350) (H(M) ,11=1, NOH) 1155 350 FO RM AT (• HEMATOCRITS: •,6(F5.4,« •)) 1156 WRITE (2,360) 1157 360 FORMAT('0LS-2 METER READINGS, AS I F ALL ON SCAL E 1, WERE:«) 1158 DO 365 M=1,NOH 1159 DO 365 NN=1,NOM 1 160 IF (DATFLG (NN, M) .EQ.O) R (NN,M) =-9. 999 1161 365 CONTINUE 1162 DO 370 M=1,NOH 1163 370 WRITE (2,380) (R(NN,M),NN=1,NOM) 1 164 380 FORMAT(» ',13 ( F 6 . 1 , ' • ) ) 1165 WRITE (2,390) 1 166 390 FORM AT('ODETR 0 SION RATES IN S**-1 WERE:') 1167 DO 400 M=1,NOH 1 168 400 WRITE (2,410) (GAMA (NN,M) ,NN=1,NOM) 1 169 410 FORM AT (' ,,13(F8.4,« «)) 1170 IF(DETEST.EQ.1 )STOP 1171 C INITIALIZE X 1172 IF(EQUC(CURVE,'P•))GOTO 430 1173 DO 420 1=1,2 1 174 420 X(I)=XSSI(I,1) 1175 GOTO 450 1176 430 DO 440 1=1,2 1177 440 X (I) =XPI (1,1) 1178 C CONVERT R TO STBESS 1179 450 DO 460 M=1,NOH 1180 DO 460 NN=1,NOM 1181 R (NN,M)=,.5261*R (NN,M) 1 182 460 CONTINUE 1183 C INITIALIZE STATISTICAL PARAMETERS 1184 N1=0 1185 BI=0 1186 RI2=0 1187 C FOEM SUM OF SQUARES OF STRESSES AND COUNT DATA POI NTS 1188 DO 470 M=1,NOH 1189 DO 470 NN=1,NOM 1190 IF (DATFLG (NN,M).EQ.0)GOTO 470 1191 N1=N1+1 1192 EI=RI+fi (NN,M) 1193 BI2=RI2+R (NN,M)**2 1194 470 CONTINUE 1195 EETUEN 1196 480 STOP 1197 END 1 $SIG DEBE PEIO=N T=15S PAGES=35 FOEM=BLANK CABDS=00 2 $R PET EE 2. G 1=PBAT2 (4 ,*L) 2=*PRINT* 4=PBATS(4,6) 3 $SIG 4 **THE FOLLOWING ARE CONSTANTS FOR STATISTICAL ANALYSI S OF 5 **THE EESULTS OF FITTING VARIOUS FUNCTIONS TO HEMOEHE OLOGICAL 6 **D-ATA. . 147 7 2 4 4 4 4 4 4 4 4 4 4 2 4 8 1 1 1 1 1 1 1 1 1 1 0 0 0 9 1 0 0 0 1 C THIS PEOGBAM FEEFOBMS THE STATISTICAL ANALYSIS OF THE EESULTS 2 C OF THE FITTING OF CONSTITUTIVE FUNCTIONS FOB BLOOD TO 3 C OBSERVATIONS. THE STANDAED DEVIATIONS AND PABAMETE E VALUES 4 C EESULTING FBOM THE FITTING ABE AVEBAGED OVEB THE S AMPLE 5 C POPULATION. 6 C**VABIABLE DEFINITIONS****************************** *********** 7 C PABTBL: FLAGS PABAMETEB TABLES (FOB EACH FUNCTION) 8 C O PREVENTS PRINTING 9 C 1 ALLOWS PRINTING 10 C ID=IDENTITY OF PERSON FROM WHICH SAMPLE WAS TAKEN; USUALLY 11 C SOCIAL INSURANCE NUMBER 12 C SAM=SAMfLE NUMBER ;SEQUENTIAL FROM 1 FOR EACH PEES ON 13 C FNFLAG: INDICATES WHICH FUNCTIONS ARE REPRESENTED IN THE DATA 14 C FN=FUNCTION NUMBEB 15 C FNC=FUNCTION NUMBEB BEAD FBOM DATA 16 C NX(FN) = NUMBEB OF PABAMERS IN FUNCTION FN 17 C NF=NUMBES OF PARAMETERS IN FUNCTION FN 18 C H1=HEMATOCBIT INDEX FOB A SAMPLE 19 C S=SAMPLE NUMBEB,SEQUENTIAL IN TEST DATA 20 C DELETE :FLAG INDICATING WHETHEB DELETIONS HAVE BE EN MADE PEICB 21 C TO LAST AVEBAGE 22 C DELEV: NUMBEB OF TIMES DELETIONS IN THE DATA HAVE EEEN MADE 23 C PAETBL =0 IF PAEMETEB S ABE NOT TO BE PBINTED 24 C =1 OTHEEWISE 25 C H=HEMATOCBIT 26 C SD=STANBABD DEVIATION OF POINTS OF A SAMPLE 27 C SDAV= AVEBAGE OVEB POPULATION OF SD VALUES 28 C SDSD= STANDABD DEVIATION OF SD VALUES W.B.T, SDAV 29 C BSQB= THE B**2 STATISTICAL PABAMETEB 30 C SD2=SUM OF SQUABES OF SD 31 C XA: AVEBAGE OF X VALUES OVER THE POPUATION 32 C S2: SUM OVER THE POPULATION OF SQUARES OF X VALUE S 33 C SDX: STANDABD DEVIATION OF X W.B.T. XA 34 SUBEOUTINE MAIN 35 IMPLICIT BEAL*4 (A- H, O-Z) , INTEGEB*4 (I-N) 36 EEAL*4 XU (4) ,XL (4) 37 INTEGEB*4 ID,SAM,FNC,NX,H1,S,FN,X1,DELETE,DELEV 38 1PABTBL,PAEAV,FNFLAG,SMAX,OBD,OBDINV,VECTI, 3 9 2 S DTEI,PB CB TB 40 COMMON/A/H (4,33),SD (13,33),SDAV(13) ,SDSD(13),X ( 4,13,33), 148 41 1ESQE (13,33) ,SD2 (13) ,RSQA (13) , XA(4, 13) , X2 (4, 13) , SDX(4, 13) , 42 2VECTOE(13), 43 3ID,SAM,FNC,NX(13) , PARTBL, FNFLAG(1 3),NFN,SMAX, 44 40ED(13) ,OEDINV (13) ,NSD (13) ,VECTI (13) , 45 5SDTBL,PBCBTB, PABAV 46 CALL FEEAD (4, 'I V: • , NX (1) , 13) 47 CALL IBEAD(4, ,I V: ' ,FNFLAG (1) ,13) 48 CALL FEEAD(4,«4I:*,SDTBL,PBOBTB,PAETBL,PAEAV) 49 SMAX=0 50 DO 50 S=1,33 51 BEAD (1,10,END=6 0)ID,SAM, (H(H1 ,S) ,H1 = 1,4) 52 SMAX=SMAX+1 53 10 FOEMAT (19,13,T13,4F6.4) 54 DO 40 FN=1,13 55 IF (FNFLAG (FN) .EQ.O) GOTO40 56 NXFN = NX (FN) 57 BEAD (1,20) FNC, (X (X1,FN,S) ,X1 = 1 ,NXFN) 58 20 FOBMAT(T18,I2,T20,4E11.4) 59 C IF THE FUNCTION NUMBEB BEAD DOES NOT MATCH THE NUM EEE 60 C EXPECTED CALL AN ERROR BOUTINE. 61 IF (FNC. NE. FN) CALL DATAEB 62 C READ STANDAED DEVIATION AND B SQUABE VALUES. 63 BEAD (1,30) FNC,SD (FN, S) , ESQE (FN,S) 64 30 FOEMAT (T18 ,12 , T21, F8. 2,F 11, 7) 65 c AS ABOVE CHECK FUNCTION NUMBER 66 IF (FNC. NE. FN) CALL DATAEB 67 40 CONTINUE 68 50 CONTINUE 69 NFN=0 70 DO 5 2 FN=1,13 71 52 NFN=NFN+FNFLAG(FN) 72 IF ((SDTBL. E Q - P ) . AND. (PBOBTE. EQ.O) ) GOTO 55 73 CALL STDDEV 74 IF(PBOBTB.EQ.1)CALL PBOB 75 55 I F ( (PAETBL.EQ.O).AND. (PABAV.EQ.O))GOTO 60 76 CALL PABTAB 77 IF (PABAV.EQ.O)GOTO 60 78 CALL EABAVG 79 60 WRITE (2,390) 80 390 FORMAT (' 1' ) 81 400 STOP 82 END 83 BLOCK DATA 84 IMPLICIT BEAL*4 (A-H,0-Z) , INTEGEB*4 (I-N) 85 INTEGER*4 ID,SAM,FNC,NX ,H1,S,FN,X 1, 86 1PARTBL,PARAV,FNFLAG,SMAX,ORD,ORDINV,VECTI, 87 2SDTEL,PBOBTB 88 COMMON/A/H(4,33),SD (13,33),SDAV(13) ,SDSD (13) ,X( 4 ,13,33), 89 1BSQR (13,33) ,SD2 (13) ,RSQA(13) , XA(4 , 13) , X2 (4, 13) , SDX(4,13), 90 2VECTOR(13), 91 3ID,SAM,FNC,NX(13),PARTBL,FNFLAG(13),NFN,SMAX, 92 40ED(13) ,ORDINV (13) ,NSD (13) ,VECTI (13) , 149 93 5SDTBL,PROBTB,PARAV 94 DATA H/132*0./,SD/429*0./,SDAV/13*0./, 95 lSDSD/13*0./,X/1716*0./,RSQR/429*0./, 96 2SD2/13*0./,RSQA/13*0./ #XA/52*0./,X2/52*0./, 97 3SDX/52*0./,NSD/13*0/ 98 END 99 SUBROUTINE DATAER 100 C**TBIS ROUTINE RECOGNIZES AN ERROR CONDITION IN THE DATA 10 1 IMPLICIT HEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 102 INTEGER*4 ID,SAM,PNC,NE,H1,S,FN,X1 , 103 1PARTBL,PARAV,FNFLAG,SMAX,OBD,OBDINV,VECTI, 104 2SDTBL,PROBTB 105 COMMCN/A/H (4,33) ,SD(13,33) ,SDAV(13) ,SDSD (13) ,X( 4,13,33), 106 1RSQR (13,33) ,SD2 (13) ,RSQA (13),XA(4,13),X2(4,13) , SDX(4,13), 107 2VECTOB (13) , 108 3ID,SAM,FNC,NX (13),PARTBL,FNFLAG(13) ,NFN,SMAX, 109 40RD (13) ,ORDINV(13) ,NSD ( 13) ,VECTI ( 13) , 110 5SDTBL,PROBTB,PARAV 111 WRITE(2,10)ID,SAM,FN 112 10 FORMAT(• FUNCTION NUMBERS OUT OF SEQ.',19,13,13 ) 113 STOP 114 END 115 SUBROUTINE COPYB(VECT) 116 C**THIS 117 IMPLICIT REAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 118 INTEGER*4 ID,SAM,FNC,NE,H1,S,FN,X 1, 119 1PARTBL,PARAV,FNFLAG,SMAX,ORD,OBDINV,VECTI, 120 2SDTBL,PROBTB 121 DIMENSION VECT(13) 122 COMMON/A/H (4,33),SD (13,33),SDAV(13) ,SDSD(13) ,X ( 4,1 3,33) , 123 1RSQR (13,33) ,SD2 (13) ,RSQA(13) , XA (4 , 13) , X2 (4, 13) , SDX(4,13), 124 2VECTOE (13) , 125 3ID,SAM,FNC,NX (13) ,PARTBL,FNFLAG(13),NFN,SMAX, 126 40RD (13) ,OBDINV (13) ,NSD (13) ,VECTI (13) , 127 5SDTEL,PROBTB,PARAV 128 FN=0 129 DO 10 K=1,NFN 130 5 FN=FN+1 131 IF (FNFLAG (FN). EQ.O) GOTO 5 132 VECTOR (K)=VECT (FN) 133 10 CONTINUE 134 EETUEN 135 END 136 SUBEOUTINE COPYI(VECT) 137 C**THIS BO0TINE COPIES A RANK 1 ABBAY CONTAINING BLAN K ELEMENTS-138 C TO AN ABRAY WITHOUT THE BLANK ELEMENTS. 139 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 140 INTEGER*4 ID,SAM,FNC,NE,H1,S,FN,X1, 141 1PARTBL,PARAV,FNFLAG,SMAX,ORD,ORDINV,VECTI,VECT, 150 142 2SDTBL,PBGBIB 143 DIMENSION VECT(13) 144 COMMON/A/H (4,33),SD (13,33),SDAV(13) ,SDSD(13) ,X( 4,13,33), 145 1E5QB (13,33) ,SD2 (13) ,BSQA(13) ,XA(4, 13) ,X2 (4, 13) , SDX(4,13) , 146 2VECIOE(13), 147 3ID,SAM,FNC,NX(13),PAETBL,FNFLAG(13),NFN,SMAX, 148 40RD (13) ,ORDINV (13) ,NSD (13) ,VECTI (13) , 149 5SDTBL,PBOBTB,PABAV 150 FN = 0 151 DO 10 K=1,NFN 152 5 FN=FN+1 153 IF (FNFLAG (FN) .EQ.O) GOTO 5 154 VECTI (K) =VECT (FN) 155 10 CONTINUE 156 EETUEN 157 END 158 SUBEOUTINE STDDEV 159 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 160 BEAL+4 XU (4) , XL (4) 161 INTEGEE*4 ID,SAM,FNC,NX,H1,S,FN,X1,DELETE,DELEV 162 ' 1PABTBL,PABAV,FNFLAG,SMAX,OBD,0EDINV,VECTI, 163 2SDTBL,PBCBTB 164 DIMENSION NVECT(13) 165 COMMON/A/H (4,33),SD (13,33) ,SDAV(13) ,SDSD(13) ,X ( 4,13,33), 166 1ESQB (13,33) ,SD2 (13) ,BSQA(13) , XA (4 ,13) , X2 (4, 13) , SDX(4, 13) , 167 2VECTOE(13), 168 3ID,SAM,FNC,NX(13),PAETBL,FNFLAG(13),NFN,SMAX, 169 40BD (13) ,CEDINV (13) ,NSD (13) ,VECTI (13) , 170 5SDTBL,PB0BTB,PABAV 171 C LOOP OVEB FUNCTIONS 172 60 DO 80 FN=1,13 173 IF (FNFLAG (FN) .EQ.0)GOTO 80 174 N=0 175 C LOOP OVER SAMPLES 176 DO 70 S=1,SMAX 177 C SKIP MISSING STANDABD DEVIATION 178 IF(SD(FN,S).EQ.O.) GOTO 70 179 N=N+1 180 C FOBM THE SUM OF STD DEV«S AND THE SUM OF B SQUARE VALUES 181 SDAV (FN) =SDAV (FN)+SD (FN,S) 182 BSQA (FN) =BSQA (FN) +ESQB (FN,S) 183 SD2 (FN)=SD2(FN)+SD (FN,S) **2 184 70 CONTINUE 185 NSD (FN) =N 186 C CALCULATE THE STANDABD DEVIATIONS OF THE STANDABD DEVIATIONS, 187 C THE THE AVEBAGES OF THE STANDABD DEVIATIONS AND TH E AVEBAGES 188 C OF THE B SQUABES. 189 SDSD (EN) =SQET ( (SD2 (FN) -SDAV (FN) ** 2/N) / (N-1) ) 151 190 SDAV (FN) =SDAV (FN)/N 191 BSQA (FN) = RSQA (FN) /N 192 80 CONTINUE 193 IF (SDTBL.EQ.O) GOTO 160 194 FN=0 195 CO 90 K= 1,NFN 196 85 FN=FN+1 197 IF (FNFLAG (FN) .EQ.O) GOTO 85 198 NVECT (K) =F N 199 90 CONTINUE 200 100 WRITE (2, 110) (NVECT (K) ,K=1 ,NFN) 20-1 110 FORMAT ('1«,///////,' PEBSON SEX HEMATOCRITS A T' , 202 1T31,13(» ',12)) 203 WBITE (2,95) 204 95 FOBMAT(8 ',T15,'WHICH MEASUREMENTS'/T15,'WERE M ADE • ) 205 DO 130 S=1,SMAX 206 CALL COPYR (SD (1,S) ) 207 WRITE(2,120) S, (H(H1,S) ,H1=1,4) , (VECTOR (FN) ,FN=1 ,NFN) 208 120 FORMAT (* » ,T4 ,12, T 10 , ' F • , T15, 3 (F 3. 2, ' , «),F3.2, T34, 13 (F6.2, • •) ) 209 130 CONTINUE 210 CALL COPYR(SDAV) 211 WRITE(2,140) (VECTOR (FN) ,FN=1,NFN) 212 140 FORM AT(* 0•,T35,13(F5.2,' •)) 213 CALL COPYR (SDSD) 214 WRIIE(2, 145) (VECTOR (FN) ,FN=1, NFN) 215 145 FORM AT (* • ,T35,13 (F5. 2, • •) ) 216 CALL COPYR (BSQA) 217 WEITE (2, 150) (VECTOB (FN) ,FN=1, NFN) 218 150 FOBM AT (' • ,T33,13 (F7. 5) ) 213 160 CALL CBDEE (SDAV) 220 IF (SDTBL.EQ. 0) BETUBN 221 CALL COPYI(OED) 222 WRITE (2,165) (VECTI (I) ,1=1,NFN) 223 165 FORMAT(' * ,T4,*OEDEBING OF •,T37,13(12,• •) ) 224 EETUEN 225 END 226 SUBEOUTINE PAETAB 227 C**THIS ROUTINE CONSTBUCTS THE TABLES, EACH OF EHICH CONTAINS 228 C ALL PABAMETEB VALUES FOB ALL PERSONS IN THE POPULA HON 229 C FOB EACH FUNCTION. 230 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 231 BEAL*4 XU(4),XL(4) 232 INTEGEB*4 ID,SAM,FNC,NX,H1,S,FN,X1,DELETE,DELEV 233 ' 1PARTBL,PARAV,FNFLAG,SMAX,ORD,ORDINV,VECTI, 234 2SDTBL,PROBTB 235 COMMON/A/H (4,33) , SD (13 ,33) ,SDAV (13) , SDSD (13) ,X( 4,13,33), 236 1RSQR (13,33) ,SD2 (13) , BSQA (13) , XA(4, 13) , X2 (4, 13) , 152 SDX(4,13), 237 2VECT0B (13) , 238 3ID,SAM,FNC,NX(13) ,PARTBL,FNFLAG(13),NFN,SMAX, 239 40BD (13) ,OBDINV (13) , NSD (13) , VECTI (13) , 240 5SDTBL,FBOBTB,PABAV 241 C LOOP OVER FUNCTIONS 242 160 DO 330 FN=1,13 243 IF (FNFLAG (FN) .EQ.O) GOTO 330 244 NXFN=NX (FN) 245 DELEV=0 246 C INITIALIZE THE PARAMETER SUMS, AVERAGE AND STANDAB D 247 C DEVIATION TO 0. 248 170 DO 180 X1=1,NXFN 249 XA (X1,FN)=0. 250 X2 (X1,FN)=0. 251 SDX (X1,FN)=0. 252 180 XA(X1,FN)=0. 253 C LOOP OVEB THE PABAMETEB NUMBEB 254 DO 200 X1=1,NXFN 255 N=0 256 DO 190 S=1,SMAX 257 C SKIP MISSING PABAMETEB VALUES 258 .IF (X (X1 ,FN ,S) . EQ. 0 . ) GOTO 190 259 C SUM THE NUMEEB OF PABAMETEBS, THEIR VALUES AND THE IB SQUABES 260 N=N+1 261 XA (X1,FN)=XA (XI,FN) +X (X 1,FN,S) 262 X2 (X1,FN)=X2(X1,FN) +X(X1,FN,S) **2 263 190 CONTINUE 264 C CALCULATE THE STANDABD DEVIATIONS AND AVEBAGE VALU ES OF 265 C FABAMETEBS. 266 IF ( (N.EQ.O).OB. (N.EQ.1)) GOTO 200 267 SDX (X1 ,FN) =SQBT ( (X2 (XI ,FN) -XA (XI , FN) **2/N) / (N-1 )) 268 XA (X1,FN)=XA(X1,FN)/N 269 200 CONTINUE 270 210 CONTINUE 271 C IF THE DELETION LEVEL IS 0 PBINT THE INITIAL PABAM ETEB TABLE 272 C FOB THE CUEBENT FUNCTION. 273 IF (PABTBL.EQ.0)GOTO 270 274 220 IF (DEIEV.NE.O)GOTO 270 275 WRITE (2 , 230)FN,DELEV 276 230 FOBM AT(* 1 *,T20,•FC N#•,12,' DELETION LEVEL=' ,12, 277 1» PARAMETER VALUES',////) 278 DO 250 S=1,SMAX 279 WRITE (2 ,240) S, (X (X1 ,FN,S) ,X1 = 1 ,NXFN) 280 240 FORMAT(1 •,T10,12,T21,4 (G 11.4,• •)) 281 250 CONTINUE 282 WRITE (2,260) (XA (X1 ,FN) ,X1 = 1 ,NXFN) 283 260 FORMAI( ,0«,I21,4 (G11.4,« «)) 284 WRITE(2,260) (SDX(X1,FN) ,X1 = 1,NXFN) 285 C INITIALIZE THE PARAMETER VALUE DELETION FLAG. 153 286 270 DELETE=0 287 DO 290 X1=1,NXFN 288 C CALCULATE UPPEE AND LOWEB BOUNDS FOE •BEPBSENTATIV E« .'; 289 C VALUES OF PARAMETERS. 290 XU (X1)=XA (X1,FN) +2.2*SDX (X1,FN) 291 XL (X1) = XA (X1,FN) -2.2*SDX (X1,FN) 292 DO 290 S=1,SMAX 293 IF (X (X1 /FN, S) . EQ. 0. ) GOTO 290 294 C CHECK WHETHEE PABAMETEB IS IN 'EEPBESENTATIVE' 295 C EANGE.. 296 IF ( (X (X1 ,FN,S) -GT. XL (X1) ) . AND. (X (X1 ,FN,S) . LT. XU (X1))) 297 1GOTO 290 298 C IF PABAliETEE IS TOO * WILD • EL LI MI NATE ALL PABAMETE BS WITH 299 C THESE VALUES OF FN S S. 300 DO 280 J=1,NXFN 301 280 X(J,FN,S)=0. 302 DELETE=1 303 290 CONTINUE 304 C I F DELETIONS•HAVE NOT OCCUBBED PBINT THE FINAL PAB AMETEB TAELE. 305 C OTHEEWISE REPEAT THE AVERAGING.. 306 IF (DELETE.EQ.0)GOTO 300 307 CELEV=DELEV+1 308 GOTO 170 309 300 CONTINUE 310 C NOW THAT DELETIONS ARE COMPLETE PRINT A PARAMETER 311 C TABLE FOE THE FUNCTION. 312 IF (PARTBL.EQ .O)GOTO 330 313 310 WBITE(2,230)FN,DELEV 314 DO 320 S=1,SMAX 315 WRITE (2,240) S, (X (X1,FN,S) ,X1 = 1 ,NXFN) 316 320 CONTINUE 317 WRITE (2,260) (XA (X1 ,FN) ,X1 = 1,NXFN) 318 WEIIE(2,260) (SDX (X 1, FN) , X1= 1, NXFN) 319 330 CONTINUE 320 RETURN 321 END 322 SUBROUTINE PARAVG 323 C PRINTS THE FINAL AVERAGE VALUES OF THE PABAMETEBS & THE 324 C STANDABD DEVIATIONS. 325 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGEB*4(I-N) 326 EEAL*4 XU (4) , XL (4) 327 INTEGEB*4 ID,SAM,FNC,NX,H1,S,FN,X1,DELETE,DELEV 328 ' 1PABTBL,PABAV,FNFLAG,SMAX,OED,OEDINV,VECTI, 329 2SDTBL,PBOBTB 330 COMMCN/A/H (4,33) ,SD(13,33) ,SDAV(13) ,SDSD(13) ,X( 4,13,33), 331 1BSQB (13,33) ,SD2(13) ,BSQA(13) , XA (4 ,13) , X2 (4 ,13) , SDX(4,13), 332 2VECTOB (13) , 333 < 3ID,SAM,FNC,NX(13),PAETBL,FNFLAG(13),NFN,SMAX, 154 334 40BD (13) ,OEDINV (13) ,NSD (13) ,VECTI (13) , 335 5SDTBL,PROBTB,PAEAV 336 340 WRITE (2,350) 337 350 FORMAT (' 1' ,////////,T 16,4 (' X •) 338 1/T16,4(' SID DEV •)/) 339 DO 380 FN=1,13 340 IF (FNFLAG (FN),. EQ.O) GOTO 380 341 NXFN=NX (FN) 342 WRITE (2,360) FN, (XA (X1,FN) ,X1= 1 ,NXFN) 343 360 FORM AT (' 0 ' ,12 ,T 16 , 4 (G11,. 4, ' •)) 344 WRITE(2,370) (SDX(X1,FN) ,X1=1,NXFN) 345 370 FORMAT (* • ,T1 6, 4 (G 11. 4, • •)) 346 380 CONTINUE 347 RETURN 348 END 349 SUBROUTINE ORDER(VECTO) 350 C**THIS ROUTINE DETERMINES THE OEDEBING OF THE FUNCTI ONS 351 C ACCORDING TO DECREASING SIANDAfiD DEVIATION OF FIT. 352 IMPLICIT REAL*4(A-H,0-2),INTEGER*4(I-N) 353 REAL*4 XU (4) ,XL (4) ,MIN 354 IN TEGER*4 ID,SAM,FNC,NX,H1,S,FN,X1,DELETE,DELEV 355 ' 1PARTBL,?ARAV,FNFLAG,SMAX,ORD,ORDINV,VECTI, 356 2SDTB1,PECBTB 357 DIMENSION VECTO (13) , VECTOC (13) 358 COMMON/A/H (4,33) ,SD (13,33) ,SDAV(13) ,SDSD(13) ,X ( 4,13,33), 359 1ESQE (13,33) ,SD2 (13) , BSQA (13) , XA(4,13) ,X2 (4, 13) , SDX (4,13) , 360 2VECTOB(13) , 361 3 ID, SAM,FNC,NX (13) , PABTBL, FNFLAG (1 3) , NFN, SMAX, 362 4 0ED (13) ,OEDINV (13) ,NSD (13) ,VECTI (13) , 363 5 SDTBL,PBOBIB,PABAV 364 DO 5 FN=1,13 365 VECTOC (FN) =VECTO (FN)' 366 5 CONTINUE 367 DO 40 J=1,NFN 368 MIN=9.E74 369 DO 30 FN=1,13 370 IF (FNFLAG (FN) .EQ.O) GOTO 30 37 1 IF (VECTOC (FN) .EQ.O.) GOTO 30 372 IF (VECTOC (FN) . LT. MIN) GOTO 10 373 GOTO 30 374 10 MINDEX=FN 375 MIN = VECTOC (FN) 376 30 CONTINUE 377 OBD(iMINDEX)=J 378 VECTOC(MINDEX)=0. 379 40 CONTINUE 380 DO 50 FN=1,13 381 IF (FNFLAG (FN) .EQ.O) GOTO 50 382 OBDINV (OED (FN) ) =FN 383 50 CONTINUE 384 EETUEN 385 END 155 386 SUBEOUTINE PEOB 387 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGEE*4 (I-N) 388 BEAL*4 XU (4) , XL (4) 389 INTEG^B*4 ID,SAM,FNC,NX,H1,S,FN,X1,DELETE,DELEV 390 ' 1PABTBL,PABAV,FNFLAG,SMAX,OED,OBDINV,VECTI, 391 2SDTBL,PBOBTB 392 COMMON/A/H (4, 33) ,S D (13, 33) , SDAV (13) ,SDSD(13) ,X( 4,13,33), 393 1ESQE (13,33) ,SD2 (13) ,BSQA (13) , XA (4 ,13) , X2 (4, 13) , SDX (4, 13) , 394 2VECTOB(13), 395 3ID,SAM,FNC,NX(13),PAETBL,FNFLAG(13),NFN,SMAX, 396 40BD (13) ,ORDINV (13) ,NSD (13) ,VECTI (13) , 397 5SDTBL,PBOBTB,PABAV 398 WBITE(2,10) 399 10 FOBMAT(•1 *//T10,'PBOBABILITIES OF COBBECTNESS 0 F APPABENT', 400 1» OEDEBINGS OF PAIBS OF AVEBAGE VALUES OF STD D EVIATIONS'///) 401 DO 30 J=2,NFN 402 I=J-1 403 I1=OBDINV(I) 404 I2=0EDINV(J) 405 U= (SDAV (I2)-SDAV ( 1 1 ) ) / 406 1SQET (2* (SDSD(I1) **2/NSD ( 1 1 ) +SDSD (12) **2/NSD (12) )) 407 GOT018 408 WBITE(2,17) SDSD ( 1 1 ) ,NSD (I1),SDSD (12) ,NSD(I2) ,SD AV (12),SDAV 409 1 (12) ,U 410 17 FOBMAT(• •,2(F8.3,I3),3F8.3) 411 18 FBO=.5*(1.+EBF(U) ) 412 WBITE (2,20)1, J,OBDINV(I) ,OEDINV(J) ,PBO 413 20 FOEMATC OBDEBS:•,12,•6',12,•; . 414 1PBOBAEILITY THAT (SDAV (• ,12, • ) < SDAV (• , 1 2 , ') ) = •,F5.3) 415 30 CONTINUE 416 EETUEN 417 END 1 C THIS PEOGEAM PLOTS T^E SPECIFIED CONSTITUTIVE FUNC TION AND 2 C THE DATA FOE ONE SAMPLE. THE USEE MAY CHOOSE FEOM TWO BANGES 3 C OF DETBUSION BATES AND FEOM STEESS AND-OB VISCOSIT Y OBDINATES. 4 C THE PROGEAM IS DOCUMENTED INTEBNALLY. 5 C**VAEIABLE DEFINITIONS****************************** ************** 6 C CASFLG ( 1 1,12) = FLAG INDICATING WHICH CASES ABE TO BE TBEATED 7 C SEE DEFINITION OF 1 1 , 1 2 8 C =0 FOE IGNOBE 9 C =1 FOE TEEAT 10 C EX = EXCHANGE COUNTEE USED IN DATA SOBT 11 C FIT = HEAL VAEIABLE FOBM OF NFIT 1 5 6 1 2 C F N = F U N C T I O N N U M B E B ; S E E F U N C T I O N C O D E B E L O W 1 3 C G A M A D E T E U S I O N B A T E S 1 4 c G A M A C = D E T E U S I O N B A T E S U S E D F O B P L O T T I N G C O N T I N U O O S C U E V E 1 5 c G A M A X M A X I M U M V A L U E O F ; G A M A C O N P L O T 1 6 c 1 1 = E L O T D E T E U S I O N B A T E I N D E X 1 7 c = 1 F O B L O W B A N G E 1 8 c = 2 F O B H I G H B A N G E 1 9 c 1 2 = S T R E S S / V I S C O S I T Y I N D E X 2 0 c 1 F O E S T E E S S 2 1 c = 2 F O E V I S C O S I T Y 2 2 c M - H E M A T O C R I T I N D E X & S Y M B O L I D E N T I F I C A T I O N N O M B E E 2 3 c N F I T N U M B E B O F S H E A H B A T E S O V E B W H I C H O P T I M I Z E D F I T W A S 2 4 C P E E F O B M E D 2 5 c N O H = N U M B E B O F H E M A T O C R I T S 2 6 c N O H N O . O F S H E A B B A T E S A T O N E H E M A T O C B I T 2 7 c N 0 M F L G = N U M E B I C A L C H E C K F L A G 2 8 c = 0 N O N U M E B I C A L C H E C K 2 9 c 1 C O M P U T E A N D P B I N T N U M E B I C A L C H E C K O F D A 3 0 1A c O M F L G — B O T A T I O M S P E E D S E L E C T I O N F L A G S , D I M E N S I O N E D F O E 2 3 1 c S E T S O F B O T A T I O N S P E E D S 3 2 c O M F I A i G 1 = W O B K I N G B O T A T I O N S P E E D S E L E C T I O N F L A G S 3 3 c P A S S = U N U S E D 3 4 c P 1 T F L G = P L O T F L A G F O B U S E W H E N T E S T I N G P E O G E A M 3 5 c = 0 F O B N O P L O T S 3 6 c = 1 F O B P L O T S 3 7 c S A M S A M P L E N O . 3 8 c S C • = E H E O M E T E E S I G N A L S C A L E F A C T O E 3 9 c S G A M A C = F U N C T I O N O F G A M A C ; U S E D T O E C O N O M I Z E C A L C O L A T I O N S 4 0 c S I = I N T E G E B V A R I A B L E C O N T A I N I N G S O C I A L I N S U R A N C E N O . 4 1 C S I L -- L O G I C A L V A R I A B L E C O N T A I N I N G S O C I A L I N S U R A N C E N U M E E B , 4 2 C ft S T A B T F L A G I N D I C A T I N G W H E T H E R I N I T I A L I Z A T I O N D A T 4 3 A c S H O U L D B E R E A D 4 4 c S T A T E = I N D I C A T E S T A T E O F D A T A S O U R C E 4 5 c F P E A K S 4 6 c N O T ( P ) S T E A D Y S T A T E 4 7 c T A U = S H E A R S T R E S S 4 8 c T A U C = S H E A R S T R E S S U S E D F O R P L O T T I N G C O N T I N U O U S C O E V E 4 9 C X S C A L = M A X V A L U E O N X A X I S 5 0 C X S X Z = L E N G T H I N I N C H E S O N P L O T F R O M 0 . T O X S C A L 5 1 C Y S C A L = M A X V A L U E O N Y A X I S 5 2 C Y S I Z = L E N G T H I N I N C H E S O N P L O T F R O M 0 . T O Y S C A L 5 3 C * * D A T A F I L E S T R U C T U R E * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *** 5 4 C N U M F L G , P L T F L G 5 5 C X S I Z ( 1 ) , X S I Z ( 2 ) , Y S I Z (1) , Y S I Z ( 2 ) 5 6 c X S C A L ( 1 ) , X S C A L ( 2 ) , Y S C A L ( 1 ) , Y S C A L ( 2 ) , G A M A X ( 1 ) , G A M A X 157 (2) 57 C OMFLG (1,1) .... (10,1) 58 c GMFLG (1,2) .... (10,2) 59 c PVISC,SI,SAfl,STATE,NFIT 60 c H(1) .... (6) 61 c E (1 ,1) .... (10,1) 62 c 63 c (1,NOH) .... (10,NOH) 64 c GAMA(1,1).,.. (10,1) 65 c 66 c (1,NOH) .... (10,NOH) 67 c FN,CASFLG (1,1), (2,1) , (1,2) . (2,2) ,X(1) .... (5) 68 c EEPEAT'- PREVIOUS LINE FOR EACH FUNCTION 69 c 0 70 c REPEAT PEEVIOUS 11 LINES FOE EACH SAMPLE 71 C**THE REMAINING DATA IS FOE PLOTTING COEVES WITHOUT DATA POINTS*** 72 C 0. 73 C FN,CASFLG ( 1 / 1 ) , ( 2 , 1 ) , (1,2) , (2, 2) , X ( 1) . . . . (5) 74 C REPEAT PREVIOUS LINE FOR EACH FUNCTION 75 ***** 76 SUBROUTINE MAIN 77 IMPLICIT REAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 78 LOGICAL EQUC 79 LOGICAL*1 STATE,SIL(9) 80 INTEGER*2 PASS,EX,SC,CASFLG,VFLG 8-1 INTEGER*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 82 1 ,START,0MFLG,0MFLG1 83 DIMENSION SGAMAC(991) 84 COMMON/A/OM (10) ,GAM A (10,6) ,PVISC,H (6) ,SOM (6) ,C ( 2,6), 85 1 R (10,6) , VISC (10,6) ,GAMAC(991) ,TAUC (991) , X(5) , 86 o \ 2 XSIZ(2) ,YSIZ(2) ,XSCAL (2) ,YSCAL(2) , GAMAX ( 87 i 3 NOH,NUMFLG,M,11,12,PLTFLG, 88 4 SI, SAM,FN,OMFLG (10,2) ,OMFLG1 (10) ,NOM,STA ET, 89 5 NNMAX,NNMAX1,NFIT,CASFLG(2,2),VFLG, 90 6 STATE 91 STAET=0 92 10 CALL DATA 1 93 VFLG=0 94 DO 40 J1=1,40 95 CALL DATA2(&10) 96 WBITE (6,13) FN , ((CASFLG (11 ,12) ,11=1,2) ,12=1,2) , ( X(I) ,1=1,4) 97 13 FORMAT (' •,512,4E11.3) 98 C LOOP OVEB STEESS/VICOSITY AND DETBUSION BATE AL TEENATIVES 99 DO 30 12=1,2 100 DO 30 11=1,2 101 IF (CASFLG (11,12) .EQ. 0) GOTO 30 102 DO 20 M=1,NOH 158 103 CALL CUBVE 104 C IF THE NUMERICAL CHECK FLAG IS 1 PBINT A SELECTION OF 105 C CALCULATED POINTS AND THE OBSEEVED POINTS. 106 IF (NUMFLG. EQ. 1) CALL NUMCHK 107 20 CONTINUE 108 IF (PLTFLG,. EQ.O) GOTO 30 109 CALL LABEL 110 30 CONTINUE 111 40 CONTINUE 112 GOTO 10 113 END 114 BLOCK DATA 115 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGER*4(I-N) 116 LOGICAL*1 STATE 117 INTEGEE*2 CASFLG,VFLG 118 INTEGER*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 119 1 ,STABT,OMFLG,OMFLG1 120 COMMON/A/OM(10) ,GAMA(10,6) ,PVISC,H(6) ,SOM (6) ,C ( 2,6), 121 1 R(10,6) ,VISC (10,6) ,GAMAC(991) ,TAUC(991) , X(5) , 122 2 XSIZ (2) , YSIZ (2) , XSCAL (2) , YSCAL (2) ,GAMAX ( 2) , 123 3 NOH,NUMFLG,M,11,12,PLTFLG, 5 124 4 SI,SAM,FN,OMFLG(10,2),0MFLG1(10),NQM,STA ET, 125 5 NNMAX,NNMAX1,NFIT,CASFLG(2,2),VFLG, 126 6 STATE 127 DATA CM/.0238,-06;. 15,. 373 ,. 94 5, 2. 38, 6. , 15.,37. 3,94.5/ 128 END 129 SUBEOUTINE DATA 1 130 IMPLICIT BEAL*4(A-H,0-Z) ,INTEGEB*4 (I-N) 131 LOGICAL EQUC 132 LOGICAL*1 STATE,SIL(9) 133 INTEGEB*2 PASS,EX,SC,CASFLG,VFLG 134 INTEGEE*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 135 1 ,STAET,OMFLG,OMFLG1 136 DIMENSION SGAMAC(991) 137 COMMON/A/OM (10) ,GAMA(10,6) ,PVISC,H(6) ,SOM(6) ,C( 2,6), 138 1 E (10,6) ,VISC (10,6) ,GAMAC(991) ,TAUC (991) , X(5), 139 2 XSIZ (2) , YSIZ (2),XSCAL (2), YSCAL (2) ,GAMAX( 2), 140 3 NOH,NUMFLG,M,11,12,PLTFLG, 141 4 SI,SAM,FN,OMFLG (10,2) ,0MFLG1 (10) ,NOM,STA ET, 142 5 NNMAX,NNMAX1,NFIT,CASFLG (2,2) ,VFLG, 143 6 STATE 144 IF (STAET.EQ.1)GOTO 20 145 CALL FEEAD(1,»21:NUMFLG,PLTFLG) 146 CALL FBEAD(1, ,2(E V) : • , XSIZ, 2 , YSIZ , 2) 147 CALL FEEAD (1, «3 (E V) : ' , XSCAL, 2 , YSCAL, 2 , GAMAX, 2) 148 DO 10 J=1,2 159 149 10 CALL IREAD (1,'R V:•,OMFLG(1,J) ,10) 150 START-1 151 20 CALL FBEAD(-2,*ENDF*,2) 152 30 CALL FBEAD (1,*B:•,PVISC,•21:•,SI,SAM,1STBING STATE 153 1 1,«I:«,NFIT,&40,&130) 154 40 IF (PVISC.NE .0-) GOTO 45 155 NOH=1 156 EETUEN 157 45 CALL FEEAD (-2 , 1 NOFILL' , .TRUE. ) 158 CALL IBEAD(1,'B V: • ,H (1) ,6,S50) 159 50 CALL FBEAD(-2,'NUMBEB',NOH) 160 DO 60 M=1,NOH 161 DO 60 NN=1,10 ! 162 B(NN,M)=-2. 163 60 GAMA (NN, M) =-1. 164 DO 7 0 1=1, NOH 165 CALL IBEAD(1,'B V: 1 , B (1,1) , 10,&7 0) 166 70 CONTINUE 167 C SOBT HEMATOCRITS AND OBSERVED SIGNALS 168 80 EX=0 169 DO 100 M=2,NOH 170 IF (H (M-1) .GT.H (M) ) GOTO 100 171 HCOPY=H (M-1) 172 H (M-1) = H (M) 173 H(M)=HCOPY 174 DO 90 NN=1,10 175 BCOPY=E (NN,M-1) 176 B(NN,M-1)=B(NN,M) 177 B (NN , M) =RCOPY 178 90 CONTINUE 179 EX=EX+1 180 100 CONTINUE 18.1 IF (EX.GT.O) GOTO 80 182 C SCALE ALL INPUTS TO SCALE 1 183 DO 110 M-1,NOH 184 SC=1 185 ECOPY=R(2,M) 186 DO 110 NN=3,10 187 IF (R (NN,M) .LT.RCOPY) SC=5*SC 188 BCOPY=E (NN, M) 189 R (NN ,M) =,.5261*SC*B (NN,M) 190 110 CONTINUE 191 DO 120 1=1,NOH 192 CALL FBEAD (1,'B V: • , G AM A (1 , 1 ) , 10 , 5120) 193 120 CONTINUE 194 EETUEN 195 130 IF(PLTFLG.EQ.O)STOP 196 CALL PLCTND ; 197 STOP 198 END 199 SUBROUTINE DATA2 (*) 200 IMPLICIT BEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 201 LOGICAL EQUC 20 2 LOGICAL*1 STATE,SIL(9) 203 INTEGEE*2 PASS,EX,SC,CASFLG,VFLG 160 204 INTEGER*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 205 1 - ,STAEI,OMFLG,OMFLG1 206 DIMENSION SGAMAC(991) 207 COMMON/A/OM (10) ,GAMA (10, 6) ,PVISC,H (6) , SOM (6) ,C( 2,6), 208 1 E (10,6) , VISC (10,6) ,GAMAC (99 1) ,TAUC (991) , X(5), 209 2 XSIZ (2) , YSIZ (2) , XSCAL (2),YSCAL (2) ,GAMAX( 2) , 210 3 NOH,NUMFLG, M, 11,12, PLTFLG, 211 4 SI,SAM,FN,OMFLG (10,2) ,0MFLG1 (10) ,NOM,STA BT 212 ' 5 NNMAX,NNMAX1, NFIT,CASFLG (-2,2) ,VFLG, 213 6 STATE 214 CALL FBEAD (1 , * 1,1* 2 V,B V : * , FN ,C ASFLG (1,1) ,4,X( 1) ,5,610,&130) 215 10 IF (FN-EC.O) BETUEN1 216 IF ( (VFLG.EQ.1) .OB. ((CASFLG (1 ,2)+CASFLG (2 , 2) ) . EQ .0))EETUEN 217 DO 20 M=1,NOH 218 DO 20 NN=1,10 219 20 VISC (NN,M)=B(NN,M)/GAMA (NN,M) 220 VFLG=1 221 EETUEN 222 130 IF (PLTFLG.EQ.0)STOP 223 CALL PLOTND 224 STOP 225 END 226 SUBEOUTINE CUBVE 227 C THIS SUBEOUTINE CONSTBUCTS A SEQUENCE OF DETEUSION BATES 228 C FOE THE SMOOTH CUBVE 229 IMPLICIT BEAL*4 (A-H,0-Z),INTEGEB*4 (I-N) 230 LOGICAL EQUC 231 LOGICAL*1 STATE,SIL(9) 232 INTEGEE*2 PASS,EX,SC,CASFLG,VFLG 233 INTEGEB*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 234 1 ,STABT,OMFLG,OMFLG1 235 DIMENSION SGAMAC (991) ,VEC1(10) ,VEC2(10) 23 6 COMMON/A/OM (10) ,G A MA (10 , 6) ,PVISC,H(6) , SOM (6) ,C ( 2,6), 237 1 B(10,6) ,VISC(10,6) ,GAMAC(991) ,TAUC (991) , X (5) , 238 2 XSIZ (2) , YSIZ (2) , XSCAL (2),YSCAL (2) , GAMAX ( 2) , 239 3 NOH,NUMFLG,M,11,12,PLTFLG, 240 4 SI,SAM,FN,OMFLG (10,2) ,0MFLG1 (10) ,NOM,STA BT, 241 5 NNMAX,NNMAX1,NFIT,CASFLG(2,2),VFLG, 242 6 STATE 243 C SELECT TEE DETEUSION BATES SCALE. 244 NNMAX=100*XSIZ (11) 245 NSTABT=2 246 NNMAX1=NNMAX-NSTABT 247 DO 10 NN=1,NNMAX1 248 GAMAC (NN)=GAMAX(11)* (NN+NSTABT)/NNMAX 161 249 10 CONTINUE 250 IF (FN.NE.8) GOTO 30 251 C DO EXTBA CALCULATION HEBE IF WOBKING ON FCN 8 252 DO 20 NN=1,NNMAX1 253 20 SGAMAC(NN)=1./(X(1)*SQBT(GAMAC(NN))) 254 30 CONTINUE 255 C BRANCH TO APPBOPBIATE FUNCTION 256 N1=FN-1 257 GOTO (60,90,120, 150, 180,230,260,3 10,360,40 2,50,4 10) ,N1 258 40 CONTINUE 259 50 CONTINUE 260 C FUNCTION 2 261 60 X2H=X (2) *H (M) 262 E1H=EXP (X (1) *H (M) ) 263 DO 80 NN=1,NNMAX 1 264 TAUC (NN) -X (4) * (E1H*GAMAC (NN) +X2H* (1. -EXP (-X (3) * GAMAC(NN)))) 265 GOTO (80,70) ,12 266 70 TAUC (NN) =TAUC (NN) /GAMAC (NN) 267 80 CONTINUE 268 GOTO 460 269 C FUNCTION 3 270 90 E1H=EXP (X (1) *H (M) ) 271 23H=X (3) *H (M) 272 DO 110 NN=1,NNM AX 1 273 TAUC (NN)=X (4) * (E1H*GAMAC (NN) +X (2) * (1. - EXP (-X3H* GAMAC (NN)))) 274 GOTO (110,100) ,12 275 100 TAUC (NN) =TAUC (NN)/GAMAC (NN) 276 110 CONTINUE 277 GOTO 460 278 C FUNCTION 4 279 120 X1H=X (1) *H (M) 280 EX3H=EXP (X (3) *H (M) ) 281 DO 140 NN=1,NNMAX1 282 TAUC (NN) =X (4) * (GAMAC (NN) *X1H*(1. - EXP 283 1 (-X (2) *GAMAC (NN) ) ) ) *EX3H 284 GOTO (140, 130) ,12 285 130 TAUC (NN) =TAUC (NN)/GAMAC (NN) 286 140 CONTINUE 287 GOTO 460 288 C FUNCTION 5 289 150 X2H=X (2) *H (M) 290 EX3H=EXP (X (3) *H (M) ) 291 DO 170 NN=1,NNMAX1 292 TAUC (NN)=X (4) * (GAMAC (NN) +X (1) * (1.-EXP 293 1 (-X2H*GAMAC (NN) ) ) ) *EX3H 294 GOTO (170,160) ,12 295 160 TAUC (NN)=TAUC (NN)/GAMAC (NN) 296 170 CONTINUE 297 GOTO 460 298 C FUNCTION 6 299 180 X2H=X (2) *H (M) : 300 EX3H=EXP (X (3) *H (M) ) 301 GOTO (190,210),12 162 302 190 DO 200 NN=1,NNMAX1 303 200 TAUC (NN)=X (4) *GAMAC (NN) * (1.+X (1) *(1.-EXP 304 1 (-X2H*GAMAC (NN) ) ) ) *EX3H 305 GOTO 460 306 210 DO 220 NN=1,NNMAX1 307 220 TAUC (NN) =X (4) * (1. + X (1) * (1.-EXP 308 1 (-X2H*GAMAC (NN) ) ) ) *EX3H 309 GOTO460 310 C FUNCTION 7 WALBUBN & SCHNECK 311 230 EXH=X (1) *EXP (X (2) *H (M) +X (4) /H (M) **2) 312 X3H= 1.-X (3) *H (M) 313 DO 250 NN=1,NNMAX1 314 TAUC (NN) =EXH*GAMAC (NN) **X3H 315 GOTO (250,240) ,12 316 240 TAUC (NN) =TAUC (NN) /GAMAC (NN) 317 250 CONTINUE 318 GOTO 460 319 C FUNCTION 8 QUEMADA 320 260 X2H=1.-X (2) *H (M) 321 X23H=t.-X (2) *X (3) *H (M) 322 GOTO (270,290) ,12 . 323 270 DO 280 NN=1,NNMAX1. 324 280 TAUC (NN)=X (4) *GAMAC (NN) * ( (1. +SGAMAC (NN) ) / 325 1 (X2H+X23H*SGAMAC (NN) ) ) **2 326 GOTO 460 327 290 DO 300 NN=1,NNMAX1 328 300 TAUC (NN)=X (4) * ( (1. + SGAMAC (NN) ) / 329 1 (X2H + X23H*SGAMAC (NN) ) ) **2 330 GOTO 460 331 C FUNCTION 9 332 310 EX4H=EXP (X (4) *H (M) ) 333 GOTO (320,340) ,12 334 320 DO 330 NN- 1,NNMAX1 335 330 TAUC (NN)=X (1) *EX4H*GAMAC (NN) * (il. + X (2) *GAMAC (NN) **2)/ 336 1 (1. +X (3) *GAMAC (NN) **2) 337 GOTO 460 338 340 DO 350 NN=1,NNMAX1 339 350 TAUC (NN)=X (1) *EX4H* (1.+X (2) *GAMAC (NN) **2) / 340 1 (1. +X (3)*GAMAC (NN) **2) 341 GOTO 460 342 C FUNCTION 10 343 360 X2H=X (2) *H (M) 344 X3H=X (3) *H (M) 345 EX4H=EXP (X (4) *H (M) ) 346 GOTO (370,390) ,12 347 370 DO 380 NN=1,NNMAX1 348 380 TAUC (NN)=X(1) *EX4H*GAMAC (NN) * (1.+X2H*GAMAC (NN) * *2)/ 349 1(1. +X3H*GAMAC (NN) **2) 350 GOTO 460 351 390 DO 400 NN= 1,NNMAX 1 352 400 TAUC (NN) =X (1) *EX4H* (1. + X2H*G AM AC (NN) **2) / 353 1 (1.+X3H*GAMAC (NN) **2) 354 GOTO 460 355 C FUNCTION 11 (HUANG) 163 3 5 6 4 0 2 D O 4 0 5 N N = 1 , N N M A X 1 3 5 7 X 2 = G A M A C ( N N ) * * X ( 2 ) 3 5 8 T A U C ( N N ) = X ( 5 ) * G A M A C ( N N ) - X ( 1 ) * X 2 * E X P ( - X ( 3 ) * X 2 ) 3 5 9 G O T O ( 4 0 5 , 4 0 4 ) ,12 3 6 C 4 0 4 T A U C ( N N ) = T A U C ( N N ) / G A M A C ( N N ) 3 6 1 4 0 5 C O N T I N U E 3 6 2 G O T O 4 6 0 3 6 3 C F U N C T I O N 1 3 3 6 4 4 1 0 G O T O ( 4 2 0 , 4 4 0 ) ,12 3 6 5 4 2 0 D O 4 3 0 N N = 1 , N N M A X 1 3 6 6 V = A L O G ( G A M A C ( N N ) ) + 5 . 3 6 7 4 3 0 T A U C ( N N ) = P V I S C * G A M A C ( N N ) * E X P ( H ( M ) * ( X ( 1 ) - X ( 2 ) * V -X ( 3 ) 3 6 8 1 * ( V - X ( 4 ) ) / ( 1 - ( V / X ( 4 ) ) * * X ( 5 ) ) ) ) 3 6 9 G O T O 4 6 0 3 7 0 4 4 0 D O 4 5 0 N N = 1 , N N M A X 1 3 7 1 V = A L O G ( G A M A C ( N N ) ) + 5 . 3 7 2 4 5 0 T A U C ( N N ) = P V I S C * E X P ( H ( M ) * ( X ( 1 ) - X ( 2 ) * V - X ( 3 ) 3 7 3 1 * ( V - X ( 4 ) ) / ( 1 - ( V / X ( 4 ) ) * * X ( 5 ) ) ) ) 3 7 4 4 6 0 D O 4 7 7 N N = 1 , N N M A X 1 3 7 5 G O T O ( 2 9 5 , 2 9 6 ) ,12 3 7 6 2 9 5 I F ( T A U C ( N N ) * Y S I Z ( 1 1 ) / Y S C A L (11) , L E , 2 9 , ) G O T O 4 7 1 3 7 7 T A U C ( N N ) = 2 9 . * Y S C A L (11) / Y S I Z (11) 3 7 8 G O T O 4 7 1 3 7 9 2 9 6 I F ( T A U C ( N N ) * Y S I Z (1 1 ) / ( 3 . * Y S C A L ( I 1 ) / X S C A L (11) ) . L E . 2 9 . ) 3 8 0 1 G O T O 4 7 1 3 8 1 T A U C ( N N ) = 2 9 . * 3 . * Y S C A L (11) / X S C A L ( I 1 ) / Y S I Z (11) 3 8 2 4 7 1 G O T O ( 4 7 5 , 4 7 6 ) ,12 3 8 3 4 7 5 I F ( T A U C ( N N ) * Y S I Z ( 1 1 ) / Y S C A L (11) . G E . - 1 0 . ) G O T O 4 7 7 3 8 4 T A U C ( N N ) = - 1 0 . * Y S C A L ( 1 1 ) / Y S I Z (11) 3 8 5 G O T O 4 7 7 3 8 6 4 7 6 I F ( T A U C ( N N ) * Y S I Z ( I 1 ) / ( 3 . * Y S C A L ( I 1 ) / X S C A L ( I 1 ) ) . G E . - 1 0 . ) 3 8 7 1 G O T O 4 7 7 3 8 8 T A U C ( N N ) = - 1 0 , . * 3 . * Y S C A L ( I 1 ) / X S C A L (11) / Y S I Z (11) 3 8 9 4 7 7 C O N T I N U E 3 9 0 I F ( P L T F L G . E Q . 0) B E T U R N 3 9 1 C S P E C I F Y A X I S L A B E L S 3 9 2 G O T O ( 4 7 0 , 4 8 0 ) ,12 3 9 3 4 7 0 I F ( M . E Q . 1 ) C A L L A L A X I S ( • B ( D W / D B ) ( S * * T 1 ) « , 1 6 , 3 9 4 1 » S H E A B S T B E S S ( M P A ) ' , 1 8 ) 3 9 5 C S P E C I F Y S C A L I N G 3 9 6 C A L L A L S I Z E ( X S I Z (11) , Y S I Z (11) ) 3 9 7 Y S C = Y S C A I ( i , 1 ) 3 9 8 C A L L A L S C A L ( 0 . , X S C A L (11) , 0 . , Y S C ) 3 9 9 G O T O 4 9 0 4 0 0 4 8 0 I F ( M • E Q . 1 ) C A L L A L A X I S ( • B ( D W / D B ) ( S * * - 1 ) ' , 1 6 , 4 0 1 1 • V I S C O S I T Y (MPA. S ) 1 , 1 7 ) 4 0 2 C A L L A L S I Z E ( X S I 2 ( I 1 ) , Y S I Z (11) ) 4 0 3 Y S C = 3 . * Y S C A L (11) / X S C A L (11) 4 0 4 C A L L A L S C A L ( 0 . , X S C A L (11) , 0 . , Y S C ) 4 0 5 4 9 0 G O T O ( 4 9 5 , 4 9 6 , 4 9 6 , 4 9 6 , 4 9 6 , 4 9 6 ) , M 4 0 6 C P L O T S M O O T H C U B V E O N N E W A X E S F O B 1 S T H E M A T O C B I T 4 0 7 4 9 5 C A L L A L G B A F ( G A M A C , T A U C , N N M A X 1 , 0 ) 4 0 8 G O T O 4 9 7 1 6 4 4 0 9 C O V E R L A Y SMOOTH C U R V E FOB S U C C E E D I N G H E M A T O C R I T S 4 1 0 4 9 6 C A L L A L G B A F ( G A M A C , T A U C , - N N M A X 1 , 0 ) 4 1 1 4 9 7 I F ( P V I S C . E C . O i ) RETURN 4 1 2 C D E T E B M I N E COORDINATES OF HEMATOCBIT L A B E L 4 1 3 C ( 1 , M ) = X S I Z ( I 1 ) *GA MAC ( N N M A X 1 ) / X S C AL (11) +. 04 4 1 4 C ( 2 , M ) = Y S I Z (11) *TAUC ( N N M A X 1 ) / Y S C - . 0 7 4 1 5 DO 5 0 0 K=1,10 4 1 6 500 O M F L G 1 ( K ) = O M F L G ( K , I 1 ) 4 1 7 C A L L COPYR (GAMA (1,M) ,VEC1) 4 1 8 GOTO ( 5 1 0 , 5 2 0 ) ,12 4 1 9 5 1 0 C A L L COPYB (B (1 ,M) , V E C 2 ) 4 2 0 GOTO 530 4 2 1 5 2 0 C A L L COPYR ( V I SC (1 , M) , V E C 2 ) 4 2 2 C PLOT OBSERVED P O I N T S 4 2 3 5 3 0 C A L L A L G B A F ( V E C 1 , V E C 2 , - N O M , - M ) 4 2 4 RETURN 4 2 5 END 4 2 6 SUBROOTINE NUMCHK 42 7 I M P L I C I T R E A L * 4 ( A - H , O-Z) , I N T E G E R * 4 ( I - N ) 4 2 8 L O G I C A L EQUC 4 2 9 L O G I C A L * 1 S T A T E , S I L ( 9 ) 4 3 0 I N T E G E R * 2 P A S S , E X , S C , C A S F L G , V F L G 431 I N T E G E R * 4 N O H , S I , S A M , F N , N U M F L G , P L T F L G 4 3 2 1 ,START,OMFLG,OMFLG1 4 3 3 D I M E N S I O N S G A M A C ( 9 9 1 ) 4 3 4 COMMON/A/OM ( 1 0 ) ,GAMA ( 1 0 , 6 ) , P V I S C , H ( 6 ) , SOM (6) ,C ( 2 , 6 ) , 4 3 5 1 R ( 1 0 , 6 ) , V I S C ( 1 0 , 6 ) , G A M A C ( 9 9 1 ) , T A U C ( 9 9 1 ) , X ( 5 ) , 4 3 6 2 X S I Z ( 2 ) , Y S I Z ( 2 ) , X S C A L ( 2 ) , Y S C A L (2) ,GAMAX( 2) , 4 3 7 T NOH,NUMFLG,M,11,12,PLTFLG, 4 3 8 4 S I , S A M , F N , O M F L G ( 1 0 , 2 ) , O M F L G 1 (10) ,NOM,STA RT, 43 9 5 N N M A X , N N M A X 1 , N F I T , C A S F L G ( 2 , 2 ) , V F L G , 4 4 0 6 S T A T E 4 4 1 INC=NNMAX/10 4 4 2 I F ( E Q U C ( S T A T E , * P 1 ) ) G O T O 20 4 4 3 WRITE ( 6 , 1 0 ) S I , S A M , F N , N F I T 4 4 4 10 FORMAT (• S I # ' , I 9 , ' S A M P L E * • , 1 2 , • F U N C T I O N * ' ,12, 4 4 5 1 • STEADY S T A T E ' , • F I T ' , 1 2 ) 4 4 6 GOTO 40 4 4 7 20 WBITE , 6 , 3 0 ) S I , S A M , F N , N F I T 4 4 8 30 FOBM AT; (' S I # ' , I 9 , ' S A M P L E * ' , 12, • F U N C T I O N * ' ,12, 4 4 9 1 • P E A K S * , ' F I T ' , 1 2 ) 4 5 0 40 GOTO ( 5 0 , 1 0 0 ) ,12 4 5 1 50 WBITE ( 6 , 6 0 ) M 4 5 2 6 0 FORMAT(• FOR H(«,I1,') S E L E C T E D D E T R U S I O N R A T E S ( S * * - 1 ) AND ', 453 1 'CORRESPONDING S T R E S S E S (MPA) ARE R E S P E C T I V E L Y ' ) 4 5 4 W R I T E ( 6 , 8 0 ) (GAMAC (NN) ,NN= 1 ,NNMAX 1,INC) 4 5 5 WRITE ( 6 , 3 0 ) (TAUC (NN) ,NN=1 ,NNMAX1 , I N C ) 4 5 6 I F ( P V I S C . E Q - 0 . ) RETURN 165 457 WBITE (6 ,70) M 45 8 C PRINT THE EXPERIMENTAL DETRUSION RATES AND STRESSE S 459 70 FORM AI (• FOB H ( M 1 , ' ) EXPERIMENTAL DETRUSION R ATESS (S**-1) •, 460 1 'AND CORRESPONDING STRESSES (MPA) ARE RE SP ECTVELY *) 461 WRITE (6,80) (GAMA (NN,M) ,NN=1 ,10) 462 WRITE(6,80) (R (NN,M) ,NN=1,10) 463 80 FORMAT (' * ,11F9.3) 464 90 FORMAT (• • ,11G9.4) 465 RETURN 466 100 WRITE (6,110) M 467 110 FORMAT(1 FOR H(*,I1,*) SELECTED DETRUSION RATES (S**-1) AND », 468 1 'CORRESPONDING VISCOSITIES (MPA.S) ARE R ESPECTIVELY') I 469 WRITE (6,80) (GAMAC (NN) , NN= 1 , NNM AX 1 ,INC) 470 WRITE (6, 90) (TAUC (NN) , NN= 1 ,NNMAX1 ,1NC) 471 IF (PVISC, EQ. 0.), RETURN 472 WRITE (6^120) M 473 C PRINT TEE EXPERIMENTAL DETRUSION RATES AND VISCOSI TIES 474 120 FORMAT (• FOR H C I l , ' ) EXPERIMENTAL DETRUSION R ATES (S**-1) ', 475 1 'AND CORRESPONDING VISCOSITIES (MPA) ARE BESPECTVELY') 476 WRITE (6,80) (GAMA (NN,M) ,NN=1,10) 477 WRITE (6 ,80) (VISC (NN,M) ,NN=1 , 10) 478 RETURN 479 END 480 C LABEL TEE PLOT 481 SUBROUTINE LABEL 482 IMPLICIT REAL*4 (A-H,0-Z) ,INTEGER*4 (I-N) 483 LOGICAL EQUC 484 LOGICAL*1 STATE,SIL(9) 485 INTEGER*2 PASS,EX,SC,CASFLG,VFLG 486 INTEGER*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 487 1 ,STARI,OMFLG,OMFLG1 488 DIMENSION SGAMAC(991) 489 COMMON/A/OM (10) ,GAMA (10,6) ,PVISC,H(6) ,SOM (6) ,C ( 2,6), 490 1 R (10,6) , VISC (10,6) ,GAMAC(991) , TAUC (991) , X(5), 491 2 XSIZ (2) , YSIZ (2) , XSCAL (2) , YSCAL (2) , GAMAX ( 2) , 492 3 NOH,NUMFLG,M,11,12,PLTFLG, 493 4 SI, SAM , FN, OMFLG (10,2) , OMFLG1 (10) ,NOM,STA RT, 494 5 NNMAX,NNMAX1,NFIT,CASFLG(2,2),VFLG, 495 6 STATE • 496 XC=. 3*XSLZ (11) 497 YC = YSIZ(I1) 498 IF (PVISC,EQ. 0. ) GOTO 20 499 C SPREAD OUT HEMATOCBIT NUMBERS IF TOO CLOSE TOGETHEB 500 DO 5 M=2>NOH 166 501 IF( (C (2,NOH-M+1)-C (2,NOH-M + 2) ) .GT.. 15) GOTO 5 502 M1=N0H-M+1 503 DO 3 M2=1,M1 504 3 C(2,M2)=C (2,M2)+.15-C (2,Ml)+C(2,M1+1) 505 5 CONTINUE 506 C LABEL THE HEMATOCRITS 507 DO 7 M=1,NOH 508 7 CALL NUMBEB (C (1, M) ,C (2, M) , . 14, H (M) , 0. , 2) 509 CALL SYMBOL(XC,YC,. 14, (SI#*,0. ,3) 510 CALL BTD (SI,SIL, 9, IDUM) 511 CALL SYMBOL(XC+-84,YC,.14,SIL,0.,9) 512 YC=YSIZ (11)-. 2 513 CALL SYMBOL(XC,YC,.14,'SAMPLE* ,0. ,6) 514 FSAM=SAM 515 CALL NUMBEB(XC+.84,YC,.14,FSAM,0,-1) 516 YC=YSIZ (11)-, 4 517 IF(EQUC(STATE,* P*) )GOTO 10 518 CALL SYMBOL(XC,YC,.14,'STEADY STATE',0.,12) 519 GOTO 20 520 10 CALL SYMBOL(XC,YC,.14,»PEAKS',0.,5) 521 20 XC=YSIZ (11)-.6 522 CALL SYMBOL(XC,YC,.14,'FCN«,0.,3) 523 FFN=FN 524 CALL NUMBEB(XC+.48,YC,.14,FFN,0.,-1) 525 IF (PVISC.EQ.O.) BETUBN 526 YC=YSIZ (11)-.8 527 CALL SYMBOL(XC,YC,.14,'FIT*,0.,3) 528 FIT=NFIT 529 CALL NUMBEB(XC+.48,YC,. 14,FIT,0, ,-1) 530 BETUBN 531 END 532 SUBEOUTINE COP YB (VECI , VECO) 533 IMPLICIT EEAL*4 (A-H,0-Z) ,INTEGEB*4 (I-N) 534 LOGICAL*1 STATE,SIL(9) 535 INTEGEE*2 CASFLG,VFLG 536 INTEGEB*4 NOH,SI,SAM,FN,NUMFLG,PLTFLG 537 1 ,STABT,OMFLG,OMFLG1 538 DIMENSION VECI (10) , VECO (1 0) 539 COMMON/A/OM (10) ,GAMA (10,6) ,PVISC, H (6) ,SOM (6) ,C ( 2,6) , 540 X(5) , 1 E (10,6) , VISC (10,6) ,GAMAC (991) ,TAUC (991) , 541 2) , 2 XSIZ(2) , YSIZ (2) , XSCAL (2), YSCAL (2) , GAMAX ( 542 3 NOH,NUMFLG,M,I1,12,PLTFLG, 543 BT, 4 SI,SAM,FN,OMFLG (10,2) ,0MFLG1 (10) ,NOM,STA 544 5 NNMAX,NNMAX 1,NFIT,CASFLG(2,2),VFLG, 545 6 STATE 546 NOM = 0 547 DO 10 K=1,10 548 IF (OMFLG1 (K) .EQ.O) GOTO 10 54 9 NOM =NCM+1 550 7EC0 (NOM)=VECI (K) 551 10 CONTINUE 552 EETUEN 167 55.3 END 554 $SIG DEBR PBIO=L T=15S PAGES=8 F0RM=11X15 CABDS=0 BET OBN=CNIB 555 $B PGBAF.G 1=PGSAD (1 1,1 5) + (34, 52) 6=*PBINT* 556 $E PLOT: Q JFLOTTIME=300 PAB=BLANK 557 $SIG 558 $SIG DEBB PRIO=N T=10S PAGES=10 FOBfcl=11X15 CABDS=0 559 $E PGEAF.G+*EBFLOT 1 = PGB AD (11 ,1 5) + (34, 52) 6=*PBINT* -560 $B PGB AF. G**IG 1=PGBAD (1 1,15)+(34,52) 561 9=*EBINT* SPEINT=*PBINT* SEBCOM=*PBINT* 562 $SIG $ 563 **THE FOLLOWING ABE CONSTANTS FOB THE PLOTTING PROGBA M* ****** 564 0 1 565 1.6667 3,. 3. 3. 566 3.5 140. 45. 1000. 3. 3 132. 567 1 1 1 1 1 1 0 0 0 0 568 0 0 0 0 1 1 1 1 1 1 569 1.549 718637846 1 PEAKS 10 570 .4235 .485 .3035 571 7 8.5 15 26.5 43.4 83.1 33.2 59.3 24.8 53.4 67.7 572 7.5 12.5 21 32.5 61 22.8 42 73.6 27. 8 60.3 78.1 573 2. 5 4 6 11 24 48.6 100 41 92.8 42.1 , 574 .0335 .0834 .2146 . 5239 1. 28 3. 254 8,1276 20.63 50. 1 1 125.5 575 .0436 .0968 .1974 .5297 1.310 3.1905 8.231 20.40 49.6 0 126.4 576 .0335 .0873 .2131 .4974 1.2665 3.201 8.07 19.99 49.34 12 5.52 577 7 0 0 0 0 0.1772E+01 0.4670E+01 0.4513E+00 .1445E-01 578 0 579 1.549 718637846 1 PEAKS 6 580 .4235 .485 .3035 581 7 8.5 15 26.5 43.4 83. 1 33.2 59.3 24.8 53.4 67.7 582 7.5 12.5 21 32.5 61 22.8 42 73.6 27. 8.60.3 78.1 583 2. 5 4 6 11 24 48.6 100 4 1 92.8 42.1 584 .0335 .0834 .2146 .5239 1.28 3.254 8.276 20.63 50.11 125.5 585 .0436 .0968 .1974 .5297-1.310 3.1905 8.231 20.40 49.6 0 126.4 586 .0335 .0873 .2131 .4974 1.2665 3.201 8.07 19.99 49.34 125.52 587 7 0 0 0 0 0.1220E+01 0.6274E+01 0.7609E+00 0.2275E-01 588 0 589 1.549 718637846 1 STEADY 10 590 . 4235 .485 ,.3035 591 5.5 7.5 13 18.5 34 66.5 27.6 59.3 24.8 53.4 67.7 592 7.5 11 16.5 26 4 2 16.3 3 3.7 67i7 27. 8 60,-3 78.1 593 2. 5 4 6 9. 5 19 37 81.2 38.4 91. 3 42.1 594 .0351 .0881 ,2153 .5319 1.312 3.186 8.073 20.17 49.85 125.88 595 .0377 .0865 .2193 .5432 1.266 3.2184 8.000 20.04 49.8 6 126.05 596 .0336 .0868 .2175 .5149 1.2727 3.205 7.924 19.74 49.2 6 125.3 1 6 8 5 9 7 2 0 0 0 0 1.965 76. 12 . 07575 2. 189 598 3 0 0 0 0 2.367 49.43 .1184 1.787 599 4 0 0 0 0 31.93 .07943 1.752 2.401 600 5 0 0 0 0 15.25 .1555 2.017 2. 118 601 6 0 0 0 0 -4.742 .9819E-03 2.674 2.370 602 7 0 0 0 0 0. 269.9E + 01 0. 3664E+01 0.3888E+00 -.4947E-0 603 2 8 0 0 0 0 .7564 .5471 3.698 2.766 604 9 0 0 0 0 8.253 .009203 .03089 1.971 605 10 0 0 0 0 5.926 .008919 .0223 2.03 606 13 0 1 0 1 6.497 .3574 .4395 6.543 30. 607 0 608 1. 549 718637846 1 STEADY 6 609 .4235 .485 .3035 610 5.5 7. 5 13 18 . 5 34 66. 5 27.6 59.3 24.8 53.4 67.7 611 7.5 11 16.5 26 42 16.3 3 3 . 7 67.7 27. 8 60.3 78. 1 612 2.5 4 6 9. 5 19 37 81.2 38.4 91. 3 42 . 1 613 .0351 .0881 .2153 .5319 1.312 3.186 8.073 2 0 . 1 7 49.85 125.88 614 .0377 .0865 .2193 .5432 1.266 3.2184 8.000 20.04 49.8 6 126.05 615 .0336 . 0 8 6 8 . 2 1 7 5 .5149 1.2727 3.205 7.924 19.74 49.2 6 125.3 616 7 1 0 0 0 0.4323E+01 0.3582E + 01 0..7979E + 00 -0.4070E-0 617 I 0 618 1.743 714771276 1 STEADY 10 619 .379 . 497 .397 620 4.5 9 21 41 50 89 25 50 21 50 621 8 16 31 55 84 25 46 87 33 70 622 .05 12 23 38 60 90 28 58 24 54 623 .0321 .0811 .2037 .5296 1.3945 3.376 8.105 20.47 50.2 6 125. 9 624 .0278 .4)801 ,.2081 . 5288 1.3694 3.474 8.32 19.93 49.8 9 124.9 1 625 .0320 .0814 .1848 .6546 1.5281 3.4394 8.8636 19.64 49 .90 123.7 626 7 0 0 0 0 0.8450E+00 0.6076E+01 0.4407E+00 0.4868E-01 627 0 628 0. 629 11 0 0 0 0 .126 .83 .209 0. 1.5 

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