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Modeling the resistance to airflow in the human lung Wiggs, Barry James Ryder 1989

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M o d e l i n g the R e s i s t a n c e t o A i r f l o w i n the Human Lung by B a r r y James Ryder Wiggs • S c . , 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 , 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( I n t e r d i s c i p l i n a r y S t u d i e s ) We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1989 © B a r r y R. Wiggs , 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of \c\ V e^{~Av .£Cvd \ ( \CA>~ S'VoCXvtV} The University of British Columbia Vancouver, Canada DE-6 (2/88) A b s t r a c t To examine t h e i n f l u e n c e o f a i r w a y smooth musc le s h o r t e n i n g and a i r w a y w a l l t h i c k n e s s on pulmonary r e s i s t a n c e a model o f the t r a c h e o b r o n c h i a l t r e e has been deve loped which a l l o w s s i m u l a t i o n s o f t h e s e mechanisms o f a i rway n a r r o w i n g . The model i s based on b o t h a s y m m e t r i c a l b r a n c h i n g system proposed by W e i b e l (61) and an asymmetr ic b r a n c h i n g scheme deve loped by H o r s f i e l d ( 2 1 ) . F l u i d mechanic e q u a t i o n s proposed by P e d l e y (49) a r e used t o c a l c u l a t e i n s p i r a t o r y r e s i s t a n c e and t o a l l o w f o r changes i n l u n g i n f l a t i o n the p r e s s u r e a r e a c u r v e s d e s c r i b e d by Lambert(32) a r e u s e d . The model i s e a s i l y implemented u s i n g a s p r e a d s h e e t and p e r s o n a l computer which a l l o w s c a l c u l a t i o n o f t o t a l and r e g i o n a l pulmonary r e s i s t a n c e . A t each g e n e r a t i o n o r o r d e r i n t h e model p r o v i s i o n i s made f o r the a i r w a y w a l l t h i c k n e s s , the maximal a i r w a y smooth musc le s h o r t e n i n g a c h i e v a b l e and an S shaped dose response r e l a t i o n s h i p t o d e s c r i b e the smooth muscle s h o r t e n i n g . Measurements o f a i rway w a l l t h i c k n e s s from 23 normal s u b j e c t s whom d i e d sudden ly and 19 a s t h m a t i c s u b j e c t s whom d i e d o f c o m p l i c a t i o n s o f t h e i r d i s e a s e a r e r e l a t e d t o i n t e r n a l a i rway p e r i m e t e r u s i n g an i t e r a t i v e r e s t r i c t e d maximum l i k e l i h o o d t e c h n i q u e . U s i n g the e s t i m a t e d r e l a t i o n s h i p f o r a i rway w a l l t h i c k n e s s i t i s p o s s i b l e t o p a r t i t i o n the changes t o c e n t r a l o r p e r i p h e r a l a i r w a y s . I t i s c o n c l u d e d t h a t the model p r o v i d e s a r e a l i s t i c q u a l i t a t i v e e s t i m a t e o f the t r a c h e o b r o n c h i a l p r e s s u r e drop t h a t may p r o v i d e v a l u a b l e i n s i g h t s i n t o the i n t e r a c t i o n o f a i rway smooth musc le s h o r t e n i n g and a i r w a y w a l l t h i c k n e s s as i m p o r t a n t c o n t r i b u t o r s t o a i rway h y p e r r e s p o n s i v e n e s s . i i T a b l e o f Content s A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f T a b l e s v i L i s t o f F i g u r e s v i i Acknowledgements i x S e c t i o n 1 I n t r o d u c t i o n S e c t i o n 1.1 I n t r o d u c t i o n 1 S e c t i o n 1.2 Normal Lung Anatomy 2 S e c t i o n 1.3 The Mechanics o f A i r f l o w i n i n t h e Lung 5 S e c t i o n 1.4 A i r w a y s R e s i s t a n c e 7 S e c t i o n 1.5 D i s e a s e s o f the Lung 11 S e c t i o n 1.6 Purpose o f the Work 12 S e c t i o n 2 G e o m e t r i c a l R e p r e s e n t a t i o n o f the Lung S e c t i o n 2 .1 I n t r o d u c t i o n 15 S e c t i o n 2.2 W e i b e l Symmetric Geometry 16 S e c t i o n 2.3 H o r s f i e l d Asymmetr ic Geometry 19 S e c t i o n 2.4 D i f f e r e n c e s i n Asymmetric v e r s u s Symmetric 21 S e c t i o n 3 F l u i d Dynamic C o n s i d e r a t i o n s S e c t i o n 3 .1 Assumpt ions 25 S e c t i o n 3.2 T u r b u l e n t Flow and B i f u r c a t i n g Systems 30 S e c t i o n 3.3 C o r r e c t i o n f o r Lung Volume 32 i i i S e c t i o n 3.4 C a l c u l a t i o n o f the t r a c h e o b r o n c h i a l p r e s s u r e drop 34 S e c t i o n 4 Measurement o f A i r w a y s R e s i s t a n c e i n Man S e c t i o n 4 .1 I n t r o d u c t i o n 40 S e c t i o n 4 .1 Measurement o f R e s i s t a n c e 40 S e c t i o n 4.2 R e g i o n a l R e s i s t a n c e 43 S e c t i o n 5 A i r w a y Smooth Musc l e and A i r w a y W a l l T h i c k n e s s S e c t i o n 5.1 I n t r o d u c t i o n 45 S e c t i o n 5.2 C a l c u l a t i o n o f C o n t r a c t e d Lumenal Diameter 46 S e c t i o n 5.3 A i r w a y W a l l T h i c k n e s s 48 S e c t i o n 5.4 E s t i m a t i o n o f the R e l a t i o n s h i p Between WA and P i 51 S e c t i o n 5.4 Dose Response o f Smooth M u s c l e 56 S e c t i o n 6 R e s u l t s I : Model F o r m u l a t i o n S e c t i o n 6.1 R e l a t i o n s h i p Between W a l l A r e a and I n t e r n a l P e r i m e t e r 59 S e c t i o n 6.2 P r o p o r t i o n o f Musc l e i n the C i r c u m f e r e n c e o f L a r g e A i r w a y s 65 S e c t i o n 6.3 Model E q u a t i o n s 66 S e c t i o n 7 R e s u l t s I I Model Response Under Normal C o n d i t i o n s 67 i v S e c t i o n 8 R e s u l t s I I I Model Response Under Abnormal (Asthmat ic ) C o n d i t i o n s 73 S e c t i o n 9 D i s c u s s i o n 77 Appendix A D e l e e s e ' s Theorem 92 Appendix B W e i b e l ' s D i s t r i b u t i o n C o r r e c t i o n f o r B i a s e d Sampl ing 94 Appendix C C a l c u l a t i o n o f C o n t r a c t e d I n t e r n a l R a d i u s 96 Appendix D R e s t r i c t e d Maximum L i k e l i h o o d E s t i m a t i o n and Gauss Program 98 Appendix E Symmetric and Asymmetric Model C a l c u l a t i o n s 112 T a b l e s 123 F i g u r e s 143 R e f e r e n c e s 199 v L i s t o f T a b l e s T a b l e l a H o r s f i e l d Asymmetric Geometry 123 T a b l e l b W e i b e l Symmetric Geometry 125 T a b l e 2 T r a n s i t i o n M a t r i x Between Models 127 T a b l e 3 L a m b e r t ' s P r e s s u r e - A r e a Parameters 129 T a b l e 4 Symmetric Model 131 T a b l e 5 Asymmtric Model 135 T a b l e 6 Maximum R e s i s t a n c e s 139 T a b l e 7 E x p e c t e d Dose f o r 50% o f Maximum R e s i s t a n c e 141 v i L i s t o f F i g u r e s F i g u r e 1 Bronchogram o f Human Lung 143 F i g u r e 2 C r o s s - s e c t i o n a l A r e a o f t h e Lung 145 F i g u r e 3 P r e s s u r e s i n t h e Lung 147 F i g u r e 4 P r e s s u r e , Flow and Volume P a t t e r n o f B r e a t h i n g 149 F i g u r e 5 Laminar V e r s u s T u r b u l e n t V e l o c i t y P r o f i l e s 151 F i g u r e 6 Diameter o f A i r w a y s V e r s u s G e n e r a t i o n 153 F i g u r e 7 S t r a h l e r O r d e r i n g System f o r R i v e r s 155 F i g u r e 8 P a t h l e n g t h s f o r Symmetric and Asymmetr ic Models 157 F i g u r e 9 P r e s s u r e - A r e a Curves 159 F i g u r e 10 S i n g l e Airway R e s i s t a n c e s 161 F i g u r e 11 Square Root o f W a l l A r e a V e r s u s I n t e r n a l P e r i m e t e r 163 F i g u r e 12 L e a s t Squares L i n e o f Each S u b j e c t 165 F i g u r e 13 S u b j e c t ' s Data and REML F i t s 167 F i g u r e 14 95% C o n f i d e n c e L i n e s f o r Group E s t i m a t e s 169 F i g u r e 15 95% C o n f i d e n c e L i n e s f o r D i f f e r e n c e Between Normals and A s t h m a t i c s 171 F i g u r e 16 Normal P r o b a b i l i t y P l o t s o f S l o p e s and I n t e r c e p t s 173 F i g u r e 17 T o t a l P r e s s u r e Drop on A i r and Hel ium-Oxygen 175 F i g u r e 18 Lambert and P e d l e y Non-Laminar P r e s s u r e A t t e n u a t i o n 177 v i i F i g u r e 19 T r a c h e o b r o n c h i a l R e s i s t a n c e V e r s u s Transpulmonary P r e s s u r e 179 F i g u r e 20 A i r w a y s R e s i s t a n c e by G e n e r a t i o n f o r I s o p l e t h s o f Transpulmonary P r e s s u r e 181 F i g u r e 21 P a r t i t i o n o f R e s i s t a n c e t o Lower A i r w a y s 183 F i g u r e 22 A i r w a y s R e s i s t a n c e by G e n e r a t i o n f o r I s o p l e t h s o f Smooth Musc l e S h o r t e n i n g 185 F i g u r e 23 T o t a l R e s i s t a n c e V e r s u s Dose 187 F i g u r e 24 R e s i s t a n c e v e r s u s Transpulmonary P r e s s u r e f o r I s o p l e t h s o f Encroachment 189 F i g u r e 25 Maximum R e s i s t a n c e f o r D i f f e r e n t Amounts o f Smooth Musc l e S h o r t e n i n g 191 F i g u r e 26 Dose Response R e l a t i o n s h i p f o r I n c r e a s i n g Amounts o f Encroachment 193 F i g u r e 27 Dose Response R e l a t i o n s h i p f o r C e n t r a l V e r u s P e r i p h e r a l A irway W a l l T h i c k e n i n g 195 F i g u r e 28 Frequency and Volume Dependence o f A i r w a y s R e s i s t a n c e 197 v i i i Acknowledgement As an i n t e r d i s c i p l i n a r y s t u d e n t the a d v i c e and gu idance I r e c e i v e d from my committee has been e x t r e m e l y i m p o r t a n t . I am g r a t e f u l t o D r . J . C . Hogg as a c t i n g as committee s u p e r v i s o r and h e l p i n g me see the i m p l i c a t i o n s o f my work. D r . Hogg a l s o r e a l i z e d t h e importance o f combin ing i n f o r m a t i o n from s e v e r a l d i s c i p l i n e s i n o r d e r t o s o l v e t h e problems encountered i n t h i s t h e s i s . D r . P . D . Pare p r o v i d e d i m p o r t a n t s u p e r v i s i o n and spent many h o u r s e x p l a i n i n g r e s p i r a t o r y mechan ic s . D r . J . B e r t e x p l a i n e d t h e f i n e r p o i n t s o f mode l ing and made me r e c o g n i z e the many s t r e n g t h s and weakness o f n u m e r i c a l a b s t r a c t i o n s o f p h y s i c a l phenomena. D r . M . T . S c h e c t e r h e l p e d i n an u n d e r s t a n d i n g o f c l i n i c a l e p i d e m i o l o g y , and D r . F . G l i c k p r o v i d e d s t a t i s t i c a l g u i d a n c e . D r . D. Hardwick i n t r o d u c e d me t o the i d e a o f I n t e r d i s c i p l i n a r y S t u d i e s and spent the t ime t o e x p l a i n t o me j u s t e x a c t l y what the s t u d y o f P a t h o l o g y i s . Many more p e o p l e have h e l p e d me but some o f t h e most v a l u a b l e h e l p has been from D r . J . W r i g h t , D r . R. Moreno, D r . A . James, D r . R Lambert and D r . C . Bosken. I am g r a t e f u l t o the L o v e l a c e B i o m e d i c a l and E n v i r o n m e n t a l R e s e a r c h I n s t i t u t e Inc f o r a c c e s s t o i m p o r t a n t m o r p h o l o g i c d a t a (morpho log ic d a t a was per formed a t the L o v e l a c e I n h a l a t i o n T o x i c o l o g y Research I n s t i t u t e s u p p o r t e d by the N a t i o n a l I n s t i t u t e o f E n v i r o n m e n t a l H e a l t h S c i e n c e s under an I n t e r a g e n c y Agreement w i t h t h e US Energy R e s e a r c h Development A d m i n i s t r a t i o n , c o n t r a c t No. DE-AC04-76EV01013) . i x During my time at the Pulmonary Research Laboratory I have had the assistance of many people, a l l of whom have helped to make my work much easier. Mr. S. Greene has provided me with excellent photography and Mr. J . Comeau has ensured that the computing equipment and programs that were so v i t a l to t h i s work were always working. I can't possibly name everyone at the Pulmonary Lab but everyone has helped, I can only look forward to a happy future with the many friends I have made. Last, but c e r t a i n l y not least, I must acknowledge my wife Veronica for keeping j u s t the r i g h t amount of sanity i n my l i f e so things could keep moving forward. x S e c t i o n 1.1 : I n t r o d u c t i o n The p r i m a r y purpose o f the r e s p i r a t o r y system i s t o s u p p l y oxygen from the atmosphere t o the pulmonary c a p i l l a r y b l o o d and t o t r a n s p o r t c a r b o n d i o x i d e from t h e b l o o d t o the atmosphere . The exchange o f t h e s e two gases i s a c c o m p l i s h e d by p a s s i v e d i f f u s i o n and i t i s t h e need f o r e f f i c i e n t d i f f u s i o n t h a t de termines t h e s t r u c t u r e o f the l u n g . The volume o f gas t r a n s p o r t e d by p a s s i v e d i f f u s i o n i s d i r e c t l y p r o p o r t i o n a l t o the s u r f a c e a r e a f o r d i f f u s i o n , and i n v e r s e l y p r o p o r t i o n a l t o the t h i c k n e s s o f the a i r - b l o o d b a r r i e r i n the l u n g . To meet gas exchange demands t h e human l u n g has e v o l v e d an a l v e o l a r - c a p i l l a r y s u r f a c e a r e a o f 50 t o 100 square meters w i t h a t i s s u e t h i c k n e s s o f 0.5 microns o r l e s s . To a c h i e v e t h i s enormous c r o s s - s e c t i o n a l a r e a i n a system which b e g i n s w i t h a narrow o r i f i c e a t the mouth r e q u i r e s a complex system o f b r a n c h i n g a i r w a y s . The a i r w a y s b r a n c h from 8 t o 23 d i v i s i o n s (60) depending on the pathway and s u p p l y a p p r o x i m a t e l y 300x l0 6 t h i n w a l l e d a l v e o l i t h a t p r o v i d e an immense s u r f a c e a r e a . The t r a n s p o r t o f gases down t h e complex network o f b r a n c h i n g tubes i n v o l v e s v a r i o u s f l u i d dynamic p r i n c i p l e s t h a t govern the magnitude o f the r e s i s t a n c e t o a i r f l o w . A n a t o m i c a l f e a t u r e s such as t h e l e n g t h , r a d i u s and b i f u r c a t i o n o f t h e i n d i v i d u a l a i rway branches a f f e c t the r e s i s t a n c e t o a i r - f l o w . F u n c t i o n a l changes such as smooth muscle s h o r t e n i n g and t h e p r o d u c t i o n o f a i r w a y s e c r e t i o n s can a l s o a f f e c t a i r w a y r e s i s t a n c e by chang ing the a i r w a y s geometry . In 1 normal s u b j e c t s t h i s r e s i s t a n c e i s not g r e a t and does no t i m p a i r b r e a t h i n g b u t i n c e r t a i n p a t h o l o g i c a l c o n d i t i o n s , most n o t a b l y asthma, t h e a n a t o m i c a l f e a t u r e s o f t h e a i r w a y s a r e a l t e r e d i n such a way as t o i n c r e a s e t h e a i r w a y r e s i s t a n c e and t h e r e f o r e t h e work o f b r e a t h i n g . The purpose o f t h i s t h e s i s i s f i r s t t o model the geometry o f t h e t r a c h e o b r o n c h i a l t r e e ; second t o a p p l y a p p r o p r i a t e f l u i d dynamic e q u a t i o n s t o c a l c u l a t e the r e s i s t a n c e o f t h e t r e e and t h i r d l y t o a l t e r the geometry o f t h e t r e e as o c c u r s i n d i s e a s e and t o examine the e f f e c t s o f these a l t e r a t i o n s on a i r f l o w r e s i s t a n c e . S e c t i o n 1.2: Normal Lung Anatomy The normal l u n g anatomy, was perhaps b e s t s t u d i e d i n the e a r l y 1960's by W e i b e l ( 5 9 , 6 0 , 6 1 ) . In h i s s t u d i e s Weibe l u t i l i z e d c a s t s o f l u n g s which were c o n s t r u c t e d by f i l l i n g the a i r passages w i t h f l u i d v i n y l c a s t i n g m a t e r i a l t h a t would e v e n t u a l l y h a r d e n . The l u n g t i s s u e c o u l d t h e n be d i s s o l v e d away r e s u l t i n g i n a c a s t o f the a i rway i n t e r i o r . The c a s t formed by t h i s t e c h n i q u e i s r emarkab ly complex as seen i n t h e bronchogram i n f i g u r e 1. To c r e a t e t h i s image a r a d i o p a q u e m a t e r i a l (here l e a d dust ) was blown i n t o the a i r w a y s and r e c o r d e d on x - r a y f i l m . P h y s i c a l measurements o f the parameters n e c e s s a r y t o c a l c u l a t e r e s i s t a n c e ( l o c a l p r e s s u r e s and f lows) a t v a r i o u s p l a c e s i n t h i s s t r u c t u r e i s i m p r a c t i c a l because o f the v a s t number o f tubes i n v o l v e d . To c i r c u m v e n t t h e s e d i f f i c u l t i e s , 2 models of the lung airways have been constructed i n an attempt to account f o r the branching nature and physical dimensions of the airways. The models that have been proposed by Weibel (60) and H o r s f i e l d (21) w i l l be described i n d e t a i l i n sections 2.2 and 2.3 of t h i s thesis and t h i s information w i l l form the anatomic basis f o r the analysis of airways function to be presented. The tracheobronchial tree, a global term used to describe the en t i r e network of airways, begins at the trachea. From the trachea the system divides into the l e f t and r i g h t main bronchi, these divide into lobar bronchi, then segmental bronchi and sub-segmental bronchi. At t h i s point the airways have divided at most four times and we have 16 major branches. There i s i n fact an asymmetrical nature to the lungs with a greater proportion of the human lung volume being located on the r i g h t side than the l e f t side. The segmental bronchi continue d i v i d i n g u n t i l approximately 17 d i v i s i o n s have occurred from the trachea. The function of these branches of the tracheobronchial tree i s to conduct gas to the exchange surface, t h i s region of the tree i s termed the conducting zone or anatomic dead space. The movement of gas through the conducting system r e s u l t s i n a f r i c t i o n a l pressure loss which constitutes a major contribution to the work of breathing. The pulmonary conducting zone i s separated from the respiratory surface by a t r a n s i t i o n a l group of airways. In the f i r s t portion of the t r a n s i t i o n zone the respiratory bronchioles have a l v e o l i which branch o f f a portion of the bronchioles walls 3 and can provide l i m i t e d gas exchange. After a variable number of branches the respiratory bronchioles gradually transform into alveolar ducts which are completely l i n e d with a l v e o l i and a f t e r a further number of d i v i s i o n s the ducts end i n b l i n d alveolar sacs. The distance from the trachea to the terminal bronchioles i s approximately 30cm, the t o t a l conducting system comprising about 150 ml of the lung volume. The distance from the respiratory bronchioles to the terminal sacs i s only about 5mm but the respiratory zone contains 3000ml of gas (62) The vast increase i n cross-sectional area (Figure 2) i s due to t h i s multiple d i v i s i o n of the airways of the lung, 8 to 23 times, to a t o t a l number of over 300 m i l l i o n a l v e o l i some V of a ' 3 millimeter i n diameter. The branches of the airways from the trachea to the respiratory bronchioles are l i n e d with s p e c i a l c i l i a t e d e p i t h e l i a l c e l l s and a mucous f l u i d layer. The mucus i s secreted by mucus glands and goblet c e l l s i n the bronchial walls and serves to capture inhaled p a r t i c l e s and to humidify the a i r . Some c e l l s have long (3-4 um) slender cytoplasmic extensions ( c i l i a ) which are powered by a f i b u l a r network and beat i n a rhythmic pattern p r o p e l l i n g the mucus and the p a r t i c l e s embedded i n i t to the e p i g l o t t i s where they are swallowed. The bronchiolar epithelium changes to a layer of cuboidal c e l l s i n the terminal bronchioles and to cuboidal c e l l s interspaced with alveolar sacs i n the respiratory bronchioles. Once the alveolar ducts are reached there i s a complete l i n i n g of alveolar sacs. The t i s s u e i n the airway wall surrounding the lumen i s 4 c o m p r i s e d o f smooth musc l e , g l a n d s , b r o n c h i a l b l o o d v e s s e l s , s u p p o r t i n g c o n n e c t i v e t i s s u e and the e p i t h e l i a l s u r f a c e l a y e r . The r e l a t i v e p r o p o r t i o n s o f these t i s s u e s i s dependent upon the p o s i t i o n i n t h e b r o n c h i a l t r e e . The smooth musc le c o m p l e t e l y s u r r o u n d s the a i r w a y s from t h e segmental b r o n c h i d i s t a l l y i n a s p i r a l f a s h i o n . When c h a l l e n g e d w i t h an a p p r o p r i a t e a g o n i s t the smooth musc le w i l l r e a c t by s h o r t e n i n g , t h u s c o n t r a c t i n g down around t h e w a l l o f t h e a i rway and u l t i m a t e l y n a r r o w i n g the a i r w a y lumen and r e d u c i n g t h e c r o s s - s e c t i o n a l a r e a a v a i l a b l e f o r a i r f l o w . S e c t i o n 1 .3: The Mechanics o f A i r f l o w i n t h e Lung The f o r c e s r e q u i r e d t o i n f l a t e the l u n g can be s e p a r a t e d i n t o t h r e e p o r t i o n s which a r e e a s i l y seen by examin ing the f o r c e s o p p o s i n g t h e t r a n s p u l m o n a r y p r e s s u r e . Transpulmonary p r e s s u r e i s t h e p r e s s u r e d i f f e r e n c e between p r e s s u r e a t the mouth and the p r e s s u r e i n the p l e u r a l space ( F i g u r e 3) . The t r a n s p u l m o n a r y p r e s s u r e , P L , i s opposed by t h e e q u a l and o p p o s i t e p r e s s u r e s which a r e e l a s t i c , P , f r i c t i o n a l o r r e s i s t i v e , P , and i n e r t i a l , P , i n n a t u r e . The i n e r t i a l ' f r ' ' i n ' p r e s s u r e i s r e l a t e d t o the volume a c c e l e r a t i o n o f gas (V) and i s i n s i g n i f i c a n t d u r i n g normal q u i e t b r e a t h i n g . I n e r t i a l p r e s s u r e drops w i l l t h e r e f o r e no l o n g e r be c o n s i d e r e d . The r e s i s t i v e p r e s s u r e d r o p s a r e p r o p o r t i o n a l t o f l o w , V, and de termined by r e s i s t a n c e w h i l e the e l a s t i c p r e s s u r e l o s s e s a r e de termined by t h e c o m p l i a n c e o f t h e l u n g and a r e p r o p o r t i o n a l t o changes i n 5 volume, V . The e q u a t i o n s o f mot ion o f t h e l u n g can t h e r e f o r e be seen t o be : P L = ( E * V ) + ( R * V ) + ( I * V) (1 .3 .1 ) where E , R and I a r e the e l a s t a n c e , r e s i s t a n c e and i n e r t a n c e , V i s t h e vo lume, V i s f low o r d V / d t , and V i s r a t e o f change o f * i d ^ A . 2 f low o r ' d t . The l u n g s a r e suspended i n the c h e s t c a v i t y w i t h a p o t e n t i a l space ( p l e u r a l space) between the t i s s u e o f the lungs and t h e c h e s t w a l l . In o r d e r f o r a i r t o f low from the s u r r o u n d i n g atmosphere i n t o the lungs a p r e s s u r e d i f f e r e n c e must be d e v e l o p e d between the the mouth and the a l v e o l i . T h a t i s , the p r e s s u r e i n t h e a l v e o l i must become n e g a t i v e w i t h r e s p e c t t o mouth p r e s s u r e . A i r w i l l c o n t i n u e t o f low i n t o t h e l u n g s u n t i l t h e p r e s s u r e i n the a l v e o l i e q u a l s mouth p r e s s u r e . Below the l u n g s i s t h e abdominal c a v i t y , and between t h i s abdominal c a v i t y and t h e c h e s t c a v i t y i s t h e p r i n c i p l e muscle o f i n s p i r a t i o n , the d iaphragm. The a c c e s s o r y musc les o f i n s p i r a t i o n a r e t h e s c a l e n e , s t e r n o m a s t o i d and i n t e r c o s t a l m u s c l e s . A t t h e b e g i n n i n g o f i n s p i r a t i o n t h e r e i s no a i r f low i n the t r a c h e o b r o n c h i a l t r e e . A t t h i s volume the l u n g i s a t i t s e q u i l i b r i u m p o s i t i o n and a l v e o l a r p r e s s u r e i s z e r o r e l a t i v e t o t h e mouth p r e s s u r e . The p r e s s u r e i n the space between t h e l u n g and c h e s t w a l l , t h e i n t r a p l e u r a l p r e s s u r e , i s about 5 cmH20 below mouth p r e s s u r e . T h i s -5 cmH20 i s the p r e s s u r e a t which a b a l a n c e o c c u r s between the tendency o f t h e c h e s t w a l l t o s p r i n g out and t h e e l a s t i c r e c o i l o f the l u n g p u l l i n g i n w a r d . To b e g i n i n s p i r a t i o n t h e musc les o f the diaphragm c o n t r a c t c a u s i n g the 6 diaphragm to move down into the abdominal cavity. The accessory muscles of i n s p i r a t i o n i n turn contract p u l l i n g the chest upward and outward. The combined r e s u l t of the action of these muscles i s to lower the pressure i n the thoracic cavity, the p l e u r a l pressure becomes more negative with respect to mouth pressure. At the s t a r t of i n s p i r a t i o n i n t r a p l e u r a l pressure i s *-5cmH20 and i t decreases to <*-8cmH20 at the end of i n s p i r a t i o n . Most of t h i s change of 3cmH20 i s used to stretch the e l a s t i c lung tissue, and when resistance i s normal only *lcmH20 at peak i n s p i r a t i o n i s communicated to the a l v e o l i . That i s , the alveolar pressure during i n s p i r a t i o n begins at 0 cmH20, f a l l s to -1 cmH20 and then returns to zero at the end of i n s p i r a t i o n . Since the only way for i n s p i r a t o r y airflow to occur i s when alveolar pressure drops below mouth pressure i t should not be sur p r i s i n g that the form of the flow signal i s exactly i n phase with the alveolar pressure. In other words, the flow at the beginning of i n s p i r a t i o n i s 0 1/sec, increases to a maximal value when the alveolar pressure reaches i t s minimum of -1 cmH20 and then f a l l s to 0 1/sec at end i n s p i r a t i o n . This ent i r e sequence i s demonstrated i n the series of graphs i n figure 4. Section 1.4: Airways Resistance P o i s e u i l l e (3) noted that the flow down a pipe was proportional to the applied pressure. When flow i s l i n e a r l y related to the pressure head laminar flow e x i s t s . In laminar 7 flow the p r o f i l e s of moving molecules are arranged i n an orderly fashion p a r a l l e l to the airway walls (figure 5). Near the airway wall the v e l o c i t y of the p a r t i c l e s i s zero and the v e l o c i t y increases to a maximum at the center of the moving stream. P o i s e u i l l e studied the pressure-flow c h a r a c t e r i s t i c s that occur with laminar flow. Under the conditions of laminar flow i n a smooth c i r c u l a r r i g i d pipe the change i n pressure down the tube i s : V 8 7} 1 AP = , where (1.4.1) TT r AP = pressure drop down the tube V = flow Tj = v i s c o s i t y of the gas 1 = length of the tube r = radius of the tube The resistance of the tube i s the change i n pressure, AP, divided by flow, V. Obviously the resistance i s very s e n s i t i v e to changes i n the c a l i b e r of the tube since i t varies inversely as the fourth power of the radius. As flow becomes higher the molecules become more random i n t h e i r movement and the v e l o c i t y stream l i n e s are no longer p a r a l l e l to the walls of the tube. Whether flow i n a tube w i l l be turbulent can be determined by the non-dimensional parameter Reynold's number, N . The formula for Reynold's number fo r f u l l y developed flow i s : 8 2 r U p N = , where (1.4.2) Re 7) ' x r and T/ are the airway radius and gas v i s c o s i t y as before U=average v e l o c i t y of the gas p=density of the gas Physical experimentation has determined that below Reynold's numbers of *2100 flow i s laminar i n smooth walled pipes and, i n such tubes, equation 1.4.1 w i l l apply for c a l c u l a t i n g AP and determining resistance. From the equation f o r N i t i s apparent that turbulent flow i s most l i k e l y to occur when there i s a high v e l o c i t y of gas flow i n a large diameter tube or airway. The tracheobronchial tree can be viewed as a series of smooth walled tubes arranged i n series and p a r a l l e l . In the human lung the branching scheme already described i s such that i n a large part of the lung the cross-sectional area of the airways a v a i l a b l e for flow i s so vast that the average v e l o c i t y i s very small, and the diameters of in d i v i d u a l airways i s small. These facts r e s u l t i n very low Reynold's numbers i n most of the lung and hence laminar flow. In central airways, most notably the trachea, main-stem bronchi, lobar and segmental bronchi, the diameters are large enough and gas v e l o c i t i e s f a s t enough for turbulent flow to e x i s t . During turbulent flow the pressure required to produce a given flow rate i s greater than under laminar conditions and the pressure drop i s nonlinearly related to flow i n these regions. Other factors i n the tracheobronchial tree that must be considered when determining the appropriate 9 f l u i d mechanics equations are alternating flow and bif u r c a t i o n s . The f l u i d dynamics of the pressure and flow relationships within the tracheobronchial tree are discussed i n more d e t a i l i n section 3. The reason to model the lung so that changes i n pressure, and airway resistance can be examined i s because these parameters can be measured i n human subjects and used to val i d a t e the t o t a l pulmonary resistance response of the model. One technique, which w i l l be discussed i n more d e t a i l l a t e r , to measure resistance involves comparing flow at the mouth measured with a pneumotachygraph to transpulmonary pressure. Transpulmonary pressure i s measured by comparing mouth and ple u r a l pressure using a d i f f e r e n t i a l pressure transducer. Pleural pressure i s estimated using an a i r f i l l e d balloon which i s inserted into the lower esophagus. Since the esophagus traverses the ple u r a l space and i s a f l a c c i d tube, pressure changes within the esophagus r e f l e c t pressure changes i n the pleu r a l space. During t i d a l breathing transpulmonary pressure swings (Pmouth-Ppieurai) represent the pressure applied to overcome the elastance, resistance and inertance of the lungs. As already discussed i n e r t i a l losses are r e l a t i v e l y unimportant at t i d a l breathing frequencies and the pressure necessary to overcome e l a s t i c forces can be separated from the r e s i s t i v e pressure drop by using information concerning the phase c h a r a c t e r i s t i c s of the r e s i s t i v e losses. 10 Section 1.5: Diseases of the Lung A c h a r a c t e r i s t i c feature of asthmatic patients i s that t h e i r airways w i l l c o n s t r i c t when exposed to inhaled i r r i t a n t s that do not e f f e c t normal subjects (26). In the l e a s t severe form of asthma the c o n s t r i c t i o n can be reversed by appropriate therapy. In the more severe cases a c e r t a i n degree of airway obstruction may already be present and be markedly enhanced by future episodes which may or may not be rev e r s i b l e . Although i t occurs rarely, severe airway obstruction may lead to the death of asthmatic patients. I t has been proposed (19,20) that the rev e r s i b l e airways obstruction i n asthmatic patients and the more severe, nonreversible mucus plugging of the airways are a r e s u l t of an inflammatory reaction i n the tracheobronchial tree. In an inflammatory reaction i n the airways the blood vessels i n the walls of the airways become d i l a t e d and f l u i d may leak from these vessels into e i t h e r the airway wall t i s s u e or even the airway lumen. These conditions are responsible f o r the i n i t i a l redness and swelling of injured t i s s u e and the l a t e r migration on inflammatory c e l l s to the s i t e of injury. A spe c i a l feature of the inflammatory reaction i n tissues that are covered by mucous (as i n the airways) i s the secretion onto the surface of mucus and a shedding of the e p i t h e l i a l layer l i n i n g the airways. The inflammatory reaction i s also accompanied by an accumulation of c e l l s at the s i t e of inflammation. I f t h i s s i t e i s i n the airway walls there can p o t e n t i a l l y be swelling of the airway 11 wall as a r e s u l t of t h i s . There i s r e l a t i v e l y l i t t l e evidence that the smooth muscle of asthmatic airways behaves abnormally. In some studies the i n v i t r o c o n t r a c t i l i t y of airway smooth muscle obtained from asthmatic patients (4,6,7,14,53,55,63) has been examined d i r e c t l y . These data suggest that the " s e n s i t i v i t y " of asthmatic airway smooth muscle to pharmacologic agents i s not d i f f e r e n t from normal. In two studies (7 and 6) the airway smooth muscle preparations of three asthmatic patients developed 2-3 times the amount of tension generated by s i m i l a r preparations from large groups of non-asthmatic subjects. Although these data suggest that the airway smooth muscle may be capable of developing more tension when stimulated maximally, the tension generated was not corrected for the cross-sectional area of smooth muscle present i n these i n v i t r o preparations. As the amount of smooth muscle i n the asthmatic airways i s increased, the observed increase i n tension could be explained by the increased amount of smooth muscle rather than abnormal muscle function. I t seems reasonable to consider the i n t e r a c t i n g e f f e c t s of changes i n airway wall thickness, possibly due to an inflammatory reaction i n the airways with otherwise normal smooth muscle shortening as an important sequence of events i n the study asthma. Section 1.6 Purpose of the Work Models are abstract descriptions of systems that usually 12 combine applicable physical laws and p r i n c i p l e s , p i c t o r i a l representation and v a r i a t i o n of system inputs that may or may not a f f e c t the system outputs. Since models are abstractions decisions must be made as to which elements w i l l be included i n the model and which w i l l be ignored. The complexity of the tracheobronchial tree forces any deta i l e d investigation into the e f f e c t s of subtle a l t e r a t i o n s of the geometry to be performed on models that simulate the actual tree. The advantage of using models i s the f i n e degree of control that i s possible on a l l parameters. The disadvantage of using a model i s that we must be very c a r e f u l of the information used to generate the model and of any conclusions made using r e s u l t s from the model. Furthermore, we can only input into a computer known relationships and information. The study of changes i n airway geometry i n asthma i s i d e a l l y suited for modeling. There e x i s t s a complex structure on which a great deal of morphometric measurements have been made. The development of a model also provides a method to t e s t f o r the e f f e c t s of changes that are occurring within the lung. I t may also be possible with t h i s model to examine the e f f e c t of subtle changes i n various parameters that can not be detected with current techniques or have not yet been investigated. In creating any model i t i s usual to begin with a purpose which i n t h i s case w i l l be to examine the resistance to airflow i n the human lung. A f t e r a purpose i s described the l e v e l of d e t a i l expected from the model must be determined. In t h i s work the model i s expected to give q u a l i t a t i v e estimates of the 13 changes i n airways resistance under various conditions and to serve as a foundation for more advanced models as further information becomes available. The next important modeling step i s to determine the conditions or r e s t r i c t i o n s under which the model i s to be used. F i n a l l y , a f t e r the model i s created, i t must be assessed for accuracy as determined by the r e s t r i c t i o n s under which i t was created. To meet these modeling goals the purpose of t h i s work i s : A) To combine current geometrical representations of the tracheobronchial tree proposed by Weibel and H o r s f i e l d with appropriate f l u i d dynamic equations from the works of Pedley, Chang and Lambert to generate a mathematical representation of a network of pipes which can be used to q u a l i t a t i v e l y estimate the pressure flow relationships i n a human lung during quiet i n s p i r a t i o n . B) To compare the pressure losses and pressure loss response to flow and gas properties between the symmetrical structure of Weibel and the asymmetrical structure of H o r s f i e l d . C) To use the H o r s f i e l d and Weibel models to investigate the consequences of various degrees and l o c a t i o n of smooth muscle shortening and airway wall thickness on the t o t a l and regional pulmonary resistance. 14 Section 2 : Geometrical Representation of the Lung Section 2.1 : Introduction In the early s i x t i e s Weibel (59) noted that because of the rapid advances being made i n physiological measurements i n man there was a need for detailed quantitative information about the lung. Therefore he set out to obtain d e t a i l e d morphologic information about the dimensions, shapes, surfaces, geometrical arrangements and the in t e r - r e l a t i o n s h i p s between these parameters i n both normal and pathological states of the lung. P r i o r to Weibel most investigations of the pulmonary structure were des c r i p t i v e i n nature. His great contribution was to develop the a n a l y t i c a l techniques required to provide quantitative information and most of what has been done since h i s c l a s s i c studies are dependent on Weibel's o r i g i n a l measurements and the techniques he developed i n the early s i x t i e s . Weibel combined d i r e c t measurements of length and diameter i n v i n y l casts of human lungs with h i s t o l o g i c a l measurements of the smaller structures on autopsy lungs. The a n a l y t i c a l techniques used by Weibel were developed by himself from the geometrical relationships of tissue and the app l i c a t i o n of multi-stage sampling techniques to random sections of structures (60) . Ho r s f i e l d , i n the l a t e s i x t i e s , expanded on the morphology of the bronchial tree investigating i t s asymmetrical nature by implementing geomorphological methods used i n r i v e r mapping (21) . 15 Section 2.2 Weibel Symmetric Geometry The f i r s t d e t a i l e d investigation of the geometry of the lung published by Weibel was to count the number of a l v e o l i . The a l v e o l i are the extreme endpoints i n the tracheobronchial tree and are the primary area of gas exchange. Using a p r i n c i p l e c a l l e d Delesse's theorem (8) i t i s possible to r e l a t e the f r a c t i o n of volume that some p a r t i c l e occupies to the f r a c t i o n of surface area the p a r t i c l e occupies of a random s l i c e through that volume (see Appendix A). In f i v e normal human lungs Weibel found that the average number of a l v e o l i i n both lungs was 296,000,000 with a c o e f f i c i e n t of v a r i a t i o n of only 4%. With t h i s information about the t o t a l number of endpoints i n the complex geometrical tree Weibel proposed possible structures to account f o r these data. Weibel modeled the airways from the trachea to the terminal elements. These terminal elements were defined to be the t o t a l number of respiratory ducts and sacs i n the lungs. Based on his previous technique for counting the a l v e o l i i t was possible for him to determine the si z e and number fo r these terminal elements. Furthermore, Weibel concluded that these ducts and sacs branch three to four times so that the l a s t four generations of any structure thought to represent the tracheobronchial tree should approximate the experimental number of terminal elements, 14*106. As seen i n the bronchogram (figure 1) of the lung the general structure can be e a s i l y interpreted as a dichotomy or a 16 d i v i s i o n of one branch (parent) into two branches (daughters). Weibel noted there i s considerable v a r i a t i o n i n the lengths and diameters of some daughter branches as well as parent branches which divide into three daughters. Regardless of these shortcomings he selected a regular and symmetric branching scheme because i t c l e a r l y described the basic shape of the lung. The r e g u l a r i t y implies that a l l branch d i v i s i o n s are dichotomies and symmetry i n t h i s context implies that daughter branches from a single parent are of exactly the same length and diameter. Using the morphometric measurements on the number of terminal elements (14 x 106) i t i s easy to see that the sum of the l a s t 20 21 22 23 four d i v i s i o n s i n a dichotomy (2 +2 +2 +2 ) c l o s e l y approximates t h i s (15xl0 6) when using a tree of twenty-three d i v i s i o n s . Understanding that a s t r i c t symmetric dichotomy w i l l not exactly describe a lung, Weibel decided that the information that would be gained by using such a simple structure would give valuable insights into the function of the lung. The term, generation, i n Weibel's model, refers to the l e v e l one has t r a v e l l e d down the tracheobronchial tree. By denoting the trachea as generation zero, which divides into the r i g h t and l e f t main bronchus, generation 1, and then into segmental bronchi, generation 2 and so on we see that the number of branches at any generation, n(z), i n t h i s regular dichotomy i s n(z)=2 z. In order to obtain the morphological information concerning lengths and diameters of the larger branches of the airways 17 Weibel borrowed casts of the human lung from Dr. A.A. Liebow (60). Using these casts i t was possible to make at lea s t p a r t i a l measurements down to the tenth generation. The d i f f i c u l t y i n working with these casts was that smaller branches were broken o f f , t h i s meant that the data concerning diameters and lengths were skewed towards larger values. This biased method by which the diameters and lengths of the branches were measured forces some assumptions to be made so that the entir e d i s t r i b u t i o n of the measurements can be completed. The technique used by Weibel to complete the d i s t r i b u t i o n s of the diameters and branches of the tracheobronchial tree and obtain more accurate estimates of mean values i s described i n appendix B. Since only generations 0 to 10 could be measured on a cast and generations 20 to 23, the terminal elements, were measured morphometrically, the res t of the airways, generations 11 to 19,would have to have t h e i r geometry determined by another method. When the log of the mean diameter i s plotted against generation a near l i n e a r r e l a t i o n s h i p i s observed for generations 0 to 10. The terminal elements, generations 20 to 23, are larger than would be predicted by t h i s l i n e a r r e l a t i o n s h i p based on the central airways ( f i g 6). Weibel choose to represent the relationships of diameter (and length) versus generation as two equations. A l i n e a r equation i s used for generations 0 to 3 and a quadratic equation for generations 4 to 23. 18 G e n e r a t i o n s z<= 3 1.8cm 12cm (2 .2 .1 ) (2 .2 .2 ) G e n e r a t i o n s z= 4 t o 23 (2 .2 .3 ) (2 .2 .4 ) W i t h t h i s s e t o f e q u a t i o n s and the dichotomous b r a n c h i n g scheme W e i b e l had c r e a t e d the f i r s t mathemat i ca l r e p r e s e n t a t i o n o f the l u n g which has been used by i n v e s t i g a t o r s f o r many y e a r s . S i m i l a r t o W e i b e l , H o r s f i e l d a l s o made h i s measurements d i r e c t l y from a c a s t o f the t r a c h e o b r o n c h i a l t r e e . The c a s t used by H o r s f i e l d was made on a s i n g l e male and was complete from the t r a c h e a down t o t h e t e r m i n a l b r o n c h i o l e s . T h i s meant t h a t each b r a n c h t y p e c o u l d be randomly sampled from the e n t i r e p o p u l a t i o n o f branches and d i d not r e q u i r e any c o r r e c t i o n f o r b i a s e d s a m p l i n g as W e i b e l d i d . H o r s f i e l d based h i s v a l u e s on 13,164 measured b r a n c h e s . A complete d e s c r i p t i o n o f t h e p a r e n t and daughter b r a n c h scheme i s g i v e n i n T a b l e l a , f o r compar i son W e i b e l ' s d a t a i s i n t a b l e l b . H o r s f i e l d has w r i t t e n s e v e r a l papers d i s c u s s i n g the s t r u c t u r e o f b r a n c h i n g systems ( 2 1 , 2 2 , 2 3 ) . A major p o r t i o n o f h i s i n v e s t i g a t i o n i n v o l v e d a scheme by which asymmetries i n the S e c t i o n 2.3 H o r s f i e l d Asymmetr ic Geometry 19 human l u n g c o u l d be r e p r e s e n t e d g e o m e t r i c a l l y so as t o improve on t h e m o r p h o l o g i c a l r e s u l t s o f W e i b e l . By implement ing the g e o m o r p h o l o g i c a l mapping t e c h n i q u e s o f Hagget t (17) and S t r a h l e r (57) , which t h e y used f o r d e s c r i b i n g r i v e r b r a n c h i n g , i t was p o s s i b l e t o c r e a t e a c o n v e n i e n t s t r u c t u r e f o r the t r a c h e o b r o n c h i a l t r e e . I n t h e s e b r a n c h i n g networks each p a r e n t b r a n c h , s tream o r tube r e g u l a r l y d i v i d e s i n t o two daughter b r a n c h e s . The d i f f e r e n c e i n t h i s geometry from t h a t o f W e i b e l ' s t e c h n i q u e i s t h a t t h e two daughter branches can have d i f f e r e n t l e n g t h s and w i d t h s . The degree t o which the daughters d i f f e r i n t h e i r geometry d e f i n e s t h e asymmetry o f the network. S t r a h l e r ' s method o f o r d e r i n g t h e branches o f a network was t o d e f i n e t h e r i v e r mouth as some h i g h v a l u e such as 50 (the r e q u i r e d v a l u e c o u l d be d e t e r m i n e d a f t e r t h e e n t i r e r i v e r was mapped). When two branches met t h e p a r e n t b r a n c h was assumed t o be one o r d e r lower than the o r d e r o f t h e lowes t o r d e r e d daughter b r a n c h ( f i g u r e 7 ) . The d i f f e r e n c e between o r d e r s o f two daughters t h a t meet i s ano ther d e t e r m i n a n t o f the asymmetry o f the network. H o r s f i e l d a l t e r e d the o r d e r i n g scheme o f S t r a h l e r by e s t a b l i s h i n g the t r a c h e a as o r d e r 31 and the a l v e o l i as o r d e r 0. A t each b i f u r c a t i o n the daughters d i v i d e d w i t h some s e t asymmetry o f D (the b r a n c h i n g d e l t a ) , u s u a l l y s e t t o 3. T h i s meant t h a t each p a r e n t , o f o r d e r X , d i v i d e d i n t o one b r a n c h o f o r d e r X - l and a n o t h e r b r a n c h o f o r d e r X - 4 . The b r a n c h i n g d e l t a o f t h i s system i s d e f i n e d as D=X-1-(X-4)=3. In t h i s s t r u c t u r e i t i s e v i d e n t t h a t i f the b r a n c h i n g d e l t a i s s e t t o 0 t h e n the 20 symmetrical scheme of Weibel w i l l r e s u l t . The central airways, the trachea to the bronchopulmonary segments or f i r s t f i v e d i v i s i o n s , can be accurately dissected and measured on human lungs. There are p r a c t i c a l reasons to describe t h i s small but aerodynamically important section of the tracheobronchial tree since i n t h i s region there are geometrical idiosyncrasies that can not e a s i l y be represented by a systematic model. Beyond t h i s section of the lung H o r s f i e l d used h i s asymmetrical branching scheme with a delta value of 3. Once down to the tenth order the branching delta value was progressively altered from 3 to 0 to r e f l e c t the increasing uniformity of the lung as one approaches the a l v e o l i . Section 2.4 Differences i n Asymmetric versus Symmetric The differences i n the two geometrical representations presented i s i n t h e i r branching structure only. I f a delta value of zero i s chosen for the H o r s f i e l d model then, except for the central airways, the symmetric Weibel tree r e s u l t s . When viewing e i t h e r a bronchogram of the lung or discussing the anatomy of the lung i t i s c l e a r that the asymmetric model i s a more f a i t h f u l representation of the true structure. Observation of the lung structure c l e a r l y shows long and short pathways, as well as daughter branches of c l e a r l y d i f f e r e n t diameters and lengths j o i n i n g . Since Weibel's model has only symmetric d i v i s i o n s the pathway length for each branch i s i d e n t i c a l . In the H o r s f i e l d design the asymmetry allows for a d i s t r i b u t i o n of 21 pathway lengths and branch d i v i s i o n s from the trachea to the a l v e o l i ( f i g 8). Why then has the vast majority of investigations of the rela t i o n s h i p s within the tracheobronchial tree r e l i e d on the symmetric geometry as described by Weibel? The answer to the obvious preference i n s e l e c t i n g the symmetric instead of the asymmetric network i s the complexity of implementing the asymmetrical design. The Weibel geometry and the associated c a l c u l a t i o n s are simple to use and understand, two important considerations f o r b i o l o g i c a l models. The more complex Ho r s f i e l d model i s very d i f f i c u l t to implement and write equations f o r as w i l l become evident i n l a t e r sections. More recently a very detailed morphological study has been performed by Raabe et a l . (51) i n which measurements of the tracheobronchial geometry, including branching angles, were made on human, dog, r a t and hamster lungs. In order to obtain accurate and complete measurements concerning the geometry of lung airways s i l i c o n e rubber casts of the conducting airways were prepared. S i l a s t i c E type s i l i c o n e rubber was slowly injected into the trachea of two human lungs obtained at autopsy. This p a r t i c u l a r casting medium was chosen because i t produces a strong f l e x i b l e cast that when suspended w i l l not deform under i t s own weight. In addition t h i s material has less than 0.1% shrinkage allowing casts corresponded to end i n s p i r a t i o n geometry to be obtained. In the case of animal casts the s i l i c o n e was injected into a closed-chest animal. Using these casts the lengths and widths were measured 22 along with a unique branch i d e n t i f i e r . This binary branch i d e n t i f i e r was used so that any branch i n the cast system could be e a s i l y i d e n t i f i e d . Using Weibel's basic b i f u r c a t i n g system the trachea i s denoted as branch 1 which divides into two daughter branches, the l e f t and r i g h t main bronchi. The larger diameter bronchi i s i d e n t i f i e d as branch 11 and the smaller diameter branch as 12. Each of these bronchi generate two daughters and these branches w i l l be coded as 111, 112, 121 and 122. The number of d i g i t s i n the branch i d e n t i f i e r therefore i d e n t i f i e s the Weibel generation and a simple computer program can be written to obtain the data for a H o r s f i e l d geometrical structure. A l l of the data for the four species studied i s avail a b l e on computer tape. When comparing calculated pressure drops and resistance values between the symmetric and asymmetric geometries i t i s important to use s t a r t i n g geometries of the same dimensions. Obviously i f the s t a r t i n g r a d i i of the symmetric model are a l l les s than those i n the asymmetric model we would expect larger pressure drops based on equation 1.4.1. I t i s possible, as previously shown by Martonen (39) to form a t r a n s i t i o n matrix between these two geometries. Table 2 shows the correspondence between the H o r s f i e l d orders and Weibel generations. This table gives the number of H o r s f i e l d branches and where they are located i f they were structured according to the symmetric model. Considering table 2 as a matrix T with a row vector R representing the column sums and a column vector C representing the row sums we can move any mean vector of airway diameters and 23 lengths between the two models. For instance, i f we have the dimensions for a Weibel model, W, the corresponding H o r s f i e l d dimensions are: H = T*W/R This gives T*W, a row vector, which when each of the 31 elements are divided by the corresponding element of R gives an estimate of H o r s f i e l d dimension. I t i s t h i s technique that was p r i n c i p a l l y used i n t h i s work to move morphometric measurements between the two geometries. 24 Section 3: F l u i d Dynamic Considerations Section 3.1: Assumptions The purpose of imposing a s t r u c t u r a l geometry on the tracheobronchial tree i s so that these defined geometries can serve as a basis f o r f l u i d dynamic equations. Each of these structures, e i t h e r Weibel or Horsfield, can be viewed as a connecting set of pipes down which f l u i d can flow. By making various assumptions regarding the shape of the pipes as well as physical properties and nature of the gas flowing during quiet i n s p i r a t i o n i t i s possible to estimate the drop i n pressure i n t h i s network. With the information regarding the pressure drop estimates i n the tracheobronchial tree i t i s then possible to calc u l a t e the resistance of the en t i r e tree as well as the lon g i t u d i n a l d i s t r i b u t i o n of t h i s resistance i n the lungs. An important parameter i n the study of f l u i d dynamic properties i s the nondimensional value Reynold's number, N . RE Reynold's number i s equal to: N = p * TJ * D RE (3.1.1) where P density of the f l u i d U f l u i d stream v e l o c i t y D diameter of the pipe f l u i d v i s c o s i t y 25 The f l u i d v i s c o s i t y (ju) i s an i n d i c a t i o n of a f l u i d s tendency to shear over t h i n sheets of i t s e l f , that i s , the a b i l i t y of the molecules to s l i d e past each other. The f l u i d density (p) i s the mass per unit volume and i s important i n determining the acceleration c h a r a c t e r i s t i c s of the f l u i d . Stream v e l o c i t y (U) i s the average a x i a l v e l o c i t y ( p a r a l l e l to the wall) i n the pipe. The diameter of the pipe i s the f i n a l piece of information required. The formula for Reynold's number i s a c t u a l l y a r a t i o of the i n e r t i a l forces to the viscous forces. I t i s also easy to see the nondimensional property of Reynold's number by examining the units of i t s components: N R E = (kg/m3) (m/s) (m)/(kg/m s) = 1 Any f l u i d which has a v i s c o s i t y w i l l sometimes flow smoothly (laminar flow) and at other times roughly (turbulent flow). At Reynold's number below 2100 the flow tends to be behave i n a laminar fashion i n a smooth walled c i r c u l a r pipe. Above values of 3,000 flow i s generally dependent on the geometrical properties of the system i n which the f l u i d i s flowing. Steady flow i n converging nozzles for instance have higher c r i t i c a l Reynold's numbers (the N at which flow becomes RE turbulent) than flow i n expansion valves. Another useful value i n the study of pipe flow i s the f r i c t i o n factor, f, also a nondimensional value. 26 f = (D/4) * (-AP/1) (3.1.2) — p u 2 This value r e l a t e s f r i c t i o n a l or head loss as a f l u i d flows i n a pipe to the k i n e t i c energy of the f l u i d and the pipe geometry (3) . A p l o t of f r i c t i o n factor against Reynold's number i s termed a Moody diagram and changes i n the slope of t h i s p l o t are useful i n determining changes i n f l u i d flow regimes. Several assumptions w i l l be used i n order to simplify the numerical equations needed to describe the pressure losses i n the tube networks that w i l l be used to describe the tracheobronchial tree. The f l u i d that flows i n t h i s network i s obviously a i r . I t w i l l be assumed that the a i r i s adiabatic, t h i s implies that there i s no heat transfer from the a i r to the surrounding lung ti s s u e , obviously t h i s d i c tates that there i s no change i n temperature of the a i r while i n the network (50) . Additional conditions imposed are that the a i r i s incompressible and flows i n a steady uniform manner as opposed to a c y c l i c a l breathing pattern. The incompressible assumption can be addressed by examining the Mach number, Ma, which i s defined as the f l u i d v e l o c i t y divided by the speed of sound i n the f l u i d . Below Mach numbers of 0.3 f l u i d flow i s considered incompressible (46) . The speed of sound i n a i r at body temperature (38°C) i s greater than 350m/s. This implies that the a i r v e l o c i t y would have to exceed 105m/s for compressible flow to e x i s t . A i r flow v e l o c i t i e s i n the lung w i l l never obtain values even close to t h i s c r i t i c a l l e v e l and hence 27 incompressible flow i s a reasonable assumption. The a i r flow w i l l also not be allowed to tra n s f e r mass, either a gain or l o s s . This condition e f f e c t i v e l y prevents the a i r to gain or lose moisture. Seely (50) has examined the temperature and water content of inspired a i r at three depths i n the nasal passage using a i r of various temperatures and r e l a t i v e humidities. He concluded that by the time the gas reached the second sampling p o s i t i o n , about h a l f way along the nasal passage, that no further change i n temperature or r e l a t i v e humidity occurs. The tracheobronchial tree w i l l also have c e r t a i n assumptions made regarding i t . As previously stated the airways are l i n e d with a mucous f l u i d layer which can p o t e n t i a l l y be alte r e d i n disease. An investigation by King (30) studied the e f f e c t of a f l u i d l i n i n g on the pressure drop i n st r a i g h t tubes. King's experiments consisted of l i n i n g Plexiglas tubes with o i l from locust bean gum which has approximately the same properties as the mucus layer of the tracheobronchial tree. The re s u l t s indicate that at low flow rates, less that 1 l i t e r per second, the addition of a f l u i d l i n i n g layer did not a l t e r the flow regime (from laminar to turbulent) and furthermore the additional pressure loss when compared to an unlined pipe could be completely described by the reduction i n cross-sectional area caused by the f l u i d . Further aspects of King's work indicated that the mucous layer could be dealt with as merely a reduction i n cross-sectional area for Reynold's number les s than 10,000 and frequencies of les s than 2Hz. As the purpose of t h i s work i s 28 t o s t u d y p r e s s u r e drops d u r i n g q u i e t i n s p i r a t i o n the mucous l a y e r w i l l be i g n o r e d except f o r i t s e f f e c t on t h e r e d u c t i o n i n c r o s s - s e c t i o n a l a r e a . A l l o f t h e branches i n the network w i l l be assumed t o be r i g i d non-de formable p i p e s w i t h c i r c u l a r c r o s s - s e c t i o n . D u r i n g normal b r e a t h i n g the a i r w a y s change v e r y s l i g h t l y i n l e n g t h but the e f f e c t o f l e n g t h changes on p r e s s u r e drops i s m i n i m a l and w i l l be i g n o r e d . The a i rway d i a m e t e r s , as w i l l be d e s c r i b e d l a t e r , change s u b s t a n t i a l l y i n d i a m e t e r as l u n g volume i s a l t e r e d . As t h e a i r f lows down the t r a c h e o b r o n c h i a l t r e e the branches i n human lungs t a p e r s l i g h t l y . Chang (5) examined the e f f e c t o f a r e d u c t i o n i n p i p e d i a m e t e r on t h e p r e s s u r e drop f o r b o t h i n s p i r a t o r y and e x p i r a t o r y a i r f l o w . He found t h a t t h e o r e t i c a l l y the p r e s s u r e l o s s e s a s s o c i a t e d w i t h f low t h r o u g h an e x p a n s i o n o r c o n s t r i c t i o n was a f u n c t i o n o f R e y n o l d ' s number, the r a t i o o f t h e d i a m e t e r s a c r o s s the expans ion o r c o n s t r i c t i o n , the r a t i o o f p i p e l e n g t h s and the r a t i o o f d i a m e t e r t o l e n g t h i n t h e d e s t i n a t i o n p i p e . F o r t u n a t e l y the e m p i r i c a l r e s u l t s demonstrated t h a t d u r i n g i n s p i r a t i o n , r e g a r d l e s s o f any o f the t h e o r e t i c a l parameters t h e r e was no a d d i t i o n a l p r e s s u r e l o s s beyond a c o n s t r i c t i o n as would be seen i n the l u n g . I sabey e t a l (25) used a s i l i c o n e a i r w a y c a s t w i t h f i v e g e n e r a t i o n s t o s t u d y v a r i o u s f low c h a r a c t e r i s t i c s i n the l u n g and f o u n d , i n s t eady f low c o n d i t i o n s , t h a t t h e d i m e n s i o n l e s s parameter R e y n o l d ' s number p r o v i d e s most o f t h e i n f o r m a t i o n r e q u i r e d t o c a l c u l a t e i n s p i r a t o r y p r e s s u r e d r o p s . Below N o f 29 4,000 the a i r flowed i n a laminar fashion. When using al t e r n a t i n g flow the Moody diagrams f o r frequencies of 0.25Hz and 0.5Hz gave curves s i m i l a r to those i n steady flow situations i n d i c a t i n g that the pressure drop under these conditions was equal to that calculated assuming steady flow. Frequency of breathing f o r quiet i n s p i r a t i o n , at 75% of t o t a l lung capacity, i s less than 0.5Hz and therefore alternating flow e f f e c t s w i l l be ignored and steady flow w i l l be assumed. Section 3.2 : Turbulent Flow and Bifu r c a t i n g Systems Examining again the equation for Reynold's number i t can be seen that when v e l o c i t i e s are high and airway dimensions large that N w i l l be large. As Reynold's number increases the flow RE w i l l become turbulent and calculations of the pressure drop using laminar flow equations w i l l no longer be v a l i d . I t i s not possible to use turbulent flow pressure drop equations as these do not account for the b i f u r c a t i n g nature of the tracheobronchial tree. Unfortunately a numerical solu t i o n for the pressure drop i n a b i f u r c a t i n g system with a r b i t r a r y branch angles and dimensions has not been made. To estimate the additional pressure drop i n a b i f u r c a t i n g and turbulent flow network of pipes empirical r e s u l t s of Pedley et a l (49) are used. Pedley examined the additional pressure drop, beyond that of assuming laminar flow, i n str a i g h t tubes by making a perspex cast model of a section of the tracheobronchial tree. In t h e i r 30 p l a s t i c model they used dimensions and geometry t y p i c a l of the larger airways of the human lung. With the cast they were able to make flow v i s u a l i z a t i o n studies and v e l o c i t y p r o f i l e measurements using hot wire techniques. To account for the additional pressure losses of turbulent flow and the disruption of flow p r o f i l e s caused by passing through a b i f u r c a t i o n Pedley and associates proposed a correction factor Z. Hypothesizing that the energy d i s s i p a t i o n would be dependent upon Reynold's number, airway diameter and airway length they formulated a parameter Z which was: where d i s the pipe diameter, I the length and C an experimental constant to be determined. Z was a correction factor which when mul t i p l i e d by the calculated laminar pressure drop determined using P o i s e u i l l e ' s equation gave the actual pressure drop measured i n t h e i r experiments. Working with several flow rates they estimated the value of C for a b i f u r c a t i n g network as seen i n human lungs was approximately 1.85. Since i t i s possible for the value of Z to drop below 1, ind i c a t i n g a pressure loss l e s s than laminar flow which i s impossible, the value of Z was used only i f i t was greater than 1. I f equation 3.2.1 gave a value of Z less than 1 then f u l l y developed laminar flow was assumed and no correction was made to P o i s e u i l l e ' s equation. An a l t e r n a t i v e correction f o r the additional pressure drop has been u t i l i z e d by Lambert (32) . In h i s work he used an (3.2.1) 31 experimentally determined equation for the additional pressure drop (beyond that of laminar flow) that was found by Reynolds (52). Using a latex cast of airways to a minimum diameter of 2mm Reynold's estimated the pressure drop under expiratory conditions as a l i n e a r function of Reynold's number times the f u l l y developed laminar pressure drop. Reynold's estimate of the pressure drop was: (1.5 + 0.0035N ) AP (3.2.2) RE Section 3.3 Correction for Lung Volume While avai l a b l e morphometric data gives average parameters for the airway dimensions length and diameter these values do not account f o r changes i n lung volume. The data of Weibel, H o r s f i e l d and Raabe were a l l made on casts of postmortem human lungs which were i n f l a t e d to a p a r t i c u l a r lung volume. In order to c a l c u l a t e the airways resistance at d i f f e r e n t lung volumes i t i s necessary to have a method of increasing or decreasing the the airway c a l i b e r of available data at various transpulmonary pressures since as the lung deflates the inte r n a l cross-sectional area of the airways reduces. At TLC the airways w i l l be as open as much as possible while at functional residual capacity (FRC) they w i l l be quite a b i t smaller. The data from a l l three data sets were made on casts i n f l a t e d to approximately 75% of t o t a l lung capacity. As w i l l be explained l a t e r , measurements of human airways resistance are t y p i c a l l y made at 32 functional residual capacity (FRC) or about 50% of TLC. I f the published airway dimensions are used d i r e c t l y , as was done by Pedley et a l (48), the calculated resistances w i l l underestimate resistance at FRC due to the i n f l a t e d geometry. To correct airway diameters i n both models equations developed by Lambert (32) to describe the pressure-area c h a r a c t e r i s t i c s of the airways were used. Lambert's equations were based on measurements of airway cross-sectional area at various transpulmonary pressures i n the postmortem human lungs of f i v e normal indiv i d u a l s studied using tantalum bronchography by Hyatt (24) . Data was available for the large airways and he extrapolated h i s r e s u l t s to the more peripheral zones of the lung. Lambert used functions of the form: A = 1.0—(1.0—a)*(1.0—P/P ) " (3.3.1) O where a and N are constants determined for each generation of the tracheobronchial tree, P , i s the pressure at which o morphological measurements were made, P the transpulmonary pressure at which the geometry i s desired and A i s the f r a c t i o n a l change i n cross-sectional area. Table 3 gives the values of a and N for the generations and orders of the two models. The curves described by equation 3.3.1 are represented gra p h i c a l l y i n figure 9. With the information that the airway dimensions were made on casts at 75% of TLC i t was assumed that t h i s corresponded to a transpulmonary pressure of 8cmH20. Using Lambert's pressure-area r e l a t i o n s h i p i t i s then possible to "deflate" the dimensions to FRC (about 5cmH O) or to " i n f l a t e " the 33 measurements to TLC (about 30cmH 0) . N 2 ' Section 3.4 Calculation of the Tracheobronchial Pressure Drop The c a l c u l a t i o n of the t o t a l tracheobronchial pressure drop involves several steps that vary s l i g h t l y between the two models. In ei t h e r model the t o t a l flow, V, i s an a r b i t r a r y value between 0 and 2000 ml/s corresponding to instantaneous flow during quiet i n s p i r a t i o n . The density and v i s c o s i t y of inspired gas are set f o r ei t h e r a i r or an 80% helium - 20% oxygen mixture as: A i r Helium-Oxygen Density g/cm P V i s c o s i t y g/cm/s 0.00113 0.000188 0.00045 0.000205 With the flow rate and physical properties of the inspired gas set i t i s further assumed that the system i s at a steady state and that the flow i s constant. In the symmetric model t h i s implies that the t o t a l flow passes through each generation and since the branches at any generation are i d e n t i c a l i n the Weibel model the flow w i l l be evenly divided between the branches. In the asymmetric model i t i s assumed that the flow w i l l p a r t i t i o n i t s e l f proportionally according to the downstream volume of lung served by any given branch. This condition i s reasonable based on the hypothesis that a l l parts of the lung 34 expand and contract i s o t r o p i c a l l y however t h i s assumption i s not s t r i c t l y true. I f t h i s technique of p a r t i t i o n i n g flow was used to estimate pressures i n i n d i v i d u a l a l v e o l i then d i f f e r e n t pressures would be predicted for a l v e o l i of the same volume (50). In order to more accurately model the tracheobronchial pressure drops some i t e r a t i v e technique would be required to balance pressure equations but t h i s i s a f a r more complex problem and has been ignored i n t h i s a p p l i c a t i o n . We are now ready to make the f i r s t c a l c u l a t i o n of the pressure drop, and hence resistance, i n each model of the tracheobronchial tree. In the symmetric model the steps are as follows. Using the morphometric values desired, (either Weibel's o r i g i n a l data, the data from Raabe et a l or Horsfield's transformed data) the value for the in t e r n a l diameter of an airway, D, i s know at 75%TLC or about 8cmH20. In addition the airway length i s known, £, from these same measurements. Using Lambert's equations (3.3.1) the airway diameters are corrected to the appropriate transpulmonary pressure to account for lung i n f l a t i o n or d e f l a t i o n from 8cmH20. The airway lengths are assumed to remain constant. In the symmetric model at each generation, z, there are 2 i d e n t i c a l airways. The cross-sectional area ava i l a b l e for airfl o w i s therefore the cross-sectional area of an in d i v i d u a l branch m u l t i p l i e d by 2 Z. The average l i n e a r v e l o c i t y i s the flow i n each generation, a preset constant, divided by the cross-sectional area available for airflow. 35 The laminar pressure drop i s calculated assuming smooth walled pipes (equation 1.4.1) and Reynold's number i s also calculated. Using Reynold's number and equation 3.3.1 to c a l c u l a t e Z the laminar pressure drop i s attenuated to account fo r the additional pressure loss i n a b i f u r c a t i n g system with non-laminar flow. The resistance of any branch, R , i s the BR pressure drop through that branch divided by the flow i n that branch. Since the branches at any given generation are a l l i d e n t i c a l the resistance of the entire generation i s eit h e r the sum of the branch resistances i n p a r a l l e l or the t o t a l pressure drop f o r that generation divided by the t o t a l flow. Both of these methods give the same r e s u l t for the resistance of a given generation. The resistance of the en t i r e tracheobronchial model i s then e i t h e r the sum of a l l the pressure drops f o r each of the generations divided by the t o t a l flow or the sum, i n s e r i e s , of the resistances for each generation. The above calculations obviously do not i n t e r a c t between generations of the tracheobronchial tree but can be performed on each generation i n turn. Because of t h i s arrangement of the cal c u l a t i o n s the steps can be e a s i l y programmed into a computer based spreadsheet where they can be automatically calculated under a v a r i e t y of conditions. For the asymmetric model the c a l c u l a t i o n of the tracheobronchial pressure drop must be made using a d i f f e r e n t technique. The associated steps for the c a l c u l a t i o n of the pressure drops i n t h i s case are as follows. 36 Determine the number of endpoints, branches of order 1, that are served by each branch i n the tree. In Table l a t h i s i s shown as the column l a b e l l e d E. I t i s important to note when ca l c u l a t i n g E that the branching pattern must be c a r e f u l l y followed as i t i s not constant over the en t i r e range of orders. Once E i s known the proportion of the t o t a l flow to a l l branches of a given order i s proportional to the number of endpoints served by that branch. For example, there are 233920 t o t a l endpoints i n the asymmetric model, each of the branches i n order 17 feed t h e i r t o t a l flow to 1760 endpoints so assuming that the flow i s proportionate to volume then each order 17 branch w i l l contain 233920* 1 0 0 P e r c e n ^ °f the t o t a l flow or about 0.752% of the t o t a l flow. The other information that must be counted by c a r e f u l l y following the orders of the daughter branches i s the number of airways with the same c h a r a c t e r i s t i c s , t h i s i s also shown i n table l a . Once we know how the flow w i l l be p a r t i t i o n e d then the correction for lung volume, given by Lambert, i s exactly the same as for the Weibel model. The cross-sectional area for airflow i s now determined using the information about the number of i d e n t i c a l l y dimensioned branches. Furthermore, the steps i n the symmetric model are repeated to obtain the pressure drop i n a sing l e branch and the resistance of that branch. The estimation of the t o t a l tracheobronchial pressure drop can not be estimated, as was done i n the symmetric model, by merely adding i n series the pressure drops of each order since fo r the asymmetric model the pressure f o r each pathway w i l l be 37 d i f f e r e n t . Instead i t i s simpler to d i r e c t l y calculate the resistance of the asymmetric tracheobronchial model using standard p a r a l l e l and series equations. As an example of the c a l c u l a t i o n consider the resistance at order 13. This order gives r i s e to two daughters, one order 12 and the other order 9. The resistance i s then the resistance of the order 13 branch i n ser i e s with the resistance of branches from orders 12 and 9 i n p a r a l l e l , denoting these values as R i 3 , R12, and R 9 the resistance including the order 13 branch i s : . R12 + R9 R l 3 + R12 . R 9 Using t h i s formula and keeping track of the daughter branches shown i n columns 3 and 4 of table l a the t o t a l tracheobronchial resistance can be calculated. Since the t o t a l pressure drop must be equal to the t o t a l flow m u l t i p l i e d by the t o t a l resistance the t o t a l pressure drop can also be estimated. 38 Summary of f l u i d mechanic Assumptions Assumption Reference A i r i n the lungs i s isothermal and 100% saturated during quiet breathing. Seely (50) A i r i s incompressable i n response to ph y s i o l o g i c a l l y reasonable pressures. Parker (46) The f l u i d l i n i n g of the lungs does not a f f e c t the pressure drop during quiet i n s p i r a t i o n . King (30) The e f f e c t of s l i g h t tapering does not add additional pressure losses. Chang (5) Representing the flow as steady state and ignoring the non-steady nature i s unimportant at low freqeuncies. Isabey (25) While the flow i s non-laminar the pressure drop can be estimated to be a laminar pressure times some value dependent upon Reynold's number. Pedley (48) Lambert (32) 39 Section 4 : Measurement of Airways Resistance i n Man Section 4.1 : Introduction One of the benefits of using a model to calculate the airways resistance i n the tracheobronchial tree i s the a b i l i t y to examine regional e f f e c t s on resistance i n various locations of the lung. In a human subject i t i s currently not possible to obtain resistance values f o r the lung other than the t o t a l tracheobronchial resistance. As previously stated the pressure drop i n the airways i s composed of r e s i s t i v e , e l a s t i c and i n e r t i a l losses. The i n e r t i a l losses are n e g l i g i b l e at normal quiet breathing frequencies. In t h i s model only the r e s i s t i v e losses are calculated, no attempt has been made to estimate the e l a s t i c pressure drop i n lung. The resistance values obtained from subjects used for comparison to the model should therefore have the e l a s t i c component removed. The procedure that was used for the data i n t h i s work to obtain airways resistance i n human subjects used a body plethysmograph, an esophageal balloon, an e l e c t r i c a l subtraction technique developed by Mead and Whittenberger (40) and a computer averaging method as described by Santamaria (54). Section 4.2 : Measurement of Resistance The f i r s t step i s to have a subject swallow a small latex balloon. This balloon i s about 7cm i n length and i s attached to a section of small diameter tubing about 50cm i n length. The 40 balloon i s passed through the nose and nasopharynx and positioned i n the lower t h i r d of the esophagus. Once i n place a small amount of a i r i s injected into the balloon to allow pressure differences to made between the balloon and mouth pressure. The pressure seen i n t h i s balloon i s used as an estimate of the i n t r a p l e u r a l pressure, P p i ' o r ^ e Pressure between the chest wall and the lung. When the balloon i s i n place the subject i s seated i n a pressure corrected volume se n s i t i v e body plethysmograph. The subject breathes through a mouth piece which i s open to atmospheric pressure, and the volume inhaled and exhaled i s measured by a displacement of box volume using a spirometer c a l i b r a t e d to produce an e l e c t r i c a l signal proportional to volume. The d e f i n i t i o n of airways resistance i s the dr i v i n g pressure to flow divided by the flow. During quiet i n s p i r a t i o n the d r i v i n g pressure i s the difference i n pressure between the mouth and the a l v e o l i , P -P , . The difference between the ao alv mouth pressure measured and the pressure i n the esophageal balloon w i l l be transpulmonary pressure, P i = P a o ~ P p l * T n ^ s pressure can now be divided into a pressure loss s o l e l y dependent upon the airways and which w i l l be a r e s i s t i v e loss, pa o ~ p a l v = p r e s i a n d a pressure drop due to the e l a s t i c properties of the lung tissue, p a i v ~ P p l = P e l * Combining the previous pressure equations transpulmonary pressure can be seen to be: P r e s + P e l = P a o " P a l v + P a l v " P p l = P a o " P p l = P l * From equation 1.3.1 we know that the e l a s t i c pressure loss i s 41 i n phase with volume and the r e s i s t i v e pressure loss i s i n phase with flow. I f we pl o t transpulmonary pressure versus flow a loop w i l l r e s u l t . The loop i s a r e s u l t of a volume dependent component of pulmonary resistance, the ti s s u e elastance. I f a portion of the signal i s subtracted that i s i n phase with volume u n t i l the loop i s closed the slope of the r e s u l t i n g r e l a t i o n s h i p i s the pulmonary resistance . When t h i s technique i s performed e l e c t r o n i c a l l y and combined with a computer to average many breaths a r e l i a b l e estimate of the t o t a l pulmonary resistance can be obtained at d i f f e r e n t flows, volumes or transpulmonary pressures. The r e l a t i o n s h i p between pressures, flows and volumes during the inspi r a t o r y and expiratory cycles are shown i n figure 4. I t can be seen that during normal quiet breathing values of 0.5 to 1.0 cmH20/l/s are reasonable values for airways resistance, the range for normal subjects i s t y p i c a l l y i n the order of 0.5 to 3 cmH20/l/s. I t i s also possible to substitute gases other than a i r , an 80% helium - 20% oxygen, He-Ox, mixture fo r example, and resistance measured under these conditions. This w i l l give a s l i g h t l y d i f f e r e n t r e l a t i o n s h i p i n the pressure-flow re l a t i o n s h i p . He-Ox i s a less dense but more viscous gas than a i r which r e s u l t s i n lower pressure drops, on the order of 70-80%, of the value compared while on a i r (38). This procedure to obtain airways resistance i n human subjects w i l l obviously include a resistance due to the upper respiratory t r a c t . These a i r passages, the nose and nasopharynx, have very complex flow patterns and hence pressure flow 42 r e l a t i o n s h i p s because of the vast changes that can occur i n the shape of these passages during breathing. This upper region w i l l create a resistance i n series with the airways resistance. Since the model presented here begins at the trachea at a point below that of the larynx the model i s expected to predict resistance values lower than observed i n normal subjects. Section 4.3 : Regional Resistance While i t i s not possible to obtain regional estimates of airways resistance i n l i v e subjects the airways resistance can be p a r t i t i o n e d between central and more peripheral airways resistance by using excised human lungs. Hogg et a l (18) used f i v e adult human lungs obtained at autopsy to study the r e l a t i o n s h i p between t o t a l airways resistance and lower airways resistance by u t i l i z i n g a technique f i r s t described by Macklem and Mead (37). The p a r t i t i o n i n g of resistance involves the use of a retrograde catheter. The catheter i s j u s t a piece of tubing b e l l e d at one end. To i n s e r t the catheter a larger catheter i s inserted into the airways u n t i l i t i s wedged i n place, since the dimension of the end of t h i s catheter i s known the s i z e of the airway where the retrograde catheter w i l l be placed i s also known. A piece of piano wire i s threaded down t h i s catheter and out through the lung surface and the wedged catheter i s removed leaving the wire i n place. The retrograde catheter i s attached to the end of the wire and can be pulled into the lung. Once i n 43 place i n the lung the entire excised lung i s placed i n a small plethysmograph which operates i n a s i m i l a r manner as discussed i n section 4.1. With the lung i n place i t was then possible to not only measure resistance from the trachea to the a l v e o l i but from the retrograde catheter to the a l v e o l i which i s a measure of lower airways resistance (here defined as airways less than 2mm i n diameter). 44 Section 5 : Airway Smooth Muscle and Airway Wall Thickness Section 5.1 : Introduction The airway walls, as mentioned i n section 1 are composed of connective ti s s u e , blood vessels, c a r t i l a g e and smooth muscle. The smooth muscle around the airways can contract which causes a narrowing of the airway lumen and hence the cross-sectional area avai l a b l e f o r airflow. Since the resistance i s inversely proportional to the airway diameter to the fourth power any changes i n airway c a l i b e r w i l l have dramatic e f f e c t s . The events that can p o t e n t i a l l y lead to an increase i n pulmonary resistance have been described by Moreno (43). Beginning with the inhalation of some agonist the amount of t h i s agonist that w i l l be deposited i n the airways w i l l be dependent upon p a r t i c l e s i z e of the agonist, the breathing pattern and the airway geometry. A f t e r the deposition of the agonist the degree of airway smooth muscle stimulation i s c o n t r o l l e d by the permeability of the epithelium (the t i s s u e between the airway and the smooth muscle) and the a b i l i t y of the lung t i s s u e to c l e a r away or de-activate the agonist. A f t e r stimulation, the extent to which the muscle w i l l shorten w i l l depend on how the agonist reacts with the receptors on the muscle, the amount of calcium present i n the medium, the load on the muscle, where the muscle i s on i t s length-tension relationship, the c o n t r a c t i l i t y of the muscle and the amount of muscle present. After contraction the external perimeter of the airway w i l l be contracted i n proportion to the amount of smooth muscle surrounding the airway. The thickness of the airway wall and the 45 presence of secretions i n the airway lumen w i l l determine the reduction i n the airway lumen and f i n a l l y , the increase i n airways resistance w i l l depend on the current flow regime. The c h a r a c t e r i s t i c feature of asthma i s excessive, re v e r s i b l e airway narrowing leading to dramatic increases i n airways resistance. Because of the rapid time course of airway narrowing observed during challenge t e s t s , where subjects are given a drug such as methacholine which i s known to stimulate smooth muscle to shorten, and the rapid recovery possible with bronchodilators, changes i n airway narrowing between normal and asthmatic subjects are attributed to differences i n airway smooth muscle. I t has been known for quite some time that the airway walls appear thickened i n the asthmatic subject r e l a t i v e to the normal subject but u n t i l recently l i t t l e quantitative i n v e s t i g a t i o n has been concerned with t h i s f a c t . Section 5.2 : Calculation of Contracted Lumenal Diameter The consequence of a thickened airway wall and i t s possible impact on resistance was c l e a r l y shown by Moreno (43). Using a singl e airway i t i s possible to calculate the change i n the radius of an airway with s p e c i f i e d values f o r the proportion of smooth muscle i n the airway circumference (PMC), the amount of airway wall that i s tiss u e (PW) and the degree of smooth muscle shortening (PMS). The sequence of calculations f o r determining the contracted diameter of the airway lumen are given i n appendix C where i t i s shown how the contracted i n t e r n a l radius 46 o f t h e a i r w a y , Ric, can be r e l a t e d t o PMS, PMC, w a l l a r e a t h a t i s t i s s u e (WA) and r e l a x e d i n t e r n a l r a d i u s , R ir , a s : Ric= (l-PMS*PMC) 2 *-( R l r 2 + W A / „ } - ^ (5 .2 .1 ) T h i s e q u a t i o n , 5 . 2 . 1 , g i v e s t h e c a l c u l a t i o n f o r t h e i n t e r n a l a i r w a y r a d i u s a f t e r some degree o f a i rway smooth muscle s h o r t e n i n g . The v a l u e Rir i s the a i rway r a d i u s a f t e r t h e volume c o r r e c t i o n o f Lambert has been a p p l i e d . A l t e r n a t i v e l y i t i s p o s s i b l e t o r e - a r r a n g e t h e v a l u e s i n 5 . 2 . 1 so t h a t the c o n t r a c t e d i n t e r n a l d i a m e t e r i s r e l a t e d t o t h e p r e - c o n t r a c t e d e x t e r n a l r a d i u s t h r o u g h t h e p r o p o r t i o n o f w a l l t h a t i s t i s s u e (PW) : An assumpt ion o f e q u a t i o n s 5 . 2 . 1 and 5 .2 .2 i s t h a t the smooth musc le i s o r i e n t e d i n a p l a n e p e r p e n d i c u l a r t o the l o n g a x i s o f t h e a i r w a y . U n f o r t u n a t e l y v e r y l i t t l e q u a n t i t a t i v e i n f o r m a t i o n e x i s t s r e g a r d i n g the a n a t o m i c a l arrangement o f the smooth musc le i n t h e t r a c h e o b r o n c h i a l t r e e . The most d e t a i l e d s t u d i e s a r e t h o s e done between 1892 and 1932 by W i l l i a m Snow M i l l e r and d i s c u s s e d i n h i s c l a s s i c monograph The Lung (42) . He found t h a t t h e smooth muscle formed bands around the a i r w a y s and t h a t t h e s e bands were always s u r r o u n d i n g t h e a i r w a y s i n such a manner t h a t c o n t r a c t i o n o f the smooth muscle would have the Ric (5 .2 .2 ) 47 maximal possible e f f e c t on airway c a l i b e r . He also showed that the smooth muscle e f f e c t i v e l y surrounds the airways of the tracheobronchial tree from the segmental bronchi to the aveolar ducts, PMC=1. Since no information was available on the proportion of smooth muscle i n the larger airways three normal human lungs were obtained at autopsy and PMC measured. Each lung was c a r e f u l l y dissected so that airways down to subsegmental branches could be i d e n t i f i e d . Airway tiss u e rings perpendicular to the airway axis were cut for processing. The t i s s u e was d e c a l c i f i e d f o r 12 hours and then cut into 5 micron sections which were stained with Masson's trichrome for l i g h t microscope viewing. Section 5.3 : Airway Wall Thickness Recent work by James et a l (27) has investigated the use of the i n t e r n a l airway perimeter defined by the folded e p i t h e l i a l surface, P i , as a method of assessing airway s i z e . In his investigation lung t i s s u e was obtained from three patients having lobar resections. The lobes were each bisected and then one h a l f placed i n a solution containing theophyline which relaxed the airway smooth muscle and the other h a l f submersed i n a s o l u t i o n containing carbachol to contract the airway smooth muscle. By measuring the airways of the adjacent surfaces, which would be the same airways for both halves, James found no s h i f t i n the d i s t r i b u t i o n of the i n t e r n a l perimeters. This implied 48 that the e p i t h e l i a l lumenal surface folded rather than shortened and that t h i s measurement, P i , could be used as a standard to describe airway s i z e . He also found that there was a very good re l a t i o n s h i p between the square root of wall area (WA) and the i n t e r n a l perimeter, P i . The importance of the r e l a t i o n s h i p between wall area and i n t e r n a l perimeter and the a b i l i t y to use the i n t e r n a l perimeter to compare airways was demonstrated by James (28) i n a study of normal and asthmatic lungs. In t h i s paper airway specimens from 18 patients who suffered from asthma and 23 subjects without asthma were qua n t i t a t i v e l y measured to obtain values f o r wall area and i n t e r n a l perimeter. James found that the airways of the asthmatic patients were markedly thicker than those of the normal subjects when airways of the same i n t e r n a l perimeter were compared. The proportion of the airway wall thickness (PW) was 25%-35% i n the asthmatics and only 10%-20% i n the normals. Figure 10 shows the possible e f f e c t of changes i n airway wall thickness i n the resistance to flow through a single tube. In the upper portion of figure 10 the proportion of the wall that i s t i s s u e i s 20% and the muscle i s assumed to completely surround the airway, PMC=1. I f the smooth muscle shortens by 30% and we assume that the wall t i s s u e i s incompressible and w i l l therefore remain constant, the resistance to airflow a f t e r contraction w i l l increase 7.6 times. I f we instead increase the value of PW from 20% of the airways to 40%, lower portion of figure 10, a modest increase i n resistance by a factor of 1.8 49 w i l l r e s u l t . When the smooth muscle of t h i s thickened airway i s shortened by 30% the f i n a l resistance i s almost 80 times the o r i g i n a l resistance. Since wall area (WA) can have a dramatic e f f e c t on the airways resistance, values f o r WA must be assigned to each branch i n the model. Using the information from James (27) that i n t e r n a l perimeter, P i , i s a r e l i a b l e marker of airway s i z e and that a good re l a t i o n s h i p e x i s t s between P i and the square root of wall area, values for WA can be estimated f o r each branch i n eit h e r the symmetric or asymmetric model. In the model the pressure-area equation 3.3.3 has been used to compute the diameter of an airway at a given transpulmonary pressure. I f we assume that at t o t a l lung capacity (about 30cmH2O) that the airways are nearly f u l l y expanded and round i n cross-section then the circumference of a c i r c l e with a diameter s p e c i f i e d by 3.3.1 w i l l equal the in t e r n a l airway perimeter as measured by James. Using t h i s value of P i , which w i l l remain constant at a l l lung i n f l a t i o n s , i t i s possible to assign wall area values to each airway branch by determining the rel a t i o n s h i p between WA and P i . 50 Section 5.4 : Estimation of the Relationship Between WA and Pi In James' (28) study comparing wall area i n normal and asthmatic subjects measurements were made of the airway i n t e r n a l perimeter and wall area on a sample of airways from subjects i n each group. This method resulted i n a d i f f e r e n t number of airways, between 1 and 23, being measured on each subject and d i f f e r e n t airway sizes being measured. His previous work (27) c l e a r l y demonstrated that when a large number of airways were measured i n one subject there was a good r e l a t i o n s h i p between the square root of wall area and i n t e r n a l perimeter. The problem with the asthmatic and normal group data i s that each subject now has few points and we wish to f i n d a r e l a t i o n s h i p that i s applicable to a group of asthmatic or normal subjects as a whole. This type of data, multiple observations on subjects within groups presents an awkward data analysis problem. By s t a t i n g that a r e l a t i o n s h i p e x i s t s between the square root of wall and i n t e r n a l perimeter we are implying the use, i n t h i s case, of l i n e a r regression. The basic assumption i s that some function of airway wall thickness i s l i n e a r l y related to i n t e r n a l perimeter for both normal and asthmatic subjects. Linear regression, as with a l l s t a t i s t i c a l analysis, have various assumptions associated with i t . The purpose of the assumptions i s to ensure that the hypothesis to be tested has known d i s t r i b u t i o n a l properties so that i t i s possible to determine r e l i a b l y when an hypothesis i s u n l i k e l y . In l i n e a r regression there are several assumptions that must be made: 51 1) The X, or independent variable, i s measured without error. In t h i s example the in t e r n a l perimeter was measured by magnifying the airway many thousands of times and t r a c i n g the int e r n a l perimeter with a d i g i t i z e r . While i t can not said that the P i i s measured without error the error involved i s small and w i l l be ignored. 2) The mean value of Y, wall area here, i s l i n e a r l y related to i n t e r n a l perimeter. In t h i s data wall area i s not l i n e a r l y related but James has shown that the square root of wall area i s l i n e a r l y related. 3) For any given value of in t e r n a l perimeter the values of square root of wall area are independent and normally d i s t r i b u t e d . I t i s t h i s assumption that i s v i o l a t e d by the current data. Each subject can be considered independent but the values for the airway wall areas and perimeters are obviously going to depend upon the p a r t i c u l a r subject, a large i n d i v i d u a l w i l l tend to have large values of P i and large wall areas. The lack of independent data values i s a serious v i o l a t i o n . 4) F i n a l l y we need to be able to assume that the residual components or the differences between the estimated wall areas and the observed wall areas are constant over the entire range of int e r n a l perimeters measured. This implies that the equation should predict no better for small airways than i t does f o r large airways. This assumption i s also a problem for the data as a whole. I f a p l o t i s made of the square root of wall area versus 52 i n t e r n a l perimeter (figure 11) and i f a simple l i n e a r regression l i n e i s f i t , the data values tend to be closer to the regression l i n e (smaller residuals) f o r smaller airways than larger airways. I t i s obvious that a simple application of l i n e a r regression i s not possible i n t h i s instance to determine the group or o v e r a l l r e l a t i o n s h i p between the square root of wall area and i n t e r n a l perimeter for a group of asthmatic or normal subjects. I f we were to pool the data the problems that could p o t e n t i a l l y a r i s e are many. I f a p a r t i c u l a r subject had many airways measured, and these airways had a good relationship, and another subject had only a few but with a poor relat i o n s h i p , the o v e r a l l regression made by ignoring the fact that there were two subjects does not r e f l e c t the doubt i n the rel a t i o n s h i p for the second subject. Another p o s s i b i l i t y i s that each of two subjects have i n fac t no p a r t i c u l a r r e l a t i o n s h i p but one subject has uniformly higher values f o r i n t e r n a l perimeter. This condition would lead to the fa l s e conclusion that for the group as a whole a rel a t i o n s h i p did e x i s t . The conclusions that one draws from the regression r e s u l t s can also be affected. Consider i f there were three subjects from whom two groups of airways were measured. In the f i r s t group 10 airways per subject were measured and i n the second group 100 airways per subject. I f the data i s i n c o r r e c t l y pooled together and a simple regression l i n e estimated i t would be concluded that for any given value of X i n group 2 that the confidence 53 with which the average value of Y i s calculated i s higher than i n group 1. This i s a f a l s e conclusion. In group two while i t i s true that f o r any given subject the average value of Y at a given value of X can be more p r e c i s e l y estimated because of the greater sampling done, there i s no more information regarding between subject v a r i a b i l i t y than with the smaller sample s i z e . Pooling of data has ignored the information regarding the number of subjects and the conclusion regarding the difference between groups analyzed i n t h i s manner would tend to f a l s e l y detect differences where none a c t u a l l y existed. Feldman (11) has addressed exactly the above problem and proposed possible solutions. I f instead of considering the problem one of many points through which a r e l a t i o n s h i p i s sought the problem i s altered to one which considers families of l i n e s within e i t h e r the normal or asthmatic subjects then a s t a t i s t i c a l l y v a l i d solution i s possible. Feldman d e t a i l e d the problem by considering four d i f f e r e n t ways that the rel a t i o n s h i p could be estimated: 1) The data for e i t h e r group could be naively pooled, NIPD, and one regression calculated for each group. This method obviously ignores intersubject v a r i a b i l i t y by l e t t i n g the between subject errors act through the residual errors. 2) A standard two stage regression, STS, could be performed where each subject i n each group has a l i n e a r regression made for t h e i r data. To compare groups the corresponding slopes and intercepts are compared using standard t e s t i n g techniques. Here within subject 54 v a r i a b i l i t y has been ignored and can only have influence by making the indi v i d u a l slopes and intercepts more var i a b l e . 3) The next technique was a weighted approach to regression, WLS. In t h i s technique the in d i v i d u a l data points f o r each subject are weighted according to how that p a r t i c u l a r subject f i t s i t s own l i n e a r regression l i n e and how that subjects estimates of the slope and intercept compare to the group average. 4) The f i n a l technique involves a procedure termed r e s t r i c t e d maximum l i k e l i h o o d , REML. In t h i s method the estimates obtained a f t e r the weighted regression approach are subjected to an i t e r a t i v e procedure which searches for the estimates which minimizes the v a r i a b i l i t y of the slopes and intercepts within a given group. The l i k e l i h o o d p r i n c i p l e i s act u a l l y a very simple concept, as stated by Berger (2) (0 refers to the s t a t i s t i c , such as the slope of a l i n e , which i s to be calculated), All the information about 9 obtainable from an experiment is contained in the likelihood function for 9 given X. Two likelihood functions for 6 (from the same or different experiments) contain the same information about 6 if they are proportional to one another. In s l i g h t l y d i f f e r e n t words the above paragraph merely states that a f t e r some experiment a p a r t i c u l a r event w i l l have been observed, i n t h i s case the measured values for wall areas and 55 perimeters. I f a model i s sp e c i f i e d , i n t h i s case we use a regression formula, then there w i l l be p a r t i c u l a r values f o r the slope and intercept that w i l l give us the maximum l i k e l i h o o d of seeing the data that was obtained, there are most c e r t a i n l y other possible values f o r the slope and intercept but they have a lower p r o b a b i l i t y of showing the data that was a c t u a l l y observed. I t i s important for t h i s method to specify a p a r t i c u l a r model and i n t h i s example a model c a l l e d random-effects regression w i l l be used. The concept of random-effects regression i s that each group, normals or asthmatics, are composed of families of l i n e s , one from each subject, and for any p a r t i c u l a r group the family of slopes and intercepts vary about some average value. I t i s the average value and an estimate f o r i t s standard error that i s important. The d i f f i c u l t l y with the weighted le a s t squares approach and es p e c i a l l y the r e s t r i c t e d maximum l i k e l i h o o d estimates i s the complex and i t e r a t i v e nature of the computation required. At t h i s time none of the common s t a t i s t i c a l packages perform the necessary cal c u l a t i o n s which means that a program must be written. The assumptions and computational procedures, as given by Feldman (11), f o r the weighted le a s t squares and r e s t r i c t e d maximum l i k e l i h o o d are detailed i n appendix D. Also given i n appendix D i s a computer program using the mathematical system Gauss(13) f o r computing the regression equations. Section 5.5 : Dose Response of Smooth Muscle 56 To provide a systematic method to simulate progressive shortening of muscle a "dose-response" equation was formulated fo r PMS, the proportion that the muscle has shortened r e l a t i v e to i t s s t a r t i n g length. Woolcock (64) proposed the equation: lo<x+0*logi Q (dose) 1 + l o « + 0*log l o(dose) ( 5 ' 4 ' 1 ) to describe the c h a r a c t e r i s t i c S shaped dose response curve. In t h i s formulation oc s h i f t s the dose response curve r e l a t i v e to the X axis with more negative values moving the curve to the r i g h t . /3 controls the slope of the response over the l i n e a r proportion of the curve where larger values give a more rapid response. 1 0cx+ e*log i o(dose) 1Qa 1 + 1 0«+P*log 1 0(dose) 1 + 10 a ( 5 4 2 ) 10 a 1 + i o a Equation 5.4.2 i s a modified form of 5.4.1 which appears more complicated but simply scales 5.4.1 so that i t equals zero at a dose of 1, has an S shape and reaches a maximum of 1. By using values of <x=-1.9 and j3=1.2 a 50% response i s obtained i n 1.5 l o g i Q doses. Smooth muscle shortening can be obtained by multiplying equation 5.4.2 by an a r b i t r a r i l y assigned maximal smooth muscle shortening, PMSmax, so that PMSobs (the observed smooth muscle shortening) w i l l range from 0 to PMSmax at some 57 simulated dose i n an S shaped manner. 58 S e c t i o n 6 : R e s u l t s I : Model F o r m u l a t i o n S e c t i o n 6.1 R e l a t i o n s h i p Between W a l l A r e a and I n t e r n a l P e r i m e t e r * U s i n g the t e c h n i q u e o f Feldman (11) and the Gauss i n s t r u c t i o n s d e t a i l e d i n appendix D; t h e d a t a from James (28) was used t o e s t i m a t e t h e l i n e a r r e l a t i o n s h i p between the square r o o t o f w a l l a r e a and i n t e r n a l p e r i m e t e r . The o r i g i n a l d a t a c o n s i s t e d o f 18 a s t h m a t i c s u b j e c t s and 23 normal s u b j e c t s . I n s p e c t i o n o f the d a t a r e v e a l e d t h a t 6 a s t h m a t i c s u b j e c t s and 4 normal s u b j e c t s had o n l y 1 o r 2 a i r w a y s measured. S i n c e i t i s n e c e s s a r y f i r s t t o c a l c u l a t e a w i t h i n s u b j e c t l i n e a r r e g r e s s i o n as t h e f i r s t s t e p i n the a n a l y s i s t h e s e s u b j e c t s were e l i m i n a t e d l e a v i n g 12 a s t h m a t i c s u b j e c t s and 19 normals w i t h a t o t a l o f 71 and 184 a i r w a y s r e s p e c t i v e l y . The o u t p u t from t h e program i s g i v e n a t the end o f t h i s s e c t i o n . The EM a l g o r i t h m was a l l o w e d t o i t e r a t e a maximum o f 50 t imes o r u n t i l s u c c e s s i v e e s t i m a t e s d i f f e r e d by l e s s t h e n 0.5%. In t h i s case t h e a l g o r i t h m d i d converge , a f t e r 37 i t e r a t i o n s , which r e q u i r e d a p p r o x i m a t e l y 2 minutes on an 80286 based computer o p e r a t i n g a t 12MHz. I t i s i n t e r e s t i n g t o note the d r a m a t i c d i f f e r e n c e s i n t h e f o u r t e c h n i q u e s . The wors t p r o c e s s , n a i v e l y p o o l i n g t h e d a t a g i v e s markedly h i g h e r v a l u e s f o r the i n t e r c e p t s and s l i g h t l y lower v a l u e s f o r t h e s l o p e t h a n the t h r e e o t h e r t e c h n i q u e s . The two s tage r e g r e s s i o n p r o c e s s g i v e s v a l u e s t h a t appear r e a s o n a b l e when compared t o the o p t i m a l s o l u t i o n s o b t a i n e d by the f i n a l two p r o c e d u r e s . 59 The we ighted e s t i m a t e s and the r e s t r i c t e d maximum l i k e l i h o o d e s t i m a t e s a r e v e r y c l o s e . T h i s i s expec ted s i n c e the t h e REML t e c h n i q u e i s i n t e n d e d t o be used as a re f inement p r o c e s s t o an a l r e a d y a p p r o p r i a t e e s t i m a t e , i n t h i s case a we ighted r e g r e s s i o n . The d i f f e r e n c e i s o b v i o u s l y i n the c o v a r i a n c e e s t i m a t e s o f t h e two p r o c e d u r e s . I t e r a t i o n o f the EM a l g o r i t h m has r e s u l t e d i n a lower v a r i a n c e o f t h e group i n t e r c e p t s and a s l i g h t l y h i g h e r v a r i a n c e o f t h e group s l o p e s t h a n t h e we ighted a p p r o a c h . F i g u r e 12 d i s p l a y s t h e l i n e a r r e g r e s s i o n l i n e s o f the square r o o t o f w a l l a r e a v e r s u s i n t e r n a l p e r i m e t e r f o r each s u b j e c t . The a s t h m a t i c s a r e shown as dashed l i n e s and t h e normals as d o t t e d l i n e s . A l l l i n e s appear t o be c o n v e r g i n g on a narrow range f o r the i n t e r c e p t w h i l e t h e r e i s more v a r i a t i o n i n the s l o p e s . The a s t h m a t i c s u b j e c t s c l e a r l y have h i g h e r v a l u e s o f w a l l a r e a f o r a g i v e n i n t e r n a l p e r i m e t e r t h a n t h e normal s u b j e c t s . F i g u r e 13 p l o t s i n d i v i d u a l d a t a , a s t h m a t i c s as c r o s s e s and normals as boxes , w i t h t h e f i n a l r e g r e s s i o n l i n e e s t i m a t e d u s i n g REML. The p o i n t s l y i n g beyond t h e edge a r e from a s m a l l number o f s u b j e c t s t h a t had l a r g e r a i r w a y s measured. C o n f i d e n c e i n t e r v a l s f o r the average v a l u e o f t h e square r o o t o f w a l l a r e a a t a g i v e n i n t e r n a l p e r i m e t e r can be e s t i m a t e d a s : y = (ai + bix) ± z [ c 2 + 2 x ^ 3 + x 2 c 2 ] 1 / 2 (6 .1 .1 ) where x i s t h e i n t e r n a l p e r i m e t e r ; ai and bi a r e t h e group 60 e s t i m a t e s o f i n t e r c e p t and s l o p e ; c^, c^, and c ^ a r e the s t a n d a r d e r r o r s from the group c o v a r i a n c e m a t r i x and z i s a s t a n d a r d normal d e v i a t e f o r the d e s i r e d c o n f i d e n c e l e v e l . F i g u r e 14 shows t h e r e s u l t s o f s e t t i n g z t o 1.96 f o r approximate 95% c o n f i d e n c e bands . H y p o t h e s i s t e s t i n g c o n c e r n i n g the e q u a l i t y o f i n t e r c e p t s , s l o p e s and o v e r a l l r e g r e s s i o n l i n e s can a l s o be made. To compare i n t e r c e p t s : (a i - a2) 2 . . . , ,~ — = % w i t h 1 d f 2 , 2 C + C ai az E q u a l i t y o f s l o p e s can be e v a l u a t e d by: (bi - b2) 2 . . . , — = x w i t h 1 d f 2 . 2 C | 3 i + °f* These two hypotheses can be compared by r e l a t i n g t h e observed t e s t s t a t i s t i c w i t h a c h i - s q u a r e d i s t r i b u t i o n w i t h one degree o f freedom. The a s s o c i a t e d v a l u e s f o r the e q u a l i t y o f i n t e r c e p t s and s l o p e s between a s t h m a t i c s and normals a r e 0.203 (NS) and 13.217 (pso.001) r e s p e c t i v e l y . A more r e l e v a n t compar i son i s whether o r no t the r e l a t i o n s h i p as a whole , t h a t i s compr i sed o f b o t h i n t e r c e p t and s l o p e , i s e q u a l between normal and a s t h m a t i c s . In o t h e r words, i s t h e average v a l u e f o r the square r o o t o f w a l l f o r a s t h m a t i c s s u b t r a c t e d from the average v a l u e f o r normals e q u a l t o z e r o . I f t h e d i f f e r e n c e between the two l i n e s i s z e r o t h e n no d i f f e r e n c e e x i s t s between t h e g r o u p s . T h i s can be t e s t e d by: 61 which i s d i s t r i b u t e d approximately as x" with 2 degrees of freedom. The value for t h i s data set i s 42.839 (p^O.001) which indicates a c l e a r difference between the groups. Using the above formula i t i s also possible to p l o t the difference between the two groups. Figure 15 shows the difference between the asthmatics and normals and c l e a r l y shows that at almost a l l values of i n t e r n a l perimeter the average difference between the two groups i s greater than zero i n d i c a t i n g that the asthmatics have more wall area than normals. Figure 16a and 16b show normal p r o b a b i l i t y p l o t s of the slopes and intercepts for the two groups. I f the values are normally d i s t r i b u t e d then the points for each groups should l i e approximately on a s t r a i g h t l i n e . While the number of points i s small there seems no reason to r e j e c t the normality assumption. Inspection of the residuals also displays a normal d i s t r i b u t i o n with no i n d i c a t i o n that the asthmatics f i t any better or worse than the normal subjects. 62 Gauss Output R e s u l t s o f f i t t i n g r e g r e s s i o n l i n e s t o grouped d a t a The f i l e used was : KAGGA.DAT T h e r e were 2.00 groups and 1.00 independent v a r i a b l e s . N a i v e l y p o o l i n g d a t a , s i n g l e r e g r e s s i o n t h r o u g h each group GROUP N INTERCEP SLOPE 1.0000 71.000 0.15810 0.12043 2.0000 184.00 0.11661 0.080176 Two s tage r e g r e s s i o n , average group s l o p e and i n t e r c e p t GROUP N INTERCEP SLOPE 1.0000 12.000 0.079272 0.13920 2.0000 19.000 0.10855 0.096534 Weighted L e a s t Squares R e g r e s s i o n — t h e f o l l o w i n g a r e i m p o r t a n t n u m e r i c a l m a t r i c e s — Dnih 0.016744 -0.0022566 -0 .0022566 0.00075656 B i a s 0.0097344 -0.0019395 -0 .0019395 0.00052616 D (Amemyia) 0.0070097 -0.00031713 -0.00031713 0.00023039 E i g e n v a l u e s o f INV(U' )*Dnih*INV(U) 1.3596 4.0962 R e s i d u a l v a r i a n c e 0.0067662 GROUP N INTERCEP SLOPE 1.0000 12.000 0.097108 0.13674 2.0000 19.000 0.10074 0.092253 63 The covariance matrix of the above parameters i s : — the square root of the diagonal elements are the parameters SE's — 0.00088490 -6.6911E-005 -6.6911E-005 2.8761E-005 0.00066153 -8.0353E-005 -8.0353E-005 3.0717E-005 Restricted maximum l i k e l i h o o d using EM algorithm f o r covariances Number of i t e r a t i o n s 37.000 F i n a l residual variance 0.0071590 GROUP N INTERCEP SLOPE 1.0000 12.000 0.089112 0.13683 2.0000 19.000 0.076277 0.096581 The covariance matrix of the above parameters i s : GROUP 1.0000 0.00044435 -0.00013236 -0.00013236 7.0034E-005 GROUP 2.0000 0.00036609 -0.00012324 -0.00012324 5.8703E-005 64 Section 6.2 Proportion of Muscle i n the Circumference of Large Airways From each autopsy lung the airways for which the proportion of muscle i n the circumference was required were the trachea, main bronchi and lobar bronchi. The other branches of the tracheobronchial tree were assumed to have a value of PMC equal to 1.0 according to the information a v a i l a b l e from M i l l e r (42). The average value of PMC i n the trachea was 0.33 and 0.61 i n the main bronchi. The r i g h t and l e f t main bronchi were d i f f i c u l t to dissect, only 3 of a possible 6 airways were obtained. PMC i n the lobar bronchi varied considerably but could e a s i l y be seen to completely surround the airway i f the s p i r a l nature of the smooth muscle was taken into account by viewing s e r i a l section s l i d e s . Because PMC was estimated to be 1 i n these airways and 0.33 i n the trachea, PMC was a r b i t r a r i l y set to 0.66 i n the r i g h t and l e f t main bronchi to l i e between these two values. Combining a l l of the previous information, the geometrical structure of e i t h e r Weibel or Horsfield, the f l u i d mechanic equations f o r pressure drops and pressure-area r e l a t i o n s h i p as well as the morphometric information regarding smooth muscle and airway wall thickness the f i n a l model formula can be made. As stated previously the model i s merely a c o l l e c t i o n of simple equations i n a computer based spreadsheet. By using a base such as Lotus 123 (35) the model i s e a s i l y accessible to most researchers. 65 Section 6.3 : Model Equations Table 4 shows the complete model for the symmetric model and table 5 the model for the asymmetric model. For both geometries the inputs are the gas density and v i s c o s i t y , a simulated dose and the t o t a l flow i n the system. The p r i n c i p l e outputs from both models are the t o t a l tracheobronchial pressure drop and the t o t a l tracheobronchial resistance. The exact equations used for eith e r the symmetric model or asymmetric model are given i n d e t a i l i n appendix E. 66 S e c t i o n 7 : R e s u l t s I I The Response o f t h e Model Under Normal C o n d i t i o n s An i n s p i r a t o r y p r e s s u r e f low c u r v e f o r e i t h e r model can be c o n s t r u c t e d by a p p l y i n g graded i n c r e a s e s i n f low between 0 and 2 1/s and p l o t t i n g t h e c a l c u l a t e d p r e s s u r e drop v e r s u s f l o w . F i g u r e 17A shows t h i s r e l a t i o n s h i p , f o r b o t h a i r and h e l i u m . The p r e s s u r e d r o p s f o r the two models a r e v e r y s i m i l a r o v e r t h i s range o f f lows b u t t h e r e i s a g r e a t e r d e c r e a s e i n p r e s s u r e a t any f low d u r i n g heliums-oxygen b r e a t h i n g i n t h e symmetric model because the R e y n o l d ' s numbers a r e h i g h e r i n t h e c e n t r a l a i rways o f the symmetr ic mode l . The shape o f the p r e s s u r e f low c u r v e s a r e s i m i l a r t o t h o s e r e p o r t e d i n normal man, as seen i f f i g u r e 17B, and t h e d e n s i t y dependence o f f low e x p r e s s e d as the d i f f e r e n c e between t h e r e s i s t a n c e u s i n g a i r and the r e s i s t a n c e u s i n g h e l i u m - o x y g e n a t 1 1/s expres sed as a p e r c e n t o f the r e s i s t a n c e on a i r i s a l s o s i m i l a r t o t h a t r e p o r t e d i n normal human s u b j e c t s (38) . The v a l u e s f o r t h i s measure o f d e n s i t y dependence a r e 32% f o r t h e symmetric model and 20% f o r the asymmetr ic model which a r e c l o s e t o the human v a l u e s (35) o f 20% t o 30%. The d i f f e r e n c e between P e d l e y ' s c o r r e c t i o n f o r t u r b u l e n t f low i n a b i f u r c a t i n g system (3 .2 .1 ) and L a m b e r t ' s c o r r e c t i o n (3 .2 .2 ) i s shown i n f i g u r e 18. L a m b e r t ' s e q u a t i o n g i v e s h i g h e r p r e s s u r e drops a t a l l f l o w s . When u s i n g L a m b e r t ' s e q u a t i o n no c o r r e c t i o n l e s s than 1.5 t imes the l a m i n a r p r e s s u r e drop i s a l l o w e d w h i l e P e d l e y ' s e q u a t i o n a l l o w s l a m i n a r f low t o govern 67 the flow regime i n about 50% of the tracheobronchial tree. I f the flow i s set at 1 l / s and the degree of lung i n f l a t i o n i s varied by a l t e r i n g the transpulmonary pressure i t i s possible to obtain both regional and t o t a l resistance curves at d i f f e r e n t "lung-volumes 1 1. In the symmetric model the t o t a l resistance ranges from 0.57 cmH20/l/s at a transpulmonary pressure of 30cmH20 to 0.89 cmH20/l/s at 5cmH20 and 2.22 cmH20/l/s at 1 cmH20. The t o t a l resistance values f o r the asymmetric model are 0.64 cmH20/l/s at a transpulmonary pressure of 30cmH 0, 0.99 cmH 0/1/s at 5cmH 0 and 2.68 cmH 0/1/s at 1 2 2 2 2 cmH20. To compare the model r e s u l t s to actual data, pl o t s of resistance versus lung volume were made. Figure 19 shows the t o t a l resistance versus transpulmonary pressure curves generated from the symmetric model and measurements of pulmonary resistance at 1 l / s on i n s p i r a t i o n from a normal 37 and a 31 year old male subject breathing at 30 breaths/minute. Pulmonary resistance was measured using the e l e c t r i c a l subtraction technique method of Mead and Whittenberger (40) and a computerized averaging c i r c u i t (54). The dashed l i n e shows the the r e l a t i o n s h i p of resistance and lung volume from a 38 year old normal subject studied by Vincent et a l (58) where lower airway resistance was measured using a trans-tracheal catheter and a d i f f e r e n t i a l pressure transducer which compared tracheal pressure to esophageal pressure at a fixed o s c i l l a t i o n frequency of 4Hz and the Mead and Whittenberger correction f o r e l a s t i c pressure losses. Figure 20 shows the resistances for the p a r a l l e l airways of 68 each generation i n the symmetric model at various transpulmonary pressures. The sharp peak at the 3 r d to 6 t h generations i s related to the r e l a t i v e l y narrow t o t a l cross-sectional area of the tracheobronchial tree i n t h i s region. While i t i s not obvious from figure 20 the r e l a t i v e contribution to t o t a l pulmonary resistance from the peripheral regions becomes greater as transpulmonary pressure i s reduced. Thus at a transpulmonary pressure of 30cmH2O 44% of the t o t a l pressure drop occurs between generations 3 and 6 while 6% of the t o t a l pressure drop occurs i n airways beyond generation 10. However, at a transpulmonary pressure of lcmH20 42% of the t o t a l pressure drop occurs between generations 3 and 6 and 19% i n the peripheral airways. A s i m i l a r r e l a t i o n s h i p between pressure drop and orders can be seen i n the asymmetric model and once again the r e l a t i v e pressure drop across peripheral airways increases at low lung volumes. Hogg et a l (28) part i t i o n e d pulmonary resistance into central and peripheral components and studied the rel a t i o n s h i p between resistance and transpulmonary pressure i n f i v e excised adult human lungs. Using a retrograde catheter they were able to measure flow, tracheal pressure, p l e u r a l pressure and bronchial pressure on excised lungs i n a plethysmograph. The bronchial pressure was measured i n airways of approximately 2mm diameter. In figure 21 the s o l i d c i r c l e s and dashed l i n e s correspond to the data obtained by Hogg et a l for the t o t a l pulmonary resistance (upper line) and lower pulmonary resistance (lower l i n e ) . The s o l i d l i n e s are the corresponding values from the 69 symmetric model for t o t a l resistance (upper line) and generations 16 and beyond (lower resistance, lower l i n e ) . Figure 22 shows the re l a t i o n s h i p between regional resistance and generation i n the symmetric model f o r various degrees of maximal smooth muscle shortening. To generate t h i s p l o t values for airway wall thickness for normal subjects were used and values of maximal smooth muscle shortening were i d e n t i c a l f o r a l l generations. Flow was set at 1 l / s and transpulmonary pressure at 5cmH20. Total pulmonary resistance increases from a value of 0.89cmH20/l/s where there i s no airway smooth muscle shortening, 2.38 cmH20/l/s at 20% muscle shortening and 12.40 cmH20/l/s at 40% smooth muscle shortening (the respective values for the asymmetric model are 0.99, 2.54 and 13.6 cmH 20/l/s). Similar to changes i n resistance associated with decreased lung volume the changes i n regional resistance associated with maximal smooth muscle shortening are f r a c t i o n a l l y greater i n the peripheral airways. With 20% smooth muscle shortening the pressure drop between generations 3 and 6 i s 1.27 cmH20 while that beyond generation 10 i s 0.23 cmH20, however at 40% smooth muscle shortening the pressure drop between generations 3 and 6 has increased 5.3 f o l d to 6.79 cmH20 while the pressure drop beyond generation 10 has increased 8.6 f o l d to 1.97cmH20. Beyond 40% smooth muscle shortening the resistance of the smallest airways becomes very large and at 46.3% shortening the airways i n generation 23 close r e s u l t i n g i n i n f i n i t e resistance. In the asymmetric model the maximal smooth muscle shortening that can occur p r i o r to complete closure of 70 a l l pathways i s 48.3%. The t a b l e below shows v a r i o u s t o t a l r e s i s t a n c e v a l u e s i n c m H 2 0 / l / s f o r a range o f maximal smooth musc le s h o r t e n i n g from 40% t o 49%. R e s i s t a n c e PMSmax Symmetric Asymmetr ic 40% 12.39 13.61 41% 14.03 15.53 42% 16.01 17.89 43% 18.44 20.87 44% 21.55 24.72 45% 25.78 29.93 46% 35.57 37.52 47% 00 50.83 48% 00 142.60 49% 00 00 F i g u r e 23 shows the "dose response c u r v e s " o f the symmetric and asymmetr ic models f o r v a r y i n g degrees o f maximal a irway smooth musc le s h o r t e n i n g . These c u r v e s were g e n e r a t e d u s i n g a f low r a t e o f 1.25 1/s a t a t r a n s p u l m o n a r y p r e s s u r e o f 5cmH20 and the v a l u e s f o r a i rway w a l l t h i c k n e s s from t h e e q u a t i o n d e t e r m i n e d i n s e c t i o n 6 f o r normal s u b j e c t s ; the dose response parameters were a=-1.9 and £ = 1 . 2 a t a l l l e v e l s . Wi th p r o g r e s s i v e l y i n c r e a s i n g smooth muscle s h o r t e n i n g t h e r e i s an i n c r e a s e i n the maximal r e s i s t a n c e a c h i e v e d b u t as l o n g as v a l u e s o f musc le s h o r t e n i n g a r e l e s s t h a n 46.3% a p l a t e a u i s a lways r e a c h e d i n the dose response c u r v e . There i s v e r y l i t t l e d i f f e r e n c e i n t h e shape o r p o s i t i o n o f the dose response c u r v e between t h e symmetric and asymmetric mode l s . S i n c e i d e n t i c a l v a l u e s o f the dose response parameters a and 0 were used t o 71 generate these dose response curves the p o s i t i o n of the curves r e l a t i v e to the x axis i s not altered by the magnitude of shortening. 72 Section 8 : Results III Model Response Under Abnormal (Asthmatic) Conditions From the r e s u l t s from section 6 regarding the r e l a t i o n s h i p between the square root of wall area and the i n t e r n a l airway perimeter i n asthmatic subjects i t i s apparent that for any airway s i z e that there i s a greater amount of wall t i s s u e between the e p i t h e l i a l c e l l layer of the lumen and the outer border of the smooth muscle layer i n asthmatic subjects than i n normal subjects. This additional t i s s u e could have a number of possible e f f e c t s but two have been considered i n the model formulation. F i r s t , the extra t i s s u e could accumulate outward from the e p i t h e l i a l c e l l layer and have no e f f e c t on the lumenal cross-sectional area available for airflow i n the absence of smooth muscle shortening. The a l t e r n a t i v e i s that a l l or some proportion of t h i s "excess" t i s s u e could encroach upon the airway lumen and reduce the cross-sectional area avai l a b l e for airflow. To account for encroachment a v a r i a b l e percentage (0-100%) of the difference i n wall area between the normal and asthmatic subject was allowed to encroach upon the lumen, t h i s factor i s termed the encroachment factor and denoted by E on most figures. Figure 24 shows the r e l a t i o n s h i p between t o t a l tracheobronchial resistance and lung i n f l a t i o n . As the amount of encroachment i s increased the resistance at a given lung volume, or transpulmonary pressure also increases. The dose response re l a t i o n s h i p has been fixed by equation 73 5.4.2 to be an S shaped rel a t i o n s h i p . As long as the d i s t a l airways do not closed completely (causing an i n f i n i t e resistance) then the t o t a l resistance w i l l plateau at some maximal dose l e v e l . Figure 25 examines t h i s p a r t i c u l a r r e l a t i o n s h i p between the maximal degree of smooth muscle shortening and the plateau resistance. Using the wall area data fo r normal subjects values of PMSmax of approximately 45% can be achieved with plateau resistances of about 30cmH2O/l/s. In contrast i f the asthmatic wall area r e l a t i o n s h i p i s used maximal smooth muscle shortening must be less than 35% or complete closure of the smaller peripheral airways r e s u l t s . I f the excess airway wall t i s s u e i s allowed to encroach upon the lumenal space then even l e s s e r degrees of PMSmax are possible. Dose response relationships using asthmatic values of wall area can be constructed exactly as was done i n figure 23 for normal subjects. Figure 26A displays the dose response re l a t i o n s h i p using asthmatic data for the r e l a t i o n s h i p of P i and wall area and graded changes i n maximal smooth muscle shortening. For these curves the degree of encroachment was set to zero. The values of PMSmax for curves A to D are 20%, 30%, 35% and 40%. The dashed l i n e on each frame of figure 26 was constructed using normal wall area values and 20% maximal smooth muscle shortening. By taking into consideration the encroachment factor and regenerating the dose response r e l a t i o n s , f a r steeper curves r e s u l t . In figure 26B the encroachment has been set to 50% for a l l curves and the maximal smooth muscle shortening was 20%, 30% 74 and 35% f o r curves A to C respectively. In figure 26C encroachment has been increased to 100%. As the encroachment increases small amounts of smooth muscle shortening which had l i t t l e e f f e c t on normal subjects (35%) causes dramatic increases i n airways resistance i n asthmatic subjects. An important feature of the model i s the a b i l i t y to examine the e f f e c t of changes i n airway wall thickness i n d i f f e r e n t regions of the tracheobronchial tree. In f i g u r e 2 7A the r e l a t i o n s h i p of airway wall thickness to P i f o r asthmatic subjects was used i n the larger central airways (generations 0 to 10) while the normal subject wall area values were used i n the peripheral airways. In t h i s figure encroachment was fixed at zero and values of 30%, 35% and 40% were used for maximal smooth muscle shortening. In t h i s case the values for resistance are marginally larger than under normal conditions and the dose response curve i s s h i f t e d s l i g h t l y to the l e f t . I f the location of the thickened airway walls i s moved to the more peripheral airways, generations 11 to 23, and the values f o r airway wall thickness f o r normal subjects are used i n the central airways, the r e s u l t s are as shown i n figure 27B. This r e s u l t s i n v i r t u a l l y no change i n baseline resistance and l i t t l e change i n the shape of the the dose response curve but a marked increase i n the plateau resistance. I f encroachment i s set to 50% the r e s u l t i s seen as figure 27C. As expected the o v e r a l l resistance values have increased but i n general the appearance of the curves i s s i m i l a r . Figure 27D displays the e f f e c t of peripheral airway wall thickening 75 with 50% lumenal encroachment. In t h i s figure there i s a dramatic s h i f t i n the shape of the dose response curve as the maximal amount of smooth muscle shortening i s altered from 30% to 35%. I t i s c l e a r that i f t h i s i s the structure of the airway wall i n some subjects that very l i t t l e smooth muscle contraction i s possible. Table 6 shows the maximum resistance possible for a vari e t y of conditions over a range of maximal smooth muscle shortening. To c a l c u l a t e these the dose response parameters a and 0 were fixe d at -1.9 and 1.2 respectively and then a large dose, 10 1 0, used to force the model to maximally c o n s t r i c t to PMSmax. As expected the maximum achievable resistance increase with increasing muscle shortening and increasing encroachment. The value i f INF i n table 6 indicates complete closure of a l l pathways and hence i n f i n i t e resistance. Table 7 displays the dose at which the 50% of the maximum resistance, as stated i n table 6, i s achieved. A value of DNE indicates that the ED g o does not e x i s t because t o t a l aitway closure occured. 76 S e c t i o n 9 : D i s c u s s i o n The aims o f t h i s t h e s i s were t o f i r s t d e v e l o p a computer based model o f t h e t r a c h e o b r o n c h i a l t r e e t h a t would compute, q u a l i t a t i v e l y , t h e p r e s s u r e drops i n the human l u n g . S e c o n d l y , d i f f e r e n c e s between symmetric and a s y m m e t r i c a l r e p r e s e n t a t i o n s o f t h e t r a c h e o b r o n c h i a l t r e e were examined. F i n a l l y , the model was used t o i n v e s t i g a t e known d i f f e r e n c e s between normal and a s t h m a t i c s u b j e c t s i n r e g a r d s t o w a l l a r e a t o e x p l o r e the p o s s i b l e i n t e r a c t i o n s between a i r w a y w a l l t h i c k n e s s and smooth musc le s h o r t e n i n g and t h e i r e f f e c t s on t r a c h e o b r o n c h i a l r e s i s t a n c e . To a c c o m p l i s h the above g o a l s i n f o r m a t i o n has been combined from many f i e l d s o f s tudy t o produce a model t h a t a l l o w s easy e x a m i n a t i o n o f changes i n v a r i o u s parameters t h a t c o u l d p o t e n t i a l l y a l t e r the r e s i s t a n c e o f the t r a c h e o b r o n c h i a l t r e e as w e l l as t o s e r v e as a f o u n d a t i o n f o r f u t u r e enhancements. R e l a t i o n s h i p Between W a l l A r e a and I n t e r n a l P e r i m e t e r The r e s t r i c t e d maximum l i k e l i h o o d e s t i m a t i o n (REML) t e c h n i q u e has been used f o r an a c c u r a t e e s t i m a t i o n o f the the r e l a t i o n s h i p between t h e square r o o t o f w a l l a r e a and i n t e r n a l p e r i m e t e r o f t h e a i r w a y s . T h i s p r o c e d u r e i s no t s u b j e c t t o a v i o l a t i o n o f assumpt ions as i n d i c a t e d i n s e c t i o n 5.4 and the assumpt ions r e q u i r e d f o r REML were i n f a c t met by t h e d a t a . The f o u r p r o c e d u r e s g i v e n f o r e s t i m a t i n g t h e r e l a t i o n s h i p 77 gave expec ted d i f f e r e n c e s . The n a i v e l y p o o l i n g method r e s u l t e d i n r e g r e s s i o n i n t e r c e p t s t h a t d i f f e r from the REML e s t i m a t e s by 50%. T h i s was because most o f t h e a i r w a y s were i n f a c t s m a l l e r than 3mm and t h i s i n f l u e n c e d the r e g r e s s i o n c a l c u l a t i o n s i g n o r i n g i n t e r s u b j e c t v a r i a b i l i t y . The two s tage r e g r e s s i o n p r o c e s s gave s l i g h t l y b e t t e r e s t i m a t e s , w i t h i n 20% o f the REML e s t i m a t e s , i n d i c a t i n g the importance o f a c c o u n t i n g f o r s u b j e c t v a r i a t i o n . The e s t i m a t e s f o r b o t h the we ighted t e c h n i q u e and REML method were v e r y c l o s e , l e s s t h a n 10% d i f f e r e n c e , and t h i s i s a l s o e x p e c t e d . The REML p r o c e d u r e i s an i t e r a t i v e p r o c e s s t h a t i s i n t e n d e d t o r e f i n e an e s t i m a t e t h a t i s a l r e a d y c l o s e t o c o r r e c t . The b e n e f i t o f t h e REML p r o c e s s i s the s l i g h t r e d u c t i o n i n o v e r a l l v a r i a n c e t h a t i t s eeks . The t e c h n i q u e w h i l e d i f f i c u l t t o implement i s easy t o now use i n a v a r i e t y o f s i t u a t i o n s where t h i s s p e c i f i c t y p e o f p o o l i n g e r r o r o c c u r s . The c r e a t i o n o f a " d i f f e r e n c e " l i n e p l o t , as was done f o r f i g u r e 15 encourages the use o f more a p p r o p r i a t e and p o w e r f u l m u l t i v a r i a t e t e c h n i q u e s . W h i l e Feldman d e s c r i b e d the t e c h n i q u e f o r two groups and a s i n g l e independent v a r i a b l e the Gauss code g i v e n i n Appendix D w i l l compute t h e r e s u l t s f o r an a r b i t r a r y number o f groups and independent v a r i a b l e s . There are s t i l l however some major problems w i t h the a n a l y s i s . As t h e s i z e o f the problem i n c r e a s e s t h e convergence t ime f o r t h e EM a l g o r i t h m i n c r e a s e s d r a m a t i c a l l y . F u t u r e work i n t o more e f f i c i e n t computat ions f o r t h i s s e c t i o n o f the p r o c e d u r e are a d v i s a b l e . The p a r t i t i o n i n g t e c h n i q u e d e s c r i b e d by Araemyia(l) and used by Feldman(11) a l s o has d i f f i c u l t i e s . I f the t e c h n i q u e 78 proposed by Amemyia t o p a r t i t i o n the c o v a r i a n c e m a t r i x i s used i t i s s t i l l p o s s i b l e t o o b t a i n a n o n - i n v e r t i b l e c o v a r i a n c e m a t r i x . The i n v e r t e d m a t r i x i s r e q u i r e d f o r use as a w e i g h t i n g m a t r i x i n t h e REML p r o c e s s . The n o n - i n v e r t a b l e c o n d i t i o n appears t o be i n d i c a t i n g an e x t r e m e l y s m a l l v a r i a n c e i n e i t h e r the average group i n t e r c e p t o r s l o p e c o e f f i c i e n t . I n t h e event o f t h i s c o n d i t i o n then the s i m p l e weighted r e g r e s s i o n i s recommended as t h e f i n a l e s t i m a t e and i t e r a t i o n o f the EM a l g o r i t h m s h o u l d not be a t t e m p t e d . I t may be p o s s i b l e t o r e - w r i t e t h e Gauss code i n s t r u c t i o n s t o use o t h e r forms o f m a t r i x i n v e r s e s such as p s e u d o - i n v e r s e s b u t t h i s would r e q u i r e a complete r e v i s i o n o f the EM p o r t i o n o f t h e program p r o v i d e d . F l u i d Mechanic E q u a t i o n s The l i m i t a t i o n s i n the c o m p u t a t i o n a l s o l u t i o n f o r the t r a c h e o b r o n c h i a l p r e s s u r e drop d e s c r i b e d by P e d l e y e t a l (48) and u t i l i z e d h e r e f o r t h i s model c a l c u l a t i o n have been p r e v i o u s l y d i s c u s s e d (47) . An i m p o r t a n t p a r t o f these e q u a t i o n s i n v o l v e s a c o r r e c t i o n f o r t h e p r e s s u r e drop i n a i r w a y s i n which t h e r e i s n o n - l a m i n a r f l o w . P e d l e y and a s s o c i a t e s based these c o r r e c t i o n s on measurements made on a l a r g e s c a l e c a s t o f a b r a n c h i n g system which had t h e r e l a t i v e d imens ions o f the t r a c h e o b r o n c h i a l t r e e . T h e i r v a l u e f o r Z i s meant t o compensate f o r a d d i t i o n a l p r e s s u r e l o s s e s r e l a t e d t o t u r b u l e n c e and the r e - e s t a b l i s h m e n t o f l a m i n a r f low f o l l o w i n g a b i f u r c a t i o n . Lambert(32) employed a d i f f e r e n t means o f c a l c u l a t i n g the a d d i t i o n a l p r e s s u r e drop based on an 79 e x p i r a t o r y f low model o f R e y n o l d . When b o t h t h e c o r r e c t i o n f a c t o r s were used the v a l u e s f o r t o t a l pulmonary r e s i s t a n c e and r e g i o n a l p r e s s u r e drop were h i g h e r f o r L a m b e r t ' s c o r r e c t i o n f a c t o r . T h i s can be e x p l a i n e d by the c o n d i t i o n s under which R e y n o l d d e v e l o p e d h i s c o r r e c t i o n f a c t o r . R e y n o l d and Lambert s t u d i e d f low under maximal e x p i r a t o r y c o n d i t i o n s . When f low pas se s t h r o u g h daughter branches i n t o a s i n g l e p a r e n t branch t h e r e i s a g r e a t e r p r e s s u r e d r o p , due t o i n c r e a s e d t u r b u l e n c e , t h a n f o r the r e v e r s e d f low c o n d i t i o n . I t i s a l s o c l e a r from e q u a t i o n 3 . 2 . 2 t h a t the minimum p r e s s u r e drop u s i n g R e y n o l d ' s e q u a t i o n w i l l be 1.5 t imes l a m i n a r f l o w . The Z f a c t o r proposed by P e d l e y a l l o w s f o r f u l l y deve loped l a m i n a r f low t o e x i s t a t low f low r a t e s i n t h e extreme l u n g p e r i p h e r y which seems r e a s o n a b l e when t h e v e l o c i t i e s , n e a r z e r o , and l a r g e c r o s s - s e c t i o n a l a r e a , about 6000 square cm, a r e c o n s i d e r e d . P e d l e y ' s c o r r e c t i o n f a c t o r Z has been used i n most p l o t s p r e s e n t e d because o f these d i f f i c u l t i e s and i t a p p e a r s , under c o n d i t i o n s o f q u i e t i n s p i r a t i o n , t h a t Z r e p r e s e n t s a more a c c u r a t e a t t e n u a t i o n o f the l a m i n a r p r e s s u r e drop t h a n R e y n o l d ' s c o r r e c t i o n . P e d l e y (6) t r i e d t o e s t i m a t e the p o t e n t i a l e r r o r i n d e t e r m i n i n g t h e p r e s s u r e drop u s i n g Z and c o n c l u d e d t h a t they tended t o u n d e r e s t i m a t e the a c t u a l energy d i s s i p a t i o n but t h a t t h i s e r r o r was l e s s t h a n 10% Lung I n f l a t i o n P e d l e y e t a l based t h e i r c a l c u l a t i o n s f o r r e g i o n a l and t o t a l 80 pulmonary r e s i s t a n c e on the l u n g geometry from s t u d i e s o f Weibe l which were o b t a i n e d from l u n g s f i x e d a t a t r a n s p u l m o n a r y p r e s s u r e o f a p p r o x i m a t e l y 75% o f t o t a l l u n g c a p a c i t y . A c t u a l pulmonary r e s i s t a n c e i s u s u a l l y measured c l o s e r t o f u n c t i o n a l r e s i d u a l c a p a c i t y which i s 50 t o 55% o f T L C . An at tempt was made t o examine f o r t h e e f f e c t s o f v a r y i n g l u n g i n f l a t i o n on pulmonary r e s i s t a n c e by employ ing r e l a t i o n s h i p s between t r a n s p u l m o n a r y p r e s s u r e and c r o s s - s e c t i o n a l a r e a d e v e l o p e d by Lambert . These e q u a t i o n s a r e based on a c t u a l r a d i o l o g i c measurements o f a i rway d i a m e t e r s o b t a i n e d a t d i f f e r e n t t r a n s p u l m o n a r y p r e s s u r e s from 5 normal human l u n g s , however the r e l a t i o n s h i p s f o r a i r w a y s beyond g e n e r a t i o n 3 a r e based on an e x t r a p o l a t i o n o f t h e c e n t r a l a i rway d a t a . Lambert made the assumpt ion t h a t the most p e r i p h e r a l a i r w a y s would behave more s i m i l a r t o l u n g parenchyma and added a f a c t o r r e l a t e d t o the a i rway w a l l t h i c k n e s s a t t h a t l e v e l t o cause t h e s e a i r w a y s t o decrease t h e i r lumenal a r e a " f a s t e r " than the cube r o o t o f l u n g volume. Symmetric v s Asymmetric S t r u c t u r e s Throughout t h i s work r e s u l t s have c o n c e n t r a t e d on u s i n g the W e i b e l s y m m e t r i c a l model r a t h e r t h a n t h e a s y m m e t r i c a l s t r u c t u r e o f H o r s f i e l d . The r e a s o n f o r t h i s i s q u i t e s i m p l e , w h i l e the two models a r e c l e a r l y d i f f e r e n t i n t h e i r i n d i v i d u a l r e p r e s e n t a t i o n o f the t r a c h e o b r o n c h i a l t r e e the r e s u l t s a r e e x t r e m e l y c l o s e f o r t h e c a l c u l a t e d p r e s s u r e drops under a v a r i e t y o f c o n d i t i o n s . When d i f f e r e n c e s between the two s t r u c t u r e s , such as s l i g h t l y h i g h e r 81 maximum r e s i s t a n c e s f o r the asymmetric model , do o c c u r t h e r e i s no change i n t h e q u a l i t a t i v e i n t e r p r e t a t i o n o f r e s u l t s . The d i f f e r e n c e s t h a t do e x i s t between these s t r u c t u r e s can be e x p l a i n e d by two f a c t o r s . F i r s t , W h i l e i t was a t tempted t o use the e x a c t same r a d i i and l e n g t h s f o r t h e a i r w a y s i n b o t h g e o m e t r i c a l r e p r e s e n t a t i o n s i t was s t i l l n e c e s s a r y t o use average v a l u e s f o r a g i v e n Weibe l g e n e r a t i o n o r H o r s f i e l d o r d e r . In the W e i b e l s t r u c t u r e f o r i n s t a n c e the r i g h t and l e f t main b r o n c h i a r e averaged w h i l e i n t h e H o r s f i e l d model the r e c o r d e d d a t a v a l u e s a r e u s e d . S e c o n d l y , t h e H o r s f i e l d model a l s o a l l o w s f o r a g r e a t e r inhomogenei ty i n the d i a m e t e r s o f the branches i n v a r i o u s pathways . The r e s u l t i s t h a t t h e r e a r e many pathways t h a t have l a r g e a i r w a y s and t h e n r a p i d l y b r a n c h t o t h e a l v e o l i . These pathways have r e l a t i v e l y lower r e s i s t a n c e s t h a n a pathway o f many b r a n c h e s . W h i l e t h e asymmetric model i s a more r e a l i s t i c model i t has not appeared t o have c o n t r i b u t e d a d d i t i o n a l i n f o r m a t i o n i n t h i s work beyond t h a t o f a symmetric mode l . Comparison W i t h Human Data In t h e c a l c u l a t i o n o f pulmonary r e s i s t a n c e u s i n g t h i s model i t was assumed t h a t the t o t a l pulmonary r e s i s t a n c e i s a i r f l o w r e s i s t a n c e t h r o u g h the c o n d u c t i n g a i r w a y s . T o t a l pulmonary r e s i s t a n c e c o n t a i n s some element o f t i s s u e r e s i s t a n c e . W h i l e t i s s u e r e s i s t a n c e has l o n g been c o n s i d e r e d t o be r e l a t i v e l y n e g l i g i b l e ( a p p r o x i m a t e l y 20% o f t o t a l pulmonary r e s i s t a n c e ) r e c e n t s t u d i e s i n dogs and r a b b i t s suggest t h a t t i s s u e r e s i s t a n c e 82 may c o n t r i b u t e a s i g n i f i c a n t l y l a r g e r f r a c t i o n t o t o t a l pulmonary r e s i s t a n c e than has been p r e v i o u s l y a p p r e c i a t e d . F r e d b e r g e t a l (12) and Ludwig e t a l (36) have used a c a p s u l e t e c h n i q u e t o p a r t i t i o n t h e dynamic p r e s s u r e drop between a i r w a y open ing and p l e u r a l s u r f a c e i n t o the p r e s s u r e drop between the a i r w a y opening and a l v e o l i and a l v e o l i and p l e u r a l s u r f a c e . E s p e c i a l l y a t low f r e q u e n c i e s , h i g h t i d a l volumes and l a r g e s t a r t i n g l u n g volumes they f i n d v a l u e s f o r t i s s u e r e s i s t a n c e t h a t approach 60% t o 70% o f t o t a l pulmonary r e s i s t a n c e r a t h e r than 10% t o 20%. To date t h e r e have been no d i r e c t measurements o f t i s s u e and a irways r e s i s t a n c e i n human s u b j e c t s o r i n e x c i s e d human l u n g s . However t h e r e i s some i n d i r e c t e v i d e n c e t o suggest t h a t t i s s u e r e s i s t a n c e c o n t r i b u t e s s i g n i f i c a n t l y l e s s t o t o t a l pulmonary r e s i s t a n c e i n man e s p e c i a l l y a t b r e a t h i n g f r e q u e n c i e s o f 20 t o 30 b r e a t h s p e r minute and f low r a t e s o f 1 l / s . R e c e n t l y K a r i y a e t a l (29) measured a i r w a y s r e s i s t a n c e u s i n g the method o f Dubois e t a l (10) and pulmonary r e s i s t a n c e u s i n g the Mead and W h i t t e n b e r g e r t e c h n i q u e (40) s i m u l t a n e o u s l y i n normal i n d i v i d u a l s a t v a r y i n g l u n g volumes b e f o r e and a f t e r i n h a l a t i o n o f m e t h a c h o l i n e . They found v a l u e s f o r t i s s u e r e s i s t a n c e o f o n l y 9% o f t o t a l pulmonary r e s i s t a n c e a t f u n c t i o n a l r e s i d u a l c a p a c i t y (FRC) and i n f a c t , a f t e r a v i t a l c a p a c i t y b r e a t h and a r e t u r n t o FRC, t o t a l t i s s u e r e s i s t a n c e c o n t r i b u t i o n t o t o t a l pulmonary r e s i s t a n c e d e c r e a s e d m a r k e d l y . F o l l o w i n g i n h a l a t i o n o f m e t h a c h o l i n e t h e f r a c t i o n a l c o n t r i b u t i o n o f t i s s u e r e s i s t a n c e t o t o t a l pulmonary r e s i s t a n c e a l s o d e c r e a s e d . A n o t h e r method o f d i s t i n g u i s h i n g between the c o n t r i b u t i o n s 83 o f a i r w a y s and t i s s u e s t o t o t a l pulmonary r e s i s t a n c e i s t o examine t h e f requency dependence, t i d a l volume dependence and l u n g volume dependence o f t o t a l pulmonary r e s i s t a n c e . In dogs i n c r e a s i n g f r e q u e n c y , d e c r e a s i n g t i d a l volume and d e c r e a s i n g s t a r t i n g l u n g volume a l l produce a decrease i n measurements o f pulmonary r e s i s t a n c e due t o a decrease i n the c o n t r i b u t i o n o f t i s s u e r e s i s t a n c e t o t o t a l pulmonary r e s i s t a n c e ( 1 2 , 3 4 , 3 6 ) . S i m i l a r f i n d i n g s a r e not a v a i l a b l e f o r man. A v a i l a b l e d a t a sugges t s t h a t the l u n g volume v e r s u s pulmonary r e s i s t a n c e r e l a t i o n s h i p c o r r e s p o n d s much more c l o s e l y t o a system where the major component o f pulmonary r e s i s t a n c e i s a i r w a y s r e s i s t a n c e . R e c e n t l y pulmonary r e s i s t a n c e as a f u n c t i o n o f f requency i n e i g h t normal human s u b j e c t s has been measured ( u n p u b l i s h e d d a t a ) . U s i n g an esophagea l b a l l o o n and the method o f Mead and W h i t t e n b e r g e r (40) c o u p l e d t o a c o m p u t e r i z e d a v e r a g i n g c i r c u i t (54) pulmonary r e s i s t a n c e was measured a t FRC w i t h a t i d a l volume o f 800 o r 1600 ml w i t h b r e a t h i n g f r e q u e n c i e s v a r y i n g between 15 and 80 b r e a t h s p e r m i n u t e . The r e l a t i o n s h i p between pulmonary r e s i s t a n c e and f requency i s shown i n f i g u r e 28. No s i g n i f i c a n t f r e q u e n c y dependent decrease i n pulmonary r e s i s t a n c e was found o v e r t h i s range o f b r e a t h i n g f r e q u e n c i e s and i n f a c t r e s i s t a n c e was s l i g h t l y l e s s a t a l l b r e a t h i n g f r e q u e n c i e s w i t h t h e h i g h e r t i d a l vo lume. These d a t a suggest t h a t o v e r t h i s range o f b r e a t h i n g f r e q u e n c i e s t h a t t i s s u e v i s c a n c e i s r e l a t i v e l y u n i m p o r t a n t i n normal humans and t h a t i n v i v o measurements o f pulmonary r e s i s t a n c e a r e l a r g e l y i n f l u e n c e d by changes i n a irway c a l i b e r . 84 The o t h e r d a t a which s u p p o r t s a i r w a y s r e s i s t a n c e as the p r i n c i p l e c o n t r i b u t o r t o t o t a l pulmonary r e s i s t a n c e i n man i s the measured and c a l c u l a t e d r e s i s t a n c e s v e r s u s d e n s i t y o f i n s p i r e d g a s e s . U s i n g t h e model a d i f f e r e n c e between t h e r e s i s t a n c e on a i r and h e l i u m - o x y g e n a t 1 l / s i s a p p r o x i m a t e l y 20% t o 30%. T h i s i s v e r y s i m i l a r t o the v a l u e s o f 20% t o 30% seen by Macklem (38) i n normal human s u b j e c t s . I f t i s s u e r e s i s t a n c e was a s i g n i f i c a n t c o n t r i b u t o r t o t o t a l pulmonary r e s i s t a n c e t h e n a l t e r i n g gas d e n s i t y would have l e s s e f f e c t on t o t a l pulmonary r e s i s t a n c e . To t h e e x t e n t t h a t t o t a l pulmonary r e s i s t a n c e i s made up o f t i s s u e r e s i s t a n c e t h e e f f e c t s on t o t a l pulmonary r e s i s t a n c e o f a l t e r i n g t h e d e n s i t y o f t h e i n s p i r e d gas w i l l be d e c r e a s e d . The f a c t t h a t a c l o s e concordance between measured and c a l c u l a t e d d e n s i t y dependence i s observed s u p p o r t s the model as an a c c u r a t e r e p r e s e n t a t i o n o f the human t r a c h e o b r o n c h i a l t r e e . The model e s t i m a t i o n o f the t o t a l t r a c h e o b r o n c h i a l p r e s s u r e drop v e r s u s f low ( f i g u r e 17B) , as e x p e c t e d , r e s u l t e d i n s l i g h t l y lower v a l u e s t h a n seen i n normal s u b j e c t s . T h i s was expec ted because t h e model i g n o r e s a l l r e s i s t i v e a i r w a y components between the mouth and t h e t r a c h e a . The shape o f the p r e s s u r e f low c u r v e does however match the human d a t a on a i r o r h e l i u m - o x y g e n . The model a l s o p a r t i t i o n s the r e s i s t a n c e i n t o c e n t r a l and p e r i p h e r a l components s i m i l a r l y t o the e x c i s e d human l u n g s s t u d i e d by Hogg(18) . A l s o , the i n c r e a s e i n r e s i s t a n c e as t r a n s p u l m o n a r y p r e s s u r e i s r educed f o l l o w s a c u r v i l i n e a r r e l a t i o n s h i p s i m i l a r t o the d a t a o f Hogg. 85 Dose Response I n c o n s t r u c t i n g t h e o r e t i c a l dose response c u r v e s u s i n g the model a r b i t r a r i l y s e l e c t e d v a l u e s f o r maximal smooth muscle s h o r t e n i n g were u s e d . There i s v e r y l i t t l e d a t a on t h e magnitude o f smooth musc le s h o r t e n i n g i n normal human l u n g s i n v i v o . A irway smooth musc le from the t r a c h e a and main stem b r o n c h i o f an imal s has a remarkab le a b i l i t y t o s h o r t e n and a t o p t i m a l l e n g t h under i s o t o n i c c o n d i t i o n s i t can s h o r t e n t o 20% o r 30% o f i t s s t a r t i n g l e n g t h . O b v i o u s l y i f a i r w a y smooth musc le i s a r r a n g e d c i r c u m f e r e n t i a l l y around the i n t r a p a r e n c h y m a l a i r w a y s t h i s degree o f a i r w a y smooth muscle s h o r t e n i n g i s i m p o s s i b l e i n v i v o s i n c e i t would r e s u l t i n w idespread a i rway c l o s u r e . In f a c t i n normal s u b j e c t s the maximum i n c r e a s e i n pulmonary r e s i s t a n c e t h a t can be a c h i e v e d u s i n g supra-maximal c o n c e n t r a t i o n s o f i n h a l e d p h a r m a c o l o g i c a g o n i s t s such as m e t h a c h o l i n e i s i n t h e o r d e r o f 8 t o 1 2 c m H 2 0 / l / s . As r e p o r t e d by Michoud (41) , G u i l l e m i (15) and Habib (16) p l a t e a u s deve lop on the dose re sponse c u r v e a f t e r i n h a l a t i o n o f c o n c e n t r a t i o n s o f m e t h a c h o l i n e i n the 32 t o 64 mg/ml range a t v a l u e s o f pulmonary r e s i s t a n c e v e r y s i m i l a r t o t h o s e o b t a i n e d u s i n g t h e model w i t h maximal smooth muscle s h o r t e n i n g between 30% and 40%. These d a t a sugges t t h a t i f the model i s r e a s o n a b l e maximal a i rway smooth musc le s h o r t e n i n g i n v i v o must be i n the o r d e r o f 30% t o 40% r a t h e r t h a n t h e 60% t o 70% which has been r e p o r t e d w i t h i s o t o n i c s h o r t e n i n g i n v i t r o . F a c t o r s which may l i m i t a i rway smooth muscle s h o r t e n i n g i n v i v o a r e e l a s t i c o r v i s c o u s l o a d s which c o n v e r t t h e c o n t r a c t i o n 86 i n v i v o t o a more i s o m e t r i c r a t h e r t h a n i s o t o n i c mode. Loads a p p l i e d by a i r w a y c a r t i l a g e i n l a r g e a i r w a y s o r l u n g e l a s t i c r e c o i l i n s m a l l a i r w a y s c o u l d be i m p o r t a n t t h e r e f o r e i n d e t e r m i n i n g t h e degree o f a i rway smooth muscle s h o r t e n i n g . In a d d i t i o n , i n t h i s model , i t has been assumed t h a t the degree o f a i r w a y smooth muscle s h o r t e n i n g i s t h e same i n a l l r e g i o n s o f t h e t r a c h e o b r o n c h i a l t r e e . A l s o , i t was assumed t h a t t h e dose response c u r v e parameters a and fl which c o n t r o l the "ra te" o f response t o an i n h a l e d a g o n i s t a r e a l s o c o n s t a n t s f o r a l l g e n e r a t i o n s . O b v i o u s l y t h i s i s a g r o s s o v e r s i m p l i f i c a t i o n . The magnitude o f smooth muscle s h o r t e n i n g i n any g e n e r a t i o n i s g o i n g t o be r e l a t e d t o the amount o f d e p o s i t e d d r u g , the s e n s i t i v i t y and r e s p o n s i v e n e s s o f the smooth musc le i n t h a t r e g i o n as w e l l as the p e r m e a b i l i t y o f the a i r w a y mucosa and s u r f a c e a r e a o v e r which the d r u g i s d i s t r i b u t e d . The shape o f the dose response c u r v e and the s l o p e w i l l be dependent upon the a r b i t r a r y v a l u e s chosen f o r oc and fi and these c o u l d be i n f l u e n c e d by t h e f a c t o r s ment ioned above . A l t h o u g h changes i n t h e s e f a c t o r s w i l l a l t e r t h e shape and p o s i t i o n o f the dose response c u r v e the maximal i n c r e a s e i n r e s i s t a n c e i s dependent s o l e l y on a i rway w a l l t h i c k n e s s and t h e maximal degree o f a i r w a y smooth muscle s h o r t e n i n g . P a t i e n t S e l e c t i o n Data from v a r i o u s s o u r c e s has been combined i n t h i s work t o o b t a i n the i n f o r m a t i o n n e c e s s a r y f o r the mode l . W e i b e l ' s geometry 87 was based on 5 human l u n g s f o r the l a r g e a i r w a y s and 3 f o r the s m a l l e r s t r u c t u r e s . H o r s f i e l d work was based on a s i n g l e l u n g c a s t . The f l u i d dynamic c a l c u l a t i o n s were e s t i m a t e d on c a s t s o f a s i n g l e l u n g t h a t i n c l u d e d o n l y a few b i f u r c a t i o n s . In these cases normal s u b j e c t s were used f o r a l l measurements. The r e l a t i o n s h i p between w a l l a r e a and i n t e r n a l p e r i m e t e r u s i n g the d a t a o f James (27) was based on normal s u b j e c t s and a s t h m a t i c s whom d i e d from c o m p l i c a t i o n s o f asthma. These c l e a r l y r e p r e s e n t t h e extreme ends o f the p o p u l a t i o n w i t h r e g a r d s t o w a l l a r e a . By s e l e c t i n g s u b j e c t s t h a t were l e s s s e v e r e a s t h m a t i c s w i t h l e s s t h i c k e n i n g o f the a i r w a y w a l l t h e r e would be a r e d u c t i o n i n the e f f e c t on a i r w a y s r e s i s t a n c e seen h e r e . S e n s i t i v i t y o f Model Parameters The model e q u a t i o n s as d e s c r i b e d i n Appendix E c o n t a i n a v a s t number o f parameters t h a t c o u l d p o t e n t i a l l y a l t e r the r e s u l t s . The g e o m e t r i c a l s t r u c t u r e s s t u d i e d were f i x e d t o be e i t h e r t h e symmetric model o f Weibe l o r asymmetr ic t r e e o f H o r s f i e l d , o b v i o u s l y many o t h e r p o s s i b l e c o m b i n a t i o n s e x i s t t h a t would a l l o w g r e a t e r inhomogenei ty and t h e r e f o r e a more r e a l i s t i c r e p r e s e n t a t i o n o f the t r a c h e o b r o n c h i a l t r e e . To o b t a i n g r e a t e r v a r i a b i l i t y i n t h e model t o account f o r i n t r a s u b j e c t v a r i a b i l i t y i n t h e geometry and d imens ions would r e q u i r e f a r l a r g e r computer r e s o u r c e s t h a n used i n t h i s work. The model i s e x t r e m e l y s e n s i t i v e t o any change i n the a irway c a l i b e r which c o u l d be a f f e c t e d by s t a r t i n g geometry, the 88 c o r r e c t i o n f o r l u n g i n f l a t i o n , the method by which encroachment i s c o n s i d e r e d and the method by which t h e c o n t r a c t e d i n t e r n a l d i a m e t e r i s c a l c u l a t e d . I t i s assumed t h a t t h i s model i s t o be used t o s t u d y a s i n g l e "hypothes i zed" s u b j e c t w i t h e i t h e r normal o r t h i c k e n e d a i r w a y w a l l s . W i t h t h i s assumpt ion i t i s r e a s o n a b l e t o a c c e p t the a v a i l a b l e g e o m e t r i c a l d a t a (21,51,61) as a t l e a s t average r e p r e s e n t a t i o n s o f a normal s u b j e c t . I t i s q u i t e l i k e l y t h a t l a r g e r p e o p l e would have l a r g e r a i r w a y s and t h e r e f o r e s m a l l e r r e s i s t a n c e s but the q u a l i t a t i v e changes seen would be the same f o r a l a r g e s u b j e c t as f o r a s m a l l s u b j e c t . I f t h e s t a r t i n g a i r w a y c a l i b e r s a r e d e c r e a s e d by 30% t h e n i n i t i a l r e s i s t a n c e i n c r e a s e s by 3 f o l d t o 2.9 which i s s t i l l i n t h e normal range . These s t a r t i n g v a l u e s however w i l l not a l l o w as g r e a t a w a l l t h i c k e n i n g o r a i rway smooth muscle s h o r t e n i n g u n t i l c l o s u r e r e s u l t s (here c l o s u r e o c c u r s a t 30% PMSmax) . The volume c o r r e c t i o n f o r l u n g i n f l a t i o n was based on e l e v e n p o i n t s i n t e n d e d t o r e p r e s e n t a l l 23 g e n e r a t i o n s o f the t r a c h e o b r o n c h i a l t r e e a t a l l l u n g i n f l a t i o n s . O b v i o u s l y more work needs t o be done the volume c h a r a c t e r i s t i c s o f the a i rways p o s s i b l y by u s i n g CAT scanners t o r e c o r d a i r w a y c a l i b e r d a t a as an e x c i s e d l u n g i s i n f l a t e d . S i n c e the a i r w a y mucosa f o l d s when t h e l u n g d e f l a t e s t h e a i r w a y s may become h a r d e r t o deform and hence may not reduce t h e i r c r o s s - s e c t i o n a l a r e a a v a i l a b l e f o r a i r f l o w as q u i c k l y as e s t i m a t e d . T h i s f o l d i n g a l s o e f f e c t s the c a l c u l a t i o n s f o r the c o n t r a c t e d i n t e r n a l r a d i u s a f t e r smooth muscle c o n t r a c t i o n as i t was assumed t h a t t h e a i r w a y s m a i n t a i n e d a c i r c u l a r shape d u r i n g c o n t r a c t i o n . The f o l d i n g o f the a irway 89 w a l l may, as w i t h changes i n l u n g i n f l a t i o n , cause l e s s o f a r e d u c t i o n i n c r o s s - s e c t i o n a l a r e a t h a n p r e d i c t e d . The parameters a and fi which c o n t r o l the shape o f the dose re sponse have no e f f e c t on maximal r e s i s t a n c e v a l u e s . However i t i s p o s s i b l e t o genera te v e r y h i g h r e s i s t a n c e v a l u e s w i t h the model which would not be p h y s i c a l l y o b t a i n a b l e . Under these c o n d i t i o n s t h e shape o f the dose response c u r v e as c o n t r o l l e d by a and fi would be i m p o r t a n t parameters i n t h e model as t h e y would d e t e r m i n e how q u i c k l y a s u b j e c t reached p h y s i o l o g i c a l l y u n a c c e p t a b l e r e s i s t a n c e l o a d s . A l l o f t h e s e f a c t o r s can have s i g n i f i c a n t e f f e c t s o f the p r e d i c t e d p r e s s u r e drop seen i n t h i s work but w i l l no t change the q u a l i t a t i v e i n t e r p r e t a t i o n o f changes i n a i r w a y w a l l t h i c k n e s s and smooth musc le s h o r t e n i n g . S i g n i f i c a n c e The model p r e s e n t e d here p r o v i d e s an approach t o mode l ing a i r w a y s s t r u c t u r e and f u n c t i o n t h a t extends t h o s e c u r r e n t l y a v a i l a b l e i n the l i t e r a t u r e . The advantage o f the model i s t h a t i t a l l o w s f o r p r e s s u r e volume c h a r a c t e r i s t i c s o f the l u n g t o be t a k e n i n t o account and f o r changes i n w a l l t h i c k n e s s and smooth musc le s h o r t e n i n g t o be s t u d i e d . T h i s model p r o v i d e s many parameters c o n c e r n i n g the f u n c t i o n o f smooth muscle and a i rway w a l l t h i c k n e s s so t h a t as more i n f o r m a t i o n c o n c e r n i n g these v a r i a b l e s becomes a v a i l a b l e i t can be e a s i l y implemented. The model has demonstrated t h a t d i f f e r e n c e s between the 90 a s y m m e t r i c a l s t r u c t u r e o f H o r s f i e l d and the symmetric s t r u c t u r e o f W e i b e l may be r e l a t i v e l y minor i n t h e i n v e s t i g a t i o n o f t o t a l t r a c h e o b r o n c h i a l r e s i s t a n c e s . C u r r e n t i n v e s t i g a t i o n i n t o asthma c o n c e n t r a t e s on the d i f f e r e n c e s b e l i e v e d t o o c c u r between normal and a s t h m a t i c s u b j e c t s i n smooth m u s c l e . T h i s model has shown t h a t marked changes i n t o t a l r e s i s t a n c e a r e p o s s i b l e by a t h i c k e n i n g o f the a i r w a y w a l l and t h a t t h i s response i s enhanced by encroachment i n t o t h e lumenal space o r by p o s i t i o n i n g t h e a i r w a y w a l l t h i c k e n i n g i n the p e r i p h e r a l as oppose t o more c e n t r a l a i r w a y s . I t may be p o s s i b l e , u s i n g e i t h e r t h i s model o r a m o d i f i c a t i o n o f i t , t o u n d e r s t a n d more f u l l y the p o t e n t i a l i n t e r a c t i o n s between a i r w a y smooth musc le and a i rway w a l l t h i c k e n i n g . 91 Appendix A U s i n g a p r i n c i p l e c a l l e d D e l e s s e ' s theorem (8) i t i s p o s s i b l e t o r e l a t e t h e f r a c t i o n o f volume t h a t some p a r t i c l e o c c u p i e s t o the f r a c t i o n o f s u r f a c e a r e a the p a r t i c l e o c c u p i e s o f a random s l i c e t h r o u g h t h a t vo lume. More f o r m a l l y s t a t e d : The fraction (-~w) of some unit volume (V) which i s occupied by a granular component which is randomly distributed in the unit volume (V) is equal to the fraction (rs) that the granular component occupies on a unit surface (S) which i s cut through the unit volume (V). This principle is true regardless of the shape of the particle. U s i n g t h i s concept i f t h e r e a r e N p a r t i c l e s , each o f some volume v d i s t r i b u t e d randomly i n a t o t a l volume V t h e n : N*v = r y * V ( A . l ) and the number o f p a r t i c l e s , n , each o f an average c r o s s - s e c t i o n a l a r e a s, on a c u t s e c t i o n o f a r e a S t h r o u g h the volume V w i l l be : n*s = r *S (A.2) s W e i b e l demonstrated t h a t the p a r t i c l e volume, v , can be r e l a t e d t o i t s mean c r o s s - s e c t i o n a l a r e a , s, by : 92 v = fi * s ( 3 / 2 ) (A.3) where 6 i s some shape parameter which i s dependent on the s t r u c t u r e under i n v e s t i g a t i o n . F o r a p e r f e c t s p h e r e , fi i s 1 .382, and the v a l u e W e i b e l de termined f o r t h e a l v e o l i was 1 .55 . I f we combine e q u a t i o n s A . 1 , A . 2 , and A . 3 and use the f a c t t h a t r s = r y = r ( D e l e s s e ' s theorem) , s e t V=l f o r a u n i t volume and S=l f o r a u n i t c r o s s - s e c t i o n t h e n : (3/2) N = — - (A.4) fi • r < 1 / 2 ) and hence we can o b t a i n the number o f a l v e o l i p e r c u b i c c e n t i m e t e r o f l u n g by d i v i d i n g the number o f a l v e o l i t r a n s s e c t i o n s p e r square c e n t i m e t e r by the shape c o e f f i c i e n t , 6, and the c o r r e s p o n d i n g d e n s i t y f r a c t i o n , r . These e q u a t i o n s a p p l y t o randomly d i s t r i b u t e d o b j e c t s i n some volume and the a l v e o l i a r e g e n e r a l l y c o n s i d e r e d t o be randomly l o c a t e d i n the parenchyma (the t i s s u e s u r r o u n d i n g the p e r i p h e r a l a i r w a y s ) . The volume o f t h e parenchyma i s about 90% o f the t o t a l l u n g c a p a c i t y and t h e r e f o r e the t o t a l number o f a l v e o l i i n the parenchymal volume can be e s t i m a t e d u s i n g e q u a t i o n A . 4 . 93 Appendix B The b i a s e d method by which W e i b e l measured t h e o r i g i n a l d i a m e t e r s and l e n g t h s o f the branches o f the t r a c h e o b r o n c h i a l t r e e f o r c e s some assumpt ions t o be made so t h a t t h e e n t i r e d i s t r i b u t i o n o f the measurements can be c o m p l e t e d . Once the d i s t r i b u t i o n s a r e completed i t i s p o s s i b l e t o o b t a i n a more a c c u r a t e e s t i m a t e o f the t r u e mean l e n g t h and d i a m e t e r . The assumpt ions r e q u i r e d f o r t h i s t e c h n i q u e a r e : 1) The measurements o f l e n g t h and d i a m e t e r f o l l o w some known d i s t r i b u t i o n , i n t h i s case t h e b i n o m i a l d i s t r i b u t i o n . 2) The minimum l e n g t h o r d i a m e t e r i s known, i n t h i s case t h e s m a l l e s t c a t e g o r y w i l l be 0.5mm. 3) A l l o f t h e d i a m e t e r s l a r g e r t h a n 2.5mm and a l l o f the l e n g t h s l a r g e r t h a n 2.0mm have been c o m p l e t e l y sampled . These assumpt ions a p p l y a t each g e n e r a t i o n i n the t r a c h e o b r o n c h i a l t r e e . The b i n o m i a l d i s t r i b u t i o n i s a d i s c r e t e d i s t r i b u t i o n t h a t de termines t h e expec ted f requency o f a c o l l e c t i o n o f o b s e r v a t i o n s as t h e y f a l l i n t o b i n s o f a s e t w i d t h between some minimum and maximum v a l u e . In t h i s case t h e d i a m e t e r s and 94 l e n g t h s a r e d i v i d e d i n t o b i n s o f 0.5mm i n w i d t h between 0.5mm and the maximum observed v a l u e . T h i s g i v e s a t o t a l o f m, (m = maximum d i a m e t e r / 0.5mm) i n t e r v a l s and any i n d i v i d u a l o b s e r v a t i o n x w i l l f a l l i n t o the x / 0 . 5 b i n . The f r e q u e n c y o f o c c u r r e n c e F(x) i n b i n x, x some i n t e g e r from 1 t o m, i s : F(x) m! p < - x ) f l , 0 . p ) x (m-x)! x! The parameter o f i n t e r e s t i n the above f o r m u l a i s p . By u s i n g t h e t h i r d assumpt ion t h a t a l l branches l a r g e r t h a n some v a l u e have been c o m p l e t e l y measured i t i s p o s s i b l e t o form an e q u a t i o n w i t h o n l y p as an unknown and s o l v e f o r p : Dm a x \ F (D)*n(z ) = sum o f a c t u a l l y measured r» o c branches >= 2.5mm D=2.5mm T h i s f o r m u l a s t a t e s t h a t p must be de termined so t h a t the measured number o f d i a m e t e r s e q u a l s the p r e d i c t e d p r o p o r t i o n o f d i a m e t e r s g r e a t e r t h a n 2.5mm t imes the number o f branches a t the c u r r e n t g e n e r a t i o n as e s t i m a t e d by 2 Z . Once p has been e s t i m a t e d t h e n t h e average d i a m e t e r (or l ength) i s n ( z ) * p . U s i n g t h i s t e c h n i q u e i t i s p o s s i b l e t o c o m p l e t e l y de termine the d i s t r i b u t i o n s o f l e n g t h s and d i a m e t e r s a t each g e n e r a t i o n and t o o b t a i n more a p p r o p r i a t e mean v a l u e s . 95 Appendix C U s i n g a s i n g l e a i rway i t i s p o s s i b l e t o c a l c u l a t e the change i n t h e r a d i u s o f an a i rway w i t h s p e c i f i e d v a l u e s f o r the p r o p o r t i o n o f smooth muscle i n the a i rway c i r c u m f e r e n c e ( P M C ) , t h e amount o f a i r w a y w a l l t h a t i s t i s s u e (PW) and t h e degree o f smooth musc le s h o r t e n i n g ( P M S ) . The sequence o f c a l c u l a t i o n s f o r d e t e r m i n i n g t h e c o n t r a c t e d d i a m e t e r o f t h e a i r w a y lumen a r e as f o l l o w s : R i r = r e l a x e d ( p r e - c o n t r a c t i o n ) i n t e r n a l r a d i u s o f the lumen Rer = r e l a x e d e x t e r n a l r a d i u s ( o u t s i d e o f muscle) WA = t i s s u e w a l l a r e a A i r = r e l a x e d lumen a r e a = rcRir2 Aer = r e l a x e d o u t e r a r e a = A i r + WA = n R i r 2 + WA = i r Re r 2 I f we now assume t h a t the c i r c u m f e r e n c e w i l l be c o n t r a c t e d a c c o r d i n g t o t h e amount o f smooth muscle i n the c i r c u m f e r e n c e and t h e amount t h e smooth muscle s h o r t e n s t h e n i t s i s easy t o see t h a t : PMC = p r o p r o t i o n o f the c i r c u m f e r e n c e t h a t c o n t a i n s ( the r e l a x e d e x t e r n a l r a d i u s ) ( the r e l a x e d e x t e r n a l c i r c u m f e r e n c e ) smooth musc le ( 0 - 1 0 0 % ) . 96 PMS = r e l a t i v e change i n l e n g t h i n t h e smooth musc le (0-100%). Cec = the e x t e r n a l c i r c u m f e r e n c e a f t e r smooth muscle c o n t r a c t i o n = ( 1 - P M C * P M S ) *Cer = ( 1 - P M C * P M S ) * 2TT/ R i r 2 + WA/TT = 2TTRec Rec = ( 1 - P M C * P M S ) / R i r 2 + WA/TT = the e x t e r n a l r a d i u s a f t e r c o n t r a c t i o n Aec = t h e e x t e r n a l a r e a a f t e r c o n t r a c t i o n = TTRec2 = T T ( 1 - P M S * P M C ) 2 * | R i r 2 + WA/TT j -A i c = lumenal a r e a a f t e r c o n t r a c t i o n = Aec - WA = TTRic 2 -.2 / A e c WA R i c = / — — - —— TT TT ( 1 - P M S * P M C ) 2 * | R i r 2 + WA/TT | ~ ^ ( C l ) T h i s e q u a t i o n , C l , g i v e s the c a l c u l a t i o n f o r t h e i n t e r n a l a i r w a y r a d i u s a f t e r some degree o f a i rway smooth muscle s h o r t e n i n g . 97 Appendix D The assumpt ions and c o m p u t a t i o n a l p r o c e d u r e s , as g i v e n by Feldman (11) , f o r the we ighted l e a s t squares and r e s t r i c t e d maximum l i k e l i h o o d a r e as f o l l o w s : 1) The square r o o t o f w a l l a r e a , SWA, and t h e i n t e r n a l p e r i m e t e r , P i , f o r each s u b j e c t ' s d a t a a r e assumed t o f o l l o w t h e r e l a t i o n s h i p : V^ WA1 = Aij + Bi j*Pi + c i j T h i s says t h a t f o r each s u b j e c t t h e r e i s a l i n e a r r e l a t i o n s h i p between v^ wS and P i . The above e q u a t i o n r e q u i r e s some r e s t r i c t i o n s t o be n u m e r i c a l l y s o l v a b l e , t h r e e c o n d i t i o n s w i l l s e t on t h e v a l u e s f o r the i n t e r c e p t s , A i j ' s , t h e s l o p e s , B i j ' s , and the e r r o r s , e i j ' s . The c o n d i t i o n s a r e t h a t t h e s l o p e s and i n t e r c e p t s f o r the p e o p l e w i t h i n each group a r e n o r m a l l y d i s t r i b u t e d about some group average v a l u e . A l s o , i t i s assumed t h a t the r e s i d u a l s a r e n o r m a l l y d i s t r i b u t e d and t h a t the s u b j e c t s i n b o t h groups f i t w i t h comparable a c c u r a c y . Wi th these assumpt ions i n p l a c e the c o m p u t a t i o n a l s o l u t i o n i s : 2) The k t h d a t a p a i r f o r the j t h s u b j e c t i n t h e i t h group w i l l be denoted a s : y , X . Where the y v a l u e s a r e the square r o o t s o f - M J k ' i jk 1 ^ w a l l a r e a and t h e X v a l u e s a r e the i n t e r n a l p e r i m e t e r s preceded by a one ( l , P i ) . The i n d i v i d u a l s l o p e s and r e g r e s s i o n s f o r each s u b j e c t i n each group i s : a = [>1J1 = ( x T X ) _ 1 X T y I J | b i j j *J ' J J J 1 J t h e average s l o p e and i n t e r c e p t f o r each group based on t h e above 98 i s t h e n : n i S i n c e i t has a l r e a d y been assumed t h a t each s u b j e c t f i t s a l i n e a r r e l a t i o n s h i p w i t h e q u a l a c c u r a c y t h e n t h e e s t i m a t e o f the v a r i a n c e o f t h e r e s i d u a l e r r o r s , s 2 can be p o o l e d a c r o s s a l l subj e c t s : 2 ni nij Z I Z (y i Jk - a i j-bij*Xijk) 2 _ 1=1 j=l k = l  Z Z (nij - 2) i=i j=i t h i s f o r m u l a mere ly adds a l l o f r e s i d u a l s , s q u a r e d , f o r each s u b j e c t and d i v i d e s t h i s v a l u e by the t o t a l number o f p o i n t s , l e s s 2, f o r each s u b j e c t . The next s t e p i s an e s t i m a t i o n o f t h e v a r i a b i l i t y o f t h e s l o p e s and i n t e r c e p t s . S i n c e t h e r e a r e two p a r a m e t e r s , a s l o p e and an i n t e r c e p t , t h e r e a r e t h r e e NOT two components t o the v a r i a b i l i t y . T h e r e i s a v a r i a n c e f o r the s l o p e and i n t e r c e p t b u t a t h i r d component a l s o e x i s t s which d e s c r i b e s how the s l o p e and i n t e r c e p t v a r y t o g e t h e r . U s i n g Fe ldman's n o t a t i o n t h i s c o v a r i a n c e m a t r i x i s : 2 ni _ _ A E Z (a i j - a i ) (aij - a i ) T 1 =1 J = 1 Dnih = Z (ni - 1) l As s t a t e d by Feldman t h e v a l u e f o r t h e v a r i a b i l i t y o f the s l o p e s 99 and i n t e r c e p t s has been b i a s e d towards an e x c e s s i v e l y l a r g e v a l u e by t h e w i t h i n - g r o u p v a r i a t i o n . The e s t i m a t e o f t h i s b i a s i s : The u s u a l e s t i m a t e f o r the v a r i a b i l i t y o f t h e r e g r e s s i o n s l o p e s A A and i n t e r c e p t s would be D=Dn ih - B but we want t o use t h i s m a t r i x , D , as a w e i g h t i n g m a t r i x . The concept can be e x p l a i n e d as f o l l o w s , i f t h e v a r i a b i l i t y o f the s l o p e i n group 1 i s A and group 2 i s B t h e n by m u l t i p l y i n g the group s l o p e s by 1 / A and 1 / B r e s p e c t i v e l y , a d d i n g these new v a l u e s t o g e t h e r and d i v i d i n g by t h e sum o f 1 / A and 1 / B we w i l l o b t a i n an average v a l u e t h a t g i v e s a g r e a t e r we ight t o the d a t a w i t h the s m a l l e r v a r i a b i l i t y . The d i f f i c u l t l y i s t h a t w h i l e the m a t r i x Dnih and B a r e i n v e r t i b l e , t h e d i f f e r e n c e between these two m a t r i c e s may not be i n v e r t i b l e . A s o l u t i o n p r o p o s e d by Amemiya (1) and used by Feldman i s t o s e p a r a t e t h e m a t r i x D i n t o an i n v e r t i b l e and n o n - i n v e r t i b l e p o r t i o n and d i s c a r d the n o n - i n v e r t i b l e p o r t i o n . The exac t d e t a i l s f o r t h i s p r o c e d u r e a r e g i v e n i n (1) o r ( 1 1 ) . A f t e r a s u i t a b l e m a t r i x i s o b t a i n e d , D A , t h e n the v a r i a n c e m a t r i x f o r any s u b j e c t i s : V i j = s 2 I + X I J D A X I J e where I i s an i d e n t i t y m a t r i x o f t h e a p p r o p r i a t e d i m e n s i o n s . I f the w e i g h t i n g m a t r i x i s now t a k e n t o be W i j = V i j - 1 t h e n the we ighted e s t i m a t e f o r the i t h group (here normals o r a s t h m a t i c s ) J = i I ( X i j X i j ) - l 100 i s : -1 r ru a i bi E x l j w i j x i j J = I E X i j W i j y u J = i and t h e c o v a r i a n c e m a t r i x f o r t h i s group i s : n-1 E xi jwi jxu j =1 The above e q u a t i o n s a r e the e s t i m a t e s f o r Fe ldman's weighted l e a s t squares approach and t h e v a r i a b i l i t y o f these parameters can be a s s e s s e d by the c o v a r i a n c e m a t r i x where the square r o o t o f t h e d i a g o n a l e lements i s t h e s t a n d a r d e r r o r o f the parameter o f i n t e r e s t . To o b t a i n t h e r e s t r i c t e d maximum l i k e l i h o o d e s t i m a t e s a c o m p l i c a t e d i t e r a t i v e p r o c e d u r e i s r e q u i r e d . L a i r d (31) has d e t a i l e d t h e s t e p s r e q u i r e d u s i n g a method c a l l e d the EM a l g o r i t h m . The EM a l g o r i t h m i s a two s t e p p r o c e d u r e where the c u r r e n t e s t i m a t e o f the s l o p e and i n t e r c e p t i s used t o compute t h e expec ted v a l u e o f t h e i n f o r m a t i o n r e q u i r e d t o make these i n i t i a l e s t i m a t e s , the E - s t e p . In the M - s t e p , u s i n g the new e s t i m a t e s o f t h e i n f o r m a t i o n r e q u i r e d t o c a l c u l a t e the s l o p e s and i n t e r c e p t s , new e s t i m a t e s a r e made t h a t maximize t h e l i k e l i h o o d f o r t h e o b s e r v e d d a t a . U n f o r t u n a t e l y the p r e c e d i n g computat ions a r e not o n l y complex b u t t h e r e i s no way o f knowing how r a p i d l y the EM a l g o r i t h m w i l l converge on an o p t i m a l s o l u t i o n . The t h e o r y p r e s e n t e d by Dempster (9) o n l y guarantees t h a t e v e n t u a l l y the EM a l g o r i t h m w i l l converge on the maximum l i k e l i h o o d e s t i m a t e , not 101 when i t w i l l c o n v e r g e . The Gauss(13) commands f o r c a l c u l a t i n g t h e p r e c e e e d i n g s s t e p s a r e : TRACE 2; * / NEW; CLOSEALL; CALL CROUTP(1 .OE-30 ) ; CALL CROUTP(ERROR(2)); / * Data must be GROUP : SUBJECT : YVAR : XVAR(s) * / F L F ^ ' K A G G A " ; / * change t h i s t o s p e c i f y c u r r e n t f i l e * / XLAB = " I n t e r n a l P e r i m e t e r " ; / * l a b e l f o r graphs * / YLAB = "Square Root o f W a l l A r e a " ; / * l a b e l f o r graphs * / OPEN D T = A F L ; NR=ROWSF(DT); DAT=READR(DT,NR); / * SORT BY GROUP THEN SUBJECT * / LET SS[2 ,1 ] = 1 2; BIAS=0; CND=0; BS=0 ; N2=0 ; / * DTT=S0RTMC(DAT, SS) ; * / NGPS=2; / * Change t h i s f o r a l a r g e r number groups * / NXVR=1; / * Change t h i s f o r more t h a n 1 X v a r i a b l e * / MAXIT=50; / * Maximum number o f i t e r a t i o n s a l l o w e d f o r EM * / PCTCVG=0.5; / * EM convergence c r i t e r i o n * / XMN=0; / * x - a x i s minimum f o r graphs * / XMX=10; / * x - a x i s maximum f o r graphs * / XST=2; / * x - a x i s s t e p s i z e f o r graphs * / YMN=2.5; / * y - a x i s minumum f o r graphs * / YMX=4.5; / * y - a x i s maximum f o r graphs * / YST=0.2; / * y - a x i s s t e p s i z e f o r graphs * / PRINT "DATA READ"; ABAR=ZER0S(1,NXVR+2); NIGP=ZER0S(1,NGPS); / * F i n d the l e a s t squares r e g r e s s i o n s c o e f f i c i e n t s * / / * f o r each s u b j e c t i n each group * / / * p l a c e r e s u l t i n form GROUP SUBJECT N INTERCEPT SLOPE RESIDUAL / * i n the m a t r i x REG * / L = l ; / * GROUP * / 1=1; / * SUBJECT * / J = l ; / * Row where s u b j e c t s t a r t s * / K=0; / * Row where s u b j e c t ends * / 102 S2=0; / * Sum o f r e s i d u a l v a r i a n c e s * / STEP1: K=K+1; I F DAT[K+1,1] NE D A T [ K , 1 ] ; / * End o f c u r r e n t group GOTO C A L C ; ENDIF; I F DAT[K+1,2] NE D A T [ K , 2 ] ; / * End o f c u r r e n t s u b j e c t GOTO C A L C ; ENDIF; I F K+l EQ NR; / * End o f d a t a K=K+1; GOTO C A L C ; ENDIF; GOTO STEP1; PRINT "MARKERS S E T " ; CALC: PRINT I ; Y = D A T [ J : K , 3 ] ; N=K-J+1; X = O N E S ( N , l ) ~ D A T [ J : K , 4 : 4 + N X V R - l ] ; B=OLSQR(Y,X); S = ( Y - X * B ) / * ( Y - X * B ) ; S2=S2+S; N2=N2+N-2; X 1 = M I N C ( X [ . , 2 : 2 + N X V R - l ] ) ; X 2 = M A X C ( X [ . , 2 : 2 + N X V R - l ] ) ; X3=MEANC(X[ . , 2 :2+NXVR- l ] ) ; X 4 = S T D C ( X [ . , 2 : 2 + N X V R - l ] ) ; Y1=MINC(Y); Y2=MAXC(Y); Y3=MEANC(Y); Y4=STDC(Y); I F J EQ 1; DESCR=(Y1-Y2~Y3~Y4~X1'~X2'~X3'~X4 ' ) ; R E G = ( L ~ I ~ N ~ B ' ~ S ) ; ELSE ; D E S C R = D E S C R | ( Y 1 ~ Y 2 ~ Y 3 ~ Y 4 ~ X 1 ' ~ X 2 ' ~ X 3 ' - X 4 ' ) ; R E G = R E G | ( L ~ I ~ N ~ B ' - S ) ; ENDIF; BS=BS+INV(X'*X); I F K==NR; BIAS=BIAS+BS/I; BS=0 ; GOTO DONE1; E L S E I F DAT[K+1,1]==L; 1=1+1; J=K+1; GOTO S T E P l ; ELSE ; BIAS=BIAS+BS/I; 103 BS=0; L=L+1; 1=1; J=K+1; GOTO STEP1; ENDIF; PRINT "INDIVIDUAL REGRESSIONS DONE"; DONE1: BIAS=BIAS*S2/(N2*NGPS); R=ROWS(REG); /* Now calcu l a t e the covariance matrix of the regression coefs */ J=l; K=0 ; STEP2: K=K+1; IF REG[K+1,1] NE REG[K,1]; /* End of a group */ GOTO CALC2; ELSEIF K+l EQ R; /*End of data */ K=K+1; GOTO CALC2; ENDIF; GOTO STEP2; CALC2: ABAR=ABAR|((MEANC(REG[J:K,4:4+NXVR]))'-(K-J+l)); IF K NE R; J=K+l; GOTO STEP2; ENDIF; ABAR=ABAR[2:2+NGPS-l,1:NXVR+1+1]; AS=REG[.,4:4+NXVR]; J=i; DO WHILE J<=R; AS[J,.]=AS[J,.]-ABAR[REG[J,l],1:1+NXVR]; NIGP[REG[J,1]]=NIGP[REG[J,1]]+l; J=J+1; ENDO ; DNIH=(AS[l:NIGP[l],.])'*(AS[1:NIGP[1],.]); DN=NIGP[l]-l; J=2; K=NIGP[1]+1; DO WHILE J<=NGPS; DNIH=DNIH+(AS[K:K+NIGP[J]-1,.])'*(AS[K:K+NIGP[J]-1,.]); DN=DN+NIGP[J]-1; J=J+1; ENDO ; DNIH=DNIH/DN; U=CHOL(BIAS); 104 L I = I N V ( U ' ) ; {E1,E2} = E I G R S 2 ( L I * D N I H * L I ' ) ; E3=E1; I F MINC(El) GE 1 .0; DAME=DNIH-BIAS J E L S E I F MAXC(El) LE 1 .0; DAME=DNIH*0.0; CND=1; E L S E ; J = l ; DO WHILE J LE NXVR+1; I F E 1 [ J ] LT 1 .0; E 1 [ J ] = 0 . 0 ; E L S E ; E 1 [ J ] = E 1 [ J ] - 1 . 0 ; ENDIF; J = J + l ; ENDO; D A M E = ( I N V ( L I ) * E 2 ) * ( D I A G R V ( E Y E ( 1 + N X V R ) , E l ) ) * ( I N V ( L I ) * E 2 ) ' ENDIF; PRINT "COVARIANCE MATRIX AND STS COMPLETE"; / * E s t i m a t i o n o f weight m a t r i c e s * / XWX=0.0; XWY=0.0; J = l ; / * C u r r e n t s u b j e c t * / K = l ; / * C u r r e n t group * / L = l ; / * C u r r e n t row where d a t a s t a r t s * / M=l; / * Row where d a t a ends f o r c u r r e n t s u b j e c t * / N=l ; / * C u r r e n t row o f m a t r i x REG d e s c r i b e s c u r r e n t s u b j e c t * DO WHILE K LE NGPS; DO WHILE J LE N I G P [ K ] ; PRINT "Weight l o o p : S u b j e c t =" J " , Group =" K; M=M+REG[N,3]-1.0; Z = 0 N E S ( R E G [ N , 3 ] , 1 ) - D A T [ L : M , 4 : 4 + N X V R - l ] ; V=(S2/N2)*EYE(REG[N,3] )+Z*DAME*Z'; W=INV(V); XWX=XWX+Z'*W*Z; XWY=XWY+Z /*W*DAT[L:M,3]; L=M+l; M=M+1; N=N+l; / * Next row o f m a t r i x REG * / J=J+1; / * Next s u b j e c t * / ENDO ; I F K EQ 1; C=INV(XWX); AB=C*XWY; ELSE ; C=C|INV(XWX); AB=AB~INV(XWX)*XWY; ENDIF; J = l ; K=K+1; 105 xwx=xwx*o.o; XWY=XWY*0.0; ENDO; / * Now t o i t e r a t e the EM a l g o r i t h m * / S2=S2/N2; SS2=S2; DDM=DAME; ITN=0; LOOP: PRINT I T N ; I F CND EQ 1; S2DI=S2*EYE(1+NXVR); ELSE ; S2DI=S2*INV(DAME); ENDIF; S4=S2*S2; K = l ; / * Group marker * / 1=1; / * S u b j e c t marker * / L = l ; / * Row where s u b j e c t d a t a s t a r t s * / M=l; / * Row where s u b j e c t d a t a ends * / R = l ; / * Row o f REG t h a t says how p o i n t s c u r r e n t s u b j e c t has NX=1+NXVR; V = l ; W=l; DO WHILE K LE NGPS; DO WHILE I LE N I G P [ K ] ; N=REG[R,3] ; / * Number o f p o i n t s f o r c u r r e n t s u b j e c t * / M=L+N-1; XN=ONES(N,1) -DAT[L:M,4:4+NXVR-1]; / * I F CND EQ 0; * / WN=(EYE(N)-XN*INV(S2DI+XN / *XN)*XN') /S2; / * E L S E ; * / / * WN=EYE(N)/S2; * / / * ENDIF; * / XW=XN'*WN; XWWX=XW*XW'; XWX=XW*XN; XWY=XW*DAT[L:M,3]; SF=SUMC(WN.*EYE(N)*ONES(N, l ) ) ; ROWW=RESHAPE(XWX,1,NX*NX)-XWY'-RESHAPE(XWWX,1,NX*NX)-SF; I F ((K EQ 1) AND (I EQ 1 ) ) ; STORE=ROWW; E L S E ; STORE=STORE|ROWW; ENDIF; 1=1+1; R=R+l; L=M+1; ENDO ; CR=INV(RESHAPE(SUMC(STORE[W:R- l , l :NX*NX]) ,NX,NX)) ; AR=CR*(SUMC(STORE[W:R-l ,NX*NX+l:NX*NX+NX])); I F K EQ 1; 106 A1=AR'; C 1=RESHAPE(CR , 1,NX*NX); E L S E ; A 1 = A 1 ( A R ' ) ; C 1 = C 1 (RESHAPE(CR,1,NX*NX)); ENDIF; 1 = 1 ; R=W; L=V; M=l; DO WHILE I LE N I G P [ K ] ; N=REG[R,3]; / * Number o f p o i n t s f o r c u r r e n t s u b j e c t */ M=L+N -1; XN=ONES(N,1 )~DAT[L:M,4:4+NXVR-l]; ROWW=STORE[R,.]; XWX=RESHAPE(ROWW[•,1:NX*NX],NX,NX); XWY=(ROWW[.,NX*NX+1:NX*NX+NX])'; XWWX=RESHAPE(ROWW[.,NX*NX+NX+1:2*NX*NX+NX],NX,NX); TRW=ROWW[.,2*NX*NX+NX+1]; BN=DAME*(XWY-XWX*AR); EN=DAT[L:M,3]-XN*(AR+BN); ADDE=EN #*EN-S4*(TRW-SUMC(SUMC((XWWX.*CR)))); ADDB=BN*BN /-DAME*XWX*(EYE(NX)-CR*XWX)*DAME; ROWW[. ,1:NX*NX]=RESHAPE(ADDB ,1,NX*NX); ROWW[.,NX*NX+1]=ADDE; STORE[R,1:NX*NX+1]=ROWW[. ,1:NX*NX+1]; R=R+1; L=M+1; 1 = 1 + 1 ; ENDO; K=K+1; l = i ; V=L; W=R; ENDO ; DE=SUMC(STORE[.,NX*NX+1])/NR; DD=RESHAPE(SUMC(STORE[.,1:NX*NX]),NX,NX)/SUMC(NIGP'); DT=MAXC(MAXC(ABS(DD) - 0 . 0 1*PCTCVG*ABS(DAME))); I F ( ( I T N LT MAXIT) AND ((ABS(DE) GT 0 . 0 1*PCTCVG*S2) OR (DT GT 0 . 0 ) S2=S2+DE; DAME=DAME+DD; ITN=ITN+ 1 ; GOTO LOOP; ENDIF; / * G e t t h e r e g r e s s i o n b y j u s t p u t t i n g a s i n g l e l i n e t h r o u g h */ / * t h e d a t a f o r e a c h s u b j e c t NIPD */ K = l ; / * Gr o u p M a r k e r */ 1=0; / * M a r k e r o f rows t h a t h o l d g r o u p d a t a */ M=l; R = l ; DO WHILE K LE NGPS; 107 L = i ; DO WHILE L LE N I G P [ K ] ; I=I+REG[R,3] ; L=L+l ; R=R+1; ENDO ; Z = O N E S ( I - M + l , 1 ) - D A T [ M : I , 4 : 4 + N X V R - l ] ; Y = D A T [ M : I , 3 ] ; B=OLSQR(Y,Z); S = ( Y - Z * B ) ' * ( Y - Z * B ) ; I F K EQ 1; R N I P = ( K ~ ( I - M + l ) ~ B ' ~ S ) ; E L S E ; R N I P = R N I P | ( K ~ ( I - M + l ) ~ B ' ~ S ) ; ENDIF; M=I+1; K=K+1; ENDO ; / * Ok, L e t ' s make a p r e t t y p i c t u r e * / / * T h i s p l o t i s the d a t a , each group as one symbol and / * the e s t i m a t e d l i n e f o r each group l i b r a r y p g r a p h ; g r a p h s e t ; X L A B E L ( X L A B ) ; Y L A B E L ( Y L A B ) ; X T I C S ( X M N , X M X , X S T , 2 ) ; Y T I C S ( Y M N , Y M X , Y S T , 2 ) ; L l = O N E S ( N R , l ) * 2 . 5 ; L2=ONES(NR, l )*7; L3=0NES(NR,1); _PSYM = D A T [ 1 : N R , 4 ] ~ D A T [ 1 : N R , 3 ] ~ D A T [ 1 : N R , 1 ] ~ L 1 - L 2 - L 3 ; _PNUMHT=0.1; _PMSGSTR = "REML F i t t o D a t a " ; _PMSGCTL= { 2.4 6.25 0.20 0 1 15 }; XMINN=MINC(DAT[1:NR,4]); XMAXX=MAXC(DAT[1:NR,4]); XVECT=SEQA(XMINN,(XMAXX-XMINN)/(50-1) ,50); Y V E C T = ( O N E S ( 5 0 , 1 ) ~ X V E C T ) * A 1 / ; X Y ( X V E C T , Y V E C T ) ; / * T h i s i s a p l o t o f t h e l i n e s f o r each s u b j e c t * / g r a p h s e t ; X L A B E L ( X L A B ) ; Y L A B E L ( Y L A B ) ; x t i c s ( X M N , X M X , X S T , 2 ) ; y t i c s ( Y M N , Y M X , Y S T , 2 ) ; _PMSGSTR = "Each s u b j e c t s l e a s t squares l i n e " ; _PMSGCTL = (2 .4 6.25 0.20 0 1 15 }; _PNUMHT=0.1; 1=1; X V C T = ( D E S C R [ . , 5 : 6 ] ) ' ; 108 YVCT=XVCT; DO WHILE I LE ROWS(REG); Y V C T [ 1 , I ] = R E G [ I , 4 ] + R E G [ I , 5 ] * X V C T [ 1 , I ] ; Y V C T [ 2 , I ] = R E G [ I , 4 ] + R E G [ I , 5 ] * X V C T [ 2 , I ] ; 1=1+1; ENDO ; _ P L T Y P E = R E G [ . , 1 ] ; _PCOLOR=l; X Y ( X V C T , Y V C T ) ; / * T h i s i s a p l o t o f each group l i n e w i t h c o n f i d e n c e i n t e r v a l s g r a p h s e t ; X L A B E L ( X L A B ) ; YLABEL(YLAB),* x t i c s ( X M N , X M X , X S T , 2 ) ; y t i c s ( Y M N , Y M X , Y S T , 2 ) ; _PMSGSTR = "REML F i t s and 95% C I s " ; _PMSGCTL = (2 .4 6.25 0.20 0 1 15 }; _PNUMHT=0.1; XVECT=SEQA(XMN,(XMX-XMN)/100,100); Z = O N E S ( 1 0 0 , 1 ) - X V E C T - ( X V E C T . * X V E C T ) ; L = i ; M=l; K = l ; LET CD[1 ,3 ] = 0.0 0.0 0 .0 ; DO WHILE L LE NGPS; C L = C 1 [ L , 1 ] ~ 2 . 0 * C 1 [ L , 2 ] ~ C 1 [ L , 4 ] ; CD=CD+CL; C G = 2 * S Q R T ( Z * C L / ) ; X = Z [ . , l : 2 ] * ( A l [ L , . ] ) ' ; I F L EQ 1; YVCT=X~(X-CG)~(X+CG); ELSE ; YVCT=YVCT-X~(X-CG)~(X+CG); ENDIF; L=L+1; ENDO ; LET S S [ 6 , 1 ] = 6 2 2 6 2 2; _PLTYPE = SS; _PCOLOR=l; X Y ( X V E C T , Y V C T ) ; I F NGPS GT 2; GOTO DPRINT; ENDIF; / * T h i s i s a p l o t o f the d i f f e r e n c e i n the r e g r e s s i o n l i n e s * / G r a p h s e t ; _PMSGSTR = " D i f f e r e n c e i n r e g r e s s i o n l i n e s " ; _PMSGCTL = {2.4 6.25 0.20 0 1 15}; 109 _PNUMHT = 0 . 1 ; SB = ( A l [ 1 , 1 ] - A l [ 2 , 1 ] ) | ( A l [ 1 , 2 ] - A l [ 2 , 2 ] ) }; X = Z [ . , 1 : 2 ] * S B ; Y V C T = X ~ ( X - 2 . 0 * S Q R T ( Z * C D ' ) ) ~ ( X + 2 . 0 * S Q R T ( Z * C D ' ) ) ; _ P L T Y P E = S S [ l : 3 , . ] ; _PCOLOR=3; _PCROSS=l; X Y ( X V E C T , Y V C T ) ; DPRINT: / * Now t o p r i n t out a l l t h e g o o d i e s ! ! * / C L S ; PRINT " I f you want hardcopy p r i n t o u t t u r n on p r i n t e r now"; F=CONS; PRINT " R e s u l t s o f f i t t i n g r e g r e s s i o n s l i n e s t o grouped d a t a " ; PRINT " " ; PRINT "The f i l e used was : "$+FL$+".DAT"; PRINT " " ; FORMAT / r d 3 , 2 ; PRINT "There were " NGPS "groups and " NXVR "independent v a r i a b l e s . 1 1 ; PRINT " " ; PRINT " " ; PRINT " N a i v e l y p o o l i n g the d a t a , s i n g l e r e g r e s s i o n t h r o u g h each group PRINT " " ; LET LL=GROUP N INTERCEPT SLOPE; FORMAT / R O 1 0 , 5 ; PRINT $ L L ' ; PRINT R N I P [ . , 1 : 3 + N X V R ] ; PRINT " " ; PRINT "Two s tage r e g r e s s i o n , average group s l o p e and i n t e r c e p t " ; PRINT " " ; PRINT $ L L ' ; PRINT R N I P [ . , 1 ] ~ A B A R [ . , N X V R + 2 ] - A B A R [ . , 1 : 1 + N X V R ] ; PRINT " " ; PRINT "Weight L e a s t Squares R e g r e s s i o n " ; PRINT " " ; PRINT " — the f o l l o w i n g a r e i m p o r t a n t n u m e r i c a l m a t r i c e s — " ; PRINT " " ; PRINT " D n i h " ; PRINT DNIH; PRINT " " ; PRINT " B i a s " ; PRINT BIAS; PRINT " " ; PRINT 1 1D (Amemyia)" ; PRINT DDM; PRINT " " ; PRINT " E i g e n v a l u e s o f I N V ( U ' ) * D n i h * I N V ( U ) " ; PRINT E 3 ; PRINT " " ; PRINT " R e s i d u a l v a r i a n c e " SS2/N2; PRINT " " ; 110 PRINT $ L L ' ; PRINT R N I P [ . ,1 ] - A B A R [ . , N X V R+2 ] - A B ' ; PRINT " " ; PRINT "The c o v a r i a n c e m a t r i x o f the above parameters i s : " ; PRINT " — the square r o o t o f the d i a g o n a l e lements a r e the parameter PRINT " 1 1 ; PRINT C ; PRINT " " ; PRINT " R e s t r i c t e d maximum l i k l i h o o d u s i n g EM a l g o r i t h m f o r c o v a r i a n c e PRINT " ••; PRINT "Number o f i t e r a t i o n s " I T N ; PRINT " " ; PRINT " F i n a l r e s i d u a l v a r i a n c e " S2; PRINT " " ; PRINT $ L L ' ; PRINT RNIP[ .,1] -ABAR[ . ,NXVR+2]~A1; PRINT " " ; PRINT "the c o v a r i a n c e m a t r i x o f the above parameters i s : " ; PRINT " " ; 1=1; DO WHILE I LE NGPS; PRINT "GROUP 1 1 I ; PRINT " " ; PRINT R E S H A P E ( C l [ I , . ] , N X , N X ) ; 1=1+1; PRINT " " ; ENDO ; PRINT " " ; PRINT " \ f " ; CLOSEALL; STOP; END; Appendix E These a r e t h e e x a c t e q u a t i o n s used i n T a b l e 4 f o r the symmetric mode l . Column 1 G e n e r a t i o n : z T h i s i s the a i rway g e n e r a t i o n as d e s c r i b e d by W e i b e l ' s A mode l . Column 2 Number o f a i r w a y s The number o f a i r w a y s a t any g i v e n g e n e r a t i o n i n the mode l . S i n c e W e i b e l ' s A model i s a d ichotomy t h i s i s always 2 Z . Column 3 R e l a x e d Diameter : D r (cm) The d i a m e t e r from morphometr ic d a t a . In t h i s example t h e s e a r e the v a l u e s de termined by W e i b e l . Column 4 I n t e r n a l P e r i m e t e r : P i (cm) T h i s i s the i n t e r n a l p e r i m e t e r o f t h e a i r w a y s as i f they had been i n f l a t e d t o TLC (30cmH2O) a c c o r d i n g t o L a m b e r t ' s e q u a t i o n s and a r e t h e r e assumed t o be p e r f e c t c i r c l e s . Column 5 Volume C o r r e c t Diameter : Dv (cm) The a i rway d i a m e t e r s , D r , i n f l a t e d o r d e f l a t e d u s i n g the e q u a t i o n s o f Lambert t o the d e s i r e d t r a n s p u l m o n a r y p r e s s u r e . ( D r * L ) L=Lambert ' s p r e s s u r e a r e a c o r r e c t i o n 112 Column 6 A i r w a y l e n g t h : £ (cm) A i r w a y l e n g t h from W e i b e l ' s d a t a . We assume t h a t t h e l e n g t h does not change w i t h l u n g i n f l a t i o n . Column 7 C r o s s - s e c t i o n a l a r e a : XSA (cm2) C r o s s - s e c t i o n a l a r e a a v a i l a b l e f o r a i r f l o w a t a g i v e n g e n e r a t i o n . n*Dc 2/4*2 z where Dc = c o n t r a c t e d i n t e r n a l d i a m e t e r (see column 16) Column 8 Average L i n e a r V e l o c i t y : V(cm/sec) T h i s i s the average l i n e a r v e l o c i t y o f gas through t h a t g e n e r a t i o n . F low/XSA Column 9 P r o p o r t i o n o f muscle i n a i rway c i r c u m f e r e n c e PMC ( f r a c t i o n ) Based on the r e s u l t s from t h r e e autopsy lungs o f normal s u b j e c t s f o r g e n e r a t i o n s 0 t o 2 and the work o f W i l l i a m Snow M i l l e r f o r g e n e r a t i o n s 3 t o 23. Column 10 W a l l A r e a WA (mm2) A r e a o f t i s s u e between the a i rway lumen and smooth musc l e . F o r normal s u b j e c t : (0 .076277+0.96581*Pi) 2 F o r a s t h m a t i c s u b j e c t s : (0 .089112+1.3783*Pi) 2 113 Column 11 P r o p o r t i o n o f w a l l t h a t i s t i s s u e : PW T h i s i s the p r o p o r t i o n p r i o r t o any smooth muscle c o n t r a c t i o n . WA/100 WA / 100+TTDV 2 / 4 Column 12 A l p h a : a T h i s i s the dose response parameter t h a t c o n t r o l s the p o s i t i o n o f t h e c u r v e r e l a t i v e t o t h e x - a x i s . A more n e g a t i v e v a l u e s h i f t s the c u r v e t o the r i g h t . Column 13 Be ta : /3 T h i s i s the dose response parameter t h a t c o n t r o l s the s l o p e o f the dose response e q u a t i o n o v e r the l i n e a r p o r t i o n o f the c u r v e . A l a r g e r v a l u e c r e a t e s a s t e e p e r r e s p o n s e . Column 14 Maximal smooth muscle s h o r t e n i n g : PMSmax ( f r a c t i o n ) T h i s i s the maximal amount the smooth musc le can s h o r t e n r e l a t i v e t o i t s s t a r t i n g l e n g t h ( a r b i t r a r i l y a s s i g n e d v a l u e ) . Column 15 Observed smooth muscle s h o r t e n i n g : PMSobs ( f r a c t i o n ) Assuming t h a t t h e smooth musc le s h o r t e n i n g a c c o r d i n g t o an S shaped dose response r e l a t i o n s h i p c o n t r o l l e d by a, and £ from 0 t o a maximum o f PMSmax we g e t , f o r some dose : 114 1 0 a + f i * l o g i o ( d o s e ) 1 Q a 1 + 1 0 « + P * l o g 1 0 ( d o s e ) 1 + 1 Q g * PMomax i o a 1 + i o a C o l u m n 1 6 C o n t r a c t e d I n t e r n a l D i a m e t e r : Dc ( cm) T h i s i s t h e i n t e r n a l d i a m e t e r o f a n a i r w a y a f t e r s o m e a m o u n t o f s m o o t h m u s c l e s h o r t e n i n g a s d e t e r m i n e d b y c o l u m n 1 5 . C o l u m n 17 R e y n o l d ' s N u m b e r : Re Re i s a n o n - d i m e n s i o n a l p a r a m e t e r t h a t i s u s e d t o d e t e r m i n e t h e a m o u n t o f t u r b u l e n c e i n t h e a i r f l o w . L i n e a r v e l o c i t y * Dc * D e n s i t y v i s c o s i t y C o l u m n 18 L a m i n a r p r e s s u r e d r o p : Piam ( c m H 2 0 ) A s s u m i n g f u l l y d e v e l o p e d l a m i n a r f l o w t h i s i s t h e p r e s s u r e d r o p i n a p i p e f o r P o i s e u i l l e e n e r g y l o s s . 8 * l e n g t h * v i s c o s i t y * ( f l o w / 2 z ) r r * 9 8 0 * D c 4 / 1 6 1 1 5 Column 19 Z e t a c o r r e c t i o n : Z T h i s i s the n o n - d i m e n s i o n a l c o r r e c t i o n e m p i r i c a l l y de termined by P e d l e y e t a l t o r e l a t e t h e a c t u a l p r e s s u r e drop i n a b i f u r c a t i n g t u r b u l e n t system as opposed t o a s t r i c t P o i s e u i l l e sys tem. 2 - * /-Sfegdr- " 2 » 1 -° = l o t h e r w i s e Column 20 Z e t a c o r r e c t e d p r e s s u r e drop : Pzeta ( cm H 20) Laminar p r e s s u r e drop c o r r e c t e d f o r the a d d i t i o n a l energy l o s s e s as c a l c u l a t e d by z e t a . Plam * Z Column 21 R e s i s t a n c e a t a g i v e n g e n e r a t i o n : Rgen ( c m H 2 0 / £ / s e c ) T h i s i s the r e s i s t a n c e o f a l l t h e a i r w a y s , assumed t o be i n p a r a l l e l , a t a g i v e n g e n e r a t i o n , a f t e r any smooth muscle c o n t r a c t i o n . Pze t a  (F low/2 z ) /1000 - jV 2 Z 116 These a r e the e x a c t e q u a t i o n s used i n T a b l e 5 f o r t h e asymmetric mode l . Column 1 O r d e r : H T h i s i s t h e a i rway g e n e r a t i o n as d e s c r i b e d by H o r s f i e l d ' s mode l . Column 2 M a j o r Daughter T h i s i s t h e l a r g e r o f two d a u g h t e r s t h a t the p a r e n t b r a n c h , column 1, d i v i d e s i n t o . Column 3 M i n o r Daughter T h i s i s the s m a l l e r o f two d a u g h t e r s t h a t the p a r e n t b r a n c h , column 1, d i v i d e s i n t o . Column 4 Downstream E n d p o i n t s E T h i s i s the number o f downstream e n d p o i n t s t h a t o b t a i n t h e i r f low d i r e c t l y from t h e c u r r e n t b r a n c h . Column 5 Flow V % T h i s i s the p e r c e n t o f the t o t a l f low t h a t w i l l pass t h r o u g h the c u r r e n t b r a n c h . E*100%/233920 Column 6 Flow as a p e r c e n t o f t r a c h e a l f low T h i s d i f f e r s from column 5 by combin ing branches o f s i m i l a r o r d e r t o g i v e t h e f low t h r o u g h an o r d e r . 117 Column 7 Number o f a i r w a y s Na The number o f a i r w a y s a t any g i v e n o r d e r i n the mode l . T h i s v a l u e must be o b t a i n e d by t r a c i n g backwards t h r o u g h the daughters from the a l v e o l i t o t h e t r a c h e a . Column 8 R e l a x e d Diameter : Dr (cm) The d i a m e t e r from morphometr ic d a t a . In t h i s example t h e s e a r e the v a l u e s d e t e r m i n e d by Weibe l and t r a n s f o r m e d u s i n g t h e t r a n s i t i o n m a t r i x d e s c r i b e d i n 5 . 4 . 2 . Column 9 I n t e r n a l P e r i m e t e r : P i (cm) T h i s i s t h e i n t e r n a l p e r i m e t e r o f t h e a i r w a y s as i f they had been i n f l a t e d t o TLC (30cmH2O) a c c o r d i n g t o L a m b e r t ' s e q u a t i o n s . Column 10 Volume C o r r e c t Diameter : Dv (cm) The a i rway d i a m e t e r s , D r , i n f l a t e d o r d e f l a t e d u s i n g the e q u a t i o n s o f Lambert t o t h e c u r r e n t t r a n s p u l m o n a r y p r e s s u r e . ( D r * L ) L=Lambert ' s p r e s s u r e a r e a c o r r e c t i o n Column 11 A i r w a y l e n g t h : £ (cm) A i r w a y l e n g t h from W e i b e l ' s d a t a . We assume t h a t the l e n g t h does not change w i t h l u n g i n f l a t i o n . 118 Column 12 C r o s s - s e c t i o n a l a r e a : XSA (cm2) C r o s s - s e c t i o n a l a r e a a v a i l a b l e f o r a i r f l o w a t a g i v e n g e n e r a t i o n . TT*Dc 2/4*Na where Dc = c o n t r a c t e d i n t e r n a l d i a m e t e r (see column 22) Column 13 Average L i n e a r V e l o c i t y : V(cm/sec) T h i s i s the average l i n e a r v e l o c i t y o f gas through t h a t g e n e r a t i o n . Flow as % o f t r a c h e a l f low / XSA Column 15 P r o p o r t i o n o f muscle i n a i rway c i r c u m f e r e n c e PMC ( f r a c t i o n ) Based on the r e s u l t s from t h r e e autopsy lungs o f normal s u b j e c t s f o r t r a c h e a , main b r o n c h i and l o b a r b r o n c h i and the work o f W i l l i a m Snow M i l l e r f o r a l l p e r i p h e r a l o r d e r s . Column 16 W a l l A r e a WA (mm2) A r e a o f t i s s u e between the a i r w a y lumen and smooth m u s c l e . F o r normal s u b j e c t : (0 .076277+0.96581*Pi) 2 F o r a s t h m a t i c s u b j e c t s : (0 .089112+1.3783*Pi) 2 Column 17 P r o p o r t i o n o f w a l l t h a t i s t i s s u e : PW T h i s i s the p r o p o r t i o n p r i o r t o any smooth muscle 119 c o n t r a c t i o n . WA/100 WA/100+rrDv2/4 Column 18 A l p h a : a T h i s i s the dose response parameter t h a t c o n t r o l s the p o s i t i o n o f t h e c u r v e r e l a t i v e t o t h e x - a x i s . A more n e g a t i v e v a l u e s h i f t s t h e c u r v e t o the r i g h t . Column 19 Beta : fi T h i s i s the dose response parameter t h a t c o n t r o l s t h e s l o p e o f the dose response e q u a t i o n over the l i n e a r p o r t i o n o f the c u r v e . A l a r g e r v a l u e c r e a t e s a s t e e p e r r e s p o n s e . Column 20 Maximal smooth muscle s h o r t e n i n g : PMSmax ( f r a c t i o n ) T h i s i s t h e maximal amount the smooth musc le can s h o r t e n r e l a t i v e t o i t s s t a r t i n g l e n g t h ( a r b i t r a r i l y a s s i g n e d v a l u e ) . Column 21 Observed smooth muscle s h o r t e n i n g : PMSobs ( f r a c t i o n ) Assuming t h a t the smooth musc le s h o r t e n i n g a c c o r d i n g t o an S shaped dose response r e l a t i o n s h i p c o n t r o l l e d by a and fi from 0 t o a maximum o f PMSmax we g e t , f o r some dose: 120 l o a + 0 * l o g i o ( d o s e ) 1Qoc 1 + 1 0 « + g * l o g i 0 ( d o s e ) 1 + 1 0 « * Fmbmax 1 + i o a Column 22 C o n t r a c t e d I n t e r n a l Diameter : Dc (cm) T h i s i s t h e i n t e r n a l d i a m e t e r o f an a i r w a y a f t e r some amount o f smooth musc le s h o r t e n i n g as de termined by column 22. + I M S }{<I-P«S-*P«C)2-PW} Column 23 R e y n o l d ' s Number : Re Re i s a n o n - d i m e n s i o n a l parameter t h a t i s used t o de termine the amount o f t u r b u l e n c e i n the a i r f l o w . L i n e a r v e l o c i t y * Dc * D e n s i t y v i s c o s i t y Column 24 Laminar p r e s s u r e drop : Piam (cmH20) Assuming f u l l y deve loped l a m i n a r f low t h i s i s the p r e s s u r e drop i n a p i p e f o r P o i s e u i l l e energy l o s s . 8 * l e n g t h * v i s c o s i t y * ( f l o w / N a ) T T * 9 8 0 * D c 4 / 16 121 Z e t a c o r r e c t i o n : Z T h i s i s the n o n - d i m e n s i o n a l c o r r e c t i o n e m p i r i c a l l y de termined by P e d l e y e t a l t o r e l a t e the a c t u a l p r e s s u r e drop i n a b i f u r c a t i n g t u r b u l e n t system as opposed t o s t r i c t l y a P o i s e u i l l e sys tem. z - J^r- * / - T S n h r - i f z ' 1 - ° = 1 o t h e r w i s e Z e t a c o r r e c t e d p r e s s u r e drop : Pzeta ( cm H 20) Laminar p r e s s u r e drop c o r r e c t e d f o r t h e a d d i t i o n a l energy l o s s e s as c a l c u l a t e d by z e t a . Plam * Z Branch R e s i s t a n c e ( c m H 2 0 / £ / s e c ) T h i s i s the r e s i s t a n c e o f a s i n g l e b r a n c h a t the g i v e n o r d e r a f t e r smooth muscle s h o r t e n i n g . P z e t a  Flow % t r a c h e a T o t a l T r a c h e o b r o n c h i a l R e s i s t a n c e T h i s i s the c a l c u l a t i o n o f t h e r e s i s t a n c e v a l u e s i n p a r a l l e l and s e r i e s t o o b t a i n t h e t o t a l r e s i s t a n c e . 122 T a b l e l a T h i s t a b l e shows t h e a s y m m e t r i c a l H o r s f i e l d geometry . Each o r d e r (31=trachea) i s shown and the two branches i t d i v i d e s i n t o are a l s o d i s p l a y e d . The column denoted E i s the number o f downstream branches from the p a r e n t b r a n c h . The l e n g t h and w i d t h i n f o r m a t i o n a r e W e i b e l ' s measured d a t a t r a n s f o r m e d u s i n g the t r a n s i t i o n m a t r i x d e t a i l e d i n t a b l e 2. 123 Table l a Asymmetrical Horsfield Geometry 1 2 3 4 5 6 7 8 Downstream Number Measured Measured Anatomic Horsfield Major Minor Branches Airways Diameter Length Branch Order Daughter Daughter E in order cm cm 0 31 Branch 1 Branch 10 233920 j 1.800 12.000 1 29 Branch 6 Branch 2 105344 1 0.871 2.483 2 27 Order 26 Order 23 44256 1 0.666 1.459 6 28 Order 27 Order 24 61088 1 0.695 1.333 10 30 Branch 13 Branch 11 128576 1 1.221 4.782 11 27 Order 26 Order 23 44256 1 0.666 1.459 13 29 Branch 15 Branch 14 84320 1 0.871 2.483 14 25 Order 24 Order 21 23232 1 0.381 1.126 15 28 Order 27 Order 24 61088 1 0.695 1.333 27 26 23 44256 2 0.666 1.459 26 25 22 32064 4 0.480 0.964 25 24 21 23232 4 0.381 1.126 24 23 20 16832 7 0.380 1.013 23 22 19 12192 11 0.368 0.951 22 21 18 8832 15 0.317 0.972 21 20 17 6400 20 0.273 0.868 20 19 16 4640 27 0.242 0.787 19 18 15 3360 38 0.225 0.737 18 17 14 2432 53 0.204 0.678 17 16 13 1760 73 0.182 0.616 16 15 12 1280 100 0.164 0.559 15 14 11 928 138 0.150 0.513 14 13 10 672 191 0.137 0.471 13 12 9 480 264 0.126 0.431 12 11 8 352 364 0.115 0.394 11 10 7 256 502 0.106 0.360 10 9 7 192 693 0.098 0.329 9 8 6 128 957 0.091 0.302 8 7 6 96 1321 0.085 0.276 7 6 6 64 2516 0.081 0.259 6 5 5 32 7310 0.073 0.226 5 4 4 16 14620 0.066 0.191 4 3 3 8 29240 0.060 0.162 3 2 2 4 58480 0.055 0.137 2 1 1 2 116960 0.051 0.117 1 1 233920 0.048 0.102 124 T a b l e l b W e i b e l ' s symmetric Model A (61) s t r u c t u r e i s shown a l o n g w i t h t h e e s t i m a t e s o f d i a m e t e r and l e n g t h as de termined by W e i b e l . 125 Table l b Symmetrical Weibel Geometry 1 2 3 4 Number of Relaxed Airway Generation Airways Diam length cm cm 0 1 1.80 12.00 1 2 1.22 4.78 2 4 0.83 1.91 3 8 0.56 0.76 4 16 0.45 1.27 5 32 0.35 1.07 6 64 0.28 0.90 7 128 0.23 0.76 8 256 0.19 0.64 9 512 0.15 0.54 10 1024 0.13 0.46 11 2048 0.11 0.39 12 4096 0.10 0.33 13 8192 0.08 0.27 14 16384 0.07 0.23 15 32768 0.07 0.20 16 65536 0.06 0.16 17 131072 0.05 0.14 18 262144 0.05 0.12 19 524288 0.05 0.10 20 1048576 0.05 0.08 21 2097152 0.04 0.07 22 4194304 0.04 0.06 23 8388608 0.04 0.05 126 T a b l e 2 T h i s t r a n s i t i o n t a b l e d i s p l a y s the correspondence between the a s y m m e t r i c a l H o r s f i e l d o r d e r s and the s y m m e t r i c a l Weibe l g e n e r a t i o n s . As e x p l a i n e d i n the t e x t i t i s p o s s i b l e t o use t h i s m a t r i x t o move morphometr ic measurements made assuming one s t r u c t u r e t o the o t h e r . M o d i f i e d from Martonen (39) . 127 Table 2 Transformation Matrix Between Geometries Total Horsfield Orders #of Airways 0 1 2 3 Weibel Generations 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 31 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 29 3 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27 4 0 0 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26 4 0 0 0 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 4 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 7 0 0 0 1 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 11 0 0 0 2 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 15 0 0 0 0 4 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 20 0 0 0 0 1 6 4 5 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 20 27 0 0 0 0 1 3 8 5 6 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 19 38 0 0 0 0 2 3 6 10 6 7 2 1 1 0 0 0 0 0 0 0 0 0 0 0 18 53 0 0 0 0 0 6 6 10 12 7 8 2 1 1 0 0 0 0 0 0 0 0 0 0 17 73 0 0 0 0 0 1 12 10 15 14 8 9 2 1 1 0 0 0 0 0 0 0 0 0 16 100 0 0 0 0 0 1 4 20 15 21 16 9 10 2 1 1 0 0 0 0 0 0 0 0 15 138 0 0 0 0 0 2 4 10 30 21 28 18 10 11 2 1 1 0 0 0 0 0 0 0 14 191 0 0 0 0 0 0 8 10 20 42 28 36 20 11 12 2 1 1 0 0 0 0 0 0 13 264 0 0 0 0 0 0 1 20 20 35 56 36 45 22 12 13 2 1 1 0 0 0 0 0 12 364 0 0 0 0 0 0 1 5 40 35 56 72 45 55 24 13 14 2 1 1 0 0 0 0 11 502 0 0 0 0 0 0 2 5 15 70 56 84 90 55 66 26 14 15 2 1 1 0 0 0 10 693 0 0 0 0 0 0 0 10 15 35 112 84 120 110 66 78 28 15 16 2 1 1 0 0 9 957 0 0 0 0 0 0 0 1 30 35 70 168 120 165 132 78 91 30 16 17 2 1 1 0 8 1321 0 0 0 0 0 0 0 1 6 70 70 126 240 165 220 156 91 105 32 17 18 2 1 1 7 2515 0 0 0 0 0 0 0 2 16 36 175 238 294 450 330 352 260 133 135 50 20 20 3 1 6 7305 0 0 0 0 0 0 0 0 6 68 177 490 770 948 1230 1012 938 702 401 318 134 60 43 8 5 14594 0 0 0 0 0 0 0 0 0 12 136 354 980 1540 1896 2460 2024 1876 1404 802 636 268 120 86 4 29016 0 0 0 0 0 0 0 0 0 0 24 272 708 1960 3080 3792 4920 4048 3752 2808 1604 1272 536 240 3 57552 0 0 0 0 0 0 0 0 0 0 0 48 544 1416 3920 6160 7584 9840 8096 7504 5616 3208 2544 1072 2 112960 0 0 0 0 0 0 0 0 0 0 0 0 96 1088 2832 7840 12320 15168 19680 16192 15008 11232 6416 5088 1 215744 0 0 0 0 0 0 0 0 0 0 0 0 0 192 2176 5664 15680 24640 30336 39360 32384 30016 22464 12832 T a b l e 3 A l p h a and n a r e the c o e f f i c i e n t s f o r e q u a t i o n 3 . 3 . 1 used t o e s t i m a t e t h e a i r w a y c a l i b e r a t any l u n g volume as a f r a c t i o n o f t h e a i r w a y c a l i b e r o b t a i n e d from c a s t s a t a d i f f e r e n t l u n g vo lume. 129 Table 3 Lambert Pressure-Area Parameters Weibel Generation alpha n 0 0.882 10.0 1 0.882 10.0 2 0.686 10.0 3 0.546 10.0 4 0.450 10.0 5 0.370 10.0 6 0.310 10.0 7 0.255 10.0 8 0.213 10.0 9 0.184 10.0 10 0.153 10.0 11 0.125 9.0 12 0.100 8.0 13 0.075 8.0 14 0.057 8.0 15 0.045 7.0 16 0.039 7.0 17 0.039 7.0 18 0.039 7.0 19 0.039 7.0 20 0.039 7.0 21 0.039 7.0 22 0.039 7.0 23 0.039 7.0 Horsfield Order alpha n 31 0.882 10.0 30 0.882 10.0 29 0.686 10.0 28 0.546 10.0 27 0.498 10.0 26 0.478 10.0 25 0.395 10.0 24 0.393 10.0 23 0.381 10.0 22 0.337 10.0 21 0.297 10.0 20 0.268 10.0 19 0.250 9.9 18 0.229 9.9 17 0.207 9.8 16 0.186 9.6 15 0.169 9.5 14 0.154 9.3 13 0.139 9.1 12 0.125 8.9 11 0.112 8.6 10 0.101 8.4 9 0.090 8.2 8 0.081 8.0 7 0.075 7.9 6 0.064 7.6 5 0.054 7.4 4 0.047 7.2 3 0.042 7.1 2 0.040 7.0 1 0.039 7.0 5 0.054 7.4 4 0.047 7.2 3 0.042 7.1 2 0.040 7.0 1 0.039 7.0 130 T a b l e 4 T h i s t a b l e , i n t h r e e s e c t i o n s , shows the s y m m e t r i c a l model as i t appears on t h e computer u s i n g L o t u s 123(35) . The formulae f o r each column a r e d e t a i l e d i n Appendix E . The g l o b a l i n p u t s are t h e f l o w , s i m u l a t e d dose , gas d e n s i t y and gas v i s c o s i t y as shown on p a r t 1 o f 3 . The p r i m a r y output i s the t o t a l t r a c h e b r o n c h i a l p r e s s u r e drop and r e s i s t a n c e , shown i n p a r t 3. T h i s t a b l e shows r e g i o n a l r e s i s t a n c e s can be e a s i l y o b t a i n e d . 131 Symmetric Model Table 4 (Part 1 of 3) Total flow= 1250 ml/s Dose= 150 Density of inspired gas= 0.00113 g/cmA3 Viscosity of inspired gas= 0.00019 g/cm/s I 2 3 4 5 6 7 8 Number Internal Volume of Relaxed Perim correct Airway Total Linear Generation Airways Diam P i Diam length X S A Velocity cm cm cm cm cm*cm cm/s 0 1 1.800 5.797 1.783 12.00 2 .19 570.03 1 2 1.221 3.933 1.210 4 .78 2 .02 619.39 2 4 0.828 2.727 0.805 1.91 1.54 809.38 3 8 0.562 1.880 0 .538 0 .76 1.17 1065.52 4 16 0.445 1.507 0.421 1.27 1.43 871.94 5 3 2 0.351 1.193 0 .330 1.07 1.75 713.66 6 64 0.281 0.957 0.262 0 .90 2 .20 568 .32 7 128 0.227 0.776 0.211 0 .76 2.84 440.85 8 256 0.186 0 .636 0.172 0.64 3.76 332.31 9 512 0.155 0.528 0 .142 0 .54 5 .12 243.96 10 1024 0.130 0.445 0 .119 0.46 7 .14 174.99 11 2048 0 .110 0.378 0.101 0 .39 10.23 122.14 12 4096 0.095 0.326 0.087 0.33 15.04 83.09 13 8192 0.083 0.285 0.076 0.27 22.58 55 .36 14 16384 0.073 0.251 0.067 0.23 34.82 35 .90 15 32768 0.066 0.225 0 .060 0 .20 55 .13 22.67 16 65536 0.059 0.204 0 .054 0 .16 89.64 13.94 17 131072 0.054 0.187 0 .050 0 .14 149.39 8.37 18 262144 0 .050 0.173 0.046 0 .12 255.36 4 .90 19 524288 0.047 0.163 0.043 0 .10 447.83 2 .79 2 0 1048576 0.045 0.155 0.041 0.08 806.03 1.55 21 2097152 0.044 0.149 0.040 0.07 1489.41 0.84 2 2 4194304 0.042 0.146 0.039 0 .06 2826.25 0.44 23 8388608 0.042 0.144 0 .038 0.05 5508.41 0.23 132 Symmetric Model Table 4 (Part 2 of 3) 1 9 10 11 12 13 14 15 Wal l Area Generation P M C Wa P W Alpha Beta PMSmax PMSobs tract nun*niin tract # # tract tract 0 0.33 32.20 0.11 -1.90 1.20 0.20 0.17 1 0.33 15.01 0.12 -1.90 1.20 0.20 0.17 2 0.67 7.34 0.13 -1.90 1.20 0.20 0.17 3 1.00 3.58 0.14 -1.90 1.20 0.20 0.17 4 1.00 2.35 0.14 -1.90 1.20 0.20 0.17 5 1.00 1.51 0.15 -1.90 1.20 0.20 0.17 6 1.00 1.00 0.16 -1.90 1.20 0.20 0.17 7 1.00 0.68 0.16 -1.90 1.20 0.20 0.17 8 1.00 0.48 0.17 -1.90 1.20 0.20 0.17 9 1.00 0.34 0.18 -1.90 1.20 0.20 0.17 10 1.00 0.26 0.19 -1.90 1.20 0.20 0.17 11 1.00 0.20 0.19 -1.90 1.20 0.20 0.17 12 1.00 0.15 0.20 -1.90 1.20 0.20 0.17 13 1.00 0.12 0.21 -1.90 1.20 0.20 0.17 14 1.00 0.10 0.22 -1.90 1.20 0.20 0.17 15 1.00 0.09 0.24 -1.90 1.20 0.20 0.17 16 1.00 0.07 0.24 -1.90 1.20 0.20 0.17 17 1.00 0.07 0.25 -1.90 1.20 0.20 0.17 18 1.00 0.06 0.26 -1.90 1.20 0.20 0.17 19 1.00 0.05 0.27 -1.90 1.20 0.20 0.17 20 1.00 0.05 0.28 -1.90 1.20 0.20 0.17 21 1.00 0.05 0.28 -1.90 1.20 0.20 0.17 22 1.00 0.05 0.29 -1.90 1.20 0.20 0.17 23 1.00 0.05 0.29 -1.90 1.20 0.20 0.17 133 Symmetric Model Table 4 (Part 3 of 3) 1 16 17 18 19 20 21 Contracted Pressure Pressure Resistance internal Reynolds Drop Drop of Generation diameter number Laminar Zeta Zeta Generation cm # cm H 2 0 # cm H 2 0 cmH20/lps 0 1.671 5725.044 0.01504 9.23 0.13887 0.1111 1 1.133 4219.871 0.01415 10.34 0.14638 0.1171 2 0.701 3410.957 0.01926 11.59 0.22315 0.1785 3 0.432 2767.370 0.02661 12.98 0.34527 0.2762 4 0.338 1770.170 0.05943 7.11 0.42226 0.3378 5 0.264 1132.403 0.06717 5.47 0.36743 0.2939 6 0.209 714.557 0.07188 4.21 0.30268 0.2421 7 0.168 445.013 0.07298 3.24 0.23658 0.1893 8 0.137 273.203 0.06997 2.50 0.17462 0.1397 9 0.113 165.520 0.06362 1.92 0.12224 0.0978 10 0.094 99.126 0.05524 1.48 0.08170 0.0654 11 0.080 58.558 0.04540 1.14 0.05170 0.0414 12 0.068 34.153 0.03546 1.00 0.03546 0.0284 13 0.059 19.713 0.02656 1.00 0.02656 0.0212 14 0.052 11.225 0.01885 1.00 0.01885 0.0151 15 0.046 6.308 0.01268 1.00 0.01268 0.0101 16 0.042 3.498 0.00810 1.00 0.00810 0.0065 17 0.038 1.916 0.00492 1.00 0.00492 0.0039 18 0.035 1.036 0.00284 1.00 0.00284 0.0023 19 0.033 0.553 0.00156 1.00 0.00156 0.0012 20 0.031 0.292 0.00081 1.00 0.00081 0.0006 21 0.030 0.152 0.00040 1.00 0.00040 0.0003 22 0.029 0.078 0.00019 1.00 0.00019 0.0002 23 0.029 0.039 0.00008 1.00 0.00008 0.0001 Total pressure drop= 2.73 cm/H20 Total Resistance— 2.18 cm/H20/l/s 134 T a b l e 5 T h i s t a b l e , i n t h r e e p a r t s , d i s p l a y s t h e a s y m m e t r i c a l ana logy t o t a b l e 4. The formulae a r e e x p l a i n e d i n d e t a i l i n Appendix E . In t h e asymmetr ic model i t i s i m p o r t a n t t o always keep t r a c k o f the d a u g h t e r branches from each p a r e n t . Note t h a t w h i l e g e n e r a t i o n s i n T a b l e 4 p r o v i d e a marker f o r d i s t a n c e from t h e t r a c h e a i n the a s y m m e t r i c a l model o r d e r does i n d i c a t e a c c u r a t e l y t h e d i s t a n c e a b r a n c h i s from t h e t r a c h e a . 135 Asymmetric Model Table 5 (Part 1 of 3) 1 2 3 4 5 6 7 8 9 Branch Order Number of Relaxed Internal Order Major 1 Major 2 E Flow Flow airways Diameter Perimeter # # # # % trachea % trachea # cm cm 31 Branch 1 Branch 10 233920 100.00 100.00 1 1.800 5.797 29 Branch 6 Branch 2 105344 45.03 81.08 1 0.871 2.803 27 Order 26 Order 23 44256 18.92 56.76 1 0.666 2.192 28 Order 27 Order 24 61088 26.11 52.23 1 0.695 2.288 30 Branch 13 Branch 11 128576 54.97 54.97 1 1.221 3.933 27 Order 26 Order 23 44256 18.92 56.76 1 0.666 2.192 29 Branch 15 Branch 14 84320 36.05 62.16 1 0.871 2.865 25 Order 24 Order 21 23232 9.93 19.86 1 0.381 1.273 28 Order 27 Order 24 61088 26.11 52.23 1 0.695 2.326 27 26 23 44256 18.92 56.76 2 0.666 2.241 26 25 22 32064 13.71 13.71 4 0.480 1.616 25 24 21 23232 9.93 19.86 4 0.381 1.291 24 23 20 16832 7.20 7.20 7 0.380 1.289 23 22 19 12192 5.21 5.21 11 0.368 1.245 22 21 18 8832 3.78 3.78 15 0.317 1.077 21 20 17 6400 2.74 2.74 20 0.273 0.931 20 19 16 4640 1.98 1.98 27 0.242 0.827 19 18 15 3360 1.44 1.44 38 0.225 0.769 18 17 14 2432 1.04 1.04 53 0.204 0.697 17 16 13 1760 0.75 0.75 73 0.182 0.623 16 15 12 1280 0.55 0.55 100 0.164 0.560 15 14 11 928 0.40 0.40 138 0.150 0.512 14 13 10 672 0.29 0.29 191 0.137 0.470 13 12 9 480 0.21 0.21 264 0.126 0.431 12 11 8 352 0.15 0.15 364 0.115 0.395 11 10 7 256 0.11 0.11 502 0.106 0.364 10 9 7 192 0.08 0.08 693 0.098 0.337 9 8 6 128 0.05 0.05 957 0.091 0.313 8 7 6 96 0.04 0.04 1321 0.085 0.292 7 6 6 64 0.03 0.03 2516 0.081 0.278 6 5 5 32 0.01 0.01 7310 0.073 0.252 5 4 4 16 0.01 0.01 14620 0.066 0.225 4 3 3 8 0.00 0.00 29240 0.060 0.205 3 2 2 4 0.00 0.00 58480 0.055 0.188 2 1 1 2 0.00 0.00 116960 0.051 0.176 1 1 0.00 0.00 233920 0.048 0.166 1 3 6 Asymmetric Model Table 5 (Part 2 of 3) 1 10 11 12 13 14 15 16 17 18 19 Volume Correct Total Flow in Linear Wal l Order diameter Length X S A airway Velocity P M C Area P W alpha beta # cm cm cm*cm ml/sec cm/sec tract m m A 2 tract # # 31 1.783 12.000 2.20 1250.00 569.26 33% 32.20 11% -1.90 1.20 29 0.863 2.483 0.51 562.93 1096.53 33% 7.75 12% -1.90 1.20 27 0.647 1.459 0.25 236.49 945.41 66% 4.81 13% -1.90 1.20 28 0.675 1.333 0.27 326.44 1197.44 66% 5.23 13% -1.90 1.20 30 1.210 4.782 1.01 687.07 679.99 33% 15.01 12% -1.90 1.20 27 0.647 1.459 0.25 236.49 945.41 66% 4.81 13% -1.90 1.20 29 0.846 2.483 0.43 450.58 1053.61 66% 8.09 13% -1.90 1.20 25 0.364 1.126 0.07 124.15 1851.74 100% 1.71 14% -1.90 1.20 28 0.665 1.333 0.22 326.44 1453.09 100% 5.40 13% -1.90 1.20 27 0.634 1.459 0.41 236.49 580.88 100% 5.02 14% -1.90 1.20 26 0.456 0.964 0.42 171.34 407.46 100% 2.68 14% -1.90 1.20 25 0.358 1.126 0.26 124.15 480.75 100% 1.75 15% -1.90 1.20 24 0.358 1.013 0.45 89.95 199.14 100% 1.75 15% -1.90 1.20 23 0.346 0.951 0.66 65.15 98.59 100% 1.64 15% -1.90 1.20 22 0.296 0.972 0.66 47.20 71.38 100% 1.25 15% -1.90 1.20 21 0.255 0.868 0.65 34.20 52.65 100% 0.95 16% -1.90 1.20 20 0.225 0.787 0.68 24.79 36.26 100% 0.76 16% -1.90 1.20 19 0.209 0.737 0.83 17.95 21.71 100% 0.67 16% -1.90 1.20 18 0.189 0.678 0.94 13.00 13.83 100% 0.56 17% -1.90 1.20 17 0.169 0.616 1.03 9.40 9.15 100% 0.46 17% -1.90 1.20 16 0.151 0.559 1.13 6.84 6.07 100% 0.38 18% -1.90 1.20 15 0.138 0.513 1.29 4.96 3.83 100% 0.33 18% -1.90 1.20 14 0.126 0.471 1.50 3.59 2.39 100% 0.28 18% -1.90 1.20 13 0.116 0.431 1.73 2.56 1.48 100% 0.24 19% -1.90 1.20 12 0.106 0.394 1.99 1.88 0.95 100% 0.21 19% -1.90 1.20 11 0.097 0.360 2.31 1.37 0.59 100% 0.18 20% -1.90 1.20 10 0.090 0.329 2.72 1.03 0.38 100% 0.16 20% -1.90 1.20 9 0.084 0.302 3.23 0.68 0.21 100% 0.14 21% -1.90 1.20 8 0.078 0.276 3.84 0.51 0.13 100% 0.13 21% -1.90 1.20 7 0.074 0.259 6.63 0.34 0.05 100% 0.12 22% -1.90 1.20 6 0.067 0.226 15.64 0.17 0.01 100% 0.10 22% -1.90 1.20 5 0.060 0.191 24.77 0.09 0.00 100% 0.09 23% -1.90 1.20 4 0.054 0.162 40.44 0.04 0.00 100% 0.08 24% -1.90 1.20 3 0.050 0.137 67.85 0.02 0.00 100% 0.07 25% -1.90 1.20 2 0.047 0.117 117.25 0.01 0.00 100% 0.06 26% -1.90 1.20 1 0.044 0.102 209.04 0.01 0.00 100% 0.06 27% -1.90 1.20 137 Asymmetric Model Table 5 (Part 3 of 3) 1 20 21 22 23 24 25 26 27 Contracted Pressure Pressure Resis internal Reynold's drop Drop of Order PMSmax PMSobs diameter Number Laminar Zeta Zeta branch # tract tract cm # c m H 2 0 # cm H 2 0 cmH20/l/s 31 20% 17% 1.672 5721 0.015 9.23 0.138 0.111 29 20% 17% 0.808 5329 0.026 13.62 0.348 0.619 27 20% 17% 0.564 3207 0.027 11.52 0.306 1.295 28 20% 17% 0.589 4240 0.028 14.16 0.400 1.224 30 20% 17% 1.134 4636 0.016 10.84 0.168 0.245 27 20% 17% 0.564 3207 0.027 11.52 0.306 1.295 29 20% 17% 0.738 4673 0.029 12.19 0.359 0.798 25 20% 17% 0.292 3252 0.150 9.50 1.424 11.473 28 20% 17% 0.535 4671 0.042 14.16 0.588 1.803 27 20% 17% 0.509 1778 0.040 8.14 0.327 1.383 26 20% 17% 0.366 896 0.072 6.03 0.434 2.536 25 20% 17% 0.287 828 0.162 4.75 0.768 6.186 24 20% 17% 0.287 343 0.106 3.22 0.340 3.780 23 20% 17% 0.277 164 0.083 2.26 0.187 2.869 22 20% 17% 0.237 102 0.114 1.63 0.185 3.926 21 20% 17% 0.203 64 0.136 1.27 0.172 5.040 20 20% 17% 0.180 39 0.147 1.00 0.147 5.918 19 20% 17% 0.166 22 0.135 1.00 0.135 7.502 18 20% 17% 0.150 12 0.135 1.00 0.135 10.395 17 20% 17% 0.134 7.36 0.141 1.00 0.141 14.969 16 20% 17% 0.120 4.37 0.145 1.00 0.145 21.234 15 20% 17% 0.109 2.52 0.140 1.00 0.140 28.137 14 20% 17% 0.100 1.44 0.132 1.00 0.132 36.827 13 20% 17% 0.091 0.81 0.124 1.00 0.124 48.520 12 20% 17% 0.083 0.47 0.120 1.00 0.120 63.785 11 20% 17% 0.077 0.27 0.112 1.00 0.112 81.781 10 20% 17% 0.071 0.16 0.105 1.00 0.105 102.813 9 20% 17% 0.066 0.08 0.087 1.00 0.087 127.730 8 20% 17% 0.061 0.05 0.081 1.00 0.081 157.191 7 20% 17% 0.058 0.02 0.062 1.00 0.062 180.129 6 20% 17% 0.052 0.00 0.041 1.00 0.041 238.269 5 20% 17% 0.046 0.00 0.027 1.00 0.027 320.631 4 20% 17% 0.042 0.00 0.017 1.00 0.017 407.673 3 20% 17% 0.038 0.00 0.011 1.00 0.011 491.601 2 20% 17% 0.036 0.00 0.006 1.00 0.006 562.265 1 20% 17% 0.034 0.00 0.003 1.00 0.003 613.049 1 3 8 T a b l e 6 T h i s t a b l e p r o v i d e s the maximum p o s s i b l e r e s i s t a n c e f o r v a r i o u s l e v e l s o f maximal smooth musc le s h o r t e n i n g . To o b t a i n these v a l u e s t h e model f low was s e t t o 1250 m l / s and a = - 1 . 9 , /3=1.2. The dose was f i x e d a t 10 1 0 t o ensure f u l l c o n t r a c t i o n o f the smooth musc le t o the d e s i r e d l e v e l . Both asymmetr ic and symmetr ic r e s u l t s a r e shown. Three d i f f e r e n t l e v e l s o f encroachment upon t h e lumen a r e d i s p l a y e d as w e l l c e n t r a l ( g e n e r a t i o n s 0-10 o r o r d e r s 31-13) o r p e r i p h e r a l l y l o c a t e d w a l l t h i c k e n i n g . A v a l u e o f INF i n d i c a t e s a i rway c l o s u r e r e s u l t i n g i n i n f i n i t e r e s i s t a n c e s . 139 Table 6 Maximum Resistance Maximal Smooth Muscle Shortening 20% 30% 35% 40% 45% Normal Symmetric 2.63 5.22 8.04 13.62 27.88 Asymmetric 2.72 5.35 8.28 14.35 31.28 Asthmatic, E=0% Symmetric 3.28 8.82 21.15 INF INF Asymmetric 3.40 9.42 23.41 INF INF Asthmatic, E=50% Symmetric 4.16 12.64 INF INF INF Asymmetric 4.38 14.23 INF INF INF Asthmatic, E=100% Symmetric 5.50 INF INF INF INF Asymmetric 5.93 41.78 INF INF INF Central Thickening E=0% Symmetric 3.21 8.25 16.41 46.48 INF Asymmetric 3.18 7.67 14.57 37.91 377.96 Peripheral Thickening E=0% Symmetric 2.70 5.80 12.77 INF INF Asymmetric 2.94 6.98 16.43 INF INF Central Thickening E=50% Symmetric 3.99 11.06 24.20 95.51 INF Asymmetric 3.81 9.79 20.15 66.08 INF Peripheral Thickening E=50% Symmetric 2.81 6.81 INF INF INF Asymmetric 3.25 9.36 INF INF INF 140 T a b l e 7 T h i s t a b l e i s s i m i l a r t o t a b l e 6 except t h a t the dose a t which 50% o f t h e maximal r e s i s t a n c e was a c h i e v e d i s d i s p l a y e d . I f the maximal r e s i s t a n c e was i n f i n i t e t h e n DNE on t h i s t a b l e i n d i c a t e s t h a t a PD does not e x i s t . 141 Table 7 Expected Dose to 50% of Maximum Resistance Maximal Smooth Muscle Shortening 20% 30% 35% 40% 45% Normal Symmetric 22 68 100 145 227 Asymmetric 20 67 101 151 250 Asthmatic, E=0% Symmetric 37 113 233 DNE DNE Asymmetric 36 119 242 DNE DNE Asthmatic, E=50% Symmetric 41 134 DNE DNE DNE Asymmetric 41 147 DNE DNE DNE Asthmatic, E=100% Symmetric 47 DNE DNE DNE DNE Asymmetric 49 520 DNE DNE DNE Central Thickening E=0% Symmetric 35 106 171 312 DNE Asymmetric 31 93 158 158 1075 Peripheral Thickening E=0% Symmetric 24 80 211 DNE DNE Asymmetric 25 97 229 DNE DNE Central Thickening E=50% Symmetric 39 118 198 446 DNE Asymmetric 33 107 180 368 DNE Peripheral Thickening E=50% Symmetric 25 101 DNE DNE DNE Asymmetric 30 132 DNE DNE DNE 142 F i g u r e 1 T h i s photograph d i s p l a y s a bronchogram o f an e x c i s e d human l u n g . The image was made by b l o w i n g l e a d d u s t i n t o the a i r w a y s v i a the t r a c h e a and t h e n an x - r a y was made o f the l u n g . The v a s t c o m p l e x i t y and a s y m m e t r i c a l n a t u r e o f the t r a c h e b r o n c h i a l t r e e i s c l e a r l y e v i d e n t i n t h i s p h o t o . 143 F i g u r e 2 T h i s p l o t shows t h e c r o s s - s e c t i o n a l a r e a o f the t r a c h e o b r o n c h i a l t r e e a c c o r d i n g t o t h e s y m m e t r i c a l g e n e r a t i o n scheme o f W e i b e l . A comparable p l o t i s not p o s s i b l e f o r the H o r s f i e l d s t r u c t u r e as o r d e r does not p r o v i d e a un ique i d e n t i f i e r i n terms o f d i s t a n c e from t h e t r a c h e a . 145 100000 i 0 5 10 15 20 25 Generation (0=trachea) F i g u r e 3 T h i s d iagram i n d i c a t e s the p r i m a r y p r e s s u r e s w i t h i n and around t h e l u n g . The d r i v i n g p r e s s u r e f o r i n s p i r a t i o n i s the d i f f e r e n c e between a l v e o l a r p r e s s u r e and mouth p r e s s u r e . 147 Figure Chest Wall Pleural space Lungs Alveoli Mouth Pressure A Transpulmonary Pressure Alveolar Pressure Intrapleural Pressure F i g u r e 4 T h i s s e t o f f i g u r e s , adapted from West (62) shows the t ime p a t t e r n s f o r l u n g volume, i n t r a p l e u r a l p r e s s u r e , a l v e o l a r p r e s s u r e and f l o w . Note t h a t a t t h e b e g i n n i n g and end o f i n s p i r a t i n f low i s z ero and t h a t a t i t s maximum a l v e o l a r p r e s s u r e i s about 1 cmH 20. T h i s v a l u e compares t o 0.89 o b t a i n e d from the mode l . 149 Figure 4 Inspiration Expiration Time 150 F i g u r e 5 T h i s d iagram demonstrates the v e l o c i t y p r o f i l e s f o r l a m i n a r and t u r b u l e n t f l o w . In l a m i n a r f low the s tream l i n e s a r e everywhere p a r a l l e l t o the p i p e w a l l s w h i l e i n t u r b u l e n t f low the s tream l i n e s a r e randomly o r i e n t e d . In l a m i n a r f low the p r e s s u r e drop from PI t o P2 i s l e s s t h a n t h a t i n the t u r b u l e n t f low p i p e . 151 Laminar Flow —> —> -> P1 P2 Turbulent Flow F i g u r e 6 T h i s f i g u r e , m o d i f i e d from W e i b e l (61) shows h i s d a t a f o r a irway d i a m e t e r as a f u n c t i o n o f g e n e r a t i o n . The open c i r c l e s w i t h e r r o r b a r s show the v a l u e s W e i b e l o b t a i n e d from c a s t s and the shaded boxes i n d i c a t e h i s r e s u l t s from morphometr ic t e c h n i q u e s . The s o l i d l i n e i s W e i b e l ' s p r e d i c t e d d i a m e t e r f o r g e n e r a t i o n s from 0 t o 3 and the dashed l i n e i s h i s p r e d i c t i o n f o r g e n e r a t i o n s 4 t o 23. 153 Generation (0=trachea) F i g u r e 7 T h i s d iagram d i s p l a y s S t r a h l e r ' s method o f o r d e r i n g f o r r i v e r b r a n c h e s . He began by d e n o t i n g the r i v e r d e l t a as some l a r g e v a l u e , h e r e 50, and when two t r i b u t a r i e s met t h e p a r e n t b r a n c h was denoted as one o r d e r l e s s t h a n the s m a l l e s t daughter b r a n c h . H o r s f i e l d m o d i f i e d t h i s t e c h n i q u e t o p r o v i d e an a s y m m e t r i c a l d e s c r i p t i o n o f t h e l u n g . 155 Figure 156 F i g u r e 8 T h i s f i g u r e shows t h e d i s t r i b u t i o n o f p a t h l e n g t h s f o r b o t h the a s y m m e t r i c a l model and symmetric mode l . F o r b o t h geometr i e s the d a t a o f W e i b e l was used f o r b r a n c h l e n g t h . S i n c e W e i b e l ' s s t r u c t u r e i s p u r e l y s y m m e t r i c a l t h e r e i s o n l y one p a t h l e n g t h p o s s i b l e w h i l e H o r s f i e l d ' s geometry a l l o w s most a i r w a y s t o b r a n c h r a p i d l y t o the a l v e o l i g i v i n g a wide range o f p a t h l e n g t h s . 157 sABMqied jo uojijodojd 158 F i g u r e 9 These c u r v e s a r e the g r a p h i c a l r e p r e s e n t a t i o n o f L a m b e r t ' s p r e s s u r e a r e a c o r r e c t i o n e q u a t i o n , 3 . 3 . 1 . The y - a x i s denotes the c r o s s - s e c t i o n a l a r e a a v a i l a b l e f o r a i r f l o w as a f r a c t i o n o f the t o t a l c r o s s - s e c t i o n a l a r e a a v a i l a b l e a t t o t a l l u n g c a p a c i t y . To use t h e s e c u r v e s t h e a v a i l a b l e morphometr ic d a t a was assumed t o be o b t a i n e d a t 75% TLC o r about ScmHMD. Each a i r w a y g e n e r a t i o n c o u l d t h e n be c o r r e c t e d t o i t ' s TLC a r e a . Once the a r e a o f any a r e a a t TLC was know the geometry c o u l d be " d e f l a t e d " t o f u n c t i o n a l r e s i d u a l c a p a c i t y , about 5cmH 0. 159 091 Fractional Cross-Sectional Area © o o p o © ro i*. b> ba b o o o o o o F i g u r e 10 T h i s f i g u r e shows the e f f e c t o f a i rway w a l l t h i c k n e s s and smooth musc le s h o r t e n i n g on the r e s i s t a n c e i n a s i n g l e t u b e . In the upper p o r t i o n o f f i g u r e 10 an a i rway w i t h smooth muscle c o m p l e t e l y s u r r o u n d i n g t h e a i r w a y s , PMC=1, and w i t h 20% a irway w a l l t h a t i s t i s s u e , PW=0.2, i s s u b j e c t e d t o a 30% s h o r t e n i n g o f t h e smooth m u s c l e , PMS=0.3. T h i s causes and i n c r e a s e i n r e s i s t a n c e due t o a d e c r e a s e i n c r o s s - s e c t i o n a l a r e a , o f 7.6 f o l d . I f t h e a i r w a y w a l l i s f i r s t t h i c k e n e d , PW=0.4, a m a r g i n a l i n c r e a s e i n b a s e l i n e r e s i s t a n c e o c c u r e s assuming t h e i n c r e a s e d w a l l t h i c k n e s s has encroached upon the lumenal s p ace . I f the smooth musc le i s now s h o r t e n e d a d r a m t i c i n c r e a s e i n r e s i s t a n c e 79.2 f o l d r e s u l t s . 161 1 6 2 F i g u r e 11 The r e l a t i o n s h i p between the square r o o t o f w a l l t h i c k n e s s and i n t e r n a l a i r w a y p e r i m e t e r i s shown. The d a t a , from James (28) d i s p l a y s each normal s u b j e c t s a i r w a y s as an open c i r c l e s and a s t h m a t i c a i r w a y s as a c r o s s . T h i s p l o t c l e a r l y shows a heavy w e i g h t i n g o f d a t a t o t h e s m a l l e r a i rways as w e l l as a tendency f o r t h e v a r i a t i o n i n w a l l t h i c k n e s s t o become g r e a t e r as the i n t e r n a l p e r i m e t e r i n c r e a s e s . These c o n d i t i o n s v i o l a t e the assumpt ions r e q u i r e d f o r l i n e a r r e g r e s s i o n . 163 3.00 2.40 CM | 1.80 CO < 1.20 CO 5 0.60 h 0.00 o Normal + Asthmatic + + + + + + 4 * + 6> t o o + o o ++ o' o o o o. o o o 8 12 + o oo o o + o 16 20 Internal Perimeter (mm) Figure 12 This figure was generated by the Gauss (13) program detailed i n appendix D. The dotted l i n e s indicate the l e a s t squares regression l i n e for each of the 19 normal subjects and the dashed l i n e s are the le a s t squares l i n e s for the 19 asthmatic subjects from the data of James (28) . Note how the l i n e s for the normal subjects tend to f a l l below those of the asthmatic subjects. a 165 Pulmonary Research Lab June 9, 1989 11:13:18 F i g u r e 13 T h i s p l o t shows the d a t a , from f i g u r e 11, w i t h the r e s t r i c t e d maximum l i k e l i h o o d e s t i m a t e f o r b o t h t h e normal and a s t h m a t i c g r o u p s . The x s c a l e has been reduced f o r c l a r i t y c a u s i n g a few p o i n t s t o be dropped from the p l o t . Normals a r e shown as open boxes w i t h a dashed l i n e and the a s t h m a t i c s as c r o s s e s w i t h a s o l i d l i n e . 167 8 9 1 F i g u r e 1 4 T h i s f i g u r e d i s p l a y s t h e r e s t r i c t e d m a x i m u m l i k e l i h o o d e s t i m a t e s f o r b o t h t h e n o r n a l s , l o w e r l i n e , a n d a s t h m a t i c s u p p e r l i n e . T h e f i n e d o t t e d l i n e s a r e t h e 95% c o n f i d e n c e i n t e r v a l s a s e x p l a i n e d i n s e c t i o n 6 . 1 1 6 9 Figure 14 3 a. o m 3 170 F i g u r e 15 T h i s f i g u r e i s the recommended d i s p l a y method sugges ted by Feldman (11) . The s o l i d l i n e denotes t h e expec ted d i f f e r e n c e between a s t h m a t i c w a l l a r e a and normal a i r w a y w a l l a r e a f o r a g i v e n i n t e r n a l a i r w a y p e r i m e t e r . A p o s i t i v e v a l u e i n d i c a t e s a s t h m a t i c s h a v i n g t h i c k e r w a l l a r e a s . The f i n e dashed l i n e s a g a i n i n d i c a t e a 95% c o n f i d e n c e i n t e r v a l on t h i s expec ted d i f f e r e n c e as s t a t e d i n s e c t i o n 6.1 171 Pulmonary Re»«orch Lab June 9. 1989 11:16:45 AM Di f ference in regress ion l ines i 1 r ~ ' r - i — i i ' • ' J ' 1 1 1 1 1 • j i_ F i g u r e 16 T h i s p a i r o f p l o t s a r e d i a g n o s t i c f i g u r e s f o r d e t e r m i n i n g whether t h e d a t a meets the c r i t e r i o n f o r the randon e f f e c t s r e g r e s s i o n mode l . The assumpt ion was t h a t t h e i n d i v i d u a l i n t e r c e p t s and s l o p e s were n o r m a l l y d i s t r i b u t e d about some mean v a l u e . These two f i g u r e s show normal p r o b a b i l i y p l o t s o f the s l o p e s and i n t e r c e p t s f o r each s u b j e c t s l e a s t square r e g r e s s i o n o f square r o o t o f w a l l a r e a a g a i n s t i n t e r n a l p e r i m e t e r . I f the d a t a a r e n o r m a l l y d i s t r i b u t e d t h e n the p o i n t s s h o u l d f a l l on a s t r a i g h t l i n e . There i s no i n d i c a t i o n , p o i n t s on the t a i l s a r e u s u a l l y i g n o r e d , t h a t these v a l u e s a r e no t n o r m a l l y d i s t r i b u t e d and hence t h e assumpt ions seem s a t i s f a c t o r y . 173 Figure 16 1 r o Normal • Asthma «o o • §. 8° •o o 2 i 1 Q 1 i i i i -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 Intercepts o Normal o B • Asthma o • o 1 h o o • o° o • o • o • o ° • - o • o J Q l i_ 2 0.00 0.05 0.10 0.15 0.20 0.25 Slope 174 F i g u r e 17 T h i s p a i r o f f i g u r e s shows the t o t a l t r a c h e o b r o n c h i a l p r e s s u r e drop on a i r and on a h e l i u m oxygen m i x t u r e . The s o l i d l i n e s a r e from the symmetric model and the dashed l i n e s from the asymmetr ic mode l . In f i g u r e 17B t h e open markers a r e from a normal 41 y e a r o l d male b r e a t h i n g a i r ( c i r c l e s ) o r h e l i u m - o x y g e n (boxes ) . W h i l e u n d e r e s t i m a t i n g the t o t a l p r e s s u r e d r o p t h e mode l , b o t h symmetric and asymmetric have the same q u a l i t a t i v e shape as a v a i l a b l e human d a t a would sugges t . 175 Figure 17 Q- Flow (ml/s) 0 200 400 600 800 1000 Flow (ml/s) 176 F i g u r e 18 T h i s f i g u r e shows the two d i f f e r e n t methods o f a t t e n u a t i n g the l a m i n a r p r e s s u r e drop t o account f o r n o n - l a m i n a r f low and b i f u r c a t i o n s . The dashed l i n e s a r e f o r t h e asymmetr ic model and t h e s o l i d l i n e s a r e f o r the symmetric mode l . The two upper l i n e s use L a m b e r t ' s c o r r e c t i o n which has a minimum i n c r e a s e o f 1.5 t i m e s the l a m i n a r p r e s s u r e d r o p . The l i n e s u s i n g P e d l e y ' s c o r r e c t i o n , t h e lower two, can have a l a m i n a r p r e s s u r e d r o p . 177 Flow (ml/s) F i g u r e 19 T h i s f i g u r e shows the model compared t o normal s u b j e c t d a t a . The s o l i d l i n e was g e n e r a t e d from the symmetric model u s i n g f low o f 1 1 / s . The s o l i d c i r c l e s and squares a r e from normal 37 and 31 y e a r o l d male s u b j e c t s b r e a t h i n g a t 30 b r e a t h s / m i n u t e and measured a t 11 / s on i n s p i r a t i o n . The dashed l i n e i s from V i n c e n t (58) where lower a i r w a y s r e s i s t a n c e was measured u s i n g a t r a n s - t r a c h e a l c a t h e t e r . 179 Figure 1 9 180 F i g u r e 20 T h i s p l o t shows the r e s i s t a n c e o f a l l a i r w a y s i n a g i v e n Weibe l g e n e r a t i o n f o r v a r i o u s i s o p l e t h s o f t r a n s p u l m o n a r y p r e s s u r e . Note t h a t as t r a n s p u l m o n a r y p r e s s u r e d e c r e a s e s , d e f l a t i o n , t h a t t h e r e s i s t a n c e i n c r e a s e s . A l s o e v i d e n t from t h i s f i g u r e i s the i n c r e a s e i n r e s i s t a n c e i n g e n e r a t i o n s 4-6 caused by a s l i g h t d e c r e a s e i n t h e c r o s s - s e c t i o n a l a r e a a v a i l a b l e f o r a i r f l o w . 181 Symmetric Generation F i g u r e 21 T h i s f i g u r e combines the d a t a o f Hogg (18) , dashed l i n e s , w i t h t h e symmetric mode l , s o l i d l i n e s , t o compare the p a r t i t i o n i n g o f t h e a i r w a y s r e s i s t a n c e t o lower a i r w a y s . The upper l i n e s a r e from Hogg e t a l ' s measurements o f t o t a l r e s i s t a n c e v e r s u s t r a n s p u l m o n a r y p r e s s u r e . The lower p a i r i s from a i r w a y s beyond a 2mm r e t r o g r a d e c a t h e t e r f o r Hogg e t a l and g e n e r a t i o n s 16 t o 23 f o r the symmetric mode l . Note t h a t the model i n c r e a s e s r e s i s t a n c e n o n l i n e a r l y as the l u n g i s d e f l a t e d . 183 Figure 21 184 F i g u r e 22 T h i s f i g u r e d i s p l a y s the r e s i s t a n c e o f a i r w a y s a t a g i v e n g e n e r a t i o n f o r i s o p l e t h s o f smooth muscle s h o r t e n i n g . Note t h a t c o n t r i b u t i o n t o t o t a l r e s i s t a n c e from t h e p e r i p h e r a l a i r w a y s i s i n c r e a s i n g as smooth muscle s h o r t e n i n g i n c r e a s e s and t h a t a t 49% PMSmax t h e r e i s complete c l o s u r e o f the s m a l l e s t a i rways r e s u l t i n g i n an i n f i n i t e r e s i s t a n c e f o r t h a t g e n e r a t i o n . 185 Figure 22 r » * < 0 i o ^ t c o c M t - © 8 / l / O Z H W O SAB/V\L|J,Bd |8||BJBd JO 8 0 U B j , S I S 9 y 186 F i g u r e 23 T h i s f i g u r e p l o t s t r a c h e o b r o n c h i a l r e s i s t a n c e a g a i n s t a s i m u l a t e d dose as d e s c r i b e d i n s e c t i o n 5 . 5 . The symmetric c u r v e s a r e s o l i d and the asymmetric c u r v e s a r e dashed . F o r a l l c u r v e s a=-1.9 and 0=1.2. There appears t o be v e r y l i t t l e d i f f e r e n c e between t h e symmetric and asymmetric response a t a l l but the h i g h e s t l e v e l s o f smooth muscle s h o r t e n i n g . 187 Figure 23 188 F i g u r e 24 T h i s f i g u r e shows the e f f e c t o f encroachment upon the lumen on t h e b a s e l i n e v a l u e o f a i rways r e s i s t a n c e i n a s t h m a t i c s u b j e c t s . The encroachment r e f e r s t o t h e l a r g e r w a l l a r e a o f the a s t h m a t i c s r e l a t i v e t o normals aand the p r o p o r t i o n o f t h i s t i s s u e t h a t i s a l l o w e d t o occupy a i rway lumen. Note t h a t a t FRC, about 8cmH 20, even 100% encroachment has o n l y a minor e f f e c t on b a s l i n e r e s i s t a n c e . 189 F i g u r e 25 T h i s f i g u r e shows the maximum o b t a i n a b l e r e s i s t a n c e as smooth musc le s h o r t e n i n g i s i n c r e a s e d . The v a l u e s f o r w a l l a r e a from normal s u b j e c t s was used as w e l l as a s t h m a t i c d a t a f o r t h r e e l e v e l s o f a i r w a y lumen encroachment . Note t h a t even w i t h no encroachment t h e maximum a l l o w a b l e smooth musc le s h o r t e n i n g f o r a s t h m a t i c s i s markedly l e s s t h a n t h a t o f n o r m a l s . 191 40 Maximum Smooth Muscle Shortening F i g u r e 26 T h i s f i g u r e shows dose response r e l a t i o n s h i p s u s i n g a s t h m a t i c s w a l l a r e a d a t a and v a r i o u s degrees o f lumenal encroachment . In f i g u r e 26A PMSmax i s i n c r e a s e d from 20% t o 40% (curves A t o D) w i t h no encroachment . In 26B PMSmax i s 20%, 30% and 35% f o r c u r v e s A t o C and encroachment i s 50%. In 26C encroachment has been i n c r e a s e d t o 100% and the v a l u e s f o r PMSmax a r e 20%, 25% and 30%. In each frame the dashed l i n e shows t h e normal w a l l a r e a response w i t h 20% maximal smooth musc le s h o r t e n i n g . F o r these p l o t s f low was s e t t o 1250 m l / s and a = - 1 . 9 , 0=1.2. 193 F i g u r e 27 These p l o t s show the e f f e c t o f a i rway w a l l t h i c k e n i n g i n j u s t t h e c e n t r a l a i r w a y s , g e n e r a t i o n s 0 t o 10, as opposed t o p e r i p h e r a l a i r w a y s w a l l t h i c k e n i n g , g e n e r a t i o n s 11 t o 23. Wi th no enroachment , p a n e l s A and B , the p e r i p h e r a l t h i c k e n i n g has a s m a l l e r e f f e c t on t r a c h e o b r o n c h i a l r e s i s t a n c e u n t i l 40% PMSmax where the p e r i p h e r a l a i r w a y s r e a c t a t much lower d o s e s . I f some encroachment i s a l l o w e d , p a n e l s C and D, t h e n t h i s e f f e c t i s enhanced w i t h t h e dose response a p p e a r i n g near normal f o r 30% PMSmax b u t w i t h any a d d i t i o n a l smooth musc le s h o r t e n i n g t o t a l c l o s u r e o f t h e a i r w a y s o c c u r s . 195 Central Changes Peripheral Changes Log Dose F i g u r e 28 These f i g u r e shows the volume and f requency dependence o f normal s u b j e c t s as d i s c u s s e d i n s e c t i o n 10. No s i g n i f i c a n t r e l a t i o n s h i p c o u l d be seen between r e s i s t a n c e and b r e a t h i n g f r e q u e n c y o r volume s u g g e s t i n g t h a t o v e r t h i s range o f the d a t a t h a t i n v i v o measurements o f pulmonary r e s i s t a n c e a r e no t i n f l u e n c e d by t i s s u e v i s c a n c e . 197 Figure 28 800ml tidal volume 2.00 0 10 20 30 40 50 60 70 80 Frequency (breaths/minute) 1600ml tidal volume 2.00 0 10 20 30 40 50 60 70 80 Frequency (breaths/minute) 1 9 8 References (1) Amemiya, Y . What s h o u l d be done when an e s t i m a t e d between-group c o v a r i a n c e m a t r i x i s no t nonnegat ive d e f i n i t e ? Amer. Stat. 39: 112-117, 1985. (2) B e r g e r , James 0. and W o l p e r t , R o b e r t L . The L i k e l i h o o d  P r i n c i p l e 2nd E d i t i o n . I n s t i t u t e o f M a t h e m a t i c a l S t a t i s t i c s , L e c t u r e Notes Volume 6. 1988. (3) Brodkey , R obe r t S. and Hershey H a r r y C . 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