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Entry flow problem of a liquid body into a suction pipette Yeung, Anthony Kwok-Cheung 1987

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Entry Flow Problem of a Liquid Body into a Suction Pipette  by  Anthony Kwok-Cheung Yeung  B.A.Sc, The University of British Columbia, 1983  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1987  ©  Anthony Kwok-Cheung Yeung, 1987  In  presenting  degree freely  at  this  the  available  copying  of  department publication  thesis  in  partial  University  of  British  for  this or  reference  thesis by  of  this  of  _  for  his thesis  scholarly  or  her  for  Date  DE-6(3/81)  DcAoUs  ,  the  requirements that the  for  Library  I further agree that permission  purposes  may  representatives.  financial  The University of British C o l u m b i a 1956 Main Mall Vancouver, Canada V6T 1Y3  of  Columbia, I agree  and study.  permission.  Department  fulfilment  It  gain shall not  be  granted  is  by  understood be  allowed  the  an  advanced  shall make for  extensive  head  that without  it  of  my  copying  or  my  written  ii ABSTRACT  The  mathematical problem o f t h e p i p e t t e  aspiration  of a  l i q u i d s p h e r e i s s t u d i e d i n t h e l o w R e y n o l d s number l i m i t . d i s t i n c t models a r e proposed f o r t h e d e f o r m i n g body. 1)  They a r e :  a l i q u i d d r o p l e t o f c o n s t a n t v i s c o s i t y , a n d 2) a v i s c o e l a s t i c  c o r t e x e n c a p s u l a t i n g an i n v i s c i d i n t e r i o r . represent energy d i s s i p a t i o n d i s t r i b u t e d the  s u r f a c e o f t h e body, r e s p e c t i v e l y .  rates vary d i f f e r e n t l y with the pipette this  These models  i n t h e i n t e r i o r and on Because t h e i n - f l o w s i z e f o r t h e two models,  i s s u g g e s t e d a s a means o f e x p e r i m e n t a l l y i d e n t i f y i n g t h e  dominant r e g i o n o f v i s c o u s d i s s i p a t i o n ,  and t h u s p r o v i d e  into the internal structure  sample.  For  of the test  insight  t h e d r o p l e t problem, t h e l i n e a r Stokes equations a r e  s o l v e d i n t h e i n t e r i o r o f t h e d e f o r m i n g body. f o r some s p e c i f i e d be  Two  stress  The  solutions,  b o u n d a r y c o n d i t i o n s on a s p h e r e , c a n  e x p r e s s e d a s i n f i n i t e sums o f L e g e n d r e p o l y n o m i a l s .  In s o l v i n g  t h e s u r f a c e flow problem, t h e c o m p l e x i t i e s o f t h e  equations n e c e s s i t a t e approximate s o l u t i o n s  by computational  means. A n u m e r i c a l p r o c e d u r e i s d e v e l o p e d w h i c h c o m p a r e s w i t h a n a l y t i c a l r e s u l t s when t h e l a t t e r i s a v a i l a b l e .  well  iii TABLE  O F  CONTENTS  Abstract  i i  L i s t of Figures  iv  Acknowledgement  v  I.  Introduction  1  II.  P i p e t t e A s p i r a t i o n of a Droplet  8  2.1) 2.2) 2.3) 2.4) III.  Two-Dimensional Membrane Mechanics 3.1) 3.2) 3.3) 3.4) 3.5) 3.6)  IV.  The Creeping Motion Equations Problem Formulation and S o l u t i o n Surface Tension E f f e c t s R e s u l t s and D i s c u s s i o n  Analysis of S t r a i n Balanace o f Forces i n a Two-Dimensional Membrane Constitutive Relations The L i n e a r Q u a s i - E l a s t i c S o l u t i o n Implementation o f Numerical Method R e s u l t s and D i s c u s s i o n  Summary  23 26 30 34 41 45 47 55  Appendix A.  S o l u t i o n t o Creeping  Appendix B.  Pipette Aspiration of C o r t i c a l S h e l l  Flow Equations  i n t h e absence o f In-Plane Shearing Appendix C.  9 12 16 19  58  S t r e s s e s 63  F i n i t e D i f f e r e n c e Equations  69  Matrix Solution t o F i n i t e D i f f e r e n c e Equations L i s t o f References  73 79  Appendix D.  IV  L I S T OF FIGURES  Number  Title  Page  1  Photograph o f an A s p i r a t e d Granulocyte  2  Schmatic Problem  3  E q u i l i b r i u m P r e s s u r e v s . P r o j e c t e d Length  3  I l l u s t r a t i o n of Droplet A s p i r a t i o n  13  f o r Droplet with I n t e r f a c i a l Tension  17  4  D r o p l e t In-Flow Rate v s . P i p e t t e Radius  20  5  H y d r o s t a t i c P r e s s u r e on a D r o p l e t S u r f a c e  22  6  The L i p i d B i l a y e r  25  7 8  The Two Modes o f Membrane Deformation D e f i n i t i o n o f Tension R e s u l t a n t s and Coordinate V a r i a b l e s  28 31  9  Two Models o f V i s c o e l a s t i c i t y  38  10  Matrix Structure of F i n i t e - D i f f e r e n c e Formulation  48  11  Time E v o l u t i o n o f C e l l w i t h *>{/>£= 0  50  12 13  Aspiration  V e l o c i t y F i e l d s o f S u r f a c e Flow Problem w i t h -vf/OL = o  51-52  In-Flow Rate v s . P i p e t t e S i z e f o r S u r f a c e Flow Model  53  14  C e l l Shapes f o r D i f f e r e n t Values o f ^/X.  54  15  D e f i n i t i o n o f Dimensions f o r t h e Problem w i t h t^/yc = 0 F i n a l Form o f Upper T r i a n g u l a r M a t r i x  66 74  16  V  ACKNOWLEDGEMENT  I would l i k e t o thank Dr. Evan Evans, my t h e s i s a d v i s o r , for  i n t r o d u c i n g me t o t h e f i e l d o f s c i e n t i f i c r e s e a r c h ,  a l l t h e encouragement and support he has g i v e n  me.  and f o r  1 I.  INTRODUCTION  M i c r o p i p e t t e a s p i r a t i o n , which i n v o l v e s the m a n i p u l a t i o n and d e f o r m a t i o n o f t e s t samples w i t h a s u c t i o n p i p e t t e , has become an important technique i n s t u d y i n g the mechanical p r o p e r t i e s of biological cells.  S i n c e i t s f i r s t a p p l i c a t i o n by M i t c h i s o n and  Swann (1954) t o t h e i r work on sea u r c h i n eggs, the technique  has  developed e x t e n s i v e l y , p a r t i c u l a r l y i n the r e s e a r c h area o f red b l o o d c e l l membranes (Rand and Burton, 1964; and Hochmuth, 1977;  Skalak e t a l . ,  1973).  Evans, 1973;  Evans  R e c e n t l y , the  m i c r o p i p e t t e technique has been adapted t o the d i r e c t measurements of weak adhesive i n t e r a c t i o n s between s u r f a c t a n t bilayers  (Evans and Needham, 1986).  A l l these a p p l i c a t i o n s  i n v o l v e measurements of s t a t i c f o r c e s i n e q u i l i b r i u m configurations.  Time dependent behaviour of b o d i e s (leukocytes)  i n a m i c r o p i p e t t e has a l s o been i n v e s t i g a t e d f o r s m a l l deformations 1984).  (Schmid-Schonbein,  et a l . ,  1981;  Chien and Sung,  In these cases, the c e l l i s t r e a t e d as a "standard  v i s c o e l a s t i c " m a t e r i a l (see Fung, 1965), which, i n essence, s o l i d body w i t h an e l a s t i c l i m i t t o deformation.  is a  The aim of  t h i s p r e s e n t work i s t o a n a l y z e the p i p e t t e a s p i r a t i o n of a l i q u i d - l i k e body i n the context of continuum mechanics.  The  body o f i n t e r e s t i s assumed t o be s u f f i c i e n t l y l a r g e t h a t i t must deform upon e n t e r i n g the p i p e t t e .  Models o f the m a t e r i a l  p r o p e r t i e s of the body can then be used t o p r e d i c t i t s deformation and flow under the g i v e n e x t e r n a l f o r c e s .  2  As t h e name i m p l i e s , t h e t h e o r y o f continuum  mechanics  r e q u i r e s t h e deforming body t o have, even when i t s s i z e i s reduced t o t h e l i m i t i n g r e s o l u t i o n o f t h e experiment, a s u f f i c i e n t number o f p a r t i c l e s t o enable a thermodynamic c h a r a c t e r i z a t i o n o f i t s macroscopic p r o p e r t i e s .  The experiment  t h a t m o t i v a t e s t h i s p r e s e n t a n a l y s i s i s t h e a s p i r a t i o n o f human white b l o o d c e l l s  ( g r a n u l o c y t e s ) i n t o m i c r o p i p e t t e s , as shown i n  f i g u r e 1, i n an attempt t o understand t h e i r behaviour are  (Evans and Kukan, 1984).  on t h e o r d e r o f IO  -3  rheological  Here, t h e c e l l s o f i n t e r e s t  cm i n diameter - a l e n g t h s c a l e t h a t i s  much l a r g e r than m o l e c u l a r dimensions.  U l t r a s t r u c t u r a l evidence  ( B e s s i s , 1973; Schmid-Schonbein  1980) has shown t h a t the  et a l . ,  g r a n u l o c y t e i s e n c a p s u l a t e d i n a plasma membrane t h a t has a r e s e r v o i r o f excess s u r f a c e area i n t h e form o f e v e n l y distributed ruffles.  T h i s membrane, when i n t h e f l a c c i d  state,  i s l i k e l y t o o f f e r l i t t l e o r no mechanical r e s i s t a n c e t o c e l l deformations  (Evans and Skalak, 1980).  Anchored t o t h e  underneath o f t h e plasma membrane i s a meshwork o f d e n s l y packed, randomly o r i e n t e d a c t i n f i l a m e n t s S t o s s e l , 1983; Amato e t a l . ,  1983).  (Southwick and  When s t i m u l a t e d t o  contract, these filaments are believed t o provide f o r the c e l l ' s a c t i v e locomotory. are  Because we a r e o n l y i n t e r e s t e d i n c e l l s t h a t  n o t "turned on", t h e c o r t i c a l meshwork i s c o n s i d e r e d here as  a passive g e l with i t s s p e c i f i c rheological properties. small s i z e of the a c t i n filaments to are  (8 nm i n diameter) enables us  t r e a t t h e g e l as a continuum; moreover, randomly o r i e n t e d on t h e c e l l  The  because t h e f i l a m e n t s  s u r f a c e , i s o t r o p y and  Figure  1:  A s p i r a t i o n o f a human g r a n u l o c y t e a t 23 C. The p i p e t t e i n n e r r a d i u s i s 2 m i c r o m e t e r s and t h e s u c t i o n p r e s s u r e i s 500 d y n e s / c m .  4  homogeneity i n the s u r f a c e plane d i m e n s i o n ) c a n b e assumed.  (excluding the thickness  In contrast to the cortex, the c e l l  i n t e r i o r i s h i g h l y heterogeneous i n s t r u c t u r e ; t h a t i s , i t contains o r g a n e l l a r bodies  l i k e the nucleus  and g r a n u l e s .  i s a c o m p o s i t e s t r u c t u r e w i t h e a c h component b e i n g itself.  To u n d e r s t a n d t h e o v e r a l l p r o p e r t y  we w i l l m o d e l t h e c e l l  i n t e r i o r as a  i s o t r o p i c a n d homogeneous s u b s t a n c e .  and s h o u l d  a continuum  of this sturcture,  three-dimensionally In doing  this,  understood that the material properties assigned bulk of the i n t e r i o r ,  This  i t is  r e f e r to the  n o t b e a t t r i b u t e d t o a n y one o f  t h e o r g a n e l l a r components.  For the a n a l y s i s i n t h i s t h e s i s , the complicated of a granulocyte  has been i d e a l i z e d as a t h r e e  structure  dimensionally  i s o t r o p i c s u b s t a n c e s u r r o u n d e d by a two d i m e n s i o n a l l y  isotropic  gel.  as  The n e x t s i m p l i f i c a t i o n i s t o i d e n t i f y t h e c e l l  e s s e s t i a l l y a l i q u i d body. flow of such c e l l s is  i s evidenced  This  i n t u r n i s e s t a b l i s h e d by a c o r t i c a l  i n t e r f a c i a l t e n s i o n between two l i q u i d s .  flow i s continuous  by t h e  i n t o m i c r o p i p e t t e s when t h e s u c t i o n  i n excess of a c e r t a i n threshold value.  pressure the  This  without  continuous pressure  threshold  s t r e s s much  like  The f a c t t h a t t h e  any a p p r o a c h t o s t a t i c e q u i l i b r i u m  i n d i c a t e s t h a t no l i m i t i n g e l a s t i c f o r c e s e x i s t i n t h e c e l l interior.  F u r t h e r , t h e c o r t i c a l t e n s i o n c a n b e shown t o b e  independent of deformation the  initial  threshold.  by l o w e r i n g t h e s u c t i o n p r e s s u r e  I n such a case,  aspirated projection of the c e l l  to  t h e f l o w c e a s e s and t h e  i n s i d e the p i p e t t e remains  5  stationary.  In g e n e r a l , both t h e c o r t i c a l s h e l l and t h e i n t e r i o r may c o n t r i b u t e t o t h e v i s c o u s r e s i s t a n c e t o flow.  The two l i m i t i n g  models proposed here a r e ones t h a t have t h e v i s c o u s  dissipation  dominated by one o f t h e r e g i o n s , w i t h t h e o t h e r r e g i o n essentially inviscid.  being  In p a r t i c u l a r , t h e models a r e : 1) a  l i q u i d d r o p l e t w i t h a constant v i s c o s i t y , r e s u l t i n g i n energy d i s s i p a t i o n d i s t r i b u t e d throughout t h e i n t e r i o r , and  2) a  v i s c o u s c o r t i c a l s h e l l e n c a p s u l a t i n g an i n v i s c i d f l u i d .  The  c o n d i t i o n o f volume c o n s e r v a t i o n , as w e l l as a c o r t i c a l are i n c o r p o r a t e d i n t o both models.  tension  A l s o , t h e v i s c o u s drag o f  the e x t e r i o r aqueous s o l u t i o n on t h e c e l l i s n e g l e c t e d because of t h e extremely slow response observed f o r c e l l e n t r y i n t o p i p e t t e s compared w i t h t h e r a p i d i n - f l o w o f water a t t h e same suction pressures 10 ). 5  ( i e : flow r a t e s t h a t d i f f e r by a f a c t o r o f  Because t h e d i s s i p a t i o n o f mechanical energy i s  v o l u m e t r i c a l l y d i s t r i b u t e d i n t h e former case and two d i m e n s i o n a l l y c o n f i n e d i n t h e l a t t e r , t h e f u n c t i o n a l dependence o f any flow r e l a t e d q u a n t i t y any  (eg., t h e c e l l e n t r y flow r a t e ) on  c h a r a c t e r i s t i c l e n g t h s c a l e (eg., t h e p i p e t t e r a d i u s )  be d i s t i n c t l y d i f f e r e n t f o r t h e two models. behaviours  These f u n c t i o n a l  can be obtained by s o l v i n g t h e above mentioned  mechanical problems i n t h e e n t i r e r e g i o n t h a t t h e c e l l The  occupies.  r e s u l t s w i l l p r o v i d e an experimenter w i t h a means t o  d i s c r i m i n a t e t h e d i f f e r e n t v i s c o u s d i s s i p a t i o n zones by performing  should  a s p i r a t i o n t e s t s with various s i z e d p i p e t t e s .  6  Continuum mechanical independent  a n a l y s e s are composed o f t h r e e  and d i s t i n c t developments.  They a r e : 1) the  q u a n t i t a t i o n o f deformation and r a t e of deformation i n r e l a t i o n t o changes i n the body's geometry; 2) the b a l a n c e o f f o r c e s w i t h i n the body, as d i c t a t e d by Newtonian mechanics; and 3) the m o d e l l i n g of the m a t e r i a l p r o p e r t i e s o f the  substance.  Knowledge o f any two of these a s p e c t s can be used t o p r e d i c t the t h i r d one.  For example, the deformation and r a t e o f deformation  o f a body i n response t o c o n t r o l l e d f o r c e s can be analyzed t o g i v e the m a t e r i a l p r o p e r t i e s (eg: e l a s t i c and v i s c o u s coefficients).  I t i s important t o note t h a t these t h r e e  developments are formulated i n terms of i n t e n s i v e  quantities,  i e : q u a n t i t i e s t h a t do not depend on the s i z e of the sample. F o r i n s t a n c e , deformation i s measured by s t r a i n , which i s a d i m e n s i o n l e s s r a t i o of the m a t e r i a l displacement t o some i n i t i a l length.  L i k e w i s e , the d i s t r i b u t i o n o f f o r c e s i s measured by i t s  intensity  ( i e : on a per u n i t area b a s i s ) c a l l e d  stress.  M a t e r i a l p r o p e r t i e s , which are c o e f f i c i e n t s r e l a t i n g the s t r e s s e s and the deformations, are t h e r e f o r e based on f u n c t i o n s as w e l l .  local  The v i r t u e of such f o r m u l a t i o n s i s t h a t  i n t r i n s i c p r o p e r t i e s of the substance can be d e f i n e d o f the n a t u r e of the experiment.  independent  In the f o l l o w i n g c h a p t e r s ,  t h e s e t h r e e developments w i l l be f o l l o w e d i n a r r i v i n g a t the e q u a t i o n s t h a t govern the flow f i e l d .  A n a l y t i c a l solutions to  t h e s e e q u a t i o n s are not a v a i l a b l e except f o r a few cases.  simplified  F o r the more g e n e r a l problem, we must be content w i t h  7  approximate  s o l u t i o n s b y n u m e r i c a l means.  These methods  will  a l s o be d i s c u s s e d i n t h e n e x t c h a p t e r s and t h e r e s u l t s w i l l presented.  be  8 II.  The  two  P I P E T T E A S P I R A T I O N OF  models proposed i n the  respectively.  DROPLET  introduction  c e l l ' s s t r e s s b e a r i n g component t o the c o r t i c a l region,  A  i n t e r i o r and  To address the  the p i p e t t e a s p i r a t i o n of a l i q u i d d r o p l e t chapter.  The  due  i n t e r f a c i a l tension,  an  t o an  droplet  attributed  first  and  the  the  problem,  i s analysed i n t h i s  i s t o have a s p h e r i c a l i n i t i a l  i n c o m p r e s s i b l e newtonian f l u i d .  geometry  i n t e r i o r i s modelled  A s i m i l a r problem i s  by Schmid-Schonbein e t a l . (1981) w i t h the c e l l as a "standard v i s c o e l a s t i c s o l i d " . used the  to  the  as  solved  i n t e r i o r treated  Because the  authors have  l i n e a r i z e d s t r a i n t e n s o r i n t h e i r a n a l y s i s , the  results  are v a l i d o n l y f o r s m a l l deformations.  Two  s i m p l i f y i n g assumptions are made here: 1)  a s p i r a t i o n , the p o r t i o n be  during  of the body e x t e r i o r t o the p i p e t t e  approximated as a s p h e r i c a l segment; and  2)  viscous  d i s s i p a t i o n i n s i d e the p i p e t t e can be n e g l e c t e d .  The  t o the  pressure  i n t e r f a c i a l tension) g r e a t l y  by the  s i t u a t i o n where the  first  assumption i s e q u i v a l e n t (created  the dynamic s t r e s s normal t o the  surface  boundary.  c r i t e r i o n can be v e r i f i e d a p o s t e r i o r i from the The  (Evans and conditions, the  Laplace exceeds  This  final  solution.  second approximation i s i n t r o d u c e d t o r e p r e s e n t the  s l i p condition  between the c e l l s u r f a c e  Kukan, 1984).  By p u t t i n g  and  can  the p i p e t t e  freewall  i n proper boundary  the p r e s e n t problem w i l l have a unique s o l u t i o n i n  form of a v e l o c i t y f i e l d .  We  s t a r t our  a n a l y s i s by  first  9  summarizing some equations r e l e v a n t t o t h e theory  o f continuum  mechanics.  2.1:  T H E C R E E P I N G MOTION  EQUATIONS  Let x^ ( i = l , 2 , 3 ) be a c o o r d i n a t e  system s e t up w i t h i n t h e  continuous body t h a t l o c a t e s every m a t e r i a l p o i n t . velocity  I f the  v ^ ( i = l , 2 , 3 ) i s continuous everywhere, then t h e r a t e o f  s t r a i n t e n s o r i s d e f i n e d as av  x v  with the property  ^  ij  =  2  (  air  avj  i  ex.  +  (2.1)  }  that  (As - A s ) 2  2  Q  Here, Ax^ i s an instantaneous  =  2  Ax  ±  Ax.  (2.2)  p o s i t i o n vector connecting  two  points that are i n f i n i t e s i m a l l y close, A s i s the absolute  length  o f Ax., and As_ l o i s t h e d i s t a n c e between t h e same two m a t e r i a l points i n the i n i t i a l configuration.  Because equation  i n v o l v e s o n l y t h e d i f f e r e n c e between a b s o l u t e  2.2  lengths, the rate  of s t r a i n t e n s o r excludes a l l r i g i d body displacements and i s t h e r e f o r e a t r u e measure o f deformation.  The 1  t r a c e of the rate of s t r a i n tensor represents  The r e p e t i t i o n o f an index w i l l imply t h a t index over i t s range.  volume  summation w i t h r e s p e c t t o  10  dilatation.  Because t h e f l u i d i s assumed i n c o m p r e s s i b l e i n t h i s  c a s e , "V\^ v a n i s h e s .  Expressed  e q u i v a l e n t l y i n terms of v e l o c i t y  components, t h e i n c o m p r e s s i b i l i y c o n d i t i o n i s dv . ax" i  =  1  < ' >  0  2  The s t r e s s v e c t o r T^ ( n ) c o r r e s p o n d i n g d e f i n e d as f o l l o w s : n o r m a l t o n. s i d e o f AA  Consider  AF  i  to a unit vector n i s  a c r o s s s e c t i o n a l a r e a AA t h a t i s  L e t AF^ b e t h e t o t a l  f o r c e e x e r t e d on t h e p o s i t i v e  ( i e : t h e s i d e on w h i c h n p o i n t s o u t w a r d ) .  t h e amount o f f o r c e w i l l d e c r e a s e limit  3  as t h e area s h r i n k s .  Obviously, In the  as AA v a n i s h e s , t h e s t r e s s v e c t o r i s g i v e n by t h e r a t i o o f  to  AA:  T  i  ( n K  \ " '  lim AA*o  =  A  i ~AA~  (2.4)  F  The s t r e s s t e n s o r , 0 ^ j , i s d e f i n e d a s t h e j  t  h  component o f t h e  s t r e s s v e c t o r o n a p l a n e whose n o r m a l i s i n t h e x ^ d i r e c t i o n :  CT  ij  Here, t h e u n i t v e c t o r s  S  T  j  (  e  i  }  ( 2  5 )  are the basis set vectors.  For a newtonian l i q u i d ,  the stress tensor i s related to the  r a t e o f s t r a i n t e n s o r by a p r o p o r t i o n a l i t y  ID  '  =  -p 5  ID  constant:  2 T) V,  (2.6)  11  ^  i s c a l l e d t h e c o e f f i c i e n t o f shear v i s c o s i t y .  hydrostatic pressure remains non-zero.  p i s introduced  The  so t h a t t h e t r a c e o f  I t c a n be v i e w e d as a Lagrange  multiplier  associated with the incompressibility constraint.  The  b a l a n c e o f f o r c e s w i t h i n a continuum i s expressed by t h e  equation  P  da i ^ -  dv^  +  a j  dt  r  X  F. i  (2.7)  w h e r e p i s t h e mass d e n s i t y o f t h e b o d y , v ^ t h e v e l o c i t y , t h e body f o r c e p e r u n i t volume.  a n d F^  Because t h e t e s t samples  (granulocytes)  a r e f r e e l y suspended d u r i n g  no b o d y f o r c e s  (or rather they are n e g l i g i b l e i n comparison t o  the  s u c t i o n f o r c e s ) , a n d h e n c e F^= 0.  equation for  2.7 r e p r e s e n t s  a newtonian f l u i d ,  represents latter  the viscous  the i n e r t i a l  experiment, there are  The l e f t h a n d s i d e o f forces while t h e term  as i n d i c a t e d i n t h e l a s t forces.  paragraph,  The r a t i o o f t h e f o r m e r t o t h e  i s known a s t h e R e y n o l d s number ( L a n d a u a n d  1982).  Lifshitz,  From t h e c h a r a c t e r i s t i c s i z e s and f l o w r a t e s o f t h e  micropipette  e x p e r i m e n t s , t h e R e y n o l d s number i s e s t i m a t e d  to  h a v e a n u p p e r b o u n d o f 10" , t h u s l e a v i n g t h e l e f t h a n d s i d e o f 6  equation  2.7 c o m p l e t e l y  negligible.  The e q u a t i o n  o f mechanical  e q u i l i b r i u m f o r our purposes i s t h e r e f o r e  da .  i l ax D  =  0  (2.8)  12  Combining equations 2.1, 2.3, 2.6, and 2.8, we a r r i v e a t t h e l i n e a r Stokes equation f o r c r e e p i n g  v p  =  nv  motions:  2  v  (2.9)  By t a k i n g t h e c u r l o f equation 2.9 and u s i n g t h e v e c t o r identity v x vp  =  o  the c r e e p i n g motion e q u a t i o n can be f u r t h e r s i m p l i f i e d t o c o n t a i n o n l y t h e v e l o c i t y term:  v  2  (v x v )  =  o  (2.10)  By d e f i n i n g t h e q u a n t i t y v x v as t h e v o r t i c i t y v e c t o r , we see t h a t t h e c r e e p i n g motion equations have been reduced t o a homogeneous L a p l a c e equation o f v o r t i c i t y .  2.2:  PROBLEM FORMULATION AND SOLUTION  E q u a t i o n 2.10 has t o be s a t i s f i e d a t every p o i n t i n s i d e t h e droplet.  I n a d d i t i o n , proper boundary c o n d i t i o n s have t o be  p r e s c r i b e d f o r t h e problem t o be w e l l posed.  Consider the  s i t u a t i o n d e p i c t e d i n f i g u r e 2. Because o f t h e body's geometry, it  i s n a t u r a l t o use s p h e r i c a l c o o r d i n a t e s (^?,e).  a z i m u t h a l angle  $  Here, t h e  drops out due t o axisymmetry w h i l e e, t h e  p o l a r angle, can be a l t e r n a t i v e l y r e p r e s e n t e d by i t s c o s i n e :  13  z  F i g u r e 2:  D e f i n i t i o n o f c o o r d i n a t e s and v a r i o u s dimensions f o r t h e p i p e t t e a s p i r a t i o n problem. Note t h a t t h e r e i s a f i n i t e r e g i o n o f c o n t a c t between the d r o p l e t and the p i p e t t e , as d e f i n e d i n eqn. 2.11.  14 £  = cos  e  In terms of £ , the o u t e r s u r f a c e into three regions F i r s t , there  of the sphere can be  t h a t are s u b j e c t e d  ( Cj> < £ <  the p i p e t t e 7[ /£  (  portion  to d i f f e r e n t stresses.  i s the p a r t i n s i d e the p i p e t t e  experiences a suction pressure A P .  divided  The  (-1  region  < £ <  C^, ) t h a t  i n contact  with  + e ) i s under a uniform compressive load  b e i n g a s m a l l q u a n t i t y ) , w h i l e the remaining s p h e r i c a l (£J, + € < £ < 1 )  i s stress free.  The  magnitude of  A  can  be r e l a t e d t o A P by r e q u i r i n g the t o t a l a x i a l f o r c e on the body be  zero.  The  e x t e r n a l l y a p p l i e d normal s t r e s s e s  collectively called  (X ( £ )  such t h a t  -1  AP  <  r + i  conditions  ^ are  r  p  < r <  £  i  a  The  s t r e s s boundary  therefore  f o r the normal s t r e s s ,  PP  (C) V J >  '  p=R  f o r zero t a n g e n t i a l s t r e s s . and  =  =  0  (2.13)  In an axisymmetric problem,  o „ are the only p o s s i b l e non-zero P  (2.12)  «(f)  and  p=R  PP  p  both p o s i t i v e q u a n t i t i e s .  o*  quantities a  <  (2.11) 0  and  c  s.  -X/e  w i t h AP  are  the  stresses  15 on t h e c o o r d i n a t e s u r f a c e p =  The 2.10)  R.  s o l u t i o n t o the Laplace equation f o r v o r t i c i t y  a s i n f i n i t e sums o f p o l y n o m i a l s i n P  c a n be e x p r e s s e d  a n g u l a r h a r m o n i c s i n L\ , a s d e v e l o p e d A).  By m a t c h i n g  by H a p p e l and  1973  (see appendix  2.13  t o t h e g e n e r a l s o l u t i o n s o f e q u a t i o n 2.10,  of the s o l u t i o n i s obtained. p = R,  they  (equation and  Brenner,  b o u n d a r y c o n d i t i o n s 2.12  In particular,  the f i n a l  and  form  on t h e b o u n d a r y  are 00  v  (O  =  p  _  ° r  v 2  -  f  X I  J  n = 3  ( n - l ) (2n-l> „ 2(n-2) (2rr+l)  R  a  2  n  P  n-l^> n n l  (2.14a)  00  v (D  =  e  v°  sin e  z  J  +  3n(n-l)  R  2(n-2)(2n +l)  n = 3  VP  a  2  n  (2.14b)  sin e  n=3 where P ( £ ) n  i s t h e L e g e n d r e p o l y n o m i a l o f d e g r e e n, and  t h e Gegenbauer p o l y n o m i a l g i v e n  n  n-2<n -  n  w i t h the degenerate  I (C) O  The  s  =-  P (C) O  condition that ^  l  V"  2n - 1  cases defined  =  (£. )  by  p  l (C)  I  ;  i (C) 1  as  =  - P^O  =  - C  = 0 a t t h e p o i n t where t h e sphere  t h e p i p e t t e i s s a t i s f i e d by s u p e r i m p o s i n g  an a x i a l  (2.16)  touches  velocity  16 onto the s e r i e s s o l u t i o n s .  The  Legendre c o e f f i c i e n t s a  are n  functions  o f AP, ^ ,  and  geometry.  The  w e l l as a d e r i v a t i o n of equations 2.14 2.3:  for a ,  as  n  are g i v e n  i n appendix A.  SURFACE TENSION EFFECTS  The not  expression  normal s t r e s s boundary c o n d i t i o n , e q u a t i o n 2.11,  i n c l u d e any  i n t e r f a c i a l tension contribution.  L a p l a c e p r e s s u r e term -2T /R ( T Q  Q  does  Adding  b e i n g the i n t e r f a c i a l  the  tension  w i t h u n i t s o f f o r c e / u n i t length) t o the r i g h t hand s i d e of i n f a c t leaves  the f i n a l e x p r e s s i o n  v e l o c i t y f i e l d , unaltered. droplet the  for a , n  and  2.11  hence the  T h i s i s because the geometry of a  ( i e : sphere) i s the e q u i l i b r i u m c o n f i g u r a t i o n c r e a t e d  i n t e r f a c i a l forces.  representing  The  v e l o c i t y f i e l d s i n equations  the balance of v i s c o u s  deform the sphere, are t h e r e f o r e  stresses against  independent of T .  by  2.14,  forces  that  Any  Q  d e v i a t i o n from t h i s e q u i l i b r i u m shape, however, w i l l be r e s i s t e d by the  i n t e r f a c i a l tension.  By c a l c u l a t i n g t h i s r e s i s t a n c e  shape changes, i t i s shown t h a t d e s p i t e of the s u r f a c e e f f e c t , the r e s u l t s from s e c t i o n 2.2 quantity thershold  Ap  to  tension  are s t i l l u s e f u l i f the  i s i n t e r p r e t e d as a p r e s s u r e i n excess of some  value.  C o n s i d e r the a s p i r a t i o n o f a l i q u i d d r o p l e t t h a t has constant i n t e r f a c i a l tension T . Q  The  pressure P  e g  a  required  to  h o l d the d r o p l e t a t s t a t i c e q u i l i b r i u m a t a p r o j e c t e d  length  d  i s c a l c u l a t e d , keeping the volume of the drop f i x e d .  Figure  3  17  Figure  3:  Suction pressure required to hold a l i q u i d d r o p l e t a t s t a t i c e q u i l i b r i u m v e r s u s the p r o j e c t e d length f o r d i f f e r e n t i n i t i a l c e l l s i z e s .  18 shows t h e p l o t o f t h e d i m e n s i o n l e s s f u n c t i o n o f d/R  f o r various c e l l  pressure,  sizes.  p  q p/T f  as a  R  e  Q  The common f e a t u r e t o  P  all  these  curves  i s the steep r i s e i n pressure  l e s s t h a n one p i p e t t e r a d i u s formed i n t h e p i p e t t e ) ,  f o rprojections  ( i e : b e f o r e a h e m i - s p h e r i c a l cap i s  f o l l o w e d by an e s s e n t i a l l y c o n s t a n t o r  even d e c r e a s i n g p r e s s u r e  level.  The p r e s s u r e  r e q u i r e d t o form  t h e h e m i s p h e r i c a l c a p c a n be c a l c u l a t e d from t h e e q u a t i o n  *  -  2  T  o  Hr  p  " ib  (2.17)  T h i s i s t h e t h r e s h o l d t h a t must be exceeded f o r f l o w t o commence. F o r p r e s s u r e s  a b o v e t h e t h r e s h o l d , t h e amount i n  A  excess of P (approximately)  i s the e f f e c t i v e pressure  balanced  against the viscous forces.  pressure  drop along t h e a s p i r a t e d l e n g t h  assumption), equation  t h i s excess pressure  2.11.  that i s  A s s u m i n g t h e r e i s no (the plug  flow  i s just the quantity  AP i n  To b e e x a c t , we w r i t e AP  =  P  -  P  (2.18)  where P i s t h e a p p l i e d s u c t i o n p r e s s u r e .  With t h i s  interpretation of AP, the v e l o c i t y f i e l d s  new  ( e q n s . 2.14a & 2.14b)  r e m a i n u n c h a n g e d , w h i l e t h e q u a n t i t y p(©) i n e q u a t i o n s h o u l d be r e p l a c e d by p(9)  - 2T /R , Q  which i s the c o r r e c t i o n f o r the Laplace  pressure.  2.14c  19 R E S U L T S  2.4:  A N D  D I S C U S S I O N  The v o l u m e t r i c flow r a t e i n t o t h e p i p e t t e i s d e r i v e d from the  relation 6  7  Q  - 2ir R  =  P  X  v  where 9  (2.19)  s i n e de  0  p  i s t h e v a l u e o f 0 a t t h e p i p e t t e entrance.  Equation  2 . 1 9 can be i n t e g r a t e d u s i n g t h e formula  S to  obtain  P  (x) d x  =  i  -  (O  I n  3  o  Q  =  0  R_  The r a t e o f growth o f t h e p r o j e c t i o n i n s i d e t h e p i p e t t e i s c a l c u l a t e d from t h e v o l u m e t r i c flow r a t e :  Q  =  TT R  2 p  L  (2.21)  A l l q u a n t i t i e s i n t h e above equations a r e made d i m e n s i o n l e s s by s c a l i n g w i t h Rp, ">j , and A p.  F i g u r e 4 shows a p l o t o f t h e  d i m e n s i o n l e s s r a t e o f e n t r y as a f u n c t i o n o f t h e p i p e t t e r a d i u s . As t h e sphere r a d i u s approaches i n f i n i t y , rate,  L ^ / C A P R ^ ) ,  the dimensionless  has a l i m i t i n g v a l u e o f 0 . 2 5 .  flow  The s i m i l a r  problem o f v i s c o u s flow from an i n f i n i t e h a l f - s p a c e i n t o an orifice  i s s o l v e d by Happel & Brenner (1973)  and T o r z e r e n e t a l .  (1984) w i t h d i f f e r e n t v e l o c i t y boundary c o n d i t i o n s a l o n g t h e  20  Figure  4:  The d i m e n s i o n l e s s flow r a t e as a f u n c t i o n of the p i p e t t e r a d i u s f o r a newtonian l i q u i d d r o p l e t .  septum: the  t h e f o r m e r u s e d t h e no s l i p b o u n d a r y c o n d i t i o n  while  l a t t e r assumed t h e v e l o c i t y be d r i v e n b y a r a d i a l l y  c o n v e r g e n t membrane ( i e : r a d i a l v e l o c i t y p r o p o r t i o n a l t o 1 / r ) . Because t h e p r e s e n t problem i n v o l v e s t h e f r e e - s l i p condition  boundary  ( e q n . 2 . 1 3 ) , we e x p e c t e d t h e f l o w r a t e t o b e  greater  t h a n t h e n o - s l i p c a s e , a n d l e s s t h a n t h a t f o r t h e membrane driven  flow.  dimensionless  I n d e e d , w i t h t h e same s c a l i n g a s u s e d a b o v e , t h e flow rates are respectively  On t h e s u r f a c e  n  = R, o u r i n i t i a l  (0.212) a n d 1/2.  assumption o f n e g l i g i b l e  normal s t r e s s i n comparison t o t h e Laplace pressure the  i n t e r f a c i a l tension)  c a n now b e v e r i f i e d .  2.14c, t h e dynamic p r e s s u r e sphere.  The r e s u l t ,  appreciable  pressure  Using  (created by equation  i s c a l c u l a t e d on t h e s u r f a c e  a s shown i n f i g u r e 5, i s t h a t t h e r e difference  implies that a l l the pressure  gradients  d r o p must be c o n c e n t r a t e d  a t t h e o r i f i c e e n t r a n c e where t h e v e l o c i t y  are large.  i s no  (relative t o the external  medium) o n t h e s e g m e n t e x t e r i o r t o t h e s u c t i o n p i p e t t e .  small region  of the  This in a  22  0  AP  -4  -8  -  F i g u r e 5:  The d i m e n s i o n l e s s h y d r o s t a t i c p r e s s u r e v s . t h e c u r v i l i n e a r d i s t a n c e on t h e s u r f a c e f = R. The s h a r p d r o p i n p r e s s u r e a t e/tc = 0.83 corresponds t o the p i p e t t e contact region.  23  I I I . TWO-DIMENSIONAL MEMBRANE MECHANICS  The  theory  o f t w o - d i m e n s i o n a l t h i n s h e l l mechanics h a s been  e x t e n s i v e l y developed by Evans and S k a l a k the t h e o r e t i c a l basis o f t h i s chapter.  (1980),  which  forms  The p r i n c i p l e s w i l l be  a p p l i e d t o the axisymmetric problem o f the p i p e t t e a s p i r a t i o n o f a cortical  shell.  The f l u i d  i n v i s c i d and incompressible, pressure. can  i n s i d e the c e l l  i s assumed t o b e  r e s u l t i n g i n a uniform  M a t e r i a l p r o p e r t i e s o f the c o r t e x , on the  be q u i t e g e n e r a l .  I n a d d i t i o n t o having  internal other  hand,  an i s o t r o p i c  i n t e r f a c i a l tension that accounts f o r the c e l l ' s s p h e r i c a l shape, the c o r t e x c a na l s o be " v i s c o e l a s t i c " .  This  i s a general  d e s c r i p t i o n o f a f a m i l y o f models t h a t are c o m b i n a t i o n s o f two b a s i c i d e a l i z a t i o n s : the v i s c o u s body. sections.  These w i l l  l i n e a r e l a s t i c body a n d t h e be d i s c u s s e d  i n more d e t a i l s i n  later  Because o f t h e s h a r p bend o b s e r v e d f o r a s p i r a t e d  c e l l s a t the  edge o f t h e p i p e t t e e n t r a n c e  a n t i c i p a t e a n e g l i g i b l e bending r i g i d i t y As  linear  ( s e e f i g u r e 1 ) , we i n the c o r t i c a l  layer.  such, the problem i s reduced t o c o n s i d e r a t i o n o f forces  act only i n the surface plane. will  The d e v e l o p m e n t o f t h i s  t h u s be based on two s i m p l i f i c a t i o n s :  neglect  that  chapter  axisymmetry and the  o f b e n d i n g moments.  I n c o n t r a s t t o a t h i n sheet o f rubber, f o r example, which has  s m a l l s c a l e s t r u c t u r e even a c r o s s  the thickness dimension, a  t w o - d i m e n s i o n a l membrane c a n h a v e i s o t r o p y c h a r a c t e r i z e d o n l y i n the  surface plane.  T h o u g h i t may seem u n u s u a l a s a n e n g i n e e r i n g  24  m a t e r i a l , such a concept of  finds i t s e l f  b i o l o g i c a l membranes.  fundamental  q u i t e common i n t h e f i e l d  As an example, c o n s i d e r t h e  c o m p o n e n t o f t h e b i o l o g i c a l c e l l membrane - t h e  lipid bilayer.  I t i s composed o f t w o l a y e r s o f l i p i d m o s a i c a s  shown i n f i g u r e 6. * B e c a u s e e a c h l i p i d m o l e c u l e o n l y o c c u p i e s a n a r e a o f a p p r o x i m a t e l y 100 A , o n a c e l l u l a r s c a l e ( i e : m i c r o m e t e r s ) , t h e s u r f a c e p l a n e o f t h e b i l a y e r c a n be c o n s i d e r e d a continuum.  A c r o s s t h e t h i c k n e s s , however, t h e r e a r e p r e c i s e l y  two m o l e c u l e s .  The p r i n c i p l e s o f c o n t i n u u m  obviously invalid  i n t h i s dimension.  i s r e f l e c t e d i n t h e mechanical  mechanics a r e  This anisotropic  structure  properties of the bilayer  membrane, a s m e a s u r e d b y Kwok a n d E v a n s ( 1 9 8 1 ) . b i l a y e r s above t h e a c y l c h a i n c r y s t a l l i z a t i o n  F o r example, temperature  e x h i b i t a very strong s t a t i c resistance t o area expansions, but h a s no s u c h r e s i s t a n c e t o s h a p e c h a n g e s u n d e r c o n s t a n t a r e a . These c o n f l i c t i n g v a l u e s o f s t a t i c r i g i d i t i e s a r e u n c h a r a c t e r i s t i c o f a t h i n membrane t h a t i s t h r e e - d i m e n s i o n a l l y isotropic.  I t c a n however be r a t i o n a l i z e d by r e c o g n i z i n g t h e  discontinuity i n the third  The  assumption  dimension.  of a two-dimensional  i z a t i o n r a t h e r than a r e s t r i c t i o n .  membrane i s a g e n e r a l -  I n general, deformations of  a t h i n s h e e t c a n be e x p r e s s e d a s a s u p e r p o s i t i o n o f two fundamental  modes: a r e a d i l a t a t i o n a n d i n - p l a n e s h e a r .  The  r e s i s t a n c e o f t h e m a t e r i a l t o t h e s e t w o modes o f d e f o r m a t i o n a r e expressed n u m e r i c a l l y as t h e "moduli" o f e l a s t i c i t y  (or the  moduli of v i s c o s i t y f o r resistance t o rates of deformation).  Figure 6:  Schematic i l l u s t r a t i o n o f a s e c t i o n o f the l i p i d b i l a y e r , which comprises o f two l a y e r s o f l i p i d molecules arranged i n two-dimensional a r r a y s .  For a three dimensionally  i s o t r o p i c s h e l l m a t e r i a l , t h e moduli  a s s o c i a t e d w i t h t h e t w o modes o f d e f o r m a t i o n  are uniquely  r e l a t e d whereas i n t h e case o f a two-dimensional m a t e r i a l , t h i s r e s t r i c t i o n i s removed and t h e two m o d u l i c a n be independent o f each other  completely  ( s e e E v a n s , 1973 f o r a more d e t a i l e d  discussion).  The  f i r s t parts o f t h i s chapter  t h e o r e t i c a l aspects numerical  will  be devoted t o t h e  o f two-dimensional t h i n s h e l l mechanics.  A  algorithm that applies theprinciples t o the pipette  aspiration of a cortical  s h e l l w i l l then be d i s c u s s e d .  3.1: ANALYSIS OF STRAIN  To  analyze  t h e deformation  o f a t h i n membrane, we  c o n c e p t u a l i z e t h e i n i t i a l ( u n d e f o r m e d ) s u r f a c e a s a g r i d o f many elemental can  squares,  w i t h each r e g i o n s m a l l enough t h a t l o c a l l y i t  be t r e a t e d as a f l a t  surface.  shape o f each i n s t a n t a n e o u s  By c o m p a r i n g t h e s i z e a n d  element t o i t s i n i t i a l  c o n f i g u r a t i o n , we c a n h a v e c o m p l e t e i n f o r m a t i o n o n t h e b o d y ' s deformation  field.  Each d i f f e r e n t i a l element c a n be o r i e n t e d so  t h a t a s q u a r e maps s i m p l y state.  This e s p e c i a l l y convenient  the p r i n c i p a l coordinate (ie:  i n t o a r e c t a n g l e i n t h e deformed  system.  o r i e n t a t i o n i s s a i d t o be i n F o r an a x i s y m m e t r i c  a s u r f a c e generated by r e v o l v i n g a m e r i d i a n  axis) with deformations p r i n c i p a l coordinates  curve  surface about an  s y m m e t r i c a b o u t t h e same a x i s , t h e a r e immediately  given - they  are i n the  27 m e r i d i o n a l and t h e azimuthal  directions.  Two q u a n t i t i e s a r e needed t o r e l a t e t h e instantaneous rectangle t o the i n i t i a l ratios to  and  square.  The s i m p l e s t a r e t h e e x t e n s i o n  A$> , which a r e t h e r a t i o s o f t h e deformed l e n g t h  t h e o r i g i n a l l e n g t h i n the m e r i d i o n a l and t h e azimuthal  directions, respectively. q u a n t i t y can be expressed  In t h e case o f axisymmetry, any as a f u n c t i o n o f one s p a t i a l v a r i a b l e  alone - namely t h e c u r v i l i n e a r p o s i t i o n s o f a m a t e r i a l p o i n t along the meridian.  T h i s v a r i a b l e i n t u r n can be u n i q u e l y  r e l a t e d t o the i n i t i a l point, s . Q  c u r v i l i n e a r d i s t a n c e o f t h e same m a t e r i a l  In t h i s manner, t h e e x t e n s i o n r a t i o s a r e g i v e n by X (s ) m o' v  X  where r ( s ) and ^ ( Q  0  s 0  *  =  =  ds  (3.1) * '  o  r  r  (3.2)  Q  ) are r e s p e c t i v e l y the r a d i a l distances  from t h e a x i s o f symmetry i n t h e instantaneous  and t h e i n i t i a l  configurations.  Although of  t h e e x t e n s i o n r a t i o s p r o v i d e a complete d e s c r i p t i o n  t h e s t r a i n f i e l d , t h e r e are o t h e r deformation v a r i a b l e s t h a t  have more r e l e v a n c e t o t h e p h y s i c s o f t h e membrane m a t e r i a l . C o n s i d e r t h e u n c o u p l i n g o f a g e n e r a l deformation fundamental modes: 1) expansion changes; and 2) deformation  i n t o t h e two  o r c o n t r a c t i o n without  a t constant area.  shape  F i g u r e 7 shows  28  L  Figure 7 :  The u n c o u p l i n g o f a g e n e r a l deformation i n t o the two independent modes.  such a case i n t h e p r i n c i p a l c o o r d i n a t e system. deformation  The a c t u a l  t h a t i n v o l v e s t h e two e x t e n s i o n r a t i o s can be broken  down i n t o two s t e p s - t h e f i r s t one c h a r a c t e r i z e d by DC , t h e f r a c t i o n a l area change, and t h e second one by "A , which measures t h e amount o f e x t e n s i o n .  These deformation v a r i a b l e s  can e a s i l y be r e l a t e d t o t h e e x t e n s i o n  ratios:  a  =  A A, - 1 m <t>  _ _ (3.3)  x  = <VV*  <-> 34  OL and ^ , form a s e t o f l i n e a r l y  The two v a r i a b l e s ,  independent f u n c t i o n s (with r e s p e c t t o ^ completely  s p e c i f i e s the geometric  ) that  f e a t u r e s o f t h e deformation.  L i k e w i s e , t h e time r a t e s o f deformation the same two modes.  and  can be seperated  into  I n an E u l e r i a n f o r m u l a t i o n ( i e : w i t h t h e  i n s t a n t a n e o u s c o o r d i n a t e s as t h e r e f e r e n c e geometry), t h e r a t e o f deformation v a r i a b l e s a r e n o t t h e time d e r i v a t i v e s o f OC and 'X / b u t o f t h e i r l o g a r i t h m s  V  a  V  s  Thus, t h e deformation axisymmetric  (see Evans and Skalak,1980):  " df < ln  -  d  dF  1+a  ,  l n X  )  ( 3  '  .  5 )  <-> 3  and r a t e o f deformation  6  o f an  body i s q u a n t i t a t e d i n terms o f f o u r i n t e n s i v e  v a r i a b l e s - each o f which can be expressed  as a f u n c t i o n o f t h e  30 i n i t i a l c u r v i l i n e a r distance s  3.2:  o  BALANCE OF FORCES IN A TWO-DIMENSIONAL  MEMBRANE  In Newtonian mechanics, t h e sum o f a l l f o r c e s and t h e sum of a l l moments a c t i n g on a body must be zero i n t h e absence o f accleration.  As mentioned e a r l i e r , t h e bending moments i n t h e  c o r t i c a l l a y e r i s assumed n e g l i g i b l e i n t h i s study.  The  membrane f o r c e r e s u l t a n t s t h e r e f o r e must a c t tangent t o t h e plane of the surface.  F o r an axisymmetric problem, these  are expressed i n terms o f two t e n s i o n r e s u l t a n t s , T  m  forces  and T^ ,as  shown i n f i g u r e 8a. These a r e i n t e n s i v e q u a n t i t i e s d e f i n e d as the  force per u n i t length i n the meridional  and t h e azimuthal  directions respectively.  From t h e c h a r a c t e r i s t i c s i z e s and flow  rates of the micropipette  experiment, i n e r t i a l e f f e c t s a r e  n e g l i g i b l e i n comparison t o o t h e r f o r c e s .  Thus, mechanical  e q u i l i b r i u m r e q u i r e s t h e balance o f t h e i n t e r n a l f o r c e s i n t h e thin shell stresses  (eg: e l a s t i c and v i s c o u s  (eg: t h e s u c t i o n p r e s s u r e ) .  forces)  against the applied  In an axisymmetric  c o n f i g u r a t i o n , t h e e q u i l i b r i u m equations f o r a membrane a r e (see Evans and Skalak, 1980)  d_ ds  T. 9r  4>  T. m  R.  m  ds + as  0  (3.7a)  (3.7b)  +  an  31  Figure 8 :  The t e n s i o n r e s u l t a n t s and c o o r d i n a t e v a r i a b l e s are i l l u s t r a t e d f o r an axisymmetric geometry. (a) The t e n s i o n r e s u l t a n t s a c t o n l y i n the m e r i d i o n a l and azimuthal d i r e c t i o n s , which are a l s o the p r i n c i p a l d i r e c t i o n s . (b) D e f i n i t i o n o f t h e c o o r d i n a t e v a r i a b l e s . R_ •  •  the m e r i d i o n a l c u r v a t u r e ,  •  i s not shown.  III  32 A l l v a r i a b l e s i n equations 3.7 are as d e f i n e d that  i n f i g u r e 8.  0^ are t h e e x t e r n a l l y a p p l i e d s t r e s s e s t h a t have  and  dimensions o f f o r c e p e r u n i t area  ( u n l i k e the t e n s i o n  r e s u l t a n t s , which have dimensions o f f o r c e / l e n g t h ) . quantities  and R^ are the r a d i i o f c u r v a t u r e  meridional  and azimuthal d i r e c t i o n s .  coordinate  v a r i a b l e s by R  =  m  d  R*  where 0 i s d e f i n e d vector  The  i n the  They can be r e l a t e d t o the  6  ds  (  sin 9  (3..9)  3  <  8  )  as the angle between the outward normal  and t h e a x i s o f symmetry.  definition  Note  I t f o l l o w s from t h i s  that  S  l  n  9  "If  =  cos 0  =  d  (3-10) r  ds  (3.11)  E q u a t i o n s 3.7 are the d i f f e r e n t i a l equations o f mechanical equilibrium.  I t i s a l s o p o s s i b l e t o cumulate the a x i a l  component o f t h e e x t e r n a l s t r e s s e s and equate t o t h e tension,  meridional  r e s u l t i n g i n a s e t o f i n t e g r a t e d e q u a t i o n s which are  a l t e r n a t i v e s t o 3.7.  They a r e  T  m  =  F  _,_ 2ir r s i n 0 z  n  (3.12a)  33 F =  ^-VT sin 9  9  (c * s  -  — 2TT  —. „ r sin 9  (3.12b) ds'  where  F (s) z  Equations  ==  s 2ir S r d s ( a c o s 9 n  3.12 a r e v a l i d  - a  f o r an a x i s y m m e t r i c  g  sin9)  geometry w i t h t h e  c u r v i l i n e a r c o o r d i n a t e s o r i g i n a t i n g from a p o l e . shown t o b e e q u i v a l e n t t o 3.7. and e x t e r i o r t o t h e c o r t i c a l c a n b e no s h e a r Also,  Because t h e f l u i d s both  s h e l l a r e assumed  shell  s u b s t i t u t e i n t o 3.12.  a e  4T 4  =  -  1  4T  s where T and T  = 0. 0* i n s i d e n  With these s i m p l i f i c a t i o n s ,  force i n equation  The e q u a t i o n s  r  r  a  ,  r  d9  (rrrr-fl Si  -rtr-R  we  3.13 a n d  o f e q u i l i b r i u m t h e n become  r £ - $2) s i n 9 ds  a (3 n sin 9 ^ J  —  there  ( r e l a t i v e t o t h e e x t e r n a l medium) t h a t i s  can r e a d i l y e v a l u a t e t h e a x i a l  =  interior  inviscid,  there i s a uniform pressure  i n d e p e n d e n t o f t h e c o o r d i n a t e s.  £  T h e y c a n be  s t r e s s e s o n t h e membrane, a n d h e n c e  f o r t h e same r e a s o n ,  the c o r t i c a l  (3.13)  = ;  - D  0  (3.14a)  u  =  0  (3.i4b)  n s i n 6 ' s i n 9 ds s  a r e r e s p e c t i v e l y t h e i s o t r o p i c and d e v i a t o r i c  t e n s i o n s d e f i n e d as  T  s  X  A  (T  m  + T ) 9  (3.15)  T  The  S  S  * <m - V T  (3.16)  above equations of e q u i l i b r i u m express the b a l a n c e of  (an e x t r i n s i c q u a n t i t y ) rigidity.  The  i n a t h i n membrane t h a t has  equations have t o h o l d r e g a r d l e s s of  s t r u c t u r e of the membrane. whether the m a t e r i a l  forces  no bending the  For example, the q u e s t i o n s  of  i s i s o t r o p i c , or whether i t i s s o l i d  or  l i q u i d , are i r r e l e v a n t .  3.3: CONSTITUTIVE RELATIONS  Any  r e l a t i o n t h a t d e s c r i b e s the p r o p e r t y  be c a l l e d a c o n s t i t u t i v e r e l a t i o n .  of a m a t e r i a l  In t h i s p r e s e n t  i n t e r e s t e d i n mathematical f u n c t i o n s t h a t r e l a t e the  can  work, we  are  intensive  deformation v a r i a b l e s ( t>C , /\) t o the i n t e n s i v e s t r e s s v a r i a b l e s ( T , T ) a t constant g  material  temperature.  i s revealed  e l a s t i c s o l i d i s one  i n these r e l a t i o n s . t h a t has  f o r c e s t h a t can be represented function).  Much i n f o r m a t i o n  conservative  about  For i n s t a n c e ,  the an  internal forces (ie:  as the g r a d i e n t  of a s c a l a r  E l a s t i c s t r e s s e s can o n l y depend on the body's  d e f o r m a t i o n and  not on i t s time r a t e .  A l i q u i d , on the  other  hand, i s c h a r a c t e r i z e d by i t s i n a b i l i t y t o s u s t a i n shear s t r e s s e s i n a s t a t e of r e s t . however, t h e r e  In the p r o c e s s of deforming,  i s i n e v i t a b l y i n t e r n a l molecular f r i c t i o n  s t r u c t u r a l changes t h a t appear m a c r o s c o p i c a l l y t o flow.  The  stresses of a viscous  and  as a r e s i s t a n c e  l i q u i d w i l l t h e r e f o r e depend  o n l y on the r a t e o f deformation v a r i a b l e s .  35  The s i m p l e s t way  t o model s o l i d and l i q u i d b e h a v i o u r s i s t o  take the f i r s t o r d e r approach.  For an e l a s t i c s o l i d undergoing  i s o t h e r m a l deformations, the mechanical work done can be shown to the  be e q u i v a l e n t t o the change i n the Helmholtz f r e e energy of body (the body i s c o n s i d e r e d here as a c l o s e d system).  It  f o l l o w s from the d e f i n i t i o n o f mechanical work t h a t the s t r e s s e s are  the d e r i v a t i v e s o f the Helmholtz f r e e energy w i t h r e s p e c t t o  the  corresponding s t r a i n v a r i a b l e s .  To o b t a i n a f i r s t o r d e r  s t r e s s - s t r a i n r e l a t i o n , the f r e e energy i s w r i t t e n as a T a y l o r expansion i n terms of the deformation v a r i a b l e s up t o the q u a d r a t i c terms the  (see Evans and Skalak, 1 9 8 0 ) .  In t h i s manner,  e l a s t i c s t r e s s e s are g i v e n by  T  T  e  e s  =  T  +  Q  K  - xX  a  (  -,-2  2  = 'A u. ( ~ ) 1 + X  v  a  effects.  K and^  a r i s e from  >  1  7  )  (3.18)  >  where t h e s u p e r s c r i p t s "e" denote e l a s t i c s t r e s s e s . c o n s t a n t i s o t r o p i c t e n s i o n which may  3  T  Q  is a  interfacial  are r e s p e c t i v e l y the i s o t r o p i c and shear  moduli o f e l a s t i c i t y .  For a t h r e e d i m e n s i o n a l l y  isotropic  m a t e r i a l , i t can be shown t h a t K = 3y£* , whereas f o r a m a t e r i a l t h a t i s a n i s o t r o p i c i n the t h i c k n e s s dimension, t h e s e two moduli are  completely unrelated.  The v i s c o u s f o r c e s i n a l i q u i d a r i s e from i n t e r n a l  friction  and  heat d i s s i p a t i o n t h a t are thermodynamically  Because of t h e i r c o m p l e x i t i e s , these  non-conservative  b e s t d e s c r i b e d by p h e n o m e n o l o g i c a l r e l a t i o n s . such r e l a t i o n s  To  forces  first  are  order,  are  -v T v  T  = =  V  K  V  (3.19) a  2 T) V  (3.20)  s where t h e  irreversible.  s T<- and ~*f  s u p e r s c r i p t s "v" d e n o t e v i s c o u s s t r e s s e s .  are r e s p e c t i v e l y the  i s o t r o p i c and  shear moduli of  viscosity.  L i k e t h e e l a s t i c s o l i d , t h e m o d u l i o f v i s c o s i t y a r e r e l a t e d by 3:1  r a t i o for a three-dimensionally  unrelated i n the two-dimensional  i s o t r o p i c m a t e r i a l , and  are  case.  I n r e a l i t y , m o s t m a t e r i a l s do n o t e x h i b i t p u r e l y e l a s t i c purely viscous characteristics. i n t r o d u c e d c a n be  instantaneous  two  first  order  deformation  l i q u i d t h a t has  or  models  "extremes"  of  1) t h e e l a s t i c s o l i d t h a t i s c a p a b l e  of  when s u b j e c t t o i m p u l s i v e f o r c e s , w i t h  m e c h a n i c a l work f u l l y r e c o v e r e d viscous  The  t h o u g h t o f as t h e i d e a l i z e d  material behaviours:  a  upon u n l o a d i n g ,  and  2)  a f i n i t e r a t e of deformation  the  and  with  no  t e n d e n c y o f r e t u r n i n g t o t h e i n i t i a l c o n f i g u r a t i o n upon removal of external forces. irrecoverable.  Work done b y  T h e s e two  first  a viscous l i q u i d  order  i n p r i n c i p l e t h e r e c a n be  an  completely  i d e a l i z a t i o n s can  combined t o model " i n t e r m e d i a t e " cases of mixed Although  is  be  behaviours.  i n f i n i t e number o f  combinations, well  we  will  i n t r o d u c e t h r e e models t h a t account  f o r most m a t e r i a l b e h a v i o u r s .  i l l u s t r a t e d metaphorically t h e e l a s t i c c o m p o n e n t and counterpart.  In the  first  e l a s t i c element i s placed represent  solids  dissipation.  The  two  models  a p i s t o n t o denote the case,  viscous  the s o - c a l l e d V o i g t model,  internal  the m a t e r i a l i s capable  energy of r e t u r n i n g to  c h a r a c t e r i z e s i t as a s o l i d ) , t h e l o a d - u n l o a d  a l w a y s show h y s t e r e s i s due  to viscous dissipation.  m o d e l , t h e t o t a l f o r c e i s t h e sum  T  =  T  =  and  the deformation  e  +  T  T  =  X  =  a  internal  curve For  will  this  +  V  (3.21)  T v  (3.22)  i s b y d e f i n i t i o n common t o b o t h  a  forces  o f t h e two c o n t r i b u t i o n s :  elements:  =  a  (3.23)  =  X  (3.24)  U n l i k e t h e V o i g t m a t e r i a l , w h i c h c a n be essentially  an  i n p a r a l l e l w i t h a v i s c o u s element t o  t h e o r i g i n a l c o n f i g u r a t i o n upon r e m o v a l o f e x t e r n a l (which  are  i n f i g u r e 9, u s i n g a s p r i n g t o d e n o t e  t h a t have a p p r e c i a b l e  Although  first  very  considered  as  a s o l i d , t h e r e a r e o t h e r s u b s t a n c e s w h i c h show  s t r e s s r e l a x a t i o n mechanisms.  c h a r a c t e r i z e d b y two  observations:  These m a t e r i a l s  are  1) an e l a s t i c r e s p o n s e t o  38  (a) Figure 9:  (b)  Metaphorical representations of v i s c o e l a s t i c models u s i n g c o m b i n a t i o n s o f s p r i n g s ( e l a s t i c component) and d a s h p o t s ( v i s c o u s component). (a) t h e V o i g t m o d e l ; (b) t h e M a x w e l l m o d e l .  i m p u l s i v e f o r c e s , and shear s t r e s s e s .  m o d e l , a s shown i n f i g u r e 9b.  ~  T  and t h e t o t a l  =  s  T  elements:  (3.25)  T  =  e s  T  (3.26)  V s  r a t e o f d e f o r m a t i o n i s d e f i n e d a s t h e sum  individual  of  the  parts:  v  s  V  =  +  V  s  < ' >  V  3  F o r t h e g e n e r a l f i r s t o r d e r c o n s t i t u t i v e r e l a t i o n s g i v e n by to  3.20,  the r a t e of deformation v a r i a b l e s f o r a Maxwell  c a n be e x p r e s s e d  =  at  in X d t  ;  =  2(T a 2  28  3.17  model  i n terms o f t h e s t r e s s e s as  d ln(l+a)  3  a  Because of  c o u p l i n g , t h e s t r e s s e s a r e common t o b o t h  T  two  flow behaviour under constant  S u c h m a t e r i a l s c a n a d e q u a t e l y be d e s c r i b e d b y  f i r s t order Maxwell the s e r i a l  2) l i q u i d  1 9T a dt  1 2  2  2  (  S  (3.29)  K  3T  ( T  + K u )*  _T_ —  9 t  dT _ s a t l  T _s 2T  »  (3  *  30)  s where  o  =  T - T  o  + K  (3.31)  40  The r a t i o  of the viscous  Maxwell model d e f i n e  are  I n p a r t i c u l a r , t h e model p r e d i c t s  response t o forces that  much s h o r t e r  l a r g e compared t o X .  that  shear.  stress  i s characterized In t h i s  ( o r t h e Bingham  by a s t r e s s m a g n i t u d e c a l l e d t h e  shear. flow  under  Beyond t h i s p o i n t ,  the  with the rate of deformation  f o r a two d i m e n s i o n a l l y  incompressible  plastic i s  T  | T  T  s'  s  equation sign  i s another  t o t h e amount o f s t r e s s i n e x c e s s o f t h e y i e l d .  The c o n s t i t u t i v e r e l a t i o n  where  constant  c a s e , t h e body b e h a v e s e l a s t i c a l l y  b e g i n s t o show l i q u i d  proportional  that  liquid.  by a t i m e  - the v i s c o p l a s t i c material  l e v e l s below t h e y i e l d  material  ideal  i s characterized  t o i t s i n t e r n a l stress relaxation, there  material)  duration  t h a t the r e l a x a t i o n time  t h e M a x w e l l model becomes an i d e a l  class of semisolid  yield  f o r a time  an  response t o durations  i n the l i m i t  W h i l e t h e M a x w e l l body related  are applied  t h a n t , and a l i q u i d  approaches zero,  in a  a r e l a x a t i o n time r that determines i t s  semi-solid behaviour. elastic  modulus t o t h e e l a s t i c m o d u l u s  as T  =  14  ~ *s  (X  M  =  2  2  77  i s the y i e l d  ~ X" )  ;  2  | V  s'  '*  |TJ < T  | T  s'  >  *s  (3.32b)  shear which i s a p o s i t i v e q u a n t i t y .  3.32b, t h e r a t e o f d e f o r m a t i o n V .  (3.32a)  e  The e l a s t i c  g  In  i s t o h a v e t h e same  c o e f f i c i e n t U. c a n be made v e r y  large to  account  f o r t h e n e g l i g i b l e y i e l d s t r a i n i n most  materials.  In the l i m i t the y i e l d shear vanishes, the  p l a s t i c becomes a  3.4:  THE  plastic ideal  liquid.  L I N E A R Q U A S I — E L A S T I C SOLUTION  I n t h e p r e v i o u s s e c t i o n s , we h a v e i n t r o d u c e d e q u a t i o n s t h a t d e s c r i b e t h e t i m e e v o l u t i o n o f a two external stresses.  d i m e n s i o n a l membrane u n d e r  In p a r t i c u l a r , the balance of  f o r c e s , a s e x p r e s s e d i n e q u a t i o n s 3.14, q u a n t i t i e s T and T  g  mechanical  m u s t be s a t i s f i e d .  The  i n these equations are c h a r a c t e r i s t i c of the  m a t e r i a l model chosen, w i t h t h e c o n s t i t u t i v e r e l a t i o n s o u t l i n e d i n e q u a t i o n s 3.17  t o 3.32.  The  as  d e f o r m a t i o n and r a t e  of  deformation v a r i a b l e s i n the c o n s t i t u t i v e r e l a t i o n s are i n turn quantified  i n 3.1  t o 3.6.  The  geometric features of the  d e f o r m i n g body a r e e x p r e s s e d i n terms o f t h e c o o r d i n a t e v a r i a b l e s i n e q u a t i o n s 3.8 a s p i r a t i o n of a c o r t i c a l be a c c o u n t e d  t o 3.11.  shell,  To m o d e l t h e  pipette  a l l t h e above e q u a t i o n s have t o  f o r o v e r t h e e n t i r e s u r f a c e a t any  i n s t a n t of time.  For a g e n e r a l c o n s t i t u t i v e model, i t i s not s u r p r i s i n g t h a t o n l y approximate  s o l u t i o n s c a n be o b t a i n e d b y n u m e r i c a l means.  However, a c l o s e d form s o l u t i o n does e x i s t i n t h e c a s e membranes w i t h no r e s i s t a n c e shear.  of  ( s t a t i c o r dynamic) t o i n - p l a n e  Here, because the shear r e s u l t a n t v a n i s h e s , the  of the c e l l  portion  e x t e r i o r t o t h e p i p e t t e must r e m a i n s p h e r i c a l  e q u a t i o n 3.14b).  I t follows that the i s o t r o p i c tension,  (see and  h e n c e o£ , i s u n i f o r m o v e r t h e d e f o r m i n g s u r f a c e a t any t i m e .  A  d e t a i l e d o u t l i n e of the s o l u t i o n t o t h i s i s o t r o p i c problem given i n appendix  B.  is  T h e s e r e s u l t s c a n be u s e d t o v e r i f y  v a l i d i t y of the numerical approach  the  t o t h e more g e n e r a l c a s e ,  where t h e s h e a r v i s c o s i t y and t h e s h e a r e l a s t i c i t y c a n be  non-  zero.  We  will  now  d e s c r i b e t h e n u m e r i c a l method f o r s o l v i n g  general c o r t i c a l  s h e l l p r o b l e m w i t h b o t h i s o t r o p i c and i n - p l a n e  shear r e s u l t a n t s . i s approximated  Here, t h e time e v o l u t i o n o f t h e f l o w p r o c e s s  by a s e r i e s o f s m a l l d i s p l a c e m e n t s , each o v e r a  short time i n t e r v a l At.  A t each time s t e p , t h e v i s c o u s  component o f t h e c o n s t i t u t i v e r e l a t i o n i s t r e a t e d e l a s t i c a l l y " by d i v i d i n g t h e v i s c o u s m o d u l i by A t . resulting solution,  t h e end o f t h e t i m e i n t e r v a l .  The  The  the c e l l  L e t t h e shape a t t i m e t  s p e c i f i e d by c y l i n d r i c a l c o o r d i n a t e s ( r , z ) , (r',z').  "quasi-  i n t h e form o f a d i s p l a c e m e n t f i e l d ,  added t o t h e o r i g i n a l geometry t o approximate  t + A t by  the  i s then  shape a t be  and t h a t a t t i m e  i n c r e m e n t i n g f u n c t i o n s a r e d e f i n e d as  Ar(s )  =  r'  - r  (3.33a)  Az(s )  =  z«  - z  (3.33b)  Q  Q  All  geometrically related variables  ( e g : Tin, s i n 6, R , e t c )  now  be w r i t t e n i n t h e same m a n n e r ; i e : f o r a n y s u c h q u a n t i t y  let  x* be  m  i t s v a l u e a t t i m e t + A t , x be t h e v a l u e a t t , a n d  can x,  43  I n g e n e r a l , A x c a n be expanded and Az. the  F o r s m a l l 4t,  s e r i e s i n terms o f A r  t h e problem i s l i n e a r i z e d by t r u n c a t i n g  series after the f i r s t  The  i n a Taylor  powers.  r a t e s o f s t r a i n a r e approximated by d i v i d i n g t h e  increments by A t .  F o r example,  the dilatory strain rate  , which  was p r e v i o u s l y w r i t t e n a s  v  =  a  5a  1 1+a  at  c a n now b e a p p r o x i m a t e d b y V where  A a  a  (l+a)At  ADC i s l i n e a r i n A r ( s ) a n d A z ( s ) . o o  With these  approximations, c o n s t i t u t i v e r e l a t i o n s that involve now become q u a s i - e l a s t i c . tension  For instance,  i n a V o i g t m a t e r i a l a t time t+At, T*  =  K (a + Aa)  =  K  =  T  a  l e t T* b e t h e i s o t r o p i c then  V  K  K +  +  +  +  viscosities  Aa  (1+a)At  AT  w h e r e T = Koi i s t h e c u m u l a t e d e l a s t i c s t r e s s u p t o t i m e t , a n d AT =  K  +  (i+a) A t ]  A  a  *  s  t  h  e  i  n  c  r  e  i  n  e  n  t  i  n  stress resultant  l i n e a r i n A r and Az) due t o p e r t u r b a t i o n  i n geometry.  e f f e c t i v e e l a s t i c m o d u l u s i s K+ (2.+a)At ' ^ - ' w h  i s dominated by t h e q u a s i - e l a s t i c term.  c h  f  o  r  (also The  small At,  44  We  see  writing an  that the  i n t e r v a l At.  amount due The  Q  The  basic perturbation  residual functions,  also written  i n the  &  =  1  a  the  e v e r y w h e r e on  e  a  s  + Ae  3.14,  + Ae.  s  =  v  0  the  cell  surface.  (3.34a) ' ,„ (3.34b)  zero,  we  N o t e t h a t by  are  satisfying  using  linear  q u a s i - e l a s t i c approach:  geometry.  of a s p h e r i c a l s h e l l  non-zero shear v i s c o s i t y , t h e volume c o n s e r v a t i o n incompressibility  however, i s n o t  there  requirement.  perturbed  For  the  force the  the  starting  To  balance  reason  for  pipette geometry)  avoid v i o l a t i n g  This an  from which the  with  with the  e q u i l i b r i u m e q u a t i o n s must sphere.  hold  the  solution consistent  arbitrary since i t defines  time i n t e r v a l At,  calculated.  i s no  requiring  Here l i e s  ( i e : the  condition, the  applied to a s l i g h t l y  are  i n equations  = 0  a  perturbed  i n the  the  and  Q  i n the  field  time  only  functions Ar(s )  condition  aspiration  over a  to contain  as d e f i n e d  plus  e q u a t i o n s o f m e c h a n i c a l e q u i l i b r i u m t h a t must  p r i m e d q u a n t i t i e s be  the  of  same manner:  e  These are  linearized  way  at time t ,  to geometric perturbation  increments are  powers o f t h e  AZ(S ). are  scheme i s a s y s t e m a t i c  a l l q u a n t i t i e s i n terms of t h e i r v a l u e s  incremental  first  numerical  be  perturbation, average v e l o c i t y viscous  stresses  45  The  o b j e c t o f t h e q u a s i - e l a s t i c approach i s t o o b t a i n  incrementing  f u n c t i o n s (Ar and Az) f o r t h e e n t i r e c e l l  T h i s i s done by s e p e r a t i n g t h e c e l l  i n t o two r e g i o n s :  surface. For the  p o r t i o n e x t e r i o r t o t h e s u c t i o n p i p e t t e , equations  3 . 3 4 a r e used  to  The method t o  s o l v e f o r t h e two f u n c t i o n s A r ( s ) and A z ( s ) . * o o  do t h i s w i l l  be d i s c u s s e d i n t h e next s e c t i o n .  The problem  i n s i d e t h e p i p e t t e i s much s i m p l e r because A r = 0 . can be seen from equation a l o n g t h e tube. equation  entrance, cell  3 . 7 a t h a t t h e a x i a l t e n s i o n i s uniform  There i s t h e r e f o r e only one f o r c e  ( i n t h e a x i a l d i r e c t i o n ) from which  evaluated.  Further, i t  AZ(S ) q  balance can be  The two s o l u t i o n s a r e matched a t t h e p i p e t t e s u b j e c t t o t h e c o n s t r a i n t o f volume c o n s e r v a t i o n .  The  shape (r,z) a t time t i s then incremented and t h e procedure  i s repeated  again.  In t h i s manner, t h e v i s c o u s flow i s  approximated by a s e r i e s o f s m a l l q u a s i - e l a s t i c displacements. 3.5:  IMPLEMENTATION OF NUMERICAL METHOD  So f a r , a l l t h e v a r i a b l e s we have i n t r o d u c e d  (eg: t e n s i o n s ,  c u r v a t u r e s , e x t e n s i o n r a t i o s , etc.) a r e continuous  i n space.  Because t h e problem i s axisymmetric, they can be w r i t t e n as f u n c t i o n s o f the instataneous  c u r v i l i n e a r d i s t a n c e s alone.  a l t e r n a t i v e approach, known as t h e Lagrangian express a l l q u a n t i t i e s ( i n c l u d i n g s i t s e l f ) initial,  formulation, i s to  i n terms o f t h e  time-independent c u r v i l i n e a r d i s t a n c e S . T h i s can be q  done because "mapping" between t h e instantaneous initial  An  space a r e assumed t o be one-to-one.  space and t h e  The next s t e p i s t o  d i s c r e t i z e the s  Q  space i n t o a number of g r i d p o i n t s .  continuous f u n c t i o n s w i l l now  be r e p r e s e n t e d by a s e r i e s of  nodal v a l u e s l o c a t e d along the m e r i d i o n a l curve. d e r i v a t i v e with respect to S  q  All  i s approximated  A  first  by d i v i d i n g the  d i f f e r e n c e between two a d j a c e n t nodal v a l u e s by the l o c a l size.  Such i s the s i m p l e s t form of numerical  c a l l e d t h e f o r w a r d - d i f f e r e n c e formula.  grid  differentiation  In d e a l i n g w i t h  c u r v a t u r e s , i t i s necessary t o e v a l u a t e second d e r i v a t i v e s of the c o o r d i n a t e v a r i a b l e s .  These q u a n t i t i e s can be r e p r e s e n t e d  by a p p l y i n g the forward d i f f e r e n c e formula t o the d e r i v a t i v e s of the v a r i a b l e s . needed t o approximate  first  A d d i t i o n a l equations are then  the f i r s t d e r i v a t i v e s themselves.  Thus,  by r e s t r i c t i n g a l l d e r i v a t i v e s t o be a t most f i r s t o r d e r (at the expense o f a d d i t i o n a l e q u a t i o n s ) , we e q u i l i b r i u m equations  can c a s t the  two  (eqns. 3 . 3 4 ) i n t o a f i n i t e d i f f e r e n c e  t h a t i n v o l v e nodal v a l u e s of o n l y two n e i g h b o r i n g p o i n t s . w i l l be summarized i n appendix  C, t h e r e are f o u r FDEs  d i f f e r e n c e equations) i n t o t a l : two to  unknowns a t each node are Ar, Az, ^ i r and 5 f A z . 4  The  As  (finite  from e q u a t i o n s 3 . 3 4 and  r e p r e s e n t the f i r s t d e r i v a t i v e s o f Ar and Az.  form  two  four  Because the  FDEs are w r i t t e n between two n e i g h b o r i n g p o i n t s , l o c a l l y t h e r e a r e f o u r e q u a t i o n s t h a t i n v o l v e e i g h t unknowns ( f o u r nodal v a l u e s a t each p o i n t ) .  Futher, because the e q u a t i o n s are l i n e a r  i n t h e i n c r e m e n t i n g f u n c t i o n s Ar and Az, they can be put i n t o a m a t r i x form.  For a g r i d of N p o i n t s , t h e r e w i l l be N - l s e t s of  f o u r a l g e b r a i c equations i n e i g h t unknowns.  These equations can  be s o l v e d g l o b a l l y i f the f o u r boundary c o n d i t i o n s (the number  o f b o u n d a r y c o n d i t i o n s must e q u a l in.  t h e number o f FDEs) a r e p u t  The m a t r i x o r g a n i z a t i o n f o r a g e n e r a l  equations  i s shown i n f i g u r e 10.  For c l a r i t y ,  a g r i d w i t h o n l y t h r e e nodes, a l t h o u g h implementation,  set of M difference t h i s i s done f o r  i n the actual  N i s t y p i c a l l y s e v e r a l hundred.  N  i s the  number o f b o u n d a r y c o n d i t i o n s a t t h e f i r s t n o d e , a n d N for  t h e end node.  c  + N  be a s l a r g e as 500.  =  M  (1986),  matrix  In our case,  M=4  and N c a n  I t i s obviously impractical to solve a set  o f 2000 l i n e a r e q u a t i o n s  al.  s  r e q u i r e d f o r a unique s o l u t i o n .  d i r e c t means.  i s that  The c o n d i t i o n o f N  is  g  by G a u s s i a n e l i m i n a t i o n o r by any  A p p e n d i x D d e s c r i b e s a m e t h o d , due t o P r e s s  that takes  i n f i g u r e 10.  full  3.6: RESULTS AND  c o d e d i n FORTRAN l a n g u a g e t h a t  a l s o be  given.  DISCUSSION  The q u a s i - e l a s t i c m e t h o d c a n b e a p p l i e d t o a n y m a t e r i a l model d i s c u s s e d flow behaviour  et  advantage o f t h e s p a r s i t y o f t h e  Subroutines  implement t h e s e methods w i l l  other  i n s e c t i o n 3.3.  of a liquid shell w i l l  shell  As a f i r s t  step, the  be i n v e s t i g a t e d .  The  a l g o r i t h m i s s e t up f o r t h e c o n s t i t u t i v e r e l a t i o n s g i v e n i n equations  3.19 a n d 3.20, w h e r e  parameters.  X. a n d ^  are the viscous  By n o r m a l i z i n g w i t h t h e f o l l o w i n g t h r e e q u a n t i t i e s :  the d i l a t o r y v i s c o s i t y  X. , t h e s u c t i o n p r e s s u r e  p i p e t t e r a d i u s R , t h e p r o b l e m has o n l y two parameters.  They a r e ^/X.  and  P, a n d t h e  dimensionless  where ^ i s t h e d e v i a t o r i c  48  N xM c  M x 2M  M x 2M Ns*M  F i g u r e 10:  Global matrix s t r u c t u r e f o r a s e t of M f i n i t e d i f f e r e n c e equations. The dimensions o f each b l o c k a r e as l a b e l l e d . For s i m p l i c i t y , t h i s i s done f o r a g r i d w i t h o n l y t h r e e nodal p o i n t s . M a t r i x elements t h a t l i e o u t s i d e o f t h e b l o c k s are z e r o s .  49 v i s c o s i t y and  R the i n i t i a l  c e l l radius.  The  v a l i d i t y of  the  numerical method i s then checked by comparing the r e s u l t s w i t h -y = 0  t h a t of the a n a l y t i c a l s o l u t i o n f o r the case appendix B).  Figure  (see  11 shows the time e v o l u t i o n of the  p r o j e c t i o n i n s i d e the tube f o r two  cell  different pipette r a d i i .  i n s t a n t a n e o u s v e l o c i t y f i e l d s are shown i n f i g u r e 12.  The  These  p l o t s c l e a r l y show the agreement of the r e s u l t s o b t a i n e d  by  t h r e e e n t i r e l y d i f f e r e n t means (the a n a l y t i c a l r e s u l t s a c t u a l l y i n v o l v e two not  methods).  Another f e a t u r e of the numerical method  shown here i s the t h a t by s e t t i n g  yj/-*-  t o zero,  the  shape remains p e r f e c t l y s p h e r i c a l , even a f t e r 30 t o 40 steps.  I t i s a l s o i n t e r e s t i n g t o l o o k a t how  the  ^ / K .  w i t h a c t u a l data r e d u c t i o n ,  time  initial  r a t e s v a r y w i t h p i p e t t e r a d i u s , as shown i n f i g u r e 13 d i f f e r e n t v a l u e s of  cell  for  Note t h a t i n o r d e r t o be  consistent  which i s p l o t t e d as L/R  vs. R P  we  are n o r m a l i z i n g  flow  /R, P  the flow r a t e s w i t h a time c o n s t a n t  defined  as (C  T  =  -P-R  s i n c e i t i s the c e l l r a d i u s R t h a t remains unchanged as parameter R /R  varies.  small shearing  s t r e s s e s can have l a r g e e f f e c t s on the o v e r a l l  flow r a t e s .  The  From t h i s p l o t , i t i s e v i d e n t  the  that very  c e l l shapes are a l s o s t r o n g l y a f f e c t e d by  shear v i s c o s i t y , as shown i n f i g u r e 14  f o r R /R  = 0.4.  the  It is  P  seen from t h e s e computed shapes t h a t c e l l f l a t t e n i n g towards the p i p e t t e i s i n d i c a t i v e of the presence of s h e a r i n g  stresses.  50  F i g u r e 11:  Comparing n u m e r i c a l r e s u l t s t o the a n a l y t i c a l s o l u t i o n of the time e v o l u t i o n of a surface flow process.  51  Figure 12(a):  Comparing the n u m e r i c a l l y c a l c u l a t e d r a d i a l v e l o c i t y f i e l d to the a n a l y t i c a l s o l u t i o n . S i s the p o s i t i o n at the p i p e t t e entrance.  52  Figure 12(b):  Comparing the n u m e r i c a l l y c a l c u l a t e d a x i a l v e l o c i t y f i e l d to the a n a l y t i c a l s o l u t i o n . S i s the p o s i t i o n at the p i p e t t e entrance.  53  Figure  13:  The i n i t i a l i n - f l o w r a t e i s p l o t t e d a s a f u n c t i o n of the dimensionless p i p e t t e radius s e v e r a l v a l u e s o f ">7 / x .  for  54  Figure  14:  C e l l s h a p e s p r e d i c t e d by t h e n u m e r i c a l m e t h o d for various values of . The i n i t i a l c e l l r a d i u s i s 2.5 R , and t h e c e l l p r o j e c t i o n s a r e 3 R i n a l l three cases.  55  IV.  The  SUMMARY AND  DISCUSSION  w o r k i n t h i s t h e s i s i s an a t t e m p t  to better  t h e p i p e t t e a s p i r a t i o n of a l i q u i d body w i t h a cortical  shell  i n t h e low Reynolds  p r e s e n t e d as a mathematical  differentiated  number s i t u a t i o n .  problem,  properties of  As m e n t i o n e d i n t h e  i n t r o d u c t i o n , b e c a u s e t h e s e c e l l s do n o t show any to  Although  the a n a l y s i s i s motivated  by our i n v e s t i g a t i o n i n t o t h e m e c h a n i c a l granulocytic white blood c e l l s .  understand  elastic  d e f o r m a t i o n , t h e y c a n be t r e a t e d , t o f i r s t o r d e r , a s  bodies.  B a s e d on b i o l o g i c a l c o n s i d e r a t i o n s , one  p o s s i b l e r e g i o n s i n which d i s s i p a t i o n may interior.  Two  model t h e c e l l entirely  limit  liquid  anticipates  two  d i f f e r e n t l e v e l s of viscous  occur: they are i n the c e l l continuum mechanical  p e r i p h e r y and  problems are posed t h a t  as h a v i n g i t s v i s c o u s d i s s i p a t i o n  i n e a c h o f t h e two  regions.  i n the  Only  first  c o n s t i t u t i v e r e l a t i o n s are used i n both cases.  dominated order These  are  equations t h a t r e l a t e the viscous s t r e s s e s to the rates of d e f o r m a t i o n by a c o n s t a n t o f p r o p o r t i o n a l i t y .  For the  m o d e l , t h i s amounts t o s o l v i n g t h e f a m i l i a r c r e e p i n g equations  i n a sphere  conditions.  subject to prescribed stress  interior  motion  boundary  N o t a b l e r e s u l t s a r e t h e v a r i a t i o n i n f l o w r a t e s as  a f u n c t i o n o f p i p e t t e r a d i u s a s shown i n f i g u r e 5, a n d  the  t h a t the net pressure drop i s concentrated i n the v i c i n i t y the o r i f i c e entrance.  As s u c h ,  fact of  flow r a t e s i n t o the tube w i l l  i n s e n s i t i v e t o t h e geometry o f t h e segment e x t e r i o r t o  the  p i p e t t e - implying the p o s s i b i l i t y of representing the  flow  be  56  p r o c e s s as  a pressure drop at the p i p e t t e entrance t h a t  proportional  t o the  problem, viscous i n s i d e the  in-flow rate.  d i s s i p a t i o n (and  In s o l v i n g the  the  suction pipette i s neglected.  f a c t t h a t the  s h o u l d be driven  This  can  equations i n s i d e the boundary c o n d i t i o n s  be  the  assumption,  experimental  observation  the p i p e t t e w a l l ) ,  e s p e c i a l l y i n c a s e s o f membrane  done by  cylinder with on  This  s l i d e s f r e e l y on  examined c r i t i c a l l y  flows.  The  cell  droplet  hence the p r e s s u r e drop)  a l t h o u g h a p p e a r s r e a s o n a b l e b a s e d on (ie:  surface  s o l v i n g the  creeping  motion  e i t h e r s t r e s s - or v e l o c i t y adjacent to the p i p e t t e w a l l .  c o r t i c a l model i s f u n d a m e n t a l l y d i f f e r e n t from  d r o p l e t model because energy d i s s i p a t i o n i s c o n f i n e d surface layer  is  plane.  By  neglecting  ( a s j u s t i f i e d by  f i g u r e 1), the  tension  deforming surface,  the  b e n d i n g moments i n t h e  within  thus s i m p l i f y i n g the  cortical  equations of  These e q u a t i o n s a r e  solved  the mechanical by  q u a s i - e l a s t i c n u m e r i c a l method w h i c h a c c o u n t s f o r t h e s t r e s s e s b o t h i n s i d e and  e x t e r n a l to the  S a t i s f a c t o r y agreement i s o b t a i n e d and  the  The  suction  between the  a n a l y t i c a l s o l u t i o n f o r the  11,12,13).  case T  g  = 0  s i t u a t i o n s where shear v i s c o s i t y can consequence of s u r f a c e  flow  magnitudes of  shear occur  be  a  viscous  pipette.  numerical r e s u l t s (see  figures  numerical procedure i s then extended t o  in-plane  the  s h a r p b e n d a r o u n d t h e p i p e t t e edge i n  r e s u l t a n t s must a c t t a n g e n t t o  equilibrium considerably.  the  non-zero.  A  i n t o a tube i s that very  flow  kinematic large  ( i e : s q u a r e s become h i g h l y  extended r e c t a n g l e s ) , e s p e c i a l l y i n regions  near the  pipette  57 entrance.  T h i s i s r e f l e c t e d i n t h e s i g n i f i c a n t changes i n f l o w  rates with the introduction of very  small values  of shear  v i s c o s i t y i n t o t h e c o n s t i t u t i v e r e l a t i o n s , a s shown i n f i g u r e 13.  The f l o w r a t e s i n f i g u r e 5 a n d 13 a r e n o r m a l i z e d  t o be  c o n s i s t e n t w i t h a c t u a l data r e d u c t i o n , which i s p l o t t e d as d(L/Rp)/d(tAP).  I n comparing t h e two t h e o r e t i c a l p l o t s ,  i t is  e v i d e n t t h a t t h e e n t r y f l o w r a t e s f o r t h e d r o p l e t model has l e s s variation with pipette radius.  T h i s may b e a t t r i b u t e d t o t h e  l o a c l i z a t i o n of the viscous d i s s i p a t i o n region near the o r i f i c e entrance,  which, f o r small values  t o t h e e x t e r n a l boundaries.  o f R /R, becomes  insensitive  I t i s therefore possible to  i d e n t i f y t h e dominant r e g i o n o f v i s c o u s d i s s i p a t i o n by performing  aspiration tests with different sized pipettes.  i n f o r m a t i o n c a n be combined w i t h u l t r a s t r u c t u r a l e v i d e n c e i n c r e a t i n g a more c o m p l e t e p i c t u r e o f b i o l o g i c a l  cells.  This  58  APPENDIX A: SOLUTION TO CREEPING MOTION EQUATIONS  I n t h i s a p p e n d i x , we p r o p o s e t o s o l v e t h e c r e e p i n g  flow  equations  v  2 -•> (v x v )  =  (2.10)  0  i n t h e case o f a s p h e r i c a l newtonian d r o p l e t s u b j e c t t o t h e s t r e s s boundary  conditions  oPP  (C)  v  p=R  % B CO  The  =  w  =  p=R  (2.12)  «(f) 0  (2.13)  a p p l i e d s t r e s s o( ( L,) i s d e f i n e d a s AP  ;  -1 < f < C P  /  p p  *  which represents a p i p e t t e suction pressure. axisymmetric and B r e n n e r  s o l u t i o n t o equation (1973).  (2.11)  P  The g e n e r a l  2.10 i s d e v e l o p e d b y H a p p e l  I n s p h e r i c a l components, t h e v e l o c i t y  f i e l d s a r egiven by  v  o  -  " I < n » ~ n=2 A  n  2 +  C  n ^  P  n-1<£>  (A.l)  59  V  6  =  1 [ n=2  n  A  n  p I 1  where t h e c o n s t a n t s A  "  2  <  +  and  C  n  n+2  >  C  equation  obtained.  (say  Gagenbaur  By c o m b i n i n g t h i s  field  i n v o l v e s t h e g r a d i e n t o f p, t h e  o n l y be d e t e r m i n e d up  solution is  pressure  t o an a d d i t i v e c o n s t a n t  ff ) : D  =  P  . „ \ f 2(2n+l) 2 [ "Vr n=2 77  The  v i s c o u s s t r e s s e s c a n be  and  velocity  t e n s o r form.  C  1 p n  expressed  - l ^ )  +  i n terms of the 2.6,  (A.3)  ff  pressure  which i s w r i t t e n i n  I n t e r m s o f s p h e r i c a l p o l a r c o o r d i n a t e s , we Lifshitz,  1982)  =  " P T  a  = P  7  j  (  +  dp  3v  ae  6V  +  _ § - - -  d  P  n-l n  1  d9  (  C  )  - )  and  J  =  v  9  n  using the  2.13)  identity  ( C )  - n(n-l) — sin 9  f r e e - s l i p b o u n d a r y c o n d i t i o n (eqn.  (A.5)  P  s u b s t i t u t i n g A . l and A . l i n t o A.5,  P  (A-4)  2  I _ _ P _ p  0  d  have  dv  °pp  By  „ n-1 n P  f i e l d s a c c o r d i n g t o eqn.  ( s e e L a n d a u and  the  ) i s the  n  (A.2)  v  an e x p r e s s i o n f o r t h e p r e s s u r e  S i n c e 2.9  d i s t r i b u t i o n can  2.15.  T- (£  0  P (/ ) i s t h e n ^ '  1  polynomial  2.9,  sin  are a r b i t r a r y .  o f d e g r e e n, and  with equation  p n  n  Legendre polynomial g i v e n by  n  becomes  (A.6)  60  2n(n-2) R " n  A  3  +  n  2(n -l) R 2  C  n _ 1  =  n  0  ;  n > 2  (A.7)  For n=2, we must have  C  while A  can be  2  =  0  (A.8)  arbitrary.  In a s i m i l a r f a s h i o n , we s u b s t i t u t e equations A . l ,A.3  into  A.4 and equate t o the e x t e r n a l normal s t r e s s e s a c c o r d i n g t o 2.12.  The r e s u l t i n g e x p r e s s i o n i s  I [ »-2, R n=2  3  i(  A  n  +  2n -6n-2_ 2B_=|  R  n-1  c  n  P  n-1^>  1 IT + a ( C ) V L  (A.9)  The r i g h t hand s i d e o f A.9 can be expanded as a sum o f Legendre polynomials  with c o e f f i c i e n t s a : n  I n n-l'f> a  P  n=l  S  "I [" + «<"]  (A.10)  Using the orthogonal property,the c o e f f i c i e n t s a  n The s e r i e s  1 1 2n-l J" T? 2 -1  P  n-1^)  n  ^  a r e g i v e n by  (A.11)  i n A.9 excludes terms a s s o c i a t e d w i t h n=l t o a v o i d  i n f i n i t e v e l o c i t i e s a t the p o l e s .  A c c o r d i n g l y , a^^ must be s e t  61 t o zero.  This r e s u l t s i n the expression  217 + AP(1+C  ) - X  =  0  (A.12)  hr  A l s o , because C =0, we see from A.9 t h a t a 2  well.  has t o v a n i s h as  2  T h i s i s i n f a c t e q u i v a l e n t t o t h e balance  of axial  forces  on t h e s p h e r i c a l body, which l e a d s t o t h e e q u a t i o n  P  C  X Combining equations Tf  =  -  2  AP — & 2C  1  (A. 13) p  +  e  A.12 and A.13, t h e i n t e g r a t i o n constant  can be r e l a t e d t o t h e s u c t i o n p r e s s u r e AP_ d C ) ( l C 2 2f +  A P by  +  p  =  +  p +  e) (A.14)  e  For n > / Z , A. 11 can be i n t e g r a t e d u s i n g t h e r e l a t i o n  S P  n  P  , dC  =  —  - P ^ 2n -1  + C  (A. 15)  to obtain  l  n  -  " W  {  A P  [ n P  V  - n-2<Vj P  x_ [p (C +e) - P . ( C e ) - P ( C ) n  p  n  2  p +  n  p  +  P . (C )] n  2  p  }  ( A  . 16)  The normal s t r e s s boundary c o n d i t i o n can now be matched a t each s e p a r a t e harmonic a c c o r d i n g t o A.9.  F o r n^-3, we have  62 2(n-2) R " n  Equations A.17  A  3  +  n  and A.7  2 n 2  -  can now  ~ n-1 6 n  2  R "! * n  c  (A.17)  =  u n  be s o l v e d s i m u l t a n e o u s l y .  terms o f the Legendre c o e f f i c i e n t s , the c o n s t a n t s A  In  and C n  A  =  (n+1)(n-1)  2  a  2(n-2)(2n +l)  n  n n  3  ^  n  C  n(n-l) 2(2n +l)  n  (A.18)  3  n  R  2  S u b s t i t u t i n g A.18  p  a  n  ^  R~  2  are:  11-1  back i n t o A . l - A.3,  the f i n a l  solutions for  = R are 00  v  =  - A  C 2  - ^ £  C  n  (n-1)(2n-l) 2(n-2)(2n l) 2  3  , R  +  a  n  P  n - l  (  C  r )  (A.19)  )  CO  v v  = e  A A  2  sin 9 sin 9  + +  ^ ^ 3  3n(n-l) 2 ' 7 2(2n^+l)(n-2  I  R  a  n  n  ( C )  (A. 20)  — n  sin 9  v  ;  (A.21) n=3  <  2n  +1  >  where the a r b i t r a r y c o n s t a n t A  2  can be i n t e r p r e t e d as an  v e l o c i t y superimposed onto the e x i s t i n g v e l o c i t y f i e l d .  axial This i s  a l l o w e d because n e i t h e r the p r e s s u r e nor the r a t e o f s t r a i n t e n s o r are a f f e c t e d .  63 APPENDIX B: PIPETTE ASPIRATION OF CORTICAL SHELL IN THE ABSENCE OF IN-PLANE SHEARING STRESSES The problem o f t h e a s p i r a t i o n o f a l i q u i d membrane can be s o l v e d a n a l y t i c a l l y when t h e r e a r e no s u r f a c e shear In  stresses.  t h i s case, t h e c o n s t i t u t i v e r e l a t i o n i s  T  while  ^  zero.  =  K V  (3.19)  a  ,the shear v i s c o s i t y i n e q u a t i o n 3.20,  i s identically  I t i s important t o r e c o g n i z e t h a t s i n c e t h e r e a r e no  s h e a r i n g s t r e s s e s , t h e c e l l geometry must remain s p h e r i c a l a t a l l times.  There a r e a c t u a l l y two seperate problems i n v o l v e d ,  they a r e : 1) for  t h e s o l u t i o n o f t h e instantaneous v e l o c i t y  a g i v e n geometry, and 2)  field  t h e time e v o l u t i o n o f t h e c e l l .  These w i l l be d e a l t w i t h by two d i f f e r e n t methods.  To o b t a i n e x p r e s s i o n s f o r t h e v e l o c i t y p r o f i l e , we  first  r e w r i t e t h e d i l a t o r y s t r a i n r a t e i n terms o f t h e v e l o c i t y components  (see Evans and Skalak,  ^  1980):  s  v  1  1  v = -dr + -frnHr" HH +v  where v  s  and v  components,  a  n  +  '  (B  1)  m * a r e t h e t a n g e n t i a l and normal v e l o c i t y a  s  respectively.  r  n  R  Because t h e c e l l  i s s p h e r i c a l with  r a d i u s R, e q u a t i o n B . l can be w r i t t e n , u s i n g t h e r e l a t i o n s=R9, as  64  V  S i n c e the c e l l  a  RTIfre  =  M  <s V  >  S i n  9  (B.2)  +  i s t o remain s p h e r i c a l , the normal v e l o c i t y ,  r e l a t i v e t o the c e n t r e of the sphere, must be a c o n s t a n t  (say  *  R).  We  s u b s t i t u t e t h i s , along w i t h equation 3.19  i n t o B.2  to  obtain (v  3! aa  s  s i n 9)  =  a sin 9  (B.3)  where the q u a n t i t y  a  i s independent of 0.  ="  2_5_ K  Equation B.3  obtain the v e l o c i t y f i e l d . c e n t r e of t h e  2  R  <-> B  can now  4  be i n t e g r a t e d t o  In a r e f e r e n c e frame f i x e d a t the  sphere  v  1 ~ cos 9 sin 9  s v  U s i n g the  -  n  =  (B.5a)  R  (B.5b)  transformations  =  v r  v  z  v  =  cos 9 + v  g  v  s  n  sin 9 - v  sin 9  n  cos 9  (B.6a)  (B.6b)  65  the v e l o c i t y  field  i n B.5 c a n b e r e w r i t t e n i n t e r m s o f t h e  r a d i a l and a x i a l components. numerical field  A l s o , t o be c o n s i s t e n t w i t h t h e  s o l u t i o n , an a x i a l v e l o c i t y  so t h a t t h e base o f t h e sphere  The f i n a l  i s added t o t h e v e l o c i t y (s=0) r e m a i n s s t a t i o n a r y .  results are  V  r  "  v„ z  =  °°  a  S6  lin"e ° C  S  9  '  +  (B.7b)  r e l a t e d i f volume  i s accounted f o r .  The t i m e e v o l u t i o n o f t h e c e l l form s o l u t i o n by r e a r r a n g i n g  c a n be e x p r e s s e d a s a c l o s e d  equation  da_ dt  =  3.19:  T (1+a)  ( B > 8 )  K.  w h e r e T a n d ot a r e b o t h u n i f o r m object  (B.7.)  (a+R) (1 - c o s 9)  The t w o p a r a m e t e r s a a n d R a r e u n i q u e l y conservation  B « n 8  over t h e deforming surface.  i s t o r e l a t e T t o 66, a n d h e n c e i n t e g r a t e e q u a t i o n  Consider  The B.8.  an a s p i r a t e d c e l l w i t h tounge l e n g t h L and a s p h e r i c a l  s e g m e n t o f r a d i u s R, a s shown i n f i g u r e 15.  The v o l u m e a n d  surface  by  area  V  i n t h i s configuration are given  =  ,  2 R  p  L  +  |  R (l C ) (2-C ) 3  2  +  p  p  +  §  .  R  3 p  (  B  - » 9  66  F i g u r e 15:  D e f i n i t i o n of v a r i o u s dimensions f o r the problem p o s e d i n a p p e n d i x B.  67 A  2TT R L + 2TT R ( i + c ) + 2TT R P P P  =  2  2  (B.10)  where  r  =  6  C O S  p  L  P  (B.ll)  G i v e n V a n d A, £  is a positive quantity.  c a n be s o l v e d f o r from  the following cubic equation: 1+c 2  2 r k  p  3  _  (1+c) 2  +  2 f k  3  p  2  (3+c) c  _  (1+c)(5+c) -2  P  2  _ -  (B.12) 0  where t h e q u a n t i t y  c  =  6V - 3R A f —  2,  R  +  (B.13)  1  3 p  f o r c o n s t a n t volume, i s a f u n c t i o n o f A a l o n e . ( o r (X. ) , we c a n s o l v e f o r X follow  u s i n g B.12.  Thus, g i v e n A  The q u a n t i t i e s R a n d L  immediately:  L  +  R  -  -h-  .  ^  '  ^  P  '  (B.15)  By r e q u i r i n g t h e m e r i d i o n a l t e n s i o n t o b e c o n t i n u o u s a c r o s s t h e p i p e t t e e n t r a n c e , t h e i s o t r o p i c t e n s i o n c a n be e x p r e s s e d i n t e r m s o f t h e s u c t i o n p r e s s u r e P and t h e i n s t a n t a n e o u s r a d i u s R  68 as _ T  PR 2 ( 1 - R /R)  =  (B.16)  p  S i n c e t h e s u c t i o n p r e s s u r e i s c o n s t a n t , T i s o n l y dependent R.  Thus, g i v e n t h e i n i t i a l  cell  size  (which e s t a b l i s h e s the  t o t a l volume) and t h e i n s t a n t a n e o u s v a l u e o f determine  t h e c o n s t a n t c i n B.13.  e q u a t i o n B.12,  from which  m a n n e r , e q u a t i o n B.8  we  c a n be  oC , we  can  Z, ^ i s t h e n s o l v e d f o r f r o m  c a n o b t a i n R,  L, and T.  In  this  i n t e g r a t e d n u m e r i c a l l y (eg: by  R u n g e - K u t t a method) t o v e r y h i g h  on  accuracy.  the  69 APPENDIX C:  FINITE  DIFFERENCE  EQUATIONS  A set of four f i n i t e d i f f e r e n c e equations every p a i r of nodal points.  A t e a c h n o d e , s a y n o d e number i  ( i = l , 2 , . . . , N , w i t h i = l c o r r e s p o n d i n g t o s=0), unknowns: A r ( i ) , A z ( i ) , ^ A r ( i ) , ^ ; A z ( i ) . d i f f e r e n c e equations are t o approximate and  The  there are four first  two  the d e r i v a t i v e s of A r  Az: i Ar ( l )  9 •=—  o  ds  5 s  where i ranges defined  Ar(i) - Ar(i-l) —»— —r i  =  1  1  (CD  Az(i)  =  Az(i) -  f r o m 2 t o N,  and h ^  Az(i-l) i  h  (C2)  i s the l o c a l g r i d  spacing  as  o t h e r two  i  -  s  o  "  ( i )  o  s  ( i - 1  )  ( C  d i f f e r e n c e e q u a t i o n s a r e b a s e d on e q u a t i o n s  By e x p r e s s i n g t h e e i g h t unknowns a t t h e two as a^  1  h  * o  h  The  i s w r i t t e n between  ( k = l , 2 , . . . , 8 ) , e q u a t i o n 3.34  nodes  *  3 )  3.34.  collectively  c a n be r e w r i t t e n  as  a da  a  k  gfT h i s i s v a l i d because  Ae  s  A£oc  a  k  • a  £  k  and  =  A£  a  -e s  s  (C4.a)  (C4.b)  are l i n e a r i z e d t o contain only  the  f i r s t powers o f t h e s o l u t i o n v e c t o r .  of the r e s i d u a l functions  From t h e d e f i n i t i o n s  (eqns. 3.14), i t f o l l o w s  4AT + o- (2R./R - 3) AR . + a R n 9' m ' 9 n 9  A(—^-) R ' m  2  x  a  Ae  4AT  s  + a  n  ( l - 2 R ^ / R ) AR^  - a ^  m  that  (C5.a)  v  (C5.b)  A  2  m The c u r v a t u r e s  and t h e i r i n c r e m e n t s a r e g i v e n  R  AR 9  1/R. m  -3 R X m m  A  m  9  sin 6  AX  A r  sin  a^r  For  5 s  o  t h e s t r e s s e s , t h e forms o f T and T  m a t e r i a l model chosen. tension  (C.7)  9  a s  az  o ds o2 a  A e  dz  a Ar as o  m a Az as  (C.6)  _ r cos 9  +  m  follows:  sin G  8 z ( dr ds^ _ 2 o os o  . 3 X m  as  a  o  a r as o  (C.8)  2  a z 2 as o  ar as  a Az _ 2 o as o  a*Ar as  d e p e n d on t h e p a r t i c u l a r  As a s i m p l e example, c o n s i d e r  resultant i n a liquid  (C.9)  2  the  (eqns. 3.19-3.20):  (CIO)  Ts  =  0  (C.ll)  71  =  A T  A T  s  increments  ^n/Na  A?" (  =  The t e n s i o n s T a n d T s t r e s s e s accumulated  AT"  A  V  +  X  m  "  (C.12)  W  A  (C.13)  W  A  a r e zero because they r e p r e s e n t  g  up t o t h e p r e s e n t c o n f i g u r a t i o n .  i n t r o d u c e d above c a n be e x p r e s s e d  =  d  cos G  Ar 9s  +  relations:  a Az 3s  sin 0  A l lthe  i n terms o f the  basic deformation variables using the following  AA_ m  elastic  (C.14)  (C.15)  A9  =  i —  , m  a  cos 6  a Ar as ( i )  _1_ h,  a Az as ( i )  1_ h.1  2  2  a Az  0  o 3  L  9 Ar s i n 6 —= 3s _  Ar as_ ( i )  a_Az_ as_  . ) '  (C.16)  (C.17)  o  (C.18) ( 1 }  O  J  S i n c e t h e r e a r e f o u r e q u a t i o n s , t h e r e must a l s o be f o u r boundary c o n d i t i o n s t o ensure a unique  solution.  I f we c h o o s e t o l o c a t e  t h e r e f e r e n c e frame a t t h e base o f t h e c e l l conditions immediately  (s=0), two boundary  follow:  Ar(l)  =  o  (C.19)  72 Az(l)  A l s o , 9 a t t h e b a s e m u s t be vanishing first  a t s=0  =  0  (C.20)  always zero.  The  requirement of  leads t o another boundary c o n d i t i o n at  the  node:  a  -(i) ds " ' A z  =  0  (C.21)  Q  At the  other  prescribed.  end  of the g r i d ,  a tangential velocity  Using the transformation  v  the  Ae  s  =  v  r  cos  0  +  v  ^  can  be  relation  z  sin 0  (C.22)  l a s t b o u n d a r y c o n d i t i o n becomes  Ar (N)  cos  0  +  Az (N)  sin G  I n a c t u a l i m p l e m e n t a t i o n , \ j f i s an  -  v  P e  At  =  0  (C.23)  i t e r a t i v e parameter t h a t  used t o s a t i s f y the volume c o n s e r v a t i o n  condition.  is  73  APPENDIX D : MATRIX SOLUTION TO F I N I T E DIFFERENCE EQUATIONS  F i g u r e 10 shows t h e m a t r i x s t r u c t u r e o f a t y p i c a l d i f f e r e n c e scheme, w h e r e a s y s t e m o f M d i f f e r e n t i a l are approximated by d i f f e r e n c e formulas v a l u e s o f two a d j a c e n t p o i n t s . o u t l i n e d by P r e s s e t a l . (1986), to  finite equations  t h a t i n v o l v e nodal  The m e t h o d o f s o l u t i o n , a s i s t o reduce t h e given  matrix  a s p e c i a l u p p e r t r i a n g u l a r f o r m , a s shown i n f i g u r e 16.  do t h i s ,  o n l y m a t r i x e l e m e n t s from two b l o c k s need be  manupilated f i g u r e 10.  a t any time.  We s t a r t w i t h t h e t o p N xM b l o c k i n c T h i s b l o c k , a l o n g w i t h t h e f i r s t N e l e m e n t s on t h e c  r i g h t hand s i d e , i s Gauss r e d u c e d u n t i l t h e f i r s t form an i d e n t i t y m a t r i x . last N  To  eliminate the f i r s t N  c  A t t h e end o f t h i s p r o c e s s ,  columns and t h e c o r r e s p o n d i n g  v e c t o r need be s t o r e d .  N  columns only the  portion of the right  hand  This i n f o r m a t i o n i s then used t o  columns i n t h e second b l o c k , which c '  d i r e c t l y underneath t h e i d e n t i t y matrix.  The r e m a i n i n g  lies  elements  o f t h i s b l o c k a r e t h e n Gauss r e d u c e d u n t i l t h e n e x t M columns (columns N + 1 last N  t o N +M) f o r m a n i d e n t i t y m a t r i x .  columns, and t h e corresponding  side are stored.  Again,  only the  p o r t i o n o f t h e r i g h t hand  This procedure i s repeated  l a s t b l o c k , w h i c h h a s d i m e n s i o n s N xM. s  u n t i l we g e t t o t h e  As b e f o r e , t h e f i r s t  N c  columns a r e e l i m i n a t e d u s i n g i n f o r m a t i o n from t h e p r e v i o u s block.  The r e m a i n i n g  part o f t h e block i s then  reduced t o an  i d e n t i t y m a t r i x , t h u s a t t a i n i n g t h e d e s i r e d form i n f i g u r e 16. Note t h a t a t t h i s stage, o n l y t h e sub-blocks t h e a l t e r e d r i g h t hand s i d e , a r e s t o r e d .  labelled  "S", and  From h e r e , t h e  74  HN hs  N  c  tM  I I  1  I  M 1  I  Figure  16:  N.  The d e s i r e d f o r m o f t h e u p p e r t r i a n g u l a r m a t r i x which minimizes storage space f o r the f i n i t e d i f f e r e n c e s o l u t i o n scheme. O n l y b l o c k s l a b e l l e d S" a r e s t o r e d . The s q u a r e b l o c k s l a b e l l e d " I " a r e i d e n t i t y m a t r i c e s , and a l l t h e remaining entries are zeros. 11  75  s o l u t i o n f o l l o w s q u i c k l y by back  Subroutines above t a s k s .  substitution.  c o d e d i n FORTRAN a r e g i v e n w h i c h i m p l e m e n t t h e  GJPP p e r f o r m s G a u s s - J o r d a n e l i m i n a t i o n  (with  p a r t i a l p i v o t i n g ) on a g i v e n m a t r i x u n t i l an i d e n t i t y m a t r i x i s formed.  The e l i m i n a t i o n o f t h e f i r s t N  columns  i n t h e sub-  c b l o c k s i s d o n e b y s u b r o u t i n e REDUCE.  UPTRI i s a d r i v e r r o u t i n e  w h i c h u s e s G J P P a n d REDUCE t o f o r m t h e u p p e r t r i a n g u l a r shown i n f i g u r e 16. FDE,  The  The u s e r h a s t o s u p p l y s u b r o u t i n e s B C 1 ,  a n d BC2, w h i c h g e n e r a t e t h e f i r s t  (Mx2M), a n d t h e l a s t final  matrix  (N xM) , t h e i n t e r m e d i a t e  (N xM) b l o c k s i n f i g u r e 1 0 , r e s p e c t i v e l y .  s o l u t i o n i s o b t a i n e d u s i n g BKSUB w h i c h p e r f o r m t h e  necessary back-substitutions.  76  C C C C  10 20  30  40 45  50 60 70 80 90 100 C  SUBROUTINE GJPP(A,MA,NA,NCI,IP) I n p u t i s m a t r i x A o f d i m e n s i o n s MA b y NA (NA.GE.MA+NC1). The f i r s t NCI columns a r e i g n o r e d . The r e s t o f t h e m a t r i x i s Gauss reduced ( w i t h p a r t i a l p i v o t i n g ) u n t i l t h e next MA c o l u m n s become a n i d e n t i t y m a t r i x . I M P L I C I T REAL*8(A-H,O-Z) DIMENSION A ( 4 , 9 ) , I P ( 4 ) , S ( 4 ) DO 20 1=1,MA S(I)=0. DO 10 J=1,MA IF(DABS(A(I,J+NC1)).GT.S(I)) S(I)=DABS(A(I,J+NC1)) CONTINUE I F ( S ( I ) . E Q . 0 . D 0 ) GOTO 80 IP(I)=I DO 70 ID=1,MA JD=ID+NC1 BIG=0. DO 30 I=ID,MA DUM=DABS(A(IP(I),JD)/S(IP(I))) IF(DUM.LE.BIG) GOTO 30 BIG=DUM IMAX=I CONTINUE IF(BIG.EQ.O.DO) GOTO 80 IDUM=IP(ID) IP(ID)=IP(IMAX) IP(IMAX)=IDUM DUM=A(IP(ID),JD) IF(DUM.EQ.l.DO) GOTO 45 DO 40 J=JD,NA A(IP(ID),J)=A(IP(ID),J)/DUM DO 60 1=1,MA I F ( ( I . E Q . I P ( I D ) ) . O R . ( A ( I , J D ) . E Q . 0 . D 0 ) ) GOTO 60 DUM=A(I,JD) DO 50 J=JD,NA A(I,J)=A(I,J)-DUM*A(IP(ID),J) CONTINUE CONTINUE GOTO 100 WRITE(*,90) FORMAT(' M a t r i x s i n g u l a r i n G J P P , p r o g r a m t e r m i n a t e d ' ) STOP CONTINUE RETURN END SUBROUTINE REDUCE(A,B,NE,NCI,IPT,NPTS) I M P L I C I T REAL*8(A-H,0-Z) DIMENSION A ( 4 , 9 ) , B ( 4 , 2 , 5 1 0 ) NS=NE-NC1 I F ( I P T . G T . N P T S ) GOTO 10 MA=NE  77  10 20  30 40 50 C  10  20 30  40 C C  NA=2*NE+1 GOTO 20 MA=NS NA=NE+1 ID0=NS I F ( I P T . E Q . 2 ) ID0=0 DO 50 J=1,NC1 ID=ID0+J DO 40 1=1,MA I F ( A ( I , J ) . E Q . 0 . D 0 ) GOTO 40 DO 30 K=1,NS A(I,NC1+K)=A(I,NC1+K)-A(I,J)*B(ID,K,IPT-1) A(I,NA)=A(I,NA)-A(I,J)*B(ID,NS+1,IPT-1) CONTINUE CONTINUE RETURN END SUBROUTINE UPTRI(B,NE,NC1,NPTS) I M P L I C I T REAL*8(A-H,0-Z) DIMENSION A ( 4 , 9 ) , B ( 4 , 2 , 5 1 0 ) , I P I V ( 4 ) DIMENSION S 0 ( 5 1 0 ) , R 0 ( 5 1 0 ) , R ( 5 1 0 ) , Z ( 5 1 0 ) NS=NE-NC1 NS1=NS+1 10=0 NA=2*NE+1 NABC=NE+1 C A L L BC1(A,NE,NC1) CALL GJPP(A,NC1,NABC,I0,IPIV) DO 10 1=1,NCI DO 10 J=1,NS1 B(I,J,1)=A(IPIV(I),NC1+J) DO 30 IPT=2,NPTS C A L L FDE(A,NE) C A L L REDUCE(A,B,NE,NCI,IPT,NPTS) CALL GJPP(A,NE,NA,NC1,IPIV) DO 20 1=1,NE DO 20 J=1,NS1 B(I,J,IPT)=A(IPIV(I),NC1+NE+J) CONTINUE IPT=NPTS+1 C A L L BC2(A,NE,NC1) C A L L REDUCE(A,B,NE,NCI,IPT,NPTS) C A L L GJPP(A,NS,NABC,NC1,IPIV) DO 40 1=1,NS B(I+NC1,1,1)=A(IPIV(I),NABC) RETURN END SUBROUTINE BKSUB(B,NE,NCI,NPTS) S o l u t i o n X(I) a t the J - t h g r i d point i s stored i n B(I,1,J) I M P L I C I T REAL*8(A-H,0-Z) DIMENSION B ( 4 , 2 , 5 1 0 ) , X ( 2 )  10  20 30 40 50  60 70 80 90 100 110 120  NS=NE-NC1 NS1=NS+1 DO 10 1=1,NS X(I)=B(I+NC1,1,1) DO 50 IPTDUM=2,NPTS IPT=NPTS+2-IPTDUM DO 30 IDUM=1,NE I=NE+1-IDUM DUM=0. DO 20 J=1,NS DUM=DUM+X(J) * B ( I , J , I P T ) B(I,NS1,IPT)=B(I,NS1,IPT)-DUM DO 40 1=1,NS X(I)=B(I,NS1,IPT) CONTINUE DO 70 IDUM=1,NC1 I=NC1+1-IDUM DUM=0. DO 60 J=1,NS DUM=DUM+X(J)*B(I,J, 1) B(I,NS1,1)=B(I,NS1,1)-DUM DO 80 1=1,NS B(I+NC1,1,NPTS)=B(I+NC1,1,1) DO 90 1=1,NCI B(I,1,1)=B(I,NS1,1) DO 120 IPT=2,NPTS DO 100 1=1,NS B(I+NC1,1,IPT-1)=B(I,NS1,IPT) DO 110 1=1,NCI B(I,1,IPT)=B(I+NS,NS1,IPT) CONTINUE RETURN END  79 L I S T OF REFERENCES Amato P h i l i p A., Unanue E m i l R., T a y l o r D. L a n s i n g . , 1983 D i s t r i b u t i o n o f A c t i n i n Spreading Macrophages: A Comparative S t u d y on L i v i n g a n d F i x e d C e l l s The J o u r n a l o f C e l l B i o l o g y , 9 6 : 7 5 0 - 7 6 1 B e s s i s M., 1973 L i v i n g B l o o d C e l l s and t h e i r U l t r a s t r u c t u r e Springer, B e r l i n C h e i n S h u , Sung K u o - L i P a u l , 1984 E f f e c t o f C o l c h i c i n e on V i s c o e l a s t i c P r o p e r t i e s o f B i o p h y s i c a l J o u r n a l 46:383-386  Neutrophils  E v a n s E v a n A., 1973 A New M a t e r i a l C o n c e p t f o r t h e Red C e l l Membrane B i o p h y s i c a l J o u r n a l 13:926-940 E v a n s E v a n A., Hochmuth R. M., 1977 A S o l i d - L i q u i d C o m p o s i t e M o d e l o f t h e Red C e l l J o u r n a l o f Membrane B i o l o g y 3 0 : 3 5 1  Membrane  E v a n s E v a n A., S k a l a k R i c h a r d , 1980 M e c h a n i c s and Thermodynamics o f Biomembranes CRC P r e s s , R o c a R a t o n , F l o r i d a E v a n s E v a n A., K u k a n B., 1984 Large Deformation, Recovery a f t e r Deformation, A c t i v a t i o n of Granulocytes B l o o d 64:1028-1035  and  E v a n s E v a n A., Needham D., 1986 P h y s i c a l P r o p e r t i e s o f L i p i d B i l a y e r Membranes: C o h e s i o n , E l a s t i c i t y , and C o l l o i d a l I n t e r a c t i o n s J o u r n a l o f P h y s i c a l Chemistry (submitted) F u n g Y. C., 1965 Foundations o f S o l i d Mechanics P r e n t i c e - H a l l , Englewood C l i f f s ,  New  Jersey  H a p p e l J o h n , B r e n n e r Howard, 1973 Low R e y n o l d s Number H y d r o d y n a m i c s P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , New  Jersey  Kwok R., E v a n s E v a n A., 1981 T h e r m o e l a s t i c i t y of Large L e c i t h i n B i l a y e r V e s i c l e s B i o p h y s i c a l J o u r n a l 35:637-652 L a n d a u L. D., L i f s h i t z E. M., F l u i d Mechanics Pergamon P r e s s , O x f o r d  1982  80 M i t c h i s o n J . M., Swann M. M., 1954 The M e c h a n i c a l P r o p e r t i e s o f t h e C e l l S u r f a c e . Elastimeter J o u r n a l o f E x p e r i m e n t a l B i o l o g y 31:443  I . The C e l l  P r e s s W. H., F l a n n e r y B. P., T e u k o l s k y S. A., V e t t e r l i n g W. T., 1986 Numerical Recipes Cambridge U n i v e r s i t y P r e s s , Cambridge R a n d R. P., B u r t o n A. C , 1964 M e c h a n i c a l P r o p e r t i e s o f t h e R e d C e l l Membrane. B i o p h y s i c a l J o u r n a l 4:115 S c h m i d - S c h o n b e i n G. W., S h i h Y. Y., C h e i n S., 1980 M o r p h o m e t r y o f Human L e u k o c y t e s B l o o d 56:866-875 S c h m i d - S c h o n b e i n G. W., Sung K. L. P., T o z e r e n H., S k a l a k R., 1 9 8 1 P a s s i v e M e c h a n i c a l P r o p e r t i e s o f Human L e u k o c y t e s B i o p h y s i c a l J o u r n a l 36:243-256 S k a l a k R., T o z e r e n A., Z a r d a R. P., C h i e n S., 1973 S t r a i n E n e r g y F u n c t i o n o f R e d B l o o d C e l l Membranes B i o p h y s i c a l J o u r n a l 13:245 S o u t h w i c k F. S., S t o s s e l T. P., 1983 C o n t r a c t i l e Proteins i n Leukocyte Function Seminars i n Hematology 20:(4)305-321 T o z e r e n H., C h e i n S., T o z e r e n A., 1984 E s t i m a t i o n o f V i s c o u s D i s s i p a t i o n i n s i d e an E r y t h r o c y t e Aspirational Entry into a Micropipette B i o p h y s i c a l J o u r n a l 45:1179-1184  during  

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