BULLETIN O F MATHEMATICALBIOLOGY VOLUm~ 38, 1976

LARGE DEFORMATION ANALYSIS OF SOME BASIC PROBLEMS IN BIOPHYSICS

[] H l ~ D ~ A r Marmara Research Institute, Division of Applied Mathematics, P.K. 141, KadikSy, Istanbul, Turkey

This study is concerned with the application of a possible form of a strain-energy function suitable for soft biological tissues. Two problems considered here axe related to simultaneous extension and inflation of cylindrical arteries, and inflation of the left ventricle under a given internal pressure. The values of the material constants are obtained via comparison of theoretical results with experimental findings. Some details concerning the wall stresses and the elastic stiffness are also given in the paper. F o r each case, it is seen t h a t experiment and theory are in good agreement.

1. Introduction. The nonlinear constitutive equations of elastometers have been studied by Mooney (1940) and Rivlin (1960). The theory developed in these works is in good agreement with experimental results obtained for vulcanized rubber which may be called "moderately elastic" materials. For soft biological tissues, however, the measurable stresses develop only after the specimen is stretched to several hundred percent of its relaxed (in vitro) length. Materials with such features will be termed "highly elastic" materials. It is therefore apparent that classical linear elasticity theory may not be adequate to describe the mechanical behavior of a class of soft biological tissues. In such cases, a suitable finite deformation theory must be used. The finite deformation theory, in analyzing the simple extension of a rabbit me~ntary, has been successfully used by Fung (1967). Fung concluded his work by stating that the strain-energy function should be exponential (in the one-dimensional case) in the stretch ratio. Some efforts at generalizing Fung's discovery to three-dimensional problems have been made by Blatz et al. 701

702

HILMI DEMIRAY

(1969), Demiray (1972, 1975), Vito (1973), Gou (1970), Snyder (1972) and many others in the current literature. Each work is good within its own right of applicability to the problems under consideration. At present we confine our attention to the derivation given by Blatz et al. (1969) and Demiray 1972), developed independently, and see the application of it to a class of problems that are deemed to be basic in biophysics. These problems are concerned with the inflation of a left ventricle and simultaneous extension and inflation of cylindrical arteries. Although the first problem has previously been worked out by Mirsky (1973), through the use of a different approach and a form of strain-energy function, the analysis employed and the form of strain-energy function introduced here are seen to be much simpler and the presentation is rather general. Also the differences between theoretical and experimental results are found to be very small as compared to those given in Mirsky (1973) who uses a modified data due to Spotnitz et al. (1966). Finally, in the last section simultaneous elongation and radial expansion of a cylindrical artery is investigated and the consistency of present result with those of experiments is shown for each particular case. Although most of the biological tissues (in particular arteries) manifest certain anisotropy in their material structure, for the sake of simplicity and practicality, here, we assume that the material under consideration is isotropie, homogeneous and incompressible. For such elastic materials the constitutive equations are given by (Eringen, 1962) 2. Theoretical Preliminaries.

aW ~ OW t kZ = P g ~ + 2 ~ c ~+2~2Bkl,

(1)

where ckl ~ GKL

~x k ~x z ~ X K ~ X L'

B kZ = Ilck~-C~m cm~, x ~ = xk(Xl~)

(2)

and W is the strain-energy function, I1, I2, and I3 = 1 are the basic invariants of the Finger deformation tensor ck~, P is the hydrostatic pressure, gk~ and G KL are the metrics in deformed and undeformed coordinates, respectively. A possible and suitable form of the strain-energy function (Blatz et al, 1969; I)emiray, 1972) may be given by B W = ~ [exp (~(I1- 3))- 1], 2a where a and fl are material constants to be determined through experimental studies.

BASIC P R O B L E M S IN BIOPHYSICS

703

I f the expression for W is introduced into (1) we obtain tk~ = p g ~ + flck~ exp (~(I1- 3)).

(3)

This stress-strain relation should satisfy appropriate equilibrium equations and properly posed boundary conditions. In the remainder of this work we present the solution of two problems and compare the result with experiments to obtain the value of material constants introduced. 3. Inflation of L e f t Ventricles. Inthis part of the study we are primarily concerned with stress distributions in ventricular wall. A precise knowledge of stress distribution in ventrieular wall is important to the physiologist for understanding the behavior of ventricular muscle, for building a realistic mathematical model of the heart as a pump. The ventricular wall is essentially composed of ordered sets of interconnected muscle fibers. The fiber orientation through the myocardial wall changes gradually from endocardium to epicardium (Streeter et al., 1969). Such a complicated structure of the heart shows that the material of ventricular wall is neither isotropie nor homogeneous; it rather has a very complicated material symmetry. It may seem reasonable to assume that the material of the left ventricle has a curvilinear orthotropy. But, even in the case of linear theory, the study of such cases is rather tedious. Moreover, a realistic model for the geometry of the left ventricle should be a thick ellipsoidal shell of revolution. In finite deformation theory, the consideration of such a geometry is not a easy task either. Because of these difficulties in the formulation of the problem (in continuum sense) we make further simplifying assumptions regarding the geometry and the material properties of the ventricle. Thus an idealized geometry, i.e. a spherical geometry, is assumed for the left ventricle. Assuming that the material under investigation obeys the conditions of isotropy, homogeneity and incompressibility, the stress-strain relation given by (3) may be used to determine the displacement and stress field of the left ventricle completely. Consider a spherical coordinate system (R, O, O) in the undeformed body, and let (r, 0, r be the corresponding spherical coordinates in the deformed body. If a spherical thick shell undergoes into a large radial deformation under an internal pressure, the finite deformation of the body is characterized by r = r(R),

0 = O,

r = O,

R / r = x.

(4)

The requirement of incompressibility of the material leads to the following form of deformation (Eringen, 1962): r = (Ra+A)I/a,

A = constant.

(5)

For this and other details of the mechanical parts of the subject the readers is referred to Eringen (1962).

704

HILMI DEMIRAY

I f the components of Finger deformation tensor for this deformation field are calculated and the result is used in (3), there follow the non-vanishing stress components trr = P + B x a e x p (a(I1-3)), I1 = x 4 + 2 / x 2, too = t ~

(6)

= P + fl/x 2 exp (a(I1-3)).

Since the sphere is subjected to an inner pressure P~ and zero pressure on the outer surface, we should have trr =

-P~

a t x = x~,

and

trr = 0

at x = Xo,

(7)

where the subscripts i and o stand for the values of a corresponding quantity on the inner and outer surfaces, respectively. The solution satisfying the boundary conditions (7) and the equilibrium equation d t r r + -l ( 2 t r r - t o o - t dr r

~r

= 0

is given by trr

= 2//|

(1 + x 8) exp ( a ( I 1 - 3))dx,

.ix o

(s) (1 + x 3) exp (c~(I1-3))dx+//(1/x2-x 4) exp ( e ( I 1 - 3)).

too = t , e = 2fl Xo

Employing the first of conditions (7) in the above equation we arrive at the following relation : P~ = 2//

(1 + x a) exp ( e ( I 1 - 3))dx.

(9)

1

This equation plays an important role in obtaining the value of material constants via comparison of present result with those of experiments. Even the value of these constants are known the above integral cannot be handled analytically; it rather must be evaluated by numerical means. Experiments on dog's left ventricle have been performed and the values of r~ and ro are measured for various pressure levels by Spotnitz et al. (1966), and the result is listed in Table I. Thus, from experimental findings we know the value of P~ and can determine x~ and xo for each measurement. Let ! u (P~, x i , X'o) and (P~, x i , Xo) be the two different sets of measurements; from Equation (9) we obtain the following relation: F(~) = P~

(1 +x3) exp ( e ( I 1 - 3 ) ) d x - P ~ i In

x

fx o ( l + x 3) exp (c~(I1- 3))dx -- 0, l'

(10)

BASIC PROBLEMS IN BIOPHYSICS

705

which involves only the unknown a. By giving several values to a and carrying out the above integral numerically, one can find a value of a for which F(a) = 0. These calculation have been done in computer and the result is found to be a = 1.256. From Equation (9) the constant fl can be calculated to obtain fl = 4550 dynes/cm 2. By resubstitution of these values of a and fl into TABLE I Pressur~RadiiData ~rtheLe~

Ventricle*

Pi Exp. ( d y n e / c m 2)

ri (cm)

ro (cm)

rm (cm)

P i Calc. ( d y n e / c m 2)

Deviation (%)

6670 13340 16000 20000 26680

2.05 2.27 2.33 2.39 2.46

3.17 3.27 3.29 3.32 3.36

2.61 2.77 2.81 2.85 2.91

6740 13240 16250 20000 26480

+ 1.00 - 0.70 + 1.50 0.00 - 0.77

A t zero pressure r t = 1.44 c m , a n d r o = 2.96 cm. *mmHgis c o n v e r t e d to d y n e s / c m 2. 50

40

"" - - -

E• Calc

.o

Y 0 2O

1 2.06

2.10

L ~.20

I

,I

2.30

2.40

~.~

rl

Figure 1.

P r e s s u r e - i n n e r radius v a r i a t i o n s for t h e left ventricle

(9) we obtain the calculated values of P~ listed in the fifth column of Table I. The percentage of the deviations between experiment and the present result is given in the last column of the table. As may be seen from the table, the results of the theory and experiments are very close, and the maximum deviation is only 1.5 per cent. The variation of P~ with r~ is depicted in Figure 1.

706

I-IILMI D E M I R A Y

I f this result is to be compared with one by Mirsky (1973), the following remarks can be made: (i) Present study uses a strain-energy function which is very simple in form and presented regardless of the geometry of the body under consideration. However, in Mirsky (1973), the relation between volume and the Laplace stress of a thick spherical shell is taken to be fundamental in his trial and error method. (ii) The procedure of determining the unknown constants a, through the use of computer, is very simple and needs a very short computer time (c.f. (10)). (iii) The deviations between experiment and the present work are less than 1.5 per cent (see Table I), while this reaches 6 per cent in Mirsky (1973). Nevertheless, although he claims that he made use of the experimental data of Spotnitz et al. (1966), the difference between the data that Mirsky had

Io

6

_l

Classical

I

--

I

4

Finite

= 2 6 , 7 0 0 dyne/crn 2

-/

\ 2-

2

\ \

I

20

~1--

--

40

"1-

60

--

--I-80

-IO0

% Wall thickness

F i g u r e 2.

T a n g e n t i a l wall stress d i s t r i b u t i o n n o r m a l i z e d t o L a p l a c e stress

(Ptri/2h) used and the one given by Spotnitz et al. is considerable. If the real data is used in Mirsky's formulation, this deviation even goes over 30 per cent. Ventricular wall stress It is of interest to know the tangential stress distribution in ventricular wall. This has been done for the pressure level P~ -- 20 mmHg (or 26,700 dynes/cm 2) and the result is plotted as a percentage of the wall thickness, Figure 2. The tangential stress is normalized to Ptrl/2h which is the average stress given by Laplace law for a sphere. There is apparently a marked difference between stress distributions based on classical theory and large deformation analysis; in particular the stress at endocardial surface is almost thirteen times greater than one predicted by the classical theory. These high stresses may be the cause of ischemia of the left ventricle (c.f. Monreo

BASIC PROBLEMS IN BIOPHYSICS

707

et al., 1972; Barnard et al., 1972). This stress decreases very rapidly as we approach to epicardial surface of the left ventricle. It may be of some interest to consider the variations of midwall tangential (or circumferential) stresses with respect to midwall tangential strain. To obtain an explicit expression for this quantity, we note that the principal tangential strain in the deformed state is given by r-R -

-

1 - x .

(11)

r

2

IO

Figure 3.

20

30

Variations of tangential stress and elastic stiffness with midwall strain (era)

Using this in (8), we obtain

too = /3 exp (~(I1- 3))

-x

exp ( I 1 - 3 ) ( l + x 3 ) d x .

+2/~

(12)

o

The variation of tool2 m . fl with respect to em = 1 - X m is illustrated in Figure 3, by a black curve. As is seen from the figure, the material becomes very stiff when strain reaches to a certain level. Ventricular elastic stiffness Another concept which we would like to discuss here is the elastic stiffness of the left ventricle. For linear elastic materials the definition of the elastic stiffness is very simple and has physical and geometrical meanings. However, for nonlinea.r elastic materials there exists no such a unique definition of this quantity (c.f. Mirsky et al., 1972; Mirsky, 1973). If the definition of Young's modulus for nonlinear stress-strain relation (in a onedimensional case) is recalled, the definition used by Mirsky et al. (1972) seems

708

I-IILMI D E M I R A Y

to be more reasonable for the present problem. Following the same path, we have E~

-

dtoo - de

-

dtoo dx

(13)

Inserting (11) in (13), the explicit expression of Et follows: Et = 2fl exp (~(I1- 3))[(1 - 2a)x a + (1 + 2~)/x 3 - 1].

(14)

I f so desired, Et can be expressed in terms of 8 through the use of (11). We are essentially interested in the variations of Et with respect to e measured on the midsurface. A plot between Et/20fl and Sm illustrates this variation in Figure 3, by a dashed curve. For small deformations, i.e. as x --~ 1 or e -+ 0, Et approaches to 2fl = 9100 dynes/cm 2. An interesting and instructive result can be drawn out of this derivation by examining the Figure 3. At large strain levels, small increments in the midwall strain cause to very large increments in the elastic stiffness. This means that after a certain strain level (which is considerably high) the material becomes very stiff against the radial expansion. This is exactly what Spotnitz et al. (1966) have observed in their experiments on dogs' left ventricle. Such a close correspondence between the mechanical model and experimental results is encouraging. In closing up this section the following remarks are in order. Considerable stress gradients take place in the endocardial layers of ventricles subjected to an internal pressure; these high stresses may be the cause of ischemia of the left ventricle (Monreo et al., 1972). Such results can only be predicted from a large deformation analysis. As observed in the experiments, the resistance of the wall to radial expansion at high stress levels becomes very large. 4. Arterial Wall Stresses. In this section we study the extension and inflation of arteries, assuming that they have a cylindrical geometry. The studies by Patel et al. (1969) and Vaishnav et al. (1972) show that the arterial wall material has anisotropie (cylindrical orthotropy) properties. However, it can be seen from their calculated results (in the linear theory) that the difference between elastic moduli in three different perpendicular directions is not very large. (In particular the u modulii in the circumferential (0) and axial (z) directions are almost the same). This may suggest us that the material of artery should be transversely isotropic where the plane of isotropy is tangent to the cylindrical surface (for such a mechanical model see Demiray (1975)). To simplify the problem further, in this study we assume that the material of artery is isotropic, homogeneous and incompressible (see Simon et al. (1972)). Consider a cylindrical coordinate system (R, | Z) in the deformed body, and let (r, 0, z) be the corresponding coordinates in the deformed body. I f a cylindrical thick shell undergoes a large axial stretch ~ and then a radial expan-

BASIC P R O B L E M S I N B I O P H Y S I C S

709

sion under an internal pressure, the finite deformation of the body is r = r(R),

0 = (9,

z = ,~Z,

R / r = x.

(15)

The requirement of incompressibility of the material yields the following form of deformation (Eringen, 1962): r = (R2/)~+B)I/2,

B =

constant.

(16)

I f the components of the Finger deformation tensor are calculated and the result is employed in (3), the nonvanishing stress components are given by x2

trr = P + fl -~

exp (a(I1- 3)),

tzz = P + fl,~2

too =

exp (u(I1- 3)),

B P + ~ 2 exp (a(I1-3)), x2

1

(17)

I1 = , ~ 2 + ~ + ~ .

Since the tube is subjected to an inner pressure Pi and zero pressure on the outer surface, we should have trr = - P i

at x = xt, and trr = 0 at x = Xo.

(18)

The solution satisfying the boundary conditions (18) and the equilibrium equation, in cylindrical coordinates, dtrr +l_ ( t r r - t o o ) dr r

= 0

may be given as follows: exp (a(I1- 3))dx,

too =

x+

exp ( ~ ( I 1 - 3 ) ) d x + ~B x~ - 7

exp (~(I1-3)),

(19)

Zo

exp (a(I1 - 3))dx + f l (~4- x~) exp (~(I1 - 3)). Using the first of conditions (18) in the above equation, the following relation is obtained P~ = - ~ J x ,

x+

exp ~(I1-3))dx.

(20)

710

HILMI DEMIRAY

I f a similar a r g u m e n t , discussed in Section 3, is used here a m o r e useful expression involving o n l y ~ m a y be given, i.e. G(a) = P c |

x+

exp (a(I1-3))dx-P~

i"

fx.( !) x+

e x p ( a ( I 1 - 3))

t] X i p

x dx = 0,

(21)

where (P6, x;, %) a n d (P~ , x , , Xo) are t w o sets of m e a s u r e m e n t s . Some tests on arteries h a v e been done b y S i m o n et al. (1972) for various s t r e t c h ratios. T h e result of their m e a s u r e m e n t s for ~ = 1.53 is given in T a b l e I I . I f the p r e s e n t result is c o m p a r e d with this e x p e r i m e n t the c o n s t a n t s ~ a n d fl are f o u n d to be: a = 1.948, fl = 9.9 x 104 d y n e s / c m 2. The l a t t e r one is the same with one f o u n d b y Simon et al. (1972). H a v i n g these m a t e r i a l constants, r

tt

TABLE II P r e s s u r e - R a d i i D a t a for an A r t e r y PC Exp. (dyne/cm 2)

re

ro

rm

(cm)

(cm)

(cm)

PC Calc. (dyne/cm 2)

Deviation (%)

33350 66700 100000 133400 200000 266800

0.348 0.396 0.425 0.442 0.467 0.483

0.401 0.445 0.473 0.485 0.510 0.524

0.375 0.420 0.449 0.464 0.489 0.582

32000 66300 100000 139000 200000 265300

- 4.0 - 0.5 0.0 + 4.0 0.0 - 0.5

At zero pressure r I = 0.312 cm, r o = 0.380 cm. f r o m (20) one can o b t a i n the calculated values of Pc, a n d the deviations b e t w e e n the e x p e r i m e n t a n d t h e n m e c h a n i c a l model. These quantities h a v e been calculated a n d the result is listed in the fifth a n d last columns of t h e table, respectively. As is seen f r o m this table, the m a x i m u m deviation is only 4 per cent, which is r e m a r k a b l y close to e x p e r i m e n t s . T h e v a r i a t i o n of P l w i t h ri is depicted in Figure 4. T h e v a r i a t i o n s of wall stresses a n d elastic stiffness can be calculated in a m a n n e r similar to one which has been done in Section 3, b u t we do not list t h e m here. I n s u m m a r y , we can d r a w the following conclusion f r o m this study. The present d e r i v a t i o n gives a suitable a n d fairly simple f o r m for the s t r a i n - e n e r g y f u n c t i o n t h a t explains certain p h e n o m e n a which h a v e been o b s e r v e d in the e x p e r i m e n t s . The solution of two f u n d a m e n t a l p r o b l e m s of biophysics are p r e s e n t e d a n d their consistency with a p p r o p r i a t e tests are checked. I t has been s h o w n t h a t for each p a r t i c u l a r case the values of m a t e r i a l c o n s t a n t s can be found so as to m a t c h the t h e o r y to e x p e r i m e n t s .

BASIC

PROBLEMS

IN BIOPHYSICS

711

25

20

~

Calc

/

15-

,r

o_ .

IO-

a2-

o

0.312 Figure 4.

1

0.352

I

0.392 r]

I

0.432

0,472

Pressure-inner radius variations for a canine abdominal aorta (2 = 1.53)

The problems treated in this work are statical in nature. However, since the external effects (both physiological and psychological) are in continuous change, the real problems are dynamical. In particular the radial motions of left ventricle and arteries are almost periodic in time. Because of the mathematical complexity of the problem, a closed form of solution may not be found although some approximate results are obtainable.

LITERATURE Blatz, P. J., B. M. Chu and H. Wayland. 1969. " O n the mechanical behavior of elastic animal tissues." Trans. Soc. Rheology, 13, 83-102. Demiray, H. 1972. " A note on the elasticity of soft biological tissues." J. Biomech., 5, 309-311. Demiray, H. 1975. " O n the constitutive equations of biological materials." J. Appl. Mech., ASME, 241 Series E, 242-243. Eringen, A. C. 1962. Nonlinear Theory of Continuous Media. New Y o r k : McGraw-Hill. Fung, Y. C. 1967. " E l a s t i c i t y of soft tissues in simple elongation." Am. J. Physiol., 213, 1532-1544. Gou, P. F. 1970. "Str,ain-energy functions for biological tissues." J. Biomech., 3, 547-550. Mirsky, I. and W. W. Parmley. 1972. "Assessment of passive elastic stiffness for isolated heart muscle and the intact h e a r t . " Circ. Res., 33, 233-243. Mirsky, I. 1973. "Ventricular and arterial wall stresses based on large deformation analysis." Biophys. J . , 13, 1141-1159. Monreo, R. G., W. J. Gamble, C. G. Lafarge, A. E. K u m a r , J. Stark, G. L. Sanders, C. P h o n p h u t m u l , and M. Davis. 1972. " T h e Anrep effect reconsidered." J. Clin. Invest. 51, 2573-2583.

712

HILMI DEMIRAY

Mooney, M. 1940. "A theory of large elastic deformation." J. Appl. Phys., 11, 582-592. Patel, D. C., J. S. Janicki and T. E. Carew. 1969. "Anisotropic elastic properties of the aorta in living dogs." Circ. Res., 25, 765-779. Rivlin, R. S. 1960. "Large elastic deformation of isotropic materials." Phil. Trans., A.240 459-508. Simon, B. 1%., A. S. Kobayashi, D. E. Strandness and C. A. Wiederhielm. 1972. "1%eevaluation of arterial constitutive laws." Circ. Res., 30, 491-500. Snyder, 1%. W. 1972. "Large deformation of isotropic biological tissue." J. Biomech., 5, 601-606. Spotnitz, H. M., E. H. Sonnenblick and D. Spire. 1966. "Relation of ultrastructure of function in the intact heart: Sarcomere structure relative to pressure volume curves of intact left ventricle of dog and cat." Circ. Res., 18, 49-66. Vaishnav, 1%. M., J. T. Youn, J. S. Janicki and J. D. Patel. 1972. "Nonlinear anisotropie properties of the canine aorta." Biophys. J., 12, 1008-1027. Vito, 1%. 1973. "A note on arterial elasticity." J. Biomech., 6, 561-564. Streeter, Jr. D. D., 1~. N. Vaishnav, D. J. Patel, H. M. Spotnitz, J. 1%oss, Jr. and E. H. Sonnenblick. 1970. "Stress distribution in the canine left ventricle during diastole and systole." Biophys. J., 10, 345-363.

RECEIVED 3-14-75 REVISED 2-20-76