A continuum approach to blood vessel growth: Axisymmetric elastic structures

J. theor. Biol. (1981) 91, 273-301

A Continuum Approach to Blood Vessel Growth: Axisymmetric Elastic Structures ALLEN Department (Received

M.

WAXMAN

of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. 10 October 1980, and in revised form 8 February

1981)

In this paper we discuss the morphogenesis of small blood vessels (venules) as the nautral consequence of physical forces prevailing during endothelial cell division. A physical model is developed in which the blood vessel is treated as a growing, thin elastic shell embedded in a viscous fluid (i.e. the surrounding tissue). It is explained how a pre-existing cylindrical vessel, induced to grow by some promoter, can buckle and thereby develop a spatially periodic structure displaying varicosity, sinuosity, and/or helicity. Growth manifests itself dynamically in terms of a “growth pressure” which disturbs any pre-existing force balance. The governing set of non-linear partial differential equations are derived, and solutions corresponding to uniform dilation are obtained. The buckled structure emerges as an instability of this time dependent basic state of uniform dilation. A linear stability analysis yields the dominant wavelength of the varicose mode; these results compare favorably with crude measurements made from the experimental literature. In the hope of uncovering the mechanism which underlies the selection of sprouting sites along a parent vessel, it is suggested that reaction and diffusion processes (between growth promoting and inhibiting substances) on buckled surfaces be coupled to the dynamical force balances discussed here. 1. Introduction A recent paper by Waxman (1981) discusses the structure which a pre-existing blood vessel can develop in response to a growth promoting substance. Various experiments on different host tissues point to a definite sequence of developmental stages comprising the vascularization process. Examples from tumor angiogenesis (Eddy & Casarett, 1973), would healing (Schoefl, 1963), organogenesis (Wolff et al., 1975), and vascularization of the uterine wall (Ramsey, 1955), seem to indicate that the first response to induced growth is a change in the shapes of individual pre-existing blood vessels (venules) with diameter less than about 100 pm. This is followed by the sprouting of new blood vessels from the pre-existing ones, elongation of these sprouts and fusion with each other to form loop capillaries. The 273

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entire process then cascades to smaller scales which leads to the formation of a capillary network, which may be further modified by interaction with the surrounding tissue. This sequence of events may be viewed as a problem in morphogenesis, and in this paper we shall concentrate on the first stage of this process, the growth and subsequent buckling of the small pre-existing blood vessels. (Various modes of vessel growth are illustrated in Fig. 1.) The point of view adopted in this work is that growth upsets the equilibrium of forces in a structure, and the evolving geometry which ensues is a consequence of the need to maintain a force balance throughout at all times. That the balance of physical forces may play a central role in morphogenesis is by no means a new concept in developmental biology. It is discussed in the works of D’Arcy Thompson (19171 and Rashevsky (19481, and is implicit in both the experiments on amphibian embryos by Beloussov, Dorfman & Cherdantzev (1975), and in the theoretical work of Greenspan (1976) on the growth and stability of tumors. These physical ideas are embodied here in the mathematical formulation of an elastic continuum theory of blood vessel growth, the development of structure emerges as a buckling instability. Similar notions of growth and instability may be found in the fluid dynamical theory of Greenspan (1976) mentioned above. The organization of this paper is as follows. In the next section we discuss some geometrical consequences of growth in a confined space, and the physical ideas which underlie the elastic shell model of a growing blood vessel. In section 3, we describe the physical laws and dimensionless parameters which govern our system. This is followed by a discussion of the analysis and its results. A comparison of the predicted modes of instability with some crude measurements from the experiments cited above, is made as well. (A detailed derivation of the governing equations, and their solution, is given in the appendix.) We conclude with some remarks in section 4 on the implications of this work and future directions of endeavor. 2. Physical Considerations

of the Model

The following geometrical considerations serve to motivate the physical model to be discussed below. Before any growth promoter is introduced the pre-existing blood vessel is assumed to possess a cylindrical geometry which derives its support from the blood fluid which flows through it. Moreover, this vessel is connected to neighboring vessels at each of its ends. This is an important point since any extension of the vessel in the axial direction (which may be induced by growth) will be impeded at its ends because of the proximity to neighboring vessels. Any axial displacements of a vessel would

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imply a motion of the neighboring vessel through the surrounding tissue medium, as well as a possible stretching of that vessel. It is likely that the resistance to such motions would be great, and thus, we must rule out any modes of growth which require axial displacements of the growing vessel’s endpoints. A second geometrical fact concerns the thickness of the vessel wall. For blood vessels of diameter less than about 100 km (i.e. venules) the wall is composed of a monolayer of flattened endothelial cells surrounded by basement membrane, pericytes, or smooth muscle cells depending on the size and role of the particular venule (cf. Rhodin, 1968). The thickness of this layer is of the order of a few microns or less, so the ratio of wall thickness to vessel radius is about one-tenth. (The dependence of our results on this thickness ratio is shown explicitly in section 3b below.) Now when an endothelial cell divides, either the parent or the daughter cells must push neighboring cells aside in order to make room for mature daughter cells in the flattened state. (In addition, surrounding basement membrane and muscle cells, if present, may have to be stretched in order to accomodate the increased surface area.) Hence, the vessel retains its thin walls, and may be considered a two-dimensional surface embedded in a three-dimensional space. As the pre-existing surface generates more surface, the vessel must alter its geometry in order to accommodate this new surface area, and it must do so without displacing its endpoints in the axial direction.

Several possible modes of growth arise and are illustrated in Fig. 1. In each case the vessel deforms to a new geometry with greater surface area, a

b

C

d

.

X-

*x-c

FIG. 1. Modes of vessel growth between fixed ends: (a) uniform dilation, structure, (c) sinuous structure, (d) helical structure. Wavelengths A are indicated: lines in (a) and (b) indicate the original cylindrical vessel of radius Ro.

(b) varicose the broken

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but with the same axial extent. Figure la shows a uniform dilation from the original cylinder of radius R. to another cylinder of larger radius. A varicose mode of growth is displayed in Fig. lb: this mode has the bead-like appearance often attributed to growing blood vessels (cf. the experimental papers cited in section 1). Figures lc and Id illustrate sinuous and helical modes, respectively, and they are characteristic of the tortuous structure which growing vessels are known to exhibit. (In LVLY~examples of these structures are illustrated in Fig. 1 of Waxman. 198 1). These various modes are all shown to display a periodicity in the axial direction with a definite wavelength A as indicated in the figure. We conjecture that they arc all possible consequences of a vessel generating equn[ amounts of surface area per unit existing area (i.e. uniform growth). and in this paper we shall concentrate on the axisymmetric modes, uniform dilation and varicosity. The fact that periodic geometries emerge from uniform growth is a consequence of the physics underlying the model vessel. In this work we adopt the point of view that biological shape is governed by a dynamic balance of forces, growth implying a slow evolution of this basic force balance. We demonstrate here that structure may emerge as the consequence of an instability of the force balance associated with a shape of simpler geometry. Indeed, this was the spirit of Greenspan’s (1976) analysis of growing spherical tumors. If growth is to be able to disturb a pre-existing force balance, then some aspect of the growth process must directly enter the dynamic balance of forces. In fact, we have already identified a mechanism by which dividing endothelial cells may exert a stress on the system; it is the flattening out of, or making room for daughter cells. As mentioned above, neighboring cells must be pushrd aside, thus, a “growth pressure” is exerted which alters any existing force balance. This “growth pressure” appears as an additional dynamical variable and, as in the dynamics of a liquid continuum. is ultimately determined by an equation of continuity (cf. the Appendix B). In this work we have chosen to model the layer of cells comprising the vessel wall as an ideal elastic continuum, the vessel itself being treated as a thin elastic shell (cf. Fliigge, 1960). Continuum modeling of tissue behavior has been used in the past by Greenspan (1976) and Philips & Steinberg (1978). When viewed as an elastic continuum, deformation of the vessel is described by displacements of material, while growth implies relative displacements between points, i.e. strains. Such strains are associated with elastic stresses as determined by the stress-strain relationship for the material. These internal stresses, along with the “growth pressure” mentioned above, as well as any external stresses on the shell wall must all be considered in the force balance which governs the structure. The shape

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which the shell assumes is that for which all forces are in balance and to which the originally cylindrical shell can deform most quickly. Conversely, we may view the growth process as the local build-up of “growth pressure”. This pressure deforms the surrounding material, thereby generating elastic stresses which result in a new force balance. As the ends of the shell are constrained in the axial direction, growth leads to an axial loading of the cylindrical shell and this load may initiate a buckling of the structure. This scenario is analogous to the thermal expansion of a heated, elastic, cylindrical shell with fixed ends, and the buckling instability which ensues (cf. Boley & Weiner, 1960). Two types of shell models have been considered. The first treats the thin shell as an isotropic, Hookean elastic solid across the entire thickness of the wall. This model is referred to as a “solid shell”. The second model is that of an “inflated shell”, i.e. a shell constructed of inner and outer elastic membranes separated by inflating liquid. The difference between these shells lies in the stress-strain relation and manifests itself in the greater bending rigidity or the inflated shell. (The details concerning these models may be found in Appendix C.) Considering that the vessel wall is actually a layer of cells, a cell being considered an elastic bag containing cytoplasm, we feel that the inflated shell model is the more realistic one. In addition to the internal shell forces discussed above, we must also take into account the stresses exerted on the deforming shell by the surrounding tissue medium. Since the overall time scale of growth is about a day, we have chosen to model the tissue as a viscous fluid. That tissue may flow on a time scale of hours is documented in the experimental work of Hickman et al. (1966), and Phillips & Steinberg (1978). On the other hand, we may ignore the stresses exerted by the blood flowing within the shell as compared to the tissue stresses due to the much larger viscosity of the tissue (Gordon et al., 1972). (The main role of the blood in this problem is to inflate the vessel to its cylindrical geometry. Viscous stresses associated with the blood flow lead to a downstream pressure drop causing a slight tapering of the vessel. Within the realm of linear elasticity theory, such tapering will not effect our results in any qualitative way.) Before going on to a description of the governing equations and discussion of the results, let us summarize here the elements comprising the model. The blood vessel is treated as an initially cylindrical, elastic shell with fixed endpoints. The shell thickness and density is taken as uniform and constant for all time. This shell is embedded in a surrounding tissue medium, of infinite extent, modeled as a viscous and incompressible fluid. The vessel is assumed to undergo uniform growth, i.e. equal production of surface area per unit existing surface area per unit time, and this growth manifests itself

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in the generation of a pressure between material elements comprising the vessel wall. When balancing forces in and on the shell, we must consider the fluid pressure and viscous stresses exerted by the surrounding tissue medium; in comparison to these we may neglect any stresses exerted by flowing blood. In addition, we have the growth pressure within the vessel wall, as well as the elastic forces associated with the displacements away from the initial cylindrical geometry. Our aim is to solve for the new, slowly evolving, vessel shape which emerges as a consequence of this force balance. 3. Growing (A)

DESCRIPTION

0~

THE

Axisymmetric

GOVERNING

Structures

EOIJATIONS

AND

PARAMETERS

Figure 2 illustrates a differential element of the shell of finite thickness h, axisymmetric about the z-axis, subject to the various tensions, moments, The middle-surface of the shell and stresses acting on its “middle-surface”. is that surface located halfway between the shell’s inner and outer boundaries. As explained in the appendix, for thin shells, the dynamics of the three-dimensional elastic structure can be reduced in an approximate way, to the dynamics of the two-dimensional middle-surface. The force balance at each point throughout the shell wall being replaced by a balance of forces on, and moments about, this middle-surface. Indicated in Fig. 2 are the longitudinal and azimuthal elastic tensions Ti and r,,, respectively, the longitudinal and azimuthal elastic bending moments Mc and M,, (associated with counter-clockwise rotations about the double-tipped arrows), the transverse elastic shear Q,, the normal stress u,, and longitudinal shear stress

FIG. 2. Differential element of axisymmetric shell of thickness h. Its middle-surface is subject to tensions T. and T,, transverse shear stress Q,. counter-clockwise bending moments MS and M!, tissue stresses v, and cq, and an isotropic “growth pressure” II.

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(rl exerted on the shell by the surrounding tissue, and the “growth pressure” II shown here as isotropic. (An isotropic growth pressure is associated with equal growth in the circumferential and longitudinal directions; growth with a preferential orientation would correspond to an anisotropic growth pressure. This is considered in more detail below.) Also shown in Fig. 2 is the radius of curvature p of the shell element in the meridional (r, z) plane; it makes an angle 4 with the axis of symmetry as indicated. The equations governing the balance of forces and moments about the middle-surface of the shell are derived in Appendix C. Utilizing the stressstrain relations for an isotropic, Hookean (linear) elastic, all the elastic tensions and moments may be expressed in terms of the Eulerian displacements of points from the initial cylindrical middle-surface. The result is two equations involving the growth pressure lI(r, t) and the two quantities &r, t) and [(z, t), the displacements in the radial and axial directions, respectively (There is no azimuthal displacement for the axisymmetric structures.) The radial displacement &z, t) also serves to define the geometry of the deformed middle-surface as is shown in Fig. 3. x

FIG. 3. Middle-surface geometries of axisymmetric structures with axial length L (all lengths in units of R,). Original cylindrical vessel has radius r, = 1; uniformly dilated vessel has radius r, = 1 + i(r); varicose vessel has radius r, = 1 + z(r) + [‘(z, t) = 1 + ((z, t) and wavelength A. Also shown are the Eulerian displacements 5, and &, of a point on the deformed middle-surface which, at time t is located at P(z), and was originally located at P(z - 5,) on the original surface.

In addition there is an equation describing the growth of the shell. It plays the role of a continuity equation for the middle-surface, stating that the rate of change of middle-surface area per unit area, following a particular element of surface, is given by a specified source function S. (The detailed derivation may be found in Appendix B.) This source function would generally depend on the concentrations of growth promoting and inhibiting substances present at the vessel surface. In this work we shall, for simplicity,

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consider the case of a source function which is a positive constant throughout space and time, i.e. uniform growth. Complementing these equations governing the growth and dynamics of the shell, are the hydrodynamic equations governing the viscous “tissue fluid” (cf. Appendix D). These two sets of equations are coupled: the motion of the shell wall providing boundary conditions for the tissue fluid motions, whereas the fluid motions generate stresses which are applied to the vessel surface. The constraint of fixed endpoints manifests itself in an additional boundary condition at the vessel’s ends, i.e. the axial displacement [( 2, f 1 must vanish at the endpoints z = 0 and z = L for all times t. The details of the analysis may be found in Appendix E-G. but here we simply describe the approach, explain the dimensionless parameters which arise, and summarize the results. The governing equations are solved in a two-stage process. The first stage focuses on the dilation of the vessel from its original cylindrical geometry, to a new cylindrical geometry of larger radius but of the same axial extent; the resulting increase of surface area being consistent with the rate of surface growth. A consequence of this growth is the establishment of various forces in and on the shell as described above. Though the solution corresponding to this uniform dilation represents an equilibrium of forces, it is not necessarily a stable equilibrium. Thus, the second stage of the analysis concerns the stability of the state of uniform dilation to infinitesimal perturbations of a varicose nature. The linearized perturbation equations admit a Fourier decomposition in terms of the wavelength of the varicosity (cf. Fig. lb). We then determine from these equations, that wavelength which becomes dominant after some characteristic period of time. Finally, we may compare this dominant wavelength with measurements of vessels displaying such varicosity. As would be expected, the wavelength which emerges from the analysis as the dominant one, depends on the quantities which characterize the material characteristics of the system as well as geometric parameters, e.g. shell thickness and radius, elastic constants, and tissue viscosity. Upon casting the governing equations in dimensionless form, we obtain the following dimensionless parameters of the system (cf. Appendix CI: 8-i

tl - VJ ‘h’/12 (1 -d’h’/4

for-‘solidshells” for “inflated shells”

and N= where

the constant

(l+uW/hE 1 (1 + v)@/2h,,,E ZJ is Poisson’s

for “solid shells” for “inflated shells”, ratio (1’ = $ for elastic elements

which

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conserve their volume when deformed), E is the Young’s modulus for the vessel wall, h is the wall thickness in units of R. (the initial cylinder radius), h, is the endothelial cell membrane thickness in units of Ro, p is the dynamic viscosity of the tissue, and S is the constant source function with magnitude S - lo-’ set-‘, i.e. S-’ - 1 day. The quantity /3 is a measure of the importance of bending moments as compared to elastic tensions, in the overall force balance. The thicker the shell wall h (measured in units of the vessel’s initial radius), the greater the resistance to bending, i.e. the shell will tend to buckle with a longer wavelength. Also note the greater bending resistance of “inflated shells” as compared to “solid shells”. In the limit h + 0, p + 0, and the shell reduces to a membrane with no bending resistance at all. The quantity N measures the relative importance of external tissue stresses to elastic stresses. Due to uncertainties associated with magnitudes of the various physical constants, we can only estimate a range for the order of magnitude of N. With E - lo’-lo* gm/cm/sec’ (Wiederhielm, 1965; Fung, 1966), h - 0.1-0.2 and h, - 10e3 based on Rot 50 km (Fung, 1966), S-lo-’ set-‘., and p - lo’-lO’gm/cm/sec (Gordon et al., 1972), we estimate N-10-‘-10-’ for solid shells and N - 1O-5-1O’ for inflated shells, though a value of N 6 10d3 is probably most reasonable in both cases. We shall find the results of interest to be insensitive to the value of N for N 5 10P3, however the size of p turns out to be significant indeed. (B)

RESULTS

FOR

THE

DOMINANT

VARICOSE

MODE

Instead of discussing the dominant wavelength A (measured in units of the vessel’s initial radius), we shall express our results in terms of the dominant wavenumber k. The two are simply related by k =277/A. The analysis in Apendix E-G yields the dominant wavenumber k = (2/p)“4 in the estimated physiological range 0
for “solid shells” for “inflated shells”.

With h = O-1, this implies a dominant varicose mode with k = 6 for “solid shells” and k = 4.5 for inflated shells”. Though the linear stability analysis yields this buckling mode as the one which grows fastest it also shows us that the buckling phenomenon is rather broadbanded. Thus, we find that all modes with 5 5 k I 7 are possible dominant modes, and so the one to be realized in a particular experiment may depend to some degree on the initial spectrum of perturbations. Hence, we should expect some inherent spread

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in the data rather than one particular wavenumber emerging as dominant. For other values of h, this broadbandedness is centered about different values of k as given by the formula above. Moreover, larger values of N increase the dominant k for fixed h (cf. Appendix E), however, we feel the relevant range of N is represented by the expression given. That is, the results are insensitive to the particular value of N for 0 1 implies a longitudinal preference. Due to the orientation of the individual endothelial cells, one might expect daughter cells to be aligned most often in the longitudinal direction. This simple generalization would allow us to investigate the affects of a growth anisotropy on the dominant varicose mode. Thus, in the equations of surface force balance we merely replace ( 7’, - II1 by ( Tl -II, 1= ( T, - cyII), and proceed with the analysis exactly as before. We find that anisotropic growth modifies the dominant wavenumber so that we must multiply the above expression for k by the factor [(l + ~7)/2a]“~ . This correction factor is applicable for values of (Y c 5, though the factor itself is quite insensitive to values of cy 2 5. Our results are summarized in Fig. 4 which illustrates the predictions of the theory and some crude measurements in the amplitude wavenumber (A-k) plane. (Both A and A = 2r/k are measured in units of R,,.) The linear stability analysis is applicable only to infinitesimal amplitude (A + 01, the dominant wavenumber being indicated by arrows pointing to the predicted values on the k-axis. These predictions are valid for values of N s 10 3 and are indicated for “solid shell” models by the arrows above the k-axis, and

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1 Hamster tumor Muscle wound Cornea lesion Rot tongue

O-I

O’Oo

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. 0 X A

-

h=O.l 0=CO a=1

I 1

7 a=caa=l h-0.2

a=~) a=1 h=O.l k

4. Amplitude A versus dominant wavenumber k for the varicose mode. Measurements from experiments are plotted along with least-squares fits to the data (solid line: linear; broken curve: exponential). Uncertainites in the data (25% in k and 50% in A) yield the region of interest lying between the two broken lines. Arrows indicate predictions of the buckling theory for A + 0. Arrows above the k-axis are for “solid shells”, arrows below are for “inflated shells”. Parameters h and a refer to the shell thickness to radius ratio, and degree of anisotropy of growth, respectively. FIG.

for “inflated shell” models by those below. Results are given for two shell thicknesses, h = O-1 and h = 0.2, and for two values of the anisotropy parameter, cr = 1 (isotropic growth) and LY+ co (a strong bias towards longitudinal growth). The data points in Fig. 4 represent measurements made from the existing experimental literature cited in section 1. These rough measurements of amplitude (indicated by A in Fig. 3) are subject to uncertainty of about SO%, and those of k to about 25 % ; this is reflected in the broken lines which bound the data points and indicate the region of interest in the A-k plane. (Measurements were made from photographic reproductions using a millimeter ruler and a magnifying glass. Error estimates are based on an assumed measuring accuracy of about a mm.) The data suggests a correlation between amplitude and wavenumber, though the need for better data is clear. The solid line represents a linear least-squares fit between k and A with k =3*3-5.5A and a correlation coefficient of 0.78; the broken curve represents a linear least squares fit between In (k) and A with

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k = 4.0 exp ( -3.4A) and a correlation coefficient of 0.85. In the limit A + 0, these two methods of extrapolation lead to a dominant wavenumber k of about 3 to 4 (implying a wavelength of about 1.S to 2 times the radius RO). Given the accuracy of the measurements, and the extrapolations based upon them. we see that the predictions for the “inflated shell” models give quite comparable results. 4. Concluding

Remarks

In this paper, we have concentrated entirely upon axisymmetric elastic shell models of growing blood vessels. The results of the linear stability analysis compare quite favorably with crude measurements made from the existing experimental literature. Still, other studies would be pertinent to this phase of the theory. A finite-amplitude analysis would facilitate comparison with observations, whereas the equations governing non-axisymmetric shells must be formulated in order to investigate the sinuous and helical buckling modes illustrated in Fig. 1. Other continuum models should also be considered. Instead of an elastically dominated surface, it is reasonable to consider a surface which exhibits some fluidity as well. (The visco-elastic liquid behavior of embryonic tissues has been studied experimentally by Phillips & Steinberg, 1978. Fluid dynamical models of growing cell cultures have been developed by Greenspan, 1976.) The sprouting of new vessels from deformed parent vessels raises new questions on the mechanism for locating sprout sites. These points are discussed in Waxman (1981) where we speculate on the existence of a diffusible growth inhibitor produced by the replicating endothelium. Thus, it may be worthwhile to study reaction-diffusion dynamics (cf. Segel & Jackson, 1972) on buckled surfaces in the hope of identifying regions of enhanced growth, i.e. sprouting sites. Nevertheless, the sprout shapeswould be determined by the need to maintain a surface force balance at all times. The resulting investigation would couple the reaction and diffusion of growth promoters and inhibitors to the force balances discussedin this paper. (The interaction between diffusion processesand dynamical processeshas been considered by Greenspan 1977a, 6, 1978, with application to fluid dynamical simulations of cell cleavage and movement. 1 In and of itself, this work may be considered one small step in a more general treatment of the morphogenesis of tissue monolayers embedded in three-dimensional space. Clearly, much work remains to be done. This researchwaspartially supportedby the National ScienceFoundation, Grant Number 78-14521 MCS.

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REFERENCES BELOUSSOV, L. V., DORFMAN, J. G. and CHERDANTZEV, V. G. (1975). J. Embryol. exp. Morph. 34,559. BOLEY, B. A. & WEINER, J. H. (1960). Theory of Thermal Stresses, New York: Wiley. EDDY, H. A. & CASARETT, G. W. (1973). Microvasc. Res. 6,63. FL~GGE, W. (1960). Stresses in Shells, chapters 5 and 6. Berlin: Springer-Verlag. FUNG, Y. C. (1966). In Biomechanics (Y. C. Fung, ed.) Amer. Sot. of Mech. Eng., pp. 151-166. GORDON, R., GOEL, N., STEINBERG, M. & WISEMAN, L. (1972). J. theor. Biol. 37,43. GREENSPAN, H. P. (1976). .Z. theor. Biol. 56,229. GREENSPAN, H. P. (1977a). Stud. appl. Math. 57,45. GREENSPAN, H. P. (19776). Z. theor. Biol. 65,79. GREENSPAN, H. P. (1978). J. theor. Biol. 70, 125. HAPPEL, J. SC BRENNER, H. (1965). Low Reynolds Number Hydrodynamics, chapters 2 and 3. Englewood Cliffs: Prentice-Hall. HICKMAN, K. E., LINDAN, O., RESWICK, J. B. & SCANLAN, R. H. (1966). In Biomedical fluid mechanics symposium, Amer. Sot. of Mech. Eng., pp. 127-147. KOITER, W. T. (1959). In The theory of thin elastic shells (W. T. Koiter, ed.), IUTAM, pp. 12-33. LEIPHOLZ, H. (1974). Theory of Elasticity, Leyden: Noordhoff International. NASH, W. A. & Ho, F. H. (1967). In Proceedings of the First International Colloquium on Pneumatic Structures (D. Feder, ed.), Int. Assoc. for Shell Structures, pp. 108-117. PHILLIPS, H. M. SC STEINBERG, M. S. (1978). J. Cell Sci. 30, 1. RAMSEY, E. M. (1955). Angiology 6,321. RASHEVSKY, N. (1948). Mathematical Biophysics, 2nd ed., Chicago: University of Chicago Press. See (1960) 3rd rev. ed., Vol. 1, chapter 27. New York: Dover. RHODIN, J. A. G. (1968). J. Ultrastructure Res. 25,452. SCHOEFL, G. I. (1963). Virchows Arch. Path. Anat. 337,97. SEGEL, L. A. & JACKSON, J. L. (1972). J. theor. Biol. 37,545. THOMPSON, D. W. (1917). On Growth and Form. Cambridge: Cambridge University Press. See (1961) abridged ed. (J. T. Bonner, ed.), chapter 4. WAXMAN, A. M. (1981). Microvasc. Res. in press. WIEDERHIELM, C. A. (1965). Fedn. Proc. 24,1075. WOLFF, J. R., GOERZ, CH., BAR, TH & G~LDNER, F. H. (1975). Microvasc. Res. 10, 373.

Notation A C D -D Dt E m F G H

amplitude of varicosity measured from experiments (in units of Ro) function defined by equation (G27b) extensional rigidity convective (material) derivative with respect to time, on the middlesurface elastic (Young’s) modulus of the vessel wall deformation measure defined by equation (F19) (circumflex is then dropped) function defined by equation (F25b) function defined by equation (F25c) function defined by equation (F25d)

286 h hm K K, k L Ml Mtl N P P

QI RO

Ll us

W W WS

A.

M.

WAXMAN

thickness of blood vessel wall (in units of R,,) thickness of endothelial cell membrane (in units of I?(,) bending stiffness modified Bessel function of order J’ wavenumber of the perturbation length of the blood vessel moment in the longitudinal direction moment in the azimuthal direction dimensionless parameter indicative of the ratio of viscous to elastic forces function defined by equation (F24d) tissue-fluid pressure transverse shear stress initial mean radius of the cylindrical blood vessel radial co-ordinate measured from the axisof symmetry of the vessel radial location of the middle-surface as a function of : and f rate of production of area per units of middle-surface tension in the longitudinal direction tension in the azimuthal direction time radial velocity of the tissue-fluid radial velocity of the middle-surface function defined by equation (F24e) axial velocity of the tissue-fluid axial velocity of the middle-surface axial co-ordinate measured along the axis of symmetry of the vessel measure of anisotropy of growth dimensionless parameter indicative of the importance of bending infinitesimal element of middle-surface area axial extent of 6A dimensionless measure of the amplitude of the wavy perturbation axial displacement of the middle-surface wavelength of the perturbation Poisson’sratio for the vessel wall radial displacement of the middle-surface growth pressure radius of curvature of the middle-surface in the meridional plane longitudinal stressexerted by the tissue-fluid on the middle-surface normal stressexerted by the tissue-fluid on the middle-surface angle between p and the z-axis (cf. Fig. 2)

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Some quantities are decomposed into uniform and wavy components, thus giving rise to symbols with overbars, primes, and circumflexes. A typical decomposition is [(z, t) = c(t) + l’(z, t) where [‘(z, t) = .&I) sin kz. Quantities followed by a comma and subscript imply partial differentiation of that quantity with respect to the subscripted variable. APPENDIX:

DERIVATION AND

OF THE GOVERNING THEIR SOLUTION

EQUATIONS

In the classical theory of thin elastic shells, the dynamics of the threedimensional structure is reduced to the dynamics of a two-dimensional surface, the so-called “middle-surface”, by referring all physical quantities which vary across the shell’s thickness to those midway between the she!l’s inner and outer surfaces via the first few terms of a Taylor expansion (Fliigge, 1960). Utilizing the Love-Kirchhoff postulates, averaging various quantities over the finite thickness, and taking the limit h/p + 0 where h is the shell thickness and p is the smallest radius of curvature of the shell, a consistent first approximation to the three-dimensional problem is obtained (Koiter, 1959). Here, the equations governing this middle-surface are derived and generalized to include the effects of an isotropic growth pressure. After completing the mathematical formulation of the model problem, the equations are solved by means of a perturbation analysis in order to determine the dominant varicose mode of buckling. (A) Kinematic

Relations

Figure 3 illustrates the middle-surface of an axisymmetric shell in its initial state with (dimensionless) radius r, = 1 (we shall measure all lengths in units of R0 and time in units of days), its dilated state with radius r, = 1 + f(t), and its buckled state with radius r, = 1 + c(t) + [‘(z, t) = 1 + c(z, t). The quantity t(z, t) is the (dimensionless) radial displacement from the initial cylinder of a point on the middle-surface which has axial location z at time C.Similarly, t(z, t) is the displacement in the axial (or z) direction of a point on the middle-surface which at time t is located at z. Thus, T(z, t) and [(z, t) are the (dimensionless) Eulerian displacements of a point which, at time t = 0, was located radially at r, = 1 and axially at z - [(z, t). For structures which are axisymmetric for all t, the displacement in the azimuthal direction vanishes. Now, consider a point which at time t has co-ordinates (r, z) = (1 + & z). After a short interval of time At, this point will have moved to a new position

288

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with co-ordinates ( 1+ 5 + A& z + AZ ), where A[ = [(z + AZ, t + At ) - c$(2. 1). and in the process will have changed its axial displacement by an amount Al = A: where A< = <(z + AZ, t + At, - il z. t). In the limit of II + 0 these relations become A[ = 5,; AZ + &lAt and A: = j.. AZ + l.lAt (where a symbol followed by a comma and a subscript implies the partial derivative with respect to that subscripted variable). However. in the limit of 1t + 0. the quantities A[/At and AZ/At tend to the radial and axial velocities, respectively, of the point itself. Thus, the kinematic relations between the (dimensionless) Eulerian velocities and displacements of points on the middle-surface are: If, = (,, + w,t. [Ala) It’, = i., + M’\(;.

IAlh)

and Da Dt

i)t

-en,

,!

talc,

‘;I:’

for the radial velocity, axial velocity, and convective derivative, respectively. (B) Equation of Continuity As in the mechanics of fluids, an equation of continuity refers to the production, disappearance, or conservation of massover relevant regions of space. For a growing shell of constant thickness and uniform density, this equation becomes, in the limit of vanishing thickness, a relation between the rate of change of middle-surface area and a source term of given strength per unit area. For a small element of middle-surface area SA, located at r, = 1 +[(z, t) with axial co-ordinate : and axial extent 8: at time t. the equation of continuity may be stated (in dimensionless variables I as R2a) where SA=2rr(l+~)(l+&‘)’

‘62

IB’b)

and S(z, t) is the rate of production of area per unit area of middle-surface. (In general, S would depend on the concentrations of growth promoting and inhibiting substances present at the shell surface.) Noting that D(Sz )/Dt = WV,,= Sz equations (A2) yield the following equation of continuity for the middle-surface, ~(1+~)(I+5.~)‘i7~rf+{(l+~)(1+~,i)”-

w,},: = {(l +or1 +&‘;,’ ‘}S(z. t). tB3)

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289

For the case of uniform growth considered in this work, S(z, t) is identically a constant and so may be set equal to unity in equation (B3) by measuring time in units of S-l. (C) Equations

of Surface Force Balance

As the timescale for blood vessel growth is much greater than the elastic pulse propagation timescale for this problem, we must require a quasisteady balance of forces on the vessel at all times (i.e. all terms associated with the inertia of the growing vessel may be neglected). Drawing heavily from the book by Fhigge (1960), we begin by balancing the forces and moments on the middle-surface of our buckled, axisymmetric shell. It should be stressed that our force and moment balancing is being done about the buckled surface and not the original cylindrical surface, as this will enable us to perform the buckling analysis by means of a simple perturbation expansion. (This is quite different than the classical approach discussed in Chapter 7 of Fliigge, 1960.) However, we must be careful to identify our forces and moments as Eulerian variables as opposed to the Lagrangian variables used in the traditional approach (cf. 02.42 of Leipholz, 1974). In Fig. 2, a small element of the axisymmetric shell of constant thickness h is shown subject to the various tensions (or stress resultants), bending moments (or stress couples), and external tissue stresses acting upon the middle-surface. The equations which describe the equilibrium of forces and moments, in the absence of any growth pressure II, are given by Fliigge (1960, cf. section 6.1.3) as, (rTI),+ -pTB cos d - rQ1 = --pm,

(C44

(rQ,),+ +pTo sin 4 + rT, = -pm,,,

(C4b)

bA4,),~ -pi%

cos 4 = prQl.

(C4c) Measuring all lengths in units of Ro, we replace r by [l +e(z, t)] and note the geometrical relations cot 4 = & and dr = (p dr$) sin 4, where the radius of curvature is given by p-l = -&= (1 + &J* 3’2 . Then transforming derivatives with respect to 4 in equations (C4) to derivati;es with respect to z, we obtain,

[Cl + 5)QAz + Te- 5,zz(l + 5W + t,:,-‘7’1 = --(I+ cW+ &:)1’2R~~n, t-b) [Cl + ~P&l,z - &zMe= Cl+ f)U + 5,:J1’*RoQ~. (C5c)

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In order to include the effects of an isotropic growth pressure, assumed uniform across the shell thickness, we must replace 7’! and To in equations (C5) by (T, - KI) and (T, -n), respectively. Here, ll represents the growth pressure integrated over the shell thickness: it is a force per unit length of middle-surface. To lowest order in h (measured in units of R,,). a pressure which is uniform across the shell thickness does not alter the moments generated. Utilizing equation (C5c) to eliminate the transverse shear force Q[ from equations (C5a) and (C5b) we obtain the two force balance equations governing the middle-surface of the axisymmetric shell: !L( 1 +I$,:,

3’*~~;‘{[(i+~)~,],z --~..-~~~+[(i+t:u-

-5,.(TH-n,=-(1+~)(1+~,,j,’ R,‘{(l+&;f)

&;(l+&(l+&:)

IC6a)

%r,,

“~[(1+5)M~],;-5,;~1i-~.j) ‘(7,-n)=

ml.,

“‘M,,},,+(T,-ll) -(I+r)(l+E,~)“‘~R,,tr,,.

(C6b)

The tensions To and T,, and moments M,, and Ml appearing in equations (C6) may be related to the displacements [(z, t) and [( 2, 1) of the middlesurface by reduced forms of the usual stress-strain relations for an isotropic, Hookean (i.e. linear) elastic solid. These relations, derived for srrzall tiisplacements from an initially cylindrical structure, may be found in Fliigge (1960, cf. section 5.1.2) in terms of Lagrangian variables. To the order of accuracy required, they are the same for Eulerian variables, and we find for axisymmetric displacements (to lowest order in h ), and

T,,=D((+v[,;) M,, = UK&;

and

T, =D(<.; +~0,

tC7a)

M, = K[,,:.

IC’7b)

where for such solid elastic shells D =lzERc,/(l

-v’)

and

h’=h’R,,D/12.

Ic’7C)

In equations (C7), D is the extensional rigidity, K is the bending stiffness, E is the elastic or Young’s modulus, and 1’ is Poisson’s ratio which, for a shell element that conserves volume under deformation, takes the value u = l/2.

At this point we should note that equations (C6) embody the force and moment balances for the exact (i.e. deformed) middle-surface, while the elastic law equations (C7) has been given in terms of linearized strains. Thus, relations (C7) are valid only for small values of 5 and & In terms of the decomposition of 5 into $+[’ (cf. Fig. 3), our analysis is valid under the conditions 1 >>]z] >>I[‘/ >>I.$‘. We shall find c= O(O*l), implying strict validity 0 for 0.1 >>I[‘/ >>O.Ol.

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The expressions for D and K given in equation (C7c) are relevant to a thin shell which is treated as an isotropic elastic across its entire thickness, i.e. a “solid shell”. Another type of shell structure to consider is that of an “inflated shell”, or a shell whose wall is composed of inner and outer elastic membranes separated (or inflated) by a liquid (Nash & Ho, 1967). This is probably a more realistic shell model for the layer comprising the wall of a small blood vessel; the cell membranes being the elastic membranes with the cytoplasm acting as the inflating liquid. We treat the “two membranes” as composed of the same material, each of thickness h, and separated by a distance h (with h and h, in units of RO). Then the total tension for the composite structure is the sum of the tensions in the two membranes, whereas the bending moment is now the difference between the outer and inner membrane tensions multiplied by the distance to the middle-surface h/2. Upon relating these tensions to the strains and curvature changes at the middle-surface, we obtain “stress-strain” relations to lowest order in h for the inflated shell model of the form of equations (C7a) and (C7b) with D = 2h,ERo/(l

- y2)

and

K = h2R,Jl/4.

CC74

The fact that the ratio K/D is larger for inflated shells by a factor of three over that for solid shells implies a greater bending resistence for inflated structures. This disparity reflects the different stress distributions that exist across these two kinds of shells. We should also note that in deforming the inflated shell, the inflating liquid is caused to flow, and this gives rise to a dissipation of energy even if the elastic membranes are treated as nondissipative. However, the dissipation due to any cytoplasmic flow may be neglected in comparison to that due to the flow of any surrounding tissue; this is a consequence of the much greater tissue viscosity. There remains to specify the normal and longitudinal stresses, u,, and (T[ exerted by the surrounding tissue fluid on the shell (cf. Fig. 3), which appear on the right-hand sides of equations (C6). Expressions for these stresses (in dimensionless units) are given in the next section. However, here we simply note that they are of order pS, with the time measured in units of S’, where w is the dynamic viscosity of the surrounding tissue when treated as a viscous liquid. Then, utilizing equations (C7) to eliminate the elastic tensions and moments from equations (C6), measuring the growth pressure II in units of D, and measuring the tissue stresses in units of &, we obtain the middlesurface equilibrium equations in their dimensionless form;

Cl- vw5zz(l+ 591)-3’2{[(1 + ZkZLI,L - &5,rzl + Kl + 5)(&z+ vs -l-h - &z(5+ &z - m = -(l-

v)N(l

+e)(l

+.$,$1’2ar,

(C8a)

292

A.

(1-v)pu1+5,:,

M.

“‘[(1+0&z],;

WAXMAN -v&&z~(l

+(5+yi;---n,-~,,,;(1+5)(1+f;~,

+‘gj

‘(&,+Y~-n, =-cl

where

p and N are dimensionless P=[

’ 2},;

-VIN(l+[)(l

+(,i,’

>V,,, (CXb)

for “solid shells” for “inflated shells”

(C8C)

for “solid shells” for “inflated shells”,

(C8d)

quantities

(1 - ~)K’h’/12 (l-v)-‘h”/4

given by

with h and h, measured in units of Ro. The significance of these dimensionless quantities is discussed in section 3a. This completes the derivation of the equations governing the growing, elastic shell. (D)

Tissue-fluid

Stresses and Boundary

Conditions

The stresses (~1 and u,, on the right-hand side of equations (C8) represent the viscous stresses and pressure exerted by the trssue-fluid on the outside of the shell. These stresses, in units of +I$ may be expressed in terms of the dimensionless fluid variables p, U, and W, the pressure, radial velocity, and axial velocity, respectively. They are given by. u/ = (1 +&:J

‘{25,&4-

w,,)+(l

-&:i)(W,,+II,:)),

(T,,={p-2(1+~,~)~l[(rt,,+,~,.)

-[,;;(W,,+II,;)]},

I *:

,‘i

(D9a)

(D9b)

where these stresses are evaluated at the middle-surface r = r, = 1 + t( z, r) (cf. Fig. 3). Since the timescale of growth is much greater than the timescale for diffusion of vorticity in the tissue-fluid. the flow is effectively a low Reynolds number flow. Treating the tissue as an incompressible fluid without sources or sinks, the dimensionless variables II, IV, and p are solutions of the Stokes-flow equations (Happel & Brenner, 1965) r~‘(ru,,),, r

-rm2u ‘(rw~,,),r

F’(m),,

Note that equations

(DlO)

+ 14.~~ = p,, + M’,,; + II’.;

= p.; = 0.

(DlOa) (DlOb) (DlOc)

imply that p is a solution of Laplace’s equation.

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The dynamics of the growing vessel-tissue system are constrained by certain physical boundary conditions and initial conditions. The vessel is taken to have dimensionless length L, thus, the axial constraint at the endpoints requires [(z,t)=O

at

z=O

and

(Dlla)

z=L

for all time. No constraint is imposed on the radial displacement 6 at these endpoints. The tissue-fluid is treated as infinite in extent in the radial direction and bounded by the vessel on the surface r = 1 +&z, t), and by impermeable, slippery plates on z = 0 and z = L. This effectively isolates the tissue surrounding the growing vessel. The tissue flow must be regular for r + cc, satisfy no-slip conditions at the vessel, and have vanishing axial velocity and tangential shear stress at the plates. Hence, u = u,

and

w = w,

where U, and w, are the middle-surface and w = w,, + u,, =0 at

at

r=l+&z,t),

(Dllb)

velocities given by equations (Al), z=O

and

z=L

(Dllc)

for all t. The initial conditions are to imply that growth begins at t = 0, i.e. the growth promoter is “turned on” at this time. Thus, the displacements must be zero everywhere initially, [(z,t)=l(z,t)=O

at

t=O.

(Dlld)

However, in what follows, we shall allow for the presence of infinitesimal perturbations in all quantities. This completes the formal statement of the problem. (E)

Uniform

Dilation

as a Basic State

We now set out to solve, by means of a perturbation expansion, the equations governing our system; equations (Al), (B3) with S = 1, (CS), (D9), (DlO), and (Dll). The solution to these equations may be decomposed into two parts, a uniform dilation plus a wavy structure. For example, the radial displacement is written as the sum of two terms, .$(z, t) = g(t) + [‘(z, t). Thus, the location of the middle-surface is r, = 1 + c+ 5’; this decomposition is illustrated in Fig. 3. The state of uniform dilation shall be treated as an O(1) basic state, while the wavy component shall be viewed as a small perturbation on this time dependent solution. We begin with the equations governing uniform dilation. In the governing equations we set 5 = f(r), II = I@,), and l= f(t) = 0 in accordance with condition (Dlla), U, = ii,(t), w, = P,(t) = 0 as implied by

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relation (Alb), u = ll(r, t), w = W(r, t) = 0 as required by conditions (Dllb and c), and p = p(r, t). Of course, all partial derivatives with respect to z then vanish. We first solve equation (B3) for c(f) subject to the initial condition (Dlld). Then equation (Alb) yields i7,1TV directly which serves as a boundary condition on U(r, t) (cf. equation (Dl 1bl). Solving equations (DlO) for ~7 and p, we obtain ii,, from expression (D9b); 5, is identically zero. Returning to equations (C8) we see that equation (C8a) vanishes identically, and that equation (C8b) yields l?(t) directly. The complete results for this time dependent basic state of uniform dilation are

II, = l+&

G, = 0,

iE12al

and Ir = (1 +

IV = 0.

f)‘/r,

r, = 0,

tEl2b)

where the tissue pressure p is measured relative to the initial tissue pressure. As the vessel dilates, it enables a greater flux of blood to pass through it if the blood pressures at the vessel ends are maintained. However, as mentioned earlier, this blood flow plays no significant role in the growth dynamics of the vessel. (F) The Equation

Governing

Perturbations

We now consider the effects of infinitesimal, way perturbations superposed on the basic state of uniform dilation given by expressions (E12). When linearizing the governing equations about this basic state, we are effectively evaluating perturbation quantities on the unperturbed middlesurface, and basic state quantities on the perturbed middle-surface. The linearized form of the continuity equation (B3) is q,,+{(l while the kinematic

conditions

+c$)rt~:l. (Ala)

rv: = <‘,,

= <‘,

rF13a)

and ~Alb) become and

rc: = [I.,

(F13b, c)

since G’, = 0. Dividing equation (F13a) by ( 1 + $1, eliminating rr*: in favor of t’ according to relation (F13b), and noting the solution (E12a) for c(t), equation (F13a) may be brought to the form

{(1+5,-‘5’+4”‘,;},, =o. The linearization

of equation

(C8a) yields, after some rearrangement

1F14) of

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terms, -~‘,,+f’,~,+[~(l+~)-(1-~)~](1+~)~1~’,2=-(1-~)N~~r

(F15a)

where we have used the fact that & is identically ation of equation (C8b) yields,

zero. The direct lineariz-

Cl- v)P(l + w,zLzZ + (5’+ d-‘9, - w - (1+ a&-

l=oc?,zz

=-(l-v)N{&,5’+(1+&;}. Making

use of relations (Dgb), (E12a), and (E12b), this becomes -.‘+d,z

+(l -~m+&‘,zzzz

+(I-43

= -(1-v)N{2(1+~)2~‘,,, Eliminating

+2t’+(l

+w,zz

+5’

+{))(TI}.

(F15b)

II’ between equations (F15a) and (F15b), find

5’3 zz -u +2m

+a-%‘,Z

-al+&‘,zzz

-PU

= N{25’,= + 2(1+ f)2)25’,zzr+ (1 + C)&

+&‘>Zzzz*

-(ri}.

(F16)

Equations (F14) and (F16) govern the displacement perturbations 5’ and 5’; it remains to express a; and (T; in terms of these quaqtities. Linearizing relations (D9a) and (D9b) about the basic state yields, d = w&S’,,

+ w’,,+

u’,,),=l+F

+ u’,,l,=l+g

(F17a)

= -4(1 +.$)-15’+{$-2u’,,-2w’,,},=~+&

(F17b)

= -w,z

+{w’,r

and u:, ={~“,lr’+pf-2u’,,-2w’,,},=,+,-

where the first term in the expression for o; arises from evaluating &,, on the perturbed middle-surface; the remaining terms are evaluated on the unperturbed surface. In order to evaluate u; and uk, it is necessary to first solve for the perturbed tissue flow governed by the (already linear) Stokesflow equations (DlO), subject to the boundary conditions (Dllb) and (Dllc) linearized about the basic state. At this point it becomes advantageous to consider the individual Fourier components which make up the spectrum of perturbations. Formally, this is achieved by Fourier analyzing the linearized equations and boundary conditions, noting that the basic state has no dependence on the axial co-ordinate z. As the individual Fourier components must satisfy the constraints imposed by the boundaries, condition (Dl la) implies that C(z, t) possesses a Fourier sine-series such that the axial wavenumber k is an

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integer multiple of r/L (where L is the vessel length in units of Ro). Since in general L >>1, the quantized wavenumbers form a nearly continuous spectrum. Therefore, we shall treat k = 25-/A as a continuous variable, and consider the dynamics of an individual Fourier mode. The mode which ultimately dominates the spectrum will be obtained in the next sub-section. Given that 5’ possessesa Fourier sine-series, consideration of the equations and boundary conditions governing small perturbations suggests the following separation of variables for a particular mode: (‘(z, t) = Fk ‘&t, cos kz. [‘(z, t) = F[([) sin !c. p’tr,

z,

t) = Fb(r, t) cos kz.

u’(r, 2, t)= ~ii(r,

II cos kz,

bv’( r, z, t)=Fff(r,

r)sin kz,

(F18a-g)

a:,(.~, t) = u?,,(t) cos k,-, fli(=, t)= EC&(~)sin kz. where F <<1 denotes the amplitude of the perturbation. In addition, we have scaled the amplitude of 6’ by the factor k ~‘. This ensures that the perturbation to the middle-surface area associated which each Fourier component is equal through O(F)), since, after all, equation (B3) implies that growth is proportional to existing surface A area. Thus, stability in an absolute sense may be judged by the size of t(t) for individual Fourier modes. However, the overall deformation of the surface should really be considered relative to the basic state which itself is growing in time, that is, rather than studying the growth of i(t) alone, a more relevant measure of deformation is the ratio &( 1 + 4). Thus, defining the deformation measure

for each member of the spectrum of perturbations, substituting relations (F18) into equations (F14), (F16), (F17), and (DlO), and dropping the cumbersome circumflex notation, we obtain the following set of equations which govern the growth of individual Fourier components:

,,&-dE dt

dr

(F2Oa)

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and -k2~+{(1+2&-~(l+~)2k2+~(1+~)2k4}E =-N{2(1+$)[l-(1+~)2k2]E+(1+~)k~“++~},

(F20b)

where (F21a)

(+/=2(1+f)E+{w,,-ku},=l+g

and (F21b)

a,, =-4k-‘E+{p-2u,,-2kw},=1+~.

The perturbed

tissue flow is governed by u,,+r-*u,r-(k2+r-2)=p,n

(F22a)

-k2w

w,, + r-h,

=-kp,

(F22b)

and r-‘(w),,

(F22c)

+ kw = 0;

these equations imply that p is a solution of pIrr + r-lp,r

(F22d)

- k2p = 0.

Equations (F22) are to be solved subject to boundary conditions (Dllb) and (Dllc) linearized about the state of uniform dilation (E12). After some simple manipulations conditions (Dll b) become, at

u(r,f)=(l+F)k-‘($+2E]

r=l+$

(F23a)

and w(r, t) = -k

-2 dE

dt

at

r=l+.$.

(F23b)

Condition (Dllc) is satisfied by virtue of the Fourier decomposition (F18). In addition, it is required that U, w, and p be regular as r + CO. The solution of equations (F22) satisfying the appropriate boundary condition is, (F24a)

p(r, t) =P(ke’)&(kr) w(r, t) = $P(ke’)rKl(kr)+

W(ke’)&(kr)

(F24b)

u(r, t) = $P(ke’)rK2(kr)+

W(ke’)Kl(kr),

(F24c)

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where

(F24d)

and K,(X) is the modified Bessel function of order j which is regular for x + co. Using these expressions to evaluate the stresses c,! and (r( according to relations (F21), substituting those into equations (F20), and eliminating f between (F20a) and (F20b), we finally obtain a single ordinary differential equation governing the deformation measure E(t) of a single Fourier component, Nd k;

F(ke’$+Glke’)E

+${t~‘~H(k.

r,E}=o.

(F25a)

where

F(.r)-[Ku(x-)+K-(x)lixKll(.r)+K,ir,l Ko(x)K?(x)-K,(s)K,(x) G(x)=

2x{Ko(x)K,,(x)+.u’K,(x)K1II)~[x~K,,(x)K*(.u)-K,(.r)K,(x)

-

lIKo(x)Kz(s))

(F25b) (F25c)

and H(k,

t)=fik’-(r’m--

l)k’+2

TV ‘.

(F25d)

Equation (F25a) may be directly integrated once and then numerically integrated after initial conditions have been specified for E and dE/dt. Since E represents any perturbation, the initial conditions are arbitrary, though they should be O( 1) to conform with our scaling. In order not to bias the spectrum of perturbations towards any particular wavenumber, we shall choose the initial conditions to be the same for all values of k. Thus, without loss of generality, we may set E(O) = 1. For values of N in the physiological range of interest, the initial value of dE/dt does not alter appreciably the results to be obtained for dE/dt = 0 at t = 0. Hence, we shall consider in detail the solution of equation (F25a) based upon the initial conditions dE E(O)= 1 and $0)

= 0.

(F26a, b)

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(G) The Dominant

299

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Varicose

Mode

A single integration of equation (F25a), utilizing the initial (F26), yields the first-order ordinary differential equation

;F(ke’)z+

(;G(ke’)+

&Z(k,

t))E = C(k),

conditions

(G27a)

where C(k)=Nk-‘G(k)+2+/3k4

(G27b)

E(0) = 1.

(G27c)

and It should be noted from their definitions in equations (F25) that, the function F(x) is positive for all x > 0, whereas H(k, f) is initially positive for all k and decreases monotonically with t, passing through zero at a finite time. Equation (G27) is, therefore, regular for all finite t 20. However, the dimensionless quantity N, defined in relation (C8d), was estimated to be quite small, with N 5 O(10P3) considered as reasonable (cf. section 3a). This would suggest taking the limit of N * 0 in equations (G27), but this limit would be singular in two ways. For one, the initial condition on dE/dr could not be satisfied, but this problem could be rectified with a “boundary layer” in time of thickness O[k-‘F(k)N] = O(lON) at t = 0. A more severe difficulty associated with the limit N +O is that equation (G27a), regular for finite N, becomes singular for N = 0 with (G28a) tending to infinity as H+ 0 monotonically. As H passes through zero, solution (G28a) indicates that E jumps to negative infinity and then tends monotonically to zero. Of course, once HJO the solution (G28a) is no longer related to the regular solution of equation (G27a) for finite N. Nevertheless, the singular solution (G28a) yields useful information bearing on the question of the dominant varicose mode. We shall first consider the singular solution in more detail before resorting to numerical integration of equation (G27a) for finite N. Though solution (G28a) is singular for all k, we can ask for which value of k does E + co at the earliest time. Thus, consider the time at which H(k, t) = 0. According to definition (F25d), it is the solution of pk4-(et-l)k*+2 where p, defined in relation

e-‘=O,

(G28b)

(C&Z), is O(h2) with h = O(O.1). Conversely,

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equation (G28b) determines those values of k for which H = 0 at a given time t. Inspection of condition (G28b) reveals that a solution exists for which t = O(h) and k = O(hC”2 ). Upon expanding the exponentials for small t and retaining only the lowest order terms, this condition becomes approximately @%k’+1 : 0. lG28C) Equation

tG28c) has solutions

which are valid for t 2 (8/J )I”. The wavenumber for which E + m tirst is then given by relation (G28d) with t = (Sp )“7, i.e. k = (2/p )l”. Substituting definition IC8c) for 8, with Y = l/2. find k = (12/h’)“J 1 (4/h?“’

for “solid shells” for “inflated shells”,

lG28e)

with the “singularity” occurring at t = (4/3 1’ ‘h for solid shells and f = 217 for inflated shells. With h = 0.1, relation (G28e) yields a dominant varicose mode with k = 6 for solid shells and k = 4.5 for inflated shells. In Fig. 5a the singular solution (G28a) for solid shells with h = 0.1 is displayed as a function of (dimensionless) time for various values of the wavenumber k. The plot is limited to 15 E I 10 with “divergence” first occurring for k = 6 at t = 0.12. By this time the basic state has dilated by about 13%. Note, howezjer, that solutions for 5 I k 5 7 are nearl!~ indistinguishahlc~. that is. the buckling phenomenon is rather broadbanded without a particular I~zuenutnher clearly dominating all others.

'5

(bi

k=6

'7

Oi2

f

t

FIG. 5. Deformation measure E(r) for “solid shells’. of thickness “singular solution” (N = 01 for various wavenumbers k. (h) the regular In both cases. k = 6 is the dominant wavenumber.

ratlo /I -7 0.1: Ia) the solution for N = 10 ‘.

GROWING

BLOOD

VESSELS

301

Returning now to equation (G27a) for finite N, we must resort to numerical integration to obtain E(t). Integrations have been carried out for solid shells with h = 0.1 and various values of the parameter N in order to determine the dominant wavenumber in each case. These solutions satisfy both of the initial conditions (F26a) and (F26b). The results for N = 10e3 are shown in Fig. 5b and look very much the same for N < 10P3. The conclusions drawn are the same as those from the singular solution (G28a); k = 6 dominates though the effect is broadbanded. These solutions remain regular for finite t, and where solution (G28a) displayed a singularity at a finite time, there is now a change over to a new time scale such that the first term in equation (G27a) is O(1) even for small N. As the value of N was increased, the dominant wavenumber was also found to increase with k = 7.5, 11, and 24, for N = 0.01, 0.1 and 1, respectively. These results are discussed further and compared to experimental measurements in section 3 of the text.