Finite element analysis of blood flow through biological tissue

Int. J. Engng Sci. Vol. 35, No. 4, pp. 375-385, 1997

Pergamon

PII: S0020-7225(96)00108-5

FINITE ELEMENT

(~) 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225•97 $17.00 + 0.00

ANALYSIS OF BLOOD FLOW THROUGH BIOLOGICAL TISSUE

w. J. VANKAN, J. M. HUYGHE, t j. D. JANSSEN and A. HUSON Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands

W. J. G. HACKING Cardiovascular Research Institute, Department of Pharmacology, University of Limburg, Maastricht, The Netherlands

W. SCHREINER Department of Medical Computer Science, University of Vienna, Austria

Abstract--Quantitative analysis of blood perfusion has been performed with a hierarchical mixture model of blood perfused biological tissue, for low Reynolds steady state Poiseuillian flow through two-dimensional, computer generated, rigid, arterial vascular trees. The mixture model describes blood flow through the vasculature by spatial and hierarchical flow components, both related to the blood pressure gradient via an extended Darcy equation. The Darcy-permeability tensor is derived from the geometry of the vascular tree, the fluid viscosity, and the vascular hierarchy which is quantified by a dimensionless parameter x0. Finite element results of fluid pressure and flow on certain levels of the hierarchy are compared to correspondingly volume averaged pressures and flows calculated by hydraulic network analysis. The correspondence between the finite element results and the network results strongly depends on the quantification of the hierarchy, which should be closely related to the hierarchical fluid pressure. Two methods for hierarchical quantification have been used, one of which is based on the network fluid pressure and the other on the segment diameters. With the first method a good correlation between x0 and fluid pressure, and consequently a good correspondence between finite element and network results, are found. The second method yields a poor correlation between x0 and fluid pressure. However, still satisfying correspondence between finite element and network results is found. ©1997 Elsevier Science Ltd. All rights reserved.

1. I N T R O D U C T I O N

Blood perfusion is responsible for the nutrition and drainage processes in biological tissue, and is a result of blood flow through a hierarchy of microvessels. It is therefore often modelled as fluid flow through a hierarchical network of tubes [1-5]. Because of the huge density of microvessels (more than 1000 capillaries per mm a tissue), it is almost impossible to explicitly describe blood ftow through each vessel. Moreover, the main interest in medicine is often the state of perfusion in a certain region of the tissue, expressed in terms of capillary pressure and flow. Therefore a continuum approach to blood perfusion, describing blood flow by averaged quantities, would be appropriate. We developed a hierarchical mixture model in which blood perfusion is described as fluid flow through deforming porous media [6]. Besides spatial components, there is a hierarchical flow component which quantifies the blood flow from arterial towards venous vessels. The vascular hierarchy is quantified by a parameter x0. A finite element formulation of the model has been implemented in the software package DIANA [7]. Degrees of freedom in the finite element model are a number of hierarchical fluid pressures. An extended Darcy equation [8] describes fluid flow through the vasculature by spatial and hierarchical components. Unlike discrete models of vascular blood flow, the extensiveness of the mixture model is independent of the complexity of the considered vascular tree, because irrespective of its complexity, the vasculature is represented by a permeability tensor function defined in each point of the tissue. Moreover, the mixture model is appropriate for finite "l"Also at the Cardiovascular Research Institute, Department of Movement Sciences, University of Limburg, Maastricht, The Netherlands. 375

376

W.J. VANKANet al.

Fig. 1. Computer generatedarterial vascular tree model (Tree 1). element formulation [7], so that integrated analysis of structural tissue mechanics (deformation, contraction) and blood perfusion can be performed. Besides the mixture model, a quantitative relationship has been derived between the geometry of the microvasculature in a representative elementary volume (which will be further referred to as the averaging volume Vavg) and the fluid viscosity on the one hand, and the permeability tensor belonging to this Vavgon the other hand [8]. The formal averaging procedure applied in [8] suggests that this relationship tends to be exact, when the number of vessels within Vavgtends to infinity. The present study investigates the reliability of the geometry-permeability relationship in the case of a limited number of vessels within //avg. For this purpose quantitative analyses of blood perfusion through rigid vasculature in non-deforming tissue has been performed with the finite element model. The results of the finite element analyses are compared to appropriately volume averaged results of corresponding network analyses of fluid flow through the vasculature. Because of the immense amount of work involved in geometric reconstruction of in vivo microcirculatory vasculature, we used computer generated models of vascular trees: a detailed vascular tree model, generated by a computer algorithm based on 'constrained constructive optimization' ~[9-11] (see Fig. 1), and a somewhat simpler vascular tree model :~ (see Fig. 2). Some biologically important properties, such as anastomoses (hierarchical shunts) and arcades (connections between hierarchically equal segments) are not included in the considered vascular trees. 2. M E T H O D S 2.1 The hierarchical mixture model

The hierarchical mixture model describes blood perfusion as fluid flow through distensible pores in a deforming porous solid. The governing conservation equations [6] are as follows: Momentum V • crelf- Vp = 0

t Developed by the sixth author. ~:Developed by the fifth author.

(1)

Blood flow through biological tissue

2~

377

k /

(a)

(b)

Fig. 2. (a) Double arterial vascular tree (Tree 2); (b) resulting vasculature a~er occlusion of the right branch.

Solid mass ~n f -- a-t- + V • ( ( l

-- n f ) ¢ )

= 0

(2)

Fluid mass a~f 0---~-+ 4 V . (gf4~f) = 0.

(3)

Equation (1) represents conservation of momentum for the total mixture. The Cauchy stress tensor of the total mixture, o', is written as a combination of the well-known effective stress, o "~fr, and hydrostatic pressure, p, of the mixture: o" = o ~fr - pl [12]. It should be noted that equation (3) represents conservation of"fluid mass for an infinitesimal part dx0 of the hierarchical range, which is quantified by a parameter x0 that runs from 0 to 1. Hence the specific hierarchical fluid volume fraction fif represents the part of the total fluid volume fraction n f that resides in dx0, and is expressed per unit x0: 1

nf = I hfdx0.

(4)

0

Equation (3) accounts for both spatial divergence of fluid flow ( V . gence of fluid flow ~,~0 ~

(/~fvf)) and hierarchical diver-

oJ], which is denoted by the four-dimensional operatora~7 and the

four-dimensional fluid velocity v e c t o r 4~f:

;

vf

4~ f

=

~f J,

(5)

in which represents the three-dimensional fluid velocity, and v~ is defined as the material time derivative of x0 with respect to the fluid [6]. Equation (2) represents the conservation of solid mass, in which the equation of saturation of the mixture (n s + n f = 1) has been substituted, and vs is the solid velocity vector, n s is the solid volume fraction and t is time. Because we restrict ourselves to analysis of rigid, non-distensible vascular trees in non-deforming tissue, the effective stress o ~ff in equation (1) and the solid velocity vS in equation (2) vanish, and the fluid volume fractions h f and n f a r e constant. Applying these restrictions and substituting

378

W.J. VANKANet al.

the appropriate form of the constitutive equation for hierarchical fluid flow, the extended Darcy equation [6] h r. 4~f = -4A'" 4VOr

(6)

into equation (3) yields the resulting governing equation: 4V • 4A"" 4V0 r = 0,

(7)

in which j ¢ is the permeability tensor and Or is the fluid pressure. In [8] Huyghe and van Campen derived a general expression f o r j ¢ in deforming porous solids, assuming Poiseuillian fluid flow on the level of the individual vessels and making use of the SlatteryWhitaker averaging theorem. In previous work, Huyghe et al. found a corresponding expression for the case of rigid vascular trees in non-deforming tissue [13]:

/if 4k

=

( d2

04xo4x~* /

_

'

(8)

where g is the curvilinear coordinate representing the distance along the vessel axis, d is the vessel diameter and r/(d) is the diameter-dependent fluid viscosity. The hierarchical and spatial positions of the vessels are expressed by the coordinate vector 4x = (xo, x). The notation ( f ) * indicates the real volume average of quantity f . Thus the permeability tensor in a point (x0, x) is uniquely defined by the fluid viscosity and the geometry of the vasculature within the Vavgsurrounding x and belonging to the hierarchical compartment 6xo surrounding x0. In the case of straight vessel segments and a diameter independent viscosity, a permeability tensor4lGol for volumetric fluid flow can be derived by taking the circular vessel cross-sections into account, and its components read [131: "IT

d4 A x i A x j

Kij - 128Vavg6X0r/..~.

l

;

i, j -- 0. . . . . 3,

(9)

where ns is the number of segments belonging to the hierarchical compartment 6xo and within the Vavg surrounding x (Fig. 3). 2.2 Quantification of the hierarchy

Analogous to the spatial averaging volume Vavg , the hierarchical compartment 6xo is a representative elementary part of the hierarchical range. The primary unknown in the hierarchical mixture model, Pf(x0, x) [equation (7)], represents the fluid pressure, averaged over the vessel segments within Vavg and 6xo. Therefore 6xo, just like Vavg, should only contain vessel segments in which the fluid pressure is within a narrow range about the average pressure Of. Consequently the x0-value should be closely related to the fluid pressure in the vasculature. On the other hand it would be preferrable to relate x0 to material properties, for example by deriving the x0-value from the microstructure of the vasculature. Several authors have been dealing with hierarchical quantification of biological tree-shaped structures [14-16]. Also Strahler ordering [17], developed in geography for modelling of rivers, is a well known quantification technique of hierarchical levels in fractai structures. However, most of these methods introduce discrete hierarchical categories of segments. In our hierarchical mixture model we essentially need a continuous representation of the hierarchy. As a first trial we use the x0 definition as presented by Huyghe in [18]: x0 is calculated in vascular branching points according to _

xo=

: ~ ~ +d2 ~/ 2d.

(10)

only if d2 < 0.95d~. Here, dl is the largest diameter and d2 is the second largest diameter in a branching point, and d. is a characteristic diameter, for example the capillary diameter. If the

Blood flow through biological tissue

379 ,, xoz i t

e

¢1 iI ¢1 xtt x2

t~.

i#

d2=dl

iI jl Vavg

/

,

XO2=XO I + A X O

t i

i

I

12

L.xL

/t "-..

_. -"

St

it

,'

l

AX2

Fig. 3. Illustration of the x0 definition, and of the contribution of a vessel segment, within averaging volume Vavg and compartment Ox0 (shaded vessels), to Jr'.

branching point is a capillary terminal node, it is assumed that d2 = dl = d,. In between these branching points the value of x0 varies linearly [18] (see Fig. 3).

2.3 Network analysis Two-dimensional, rigid, arterial vascular trees in square domains are generated by computer algorithms. Blood flow through each of the segments in these trees is calculated by network analysis. Pressure boundary conditions are prescribed at the free ends of the terminal segments: high arterial pressure at the inflow and low pressure at the capillary outflows. According to Poiseuille's law the pressure-flow relation in each segment is Q-

rrd 4 1280/AP,

(11)

where Q is the volumetric flow and AP the pressure drop in the segment. A constant blood viscosity of approximately r/= 0.0037 Pa s is used. From the connectivity of the vessel segments a system of linear equations can be derived for the pressure flow relation in the total network.

2.4 Finite element analysis The problem definition of the network analysis, including boundary conditions, is translated to a finite element formulation. For this purpose the hierarchy of the vasculature is translated to an x0 range, which is scaled to [0,1] and divided into a number of compartments of equal size 6xo. The square domain of the vascular tree is divided into a number of square finite elements. The numbers of finite elements and compartments are optional, although the finite element software can only deal with up to five compartments. Permeability tensors are calculated according to equation (9) for each compartment of each finite element. Thus we obtain a three-dimensional field of permeability tensors, piecemeal constant per finite element and per compartment (see Fig. 4). Volume averages of the network pressure at the lowest (arterial) and highest (capillary) hierarchical level are prescribed as boundary conditions in the nodal points of the finite element model.

380

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,QJ

.....

/t

,

/I

x:: XtlI

VASCULARTREE

) ~ 0 X2

~rmal*l_

.~K

XI COMPARTMENTS

FINITEELEMENTMESH

Fig. 4. Illustration of the finite element problem definition. Hierarchical fluid pressures on the boundaries x0b, of the compartments (Fig. 4), being degrees of freedom in the finite element model, are calculated in the nodal points of the mesh. These fluid pressures actually represent real volume averages of fluid pressure [19]. The corresponding volume averages of the network pressure are calculated by summation over all the segments (nsi) in the Vavg surrounding nodal point I, and within the x0-range Ix b, - (fix0/2), x b, + (fix0/2)]: P/w, = ~-~ 1 Z Pseg l?seg, nsi

(12)

where Pseg represents the average pressure in a segment, Vsegis the volume of a segment and vf is the total fluid volume within Vavg: vf = E

~'~'rseg "

(13)

IL~"i The nodal values of fluid flow in the finite element model represent the integral over the domain of the nodal shape function qb/ multiplied with the fluid flux passing a compartment boundary x0b, per unit of volume. The corresponding value of the network flow is obtained by summation of Xb the nsi segment flows passing the compartment boundary Qseg(0t), within Vavgsurrounding nodal point I, multiplied by the nodal shape function value at the position of the segment qb1(x):

Olw, = 2 ~1(x) Oseg(Xbo)"

(14)

tl,~'i

The schema in Fig. 5 summarizes all the steps involved in the analysis. 3. R E S U L T S 3.1 Translation o f discrete network to finite element model

The vascular tree in Fig. 1 contains 7999 vessel segments and is embedded in a square piece of tissue of area 7.854 × 10-3 m 2 with a mass of 100 g. According to the network analysis, it conveys a total blood flow of 1.133 × 10- 5 m 3s- l, which results from the network pressure boundary conditions of 13.3 KPa at the arterial inflow and 8.0 KPa at the terminal capillary outflows, The blood perfused, square piece of tissue is modelled by a finite element mesh of 8 × 8 equal, square linear elements, each containing five fluid compartments of 6x0 = 0. 2. Nodal boundary conditions for fluid pressure are derived from volume averages of the pressure boundary conditions of the network analysis according to equation (12). A circular averaging volume with a diameter of twice the element edge size is applied to achieve smooth variation of the element permeability tensors. In the translation of the network to the finite element model, we used as a first trial an x0 definition which is based on the network pressure: x0 = -Pnw. Contour plots of pre-capillary fluid

Blood flow through biological tissue

381

generation I blood pressure of vascular boundary tree conditions

network simulation

1

volume averaging of network results

translation to finite element formulation

finite element I simulation

~ confrontation Fig. 5. Schema of the analyses. pressure and capillary flow, from both the finite element and the network simulations, are given in Fig. 6. Fluid pressure is given at the pre-capillary level (x0 = 0. 8), because capillary (x0 = 1.0) fluid pressure is prescribed as boundary condition. A total finite element flow of 1.089 x 10 -5 m3s -I is calculated, which is 96% of the total network flow. Next we investigated the quality of the finite element model in the case that x0 was derived from structural parameters according to equation (10). The finite element simulation gives a total flow of 0.705 x 10 -5 m3s -l , which is 62%, of the total network flow. Contour plots of pre-capillary fluid pressure and capillary flow, from both the finite element and the network simulations, are given in Fig. 7. A rather wide range of fluid pressures at the pre-capillary level is found in both the finite element and the network analysis. This is due to the poor correspondence of the x0 values to the fluid pressure (Fig. 8). Mainly at the higher x0 levels a wide range of fluid pressures is found.

3.2 Macroscopic changes in blood perfusion The finite element model has also been used for analysis of macroscopic changes of blood perfusion due to an occlusion of a feeding artery. The vasculature consists of two independent vascular trees [Fig. 2(a)] in a square piece of tissue of dimensionless size 0 < (Xl, x2) - 1. Each tree contains 300 vessel segments with diameters ranging from 3. 125 x 10 -3 for all the capillary segments to 23.89 × 10 -3 for the feeding artery. The x0 definition of equation (10) is used and the resulting x0-range is scaled to [0, 1]. Dimensionless pressure boundary conditions are prescribed at the feeding arteries (P = 1.0) and at the capillaries (P = 0.0). A fair relation between the x0 value and the fluid pressure exists in this tree (Fig. 8). A finite element mesh of 8 x 8 equal, square linear elements, each containing five fluid compartments, is used. Volume averages of the pressure boundary conditions of the network analysis, calculated according to equation (12), are prescribed as arterial and capillary fluid pressure boundary conditions. The patterns of pre-capillary pressure and capillary flow from the network analysis are predicted by the finite element model (Fig. 9). Good correspondence between total flow in the finite element analysis (1.126 × 10-3) and network analysis (1.117 x 10-3) is found. Occlusion of the right feeding artery of the double tree results in the vasculature given in Fig. 2(b). Total network flow in the occluded vasculature is 0. 58 × 10-3, and the finite element flow is 0. 535 x 10 -3. The changes in the patterns of pre-capillary pressure and capillary flow from the network analysis are predicted by the finite element model (Fig. 9).

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Capillary pres. (left) and f l o w (right) l r o m F E (upper) and netw, (lower)

Fig. 6. Pressure (Pa) and flow (m3s -1) patterns from finite element (FE) and network (NW) simulations in which x0 = -P,w.

4. D I S C U S S I O N The hierarchical mixture model has been used in quantitative analysis of fluid flow through vascular tree models. The Darcy permeability was derived from the microstructure of the vasculature. For this purpose the vascular hierarchy must be adequately quantified, and should as such be closely related to the fluid pressure in the hierarchy. A hierarchical quantification method, based on microvascular fluid pressures~ yielded satisfying results in the finite element analysis. Microstructural material properties are an alternative basis for hierarchical quantification, and a method which relates the hierarchical parameter x0 to microvascular diameters in branching points was used. The considered vascular trees show a large variation in fluid pressure per hierarchical level (Fig. 8). This is due to the poor correlation between fluid pressure and segment diameter, the latter of which is the basis of the structural x0 definition. Experimental measurements of blood pressure distributions in skeletal muscle microvasculature have shown a much tighter correlation between blood pressure and vessel diameter, mainly in the arteriolar and capillary range [20]. This supports the applicability of the mixture model to physiological microvascular blood perfusion. The finite element results are very sensitive to the employed x0 definition. A good correspondence between x0 and fluid pressure is essential for the model. However, even if this requirement is not completely fulfilled, the finite element model quantitatively predicts regional distributions of blood perfusion in terms of pressure and flow, hereby accounting for local microvascular properties. The mixture model deals with averaged pressures and flows, representative in a local region. Because the representation of regional fluid pressure and flow by means of volume averaged values

Blood flow through biological tissue FEMGEN/FEMVIEW

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Fig. 9. Pressure and flow patterns from finite element (FE) and network (NW) simulations of (a) the double tree; (b) the occluded tree.

Blood flow through biological tissue

385

Acknowledgements--We thank Dr E Neumann of the Department of Cardio-thoraeic Surgery, University of Vienna, Austria, for her support in making the data of the vascular tree available. We thank Professor D. Slaaf of the Cardiovascular Research Institute, Department of Biophysics, University of Limburg, for useful discussions on this topic.

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(Received6 March 1996; accepted 5 September 1996)

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