Numerical simulation of blood flow and interstitial fluid pressure in ...

Acta Mech Sin (2007) 23:477–483 DOI 10.1007/s10409-007-0098-x

RESEARCH PAPER

Numerical simulation of blood flow and interstitial fluid pressure in solid tumor microcirculation based on tumor-induced angiogenesis Gaiping Zhao · Jie Wu · Shixiong Xu · M. W. Collins · Quan Long · Carola S. König · Yuping Jiang · Jian Wang · A. R. Padhani

Received: 15 September 2006 / Revised: 23 May 2007 / Accepted: 23 May 2007 / Published online: 5 September 2007 © Springer-Verlag 2007

Abstract A coupled intravascular–transvascular–interstitial fluid flow model is developed to study the distributions of blood flow and interstitial fluid pressure in solid tumor microcirculation based on a tumor-induced microvascular network. This is generated from a 2D nine-point discrete mathematical model of tumor angiogenesis and contains two parent vessels. Blood flow through the microvascular network and interstitial fluid flow in tumor tissues are performed by the extended Poiseuille’s law and Darcy’s law, respectively, transvascular flow is described by Starling’s law; effects of the vascular permeability and the interstitial hydraulic conductivity are also considered. The simulation results predict the heterogeneous blood supply, interstitial hypertension and low convection on the inside of the tumor, which are consistent with physiological observed facts. These results may provide beneficial information for anti-angiogenesis treatment of tumor and further clinical research.

The project supported by the National Natural Science Foundation of China (10372026). G. Zhao · J. Wu · S. Xu (B) Department of Mechanics and Engineering Science, Fudan University, Shanghai 200433, China e-mail: [email protected] M. W. Collins · Q. Long · C. S. König Brunel Institute for Bioengineering, School of Engineering and Design, Brunel University, Uxbridge, Middlesex, UK Y. Jiang · J. Wang Department of Neurology, Huashan Hospital, Fudan University, Shanghai 200040, China A. R. Padhani Paul Strickland Scanner Centre, Mount Vernon Hospital, Rickmansworth Road, Northwood, Middlesex HA6 2RN, UK

Keywords Solid tumor · Blood flow · Interstitial pressure · Angiogenesis · Numerical simulation

1 Introduction A major obstacle to understand blood transport in tumors is the heterogeneous architecture of the tumor microvasculature. Tumors are known to contain many tortuous vessels, shunts, vascular loops, widely variable intervascular distances, and large avascular areas [1]. In addition, tumor vessels are more leaky than normal vessels and this may enhance the efficiency of fluid exchange between the microvascular and the interstitial space. The microvascular network supplies blood and nutrients for continued development of the tumor and provides the initial route for invading cancer cells to escape from the primary tumor and form metastases. Furthermore, chemotherapeutic drugs will be delivered through the capillary network and interstitial space with anisotropic conductivity to the tumor [2]. All of these processes depend crucially upon the blood flow within the microvascular network and the interstitial fluid flow in the solid tumor, which have predominant roles for solid tumor growth, metastasis and therapy. In recent years, several mathematical models of tumorinduced angiogenesis and hemodynamics have been studied to improve understanding of the vascular architecture and microcirculatory dynamics in a solid tumor [3–9]. Mcdougall et al. [6] simulated blood flow and drug delivery through the vascular network from a nearby parent vessel to the tumor surface via an associated capillary bed generated from their mathematical model of tumor-induced angiogenesis. On the basis of the work of Mcdougall et al. and Stephanou et al. [7] studied the efficiency of drug delivery in 2D and 3D

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tumor-induced vasculatures, evaluated the influence of key parameters upon uptake by the tumor and examined how the removal of certain capillaries affected the distribution of blood flow in the system. Further Stephanou et al. [8] modeled an adaptive vasculature associated with tumor-induced angiogenesis and considered how this adaptive remodeling affected the supply of oxygen and drugs to the tumor cells. However, all these models focused on exterior capillary networks of the tumor emerging from a parent vessel. Zheng et al. [9] first incorporated a realistic model of angiogenesis inside the tumor with a continuum model of tumor growth for simulating tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite element method, but they did not investigate the flow of blood through the network in the coupled model. However, few models of coupling between the blood flow within tumor microvascular networks and the interstitial fluid flow in tumor tissues have been studied up to now. We [10] studied the blood flow and interstitial fluid flow in a tumorinduced capillary network emerging from a nearby parent vessel, which was generated from a five-point discrete mathematical model of tumor angiogenesis. On the basis of the physiological facts that tumor microvasculatures have different growth directions and emerge from multi host vessels near the tumor, in this paper, we report an extension of the previous network model to include two parent vessels and nine motion directions of an individual capillary sprout in a microvascular network. This network is generated from a discrete mathematical model of tumor-induced angiogenesis. Interstitial, transvascular, and intravascular fluid transport are simulated in this network with pruning of immature capillaries. Moreover, we also further consider the effects of high vascular permeability and interstitial hydraulic conductivity on the tumor microcirculation.

2 Mathematical models 2.1 Tumor-induced angiogenesis This model is based on a discrete mathematical model of tumor angiogenesis [5], which describes the formation of a capillary network in response to chemical stimuli released by a solid tumor. The model assumes that the migration of endothelial cells is influenced mainly by three factors: random motility, chemotaxis in response to tumor angiogenic factor (TAF) released by the tumor and haptotaxis in response to fibronectin (FN) gradients in the extracellular matrix. All parameters and equations are expressed in nondimensional forms in the model. The endothelial cell flux J n is given by J n = J random + J chemo + J hapto ,

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(1)

where J random = −D∇n denotes the random motility flux, n is the endothelial cell density per unit area, and D is the diffusion coefficient. Considering the different mechanical environments between the inside and the outside of the tumor, we assume [11] D=

D0 D0 (r/rt )2

(r > rt ), (r ≤ rt ),

where D0 is a scalar parameter, r the distance to the tumor center and rt the tumor radius. J chemo = χ (c)n∇c denotes the chemotactic flux, where χ (c) = χ /(1 + αc) is a chemotactic function, reflecting the decrease in chemotactic sensitivity with increased TAF concentration c. The coefficients χ and α are positive constants. J hapto = ρ0 n∇ f denotes the haptotactic flux, where the coefficient ρ0 characterizes haptotactic cell migration and f is the fibronectin concentration. The conservation equation for the endothelial cell density is ∂n + ∇ · J n = 0. ∂t

(2)

The TAF concentration c and the fibronectin concentration f satisfy equations of the form ∂f = βn − γ n f, ∂t ∂c = −ηnc, ∂t

(3) (4)

where β is the rate at which fibronectin is produced by migrating endothelial cells, and γ and η are coefficients describing the rates of TAF uptake and fibronectin degradation by the endothelial cells, respectively. β, γ and η are positive constants. In the present model, the dimensionless simulation domain is [0,2]×[0,2], which is equivalent to [0, 2 mm]×[0, 2 mm]. Two parent vessels are located on the two edges. Normal tissue surrounds the tumor, which is located at the center (1.0, 1.0) and has a dimensionless radius of 0.5. On the basis of perfusion rates [12], the tumor is divided into three regions: a well vascularized region T1, a semi-necrotic region T2 and a necrotic region T3. The resultant schematic modeling of tumor angiogenesis is shown in Fig. 1. No-flux boundary conditions are ζ · ( J random + J chemo + J hapto ) = 0,

(5)

where ζ is an outward unit normal vector. According to Ref. [11] and considering the positions of the two parent vessels in the model, the initial concentration

Numerical simulation of blood flow and interstitial fluid pressure in solid tumor microcirculation

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Fig. 1 Schematic modeling of tumor angiogenesis. T1 Well vascularized region: 0.3 ≤ r < 0.5; T2 semi-necrotic region: 0.1 ≤ r < 0.3; T3 necrotic region: 0.0 ≤ r < 0.1, r = (x − 1.0)2 + (y − 1.0)2

profiles of the TAF and fibronectin have the forms: ⎧ 1 0 ≤ r ≤ 0.3, ⎨ c(x, y, 0) = (v − r )2 ⎩ 0.3 ≤ r, v − 0.4771 f (x, y, 0) = κ · e−x

2 /ε 1

+ κ · e−(x−2)

Fig. 2 Nine possible motion directions of an individual endothelial cell (P0 –P8 )

(6)

2 /ε , 1

2.2 Blood flow through the microvascular network

(x, y) ∈ [0, 2] × [0, 2],

(7)

where v, κ and ε1 are positive constants. The parameter values used for the simulations are as follows [5–7]: D0 = 0.00035, χ = 0.38, α = 0.6, ρ0 = 0.34, η = 0.1,

β = 0.05, γ = 0.1, ν = 1.07, κ = 0.75,

ε1 = 0.45. By using the nine-point finite difference scheme, we obtained discretized equations as follows q+1

q

q

q

q

nl,m = nl,m P0 + nl+1,m P1 + nl−1,m P2 + nl,m+1 P3 q

q

q

+nl,m−1 P4 + nl+1,m+1 P5 + nl+1,m−1 P6 q

q

+ nl−1,m+1 P7 + nl−1,m−1 P8 , q+1

q

q

q

fl,m = fl,m (1 − kγ nl,m ) + kβnl,m , q cl,m

=

q q cl,m (1 − kηnl,m ),

tissues after capillary sprouts reach the tumor boundary and penetrate into it.

The flow rate in capillary elements among adjacent nodes is approximately described by Poiseuille’s law [7] Q=

(9) (10)

where the subscripts l and m specify the spatial locations and the superscript q specifies the time steps. The migration of an individual endothelial cell located at the capillary sprout is determined by the set of coefficients P0 –P8 , which correspond to nine directions: stationary, moving vertically up or down, to right, left, upper right, lower right, upper left and lower left (Fig. 2). The phenomena of branching and anastomosis of capillary sprouts are also considered in the model (see Ref. [5] for details). These coefficients Pn (n = 0, . . . , 8) will be adjusted according to the features of solid tumor

(11)

where µ is the apparent blood viscosity, pv the blood pressure drop across the vessel, and R and L the radius and length of the vessel, respectively. To calculate the flow through a given microvascular network of interconnected capillary elements, it is simply necessary to conserve mass at each junction where the capillary elements meet. Hence, for each node the following expression can be written N

(8)

π R 4 pv , 8µ L

Q (i, j),k = 0,

(12)

k=1

where the index k refers to adjacent nodes and N = 8 in a fully connected regular two-dimensional grid. For given inlet and outlet pressures pin and pout , respectively, for each parent vessel, the pressure of the blood flow at each node within the entire interconnected microvascular network can be found numerically. 2.3 Interstitial fluid flow in the tumor Considering the tumor tissue as an isotropic porous medium, its interstitial fluid flow is described by Darcy’s law [13] u = −K ∇ pi ,

(13)

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where u is the interstitial fluid velocity, pi the interstitial fluid pressure, and K the hydraulic conductivity coefficient of the interstitium. The extravasation flux is described by Starling’s law qe =

L pS [( pv − pi ) − σ (πv − πi )], V

(14)

where qe is the volumetric flow rate out of the vasculature per unit volume of tissue, L p the hydraulic conductivity of the microvascular wall, S/V the surface area per unit volume for transport in the tumor, pv the microvascular pressure, σ the average osmotic reflection coefficient for plasma proteins, and πv and πi are the vascular and interstitial osmotic pressures, respectively. Mass conservation at each junction where the interstitial fluid pressure satisfies equation ∇ 2 pi =

ε2 ( pi − pe ) · A, R2

(15)

where ε = R L p S/K V and pe = pv − σ (πv − πi ). In these entities ε is the ratio of interstitial to vascular resistances to fluid flow and pe is the effective pressure. When the parameters S/V and R are given, the ratio ε reflects the coupling effect of tumor vascular permeability and interstitial hydraulic conductivity. A is the matrix obtained from numerical simulation of tumor angiogenesis, which describes a heterogeneous blood supply inside the tumor. Matrix A is composed of two elements 1 and 0, which denotes vasculature and no vasculature, respectively. The following conditions hold no-flux at the center of the tumor: ∇ pi |r =0 = 0 continuity of interstitial pressure and velocity between tumor and normal tissue: pi |r =r − = pi |r =r + , t

and

t

− (K ∇ pi )|r =r − = −(K ∇ pi )|r =r + . t

t

Table 1 shows values for those parameters used in the simulations.

3 Simulation results 3.1 Tumor microvascular network growth An example of the spatiotemporal evolution of the tumor microvascular network growth is shown in Fig. 3. Two parent vessels are located on the upper and lower boundaries of the spatial domain. The tumor surrounded by normal tissue is located at the center and comprises three regions: outermostwell-vascularized, middle-semi-necrotic of the domain and innermost-necrotic. Acting on the eight initial capillary sprouts emerging from the parent vessels, the randomizing component of the discrete model, results in seven of them forming a successful microvascular network. Figure 3 also shows details of the development process of the network. During the initial stage the capillary sprouts from the parent vessels are almost parallel. Once the sprouts reach a certain distance from the parent vessel, however, they tend to be attracted to each other and a modest amount of branching and anastomosis occurs. The migratory path taken by each sprout tip does not actually emerge with its neighbors before the tumor is reached. The first penetration of the tumor by a single sprout tip occurs at the 8th day of vessel growth (Fig. 3a), closely followed by several others. After penetration by the tips, branching and anastomosis of the sprouts rapidly increase, and form blood perfusion loops. Subsequently, the abundant blood perfusion in the well vascularized region of the tumor results from what is an extensive new capillary bed. In the semi-necrotic region less tortuous capillaries still give a fairly substantial blood perfusion, but in the necrotic region there are hardly any capillary sprouts because of the lack of nutrients and oxygen required for endothelial cell survival and proliferation. These results are in agreement with that observed in experiments (see Fig. 4). Finally, the capillary network takes on a steady structure due to the slower migration of sprout tips with time, and this occurs at 28 simulation days in our model (Fig. 3b). The resultant microvascular network in Fig. 3b is adopted to perform fluid flow calculations and any immature capillaries (bottom left in Fig. 3b) are pruned before the calculations are made in this study. 3.2 Blood flow through microvascular network

Table 1 Baseline parameter values used in simulations [15] Parameter

Tumor tissue values

Normal tissue values

L p /(m · (Pa s)−1 )

2.1 × 10−11

2.7 × 10−12

K /(m (Pa s)−1 )

3.1 × 10−12

6.4 × 10−12

× 103

1.33 × 103

πv /Pa

2.0

πi /Pa

2.67 × 103

2.67 × 103

S/V /mm−1

200

70

σ

0.82

0.91

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The pressure profiles of blood flow through each vessel segment in the microvascular network are shown in Fig. 5. Blood enters the network from the two influx ends of the parent vessels, distributes throughout the microvascular network, and leaves from the two efflux ends. In a given microvascular network the blood flow through it is primarily controlled by the pressure difference between the inlet and outlet pressures and by geometric and viscous resistances. Hence, in the simulations the inlet pressure pin and outlet pressure pout across


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Fig. 3 Spatiotemporal evolution of a developing microvascular network. a 8 days; b 28 days

Fig. 4 Experimental result of blood vessels feeding a tumor in the skin (The above image is from Dr. Steven Stacker and Dr. Marc Achen from the Ludwig Institute for Cancer Research at www.ludwig. edu.au/angiogenesis)

depends on the difference between the inlet and outlet pressures. Moreover, the blood flow pressure within some of the daughter vessels connecting the parent vessels to the tumor surface drops monotonically from the branching points to the tumor. This provides blood perfusion for tumor growth and development and simultaneously may also transport nutrients and therapeutic agents to the tumor. The blood flow in other daughter vessels outside the tumor, however, shows inverse flow directions and remove nutrients, drugs and metabolic material from the solid tumor. The blood flow distribution within the tumor itself is very nonuniform and highly heterogeneous. The figure shows that the complex and chaotic tumor microvascular network leading to this heterogeneity makes the blood flow pressure vary significantly less in the whole interior of the tumor compared to its exterior and leaves a scarcely perfused area in its center. This area is thus hypoxic and often necrotic, but at the same time an area where drug delivery is particularly ineffective [14]. Hence the heterogeneity of the blood flow in the tumor is intimately connected with the abnormal topological and morphological structures of its microvascular network. The latter also influences the distribution of interstitial fluid pressure in the tumor.

3.3 Interstitial fluid pressure in the tumor

Fig. 5 Blood pressure distribution in the microvascular network. pin and pout denote influx ends and efflux ends of blood flow through the microvascular network, respectively

each parent vessel are kept fixed at 3,325 Pa (25 mmHg) and 2,128 Pa (16 mmHg), respectively, in accordance with physiological values at the capillary scale. The diameter of each microvascular segment is 10 µm, and the diameter ratio of parent to microvascular vessels is 2 [6]. The results of our simulations show that the pressure-flow varies almost linearly from the influx end to the efflux end in each parent vessel. This trend is irrespective of the numbers of branching vessels of each parent vessel, and only

Figure 6 shows the variations in interstitial fluid pressure (IFP) in the solid tumor for a range of values of ε. The dimensionless parameter ε = R L p S/K V is a measure of the ratio of interstitial to vascular resistances to fluid flow and controls the physical phenomena of the fluid movement. An increase in ε may be the result of an increase in the tumor radius R; an increase in the exchange surface area per unit volume S/V ; and an increase in the ratio of the hydraulic conductivity of the microvascular wall to that of the interstitial space L p /K . Figure 6 shows that interstitial fluid pressure is elevated throughout the tumor tissue as ε increases except at the periphery of the tumor, where the pressure is nearly equal to the normal tissue pressure. This phenomenon is due to the enhanced fluid exchange between the vascular and interstitial space

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Fig. 6 Distribution of interstitial fluid pressure for different values of ε in solid tumor. The contour values represent the normalized IFP pi / pin , ε0 denotes a value of ε that is obtained by using the baseline parameter of values given in Table 1. a ε = 0.04ε0 ; b ε = 0.2ε0 ; c ε = ε0 ; d ε = 5ε0

mediated by the high leakiness of the tumor microvasculature and the accumulated fluid in the tumor interstitium. However, the enhancement of transvascular fluid flux and the accumulation of the extravasation make the variation value of IFP to decline gradually with the increase of ε (see contour values from Fig. 6a to d). Moreover, the region of high-pressure gradually augments from near the tumor’s center to its periphery and becomes flater. As noted by Jain [15] for low values of ε, there is much less resistance to filtration in the interstitium, hence, the pressure gradient is relatively flat. However, for large values of ε, there is much greater resistance to interstitial fluid transport than for transcapillary flow, resulting in a very sharp pressure gradient near the outer edge of the tumor. It is noteworthy that as a consequence of the enhanced transvascular fluid exchange, the pressure gradients are flattened near the center of the tumor. This indicates a reduced fluid flow velocity in the central area, leading to a barrier to both cancer detection and treatment since relevant agents can not reach the target cells in the tumor sufficiently quickly. In contrast, the pressure at the periphery of the tumor rapidly drops resulting in a high pressure gradient. Under these conditions, fluid flow is diverted away from the center of the tumor to a more peripheral path, and therefore the extravasation is highest at the tumor periphery [15]. Furthermore, it can be observed that the interstitial fluid pressures closely track the local vascular pressures throughout the tumor for this configuration (see Fig. 6c). These simulation results indicate that the microvascular network distribution,

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the capillary wall characteristics and interstitial conductivity play important roles in tumor microcirculation.

4 Concluding comments Simulations of microvascular networks based on the discrete mathematical model as employed here demonstrate the use of a new tool for studying blood flow and transport of interstitial fluid containing nutrients and chemotherapeutic drugs in a tumor. Moreover, the observed heterogeneity is better matched than with the widely used models based on a small number of vascular reconstructions and a regular array of vessels. In consistency with the complex and chaotic tumor microvascular network, we introduce a matrix A to describe the heterogeneous blood supply and then present a coupled intravascular–transvascular–interstitial fluid flow model to study the hemodynamics of solid tumor microcirculation. This model incorporates the effects of vessel distribution, intravascular distance, vascular permeability and interstitial conductivity and is able to integrate the microscopic variables (blood flow within microvascular network) with the macroscopic (interstitial fluid flow). The simulation results confirm the heterogeneous blood supply, high interstitial pressure, low convection flow inside the tumor and the coupling effects of flow between the vascular and interstitial space in tumors. However, the present study has some assumptions that are somewhat idealistic in the context of our calculations. Firstly,


the microvascular and parent vessel diameters are considered constants, which will influence the blood flow through the microvascular network as well as the interstitial flow. Secondly, the values used for microvascular and parent vessel diameters are equivalent to the size of one or two red blood cells. Hence, the blood flowing through the vessels does not behave as a continuum and the viscosity for the different rates of flow in the parent vessels and microvasculatures is not constant. In other words, the rheological properties of blood should be addressed in further developing the current model. Finally, the flow rate in capillary elements among adjacent nodes is approximately described by Poiseuille’s law, which assumes that the fluid exchange across the vessel wall is negligible and the flow occurs mainly inside the vessel. Because the fraction of transmural flow is negligible and the axial flow is dominated by the control parameter for the fluid leakage, this results in a low value for the ratio of the axial flow resistance to the transmural flow resistance [14]. Hence, in the present simulations, the calculation of the blood flow through the microvascular network is likely to introduce some error. As pointed out in this paper, elevated interstitial fluidpressure is a major physiological barrier to the delivery of chemotherapeutic drugs. The results of our simulations indicate that the control parameter ε directly influences the formation of elevated interstitial fluid pressure in the tumor. In addition, a variation in ε may be a result of a variation in the tumor radius, in the exchange surface area per unit volume, and in the ratio of the hydraulic conductivity of the microvascular wall to that of the interstitial space. Taking this mechanism as a basis, it is possible to postulate that the behavior of interstitial fluid pressure could be modified with variations of these factors for the better delivery of chemotherapeutic drugs. This approach holds the promise of yielding strategies for improved cancer therapy. Furthermore, anti-angiogenic strategies for cancer therapy could be combined with this model based on the fact that several anti-angiogenesis physical and chemical agents may lead to a reduction in tumor blood flow or a decrease in vascular permeability and interstitial transport rate. Also, it might inhibit the production of tumor angiogenic factors. In clinical diagnosis, measurements of the tumor microvascular network and blood flow are common methods for assessing the efficacy of antiangiogenic drugs. Hence, this work may provide beneficial information for anti-angiogenic strategies for cancer therapy and for further clinical research. To further understand the underlying mechanisms of tumor angiogenesis and anti-angiogenesis and to extend research of the hemodynamics of solid tumor microcirculation, more realistic features need to be incorporated within the model: the neoplasm anatomy, physiology, the mechanism of angiogenesis and anti-angiogenesis, and the coupling effects of various factors.

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Acknowledgments The Royal Society of UK’s Short Visit Scheme supported a visit of Shixiong Xu to Brunel University in 2006.

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