**3. Simulation results**

### **3.1. 3D blood perfusion of metastatic tumor**

We simulated the evolution of blood flow pressure in the presence/absence of angiostatin for 14 days representing the typical timescale for tumor vasculature to grow. **Figure 3** shows 3D Numerical Study of Metastatic Tumor Blood Perfusion and Interstitial Fluid Flow Based… http://dx.doi.org/10.5772/intechopen.78949 73

∇⋅ *u<sup>i</sup>* = *ϕ<sup>v</sup>* − *ϕ <sup>L</sup>* (5)

the lymphatic drainage term, which is proportional to the pressure difference between the inter-

is the ratio of interstitial to vascular resistances to fluid flow. *LpL* is the hydraulic permeability

and given a uniform value for normal tissue referring to Baxter and Jain [13]. The continuity of pressure and flux on the interconnected boundary between the tumor and normal tissue

We simulated the evolution of blood flow pressure in the presence/absence of angiostatin for 14 days representing the typical timescale for tumor vasculature to grow. **Figure 3** shows

**Table 1** shows the values of the parameters used in the microcirculation simulations.

( S \_\_*<sup>v</sup>*

*Lpv*

**Parameter Name Value Parameter Name Value**

8.7 × 10<sup>−</sup><sup>5</sup> T

19.8T mmHg 20<sup>N</sup> mmHg

17.3T mmHg 10<sup>N</sup> mmHg

Subscript "N" and "T" represents the values in normal and tumor tissues, respectively.

0.91<sup>N</sup>

Mass conservation at each junction where the interstitial fluid pressure satisfies equation:

)) is the fluid source term leaking from blood vessels. *ϕ <sup>L</sup>* <sup>=</sup>

*<sup>R</sup>*<sup>2</sup>(*pi* <sup>−</sup> *pev*) <sup>⋅</sup> *<sup>B</sup>* (6)

/V is assumed zero for tumor tissue,

2.5 × 10<sup>−</sup><sup>7</sup>

2.5 × 10<sup>−</sup><sup>7</sup>

50<sup>T</sup> cm<sup>−</sup><sup>1</sup> 50<sup>N</sup> cm<sup>−</sup><sup>1</sup>

0<sup>T</sup> 1/mmHg s 1.0 × 10<sup>−</sup><sup>4</sup>

1.86 × 10<sup>−</sup><sup>6</sup>

3.6 × 10−8

is the effective pressure and *<sup>α</sup>* <sup>=</sup> *<sup>R</sup>*<sup>√</sup>

/*V* are the surface areas of blood vessel wall and lymphatic

, and *κN* are the hydraulic conductivity coefficients of nor-

coefficient of interstitium

volume for transport in

*κ*<sup>a</sup> Hydraulic conductivity

<sup>V</sup>)<sup>a</sup> Surface area per unit

interstitium

*<sup>V</sup>* <sup>c</sup> Absorption capacity of lymphatic system

<sup>a</sup> Hydraulic permeability of vascular wall

<sup>L</sup>*pL SL* \_\_\_\_\_ *<sup>V</sup>* (*pi*

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (*<sup>L</sup> pv Sv* <sup>+</sup> *LpL SL*)/*<sup>V</sup>*

<sup>T</sup> cm2 /mmHg s

<sup>N</sup> cm2 /mmHg s

<sup>N</sup> 1/mmHg s

<sup>T</sup> cm/mmHg s

<sup>N</sup> cm/mmHg s

<sup>−</sup> *pL*) is

where *ϕ<sup>v</sup>* <sup>=</sup>

where *pev* <sup>=</sup> (

*<sup>Γ</sup>*: *pi*|*Γ*<sup>−</sup> <sup>=</sup> *pi*|*Γ*<sup>+</sup>

*σt*

*πv*

*pL*

*πi*

a

b

c

Jain et al. [5].

Zhao et al. [25].

Stephanou et al. [18].

<sup>L</sup>*pv Sv* \_\_\_\_\_ *<sup>V</sup>* (*pv* <sup>−</sup> *pi*

72 Highlights on Hemodynamics

stitium and the lymphatics.

*Lpv Sv*(*pv* − *σ<sup>t</sup>*

of lymphatic vessel wall. *Sv*

<sup>a</sup> Average osmotic

<sup>a</sup> Colloid osmotic

<sup>a</sup> Colloid osmotic

**3. Simulation results**

**3.1. 3D blood perfusion of metastatic tumor**

**Table 1.** Baseline parameter values used in the simulations.

− *σ<sup>t</sup>* (*π<sup>v</sup>* − *π<sup>i</sup>*

<sup>∇</sup><sup>2</sup> *pi* <sup>=</sup> *<sup>α</sup>*<sup>2</sup> \_\_

(*π<sup>v</sup>* − *π<sup>i</sup>*

, −*κ<sup>T</sup>* <sup>∇</sup> *pi*|*Γ*<sup>−</sup> <sup>=</sup> <sup>−</sup>*κ<sup>N</sup>* <sup>∇</sup> *pi*|*Γ*<sup>+</sup>

mal tissue and tumor tissue, respectively.

reflection coefficient for plasma proteins

<sup>c</sup> Lymphatic pressure 0.5N mmHg <sup>L</sup>*pL <sup>S</sup>* \_\_\_\_\_*<sup>L</sup>*

pressure of plasma

pressure of interstitium

*μ*<sup>b</sup> Blood viscosity 1.0 cP

)) <sup>+</sup> *LpL SL pL*)/

vessel wall per unit volume of tissue. In the model, *LpL SL*

/*V* and *SL*

, *κ<sup>T</sup>*

(

*Lpv Sv* <sup>+</sup> *LpL SL*)

**Figure 3.** Simulations of blood pressure through 3D microvascular networks over time: (a–c) example simulation with angiostatin; (d–f) example simulation without angiostatin. Simulation domain is [0, 1] × [0, 1] × [0, 1]. Blood enters the networks at the end of the parent vessel (*x* = 0, *y* = 0, *z* = 0.5), distributes throughout the capillary network, leaves from the end of the parent vessel (*x* = 0, *y* = 1, *z* = 0.5).

the snapshots of the pressure profiles of blood flow through each vessel segment in a threedimensional microvascular networks. We keep the inlet pressure and outlet pressure across parent vessel fixed at 25 and 16 mmHg [25] in the simulation, in accordance with physiological values at the capillary scale. **Figure 3** highlights a direct comparison of blood pressure distributions (**Figure 3a**–**c** shows the blood pressure distribution in the presence of angiostatin, **Figure 3d**–**f** shows the blood pressure distribution in the absence of angiostatin). We observe that the overall blood pressure is higher in the presence of angiostatin than that in the absence of angiostatin over the same growth duration. The blood flow distribution is complex and chaotic which makes the variety of blood pressure small in the interior of the metastatic tumor compared to its exterior, contributing to the difficulties of efficient drug delivery in metastatic tumor. In the presence of angiostatin, the pressure-flows within some of the daughter vessels are elevated from the branching points to the metastatic tumor surface which provides effective blood perfusion and thus efficient therapeutic agents to the tumor. The simulation results indicate that blood perfusion varies significantly with the complex and chaotic three-dimensional microvascular networks inside and outside the metastatic tumor. The poor blood perfusion can be improved through the increased intravascular pressure with the presence of angiostatin. These results suggest that the inhibitory effect of angiostatin can affect the distribution of blood flow pressure and improve drug delivery to tumor.

#### **3.2. 3D interstitial fluid flow of metastatic tumor**

**Figure 4** shows the distribution of interstitial fluid pressure (IFP) within the metastatic tumor under the two mentioned situations. From the simulation results, we obtain that maximum IFP near the tumor center significantly dropped from 3.3, 11.48, and 11.53 to 0, 4.7, and 10.3 mmHg with the presence of angiostatin at t = 3, 7 and 14 days, respectively, which indicated the IFP plateau is well relieved. As the growth days increase, IFP gradually elevates throughout the 3D metastatic tumor and the high pressure zone is at the center of the tumor and diminishes to the periphery and later becomes flatter. Comparing **Figure 4a**–**c** to **Figure 4d**–**f**, we come to conclude that angiostatin decreases the high IFP in the tumor, thus with the lower transvascular pressure in the 3D heterogeneous capillary networks, leading to an significantly improved situation for interstitial convection which plays a significant role in nonuniform distribution of drug delivery to the metastatic tumor. These results provide important references for cancer prevention and treatment. Furthermore, antiangiogenic therapies can normalize tumor vasculature and microenvironment, at least transiently in both preclinical and clinical settings [5].

3D Numerical Study of Metastatic Tumor Blood Perfusion and Interstitial Fluid Flow Based…

http://dx.doi.org/10.5772/intechopen.78949

75

The inhibitory effect of angiostatin on the growth of metastatic tumor has been observed in some clinical and experimental malignancies. In this chapter, we develop three-dimensional mathematical models describing the metastatic tumor microvasculature and microenvironment to investigate the inhibitory effect of antiangiogenic factor angiostatin secreted by the primary tumor on metastatic tumor angiogenesis, blood perfusion, and interstitial fluid flow. Simulation results demonstrate that angiostatin has an obvious impact on the morphology, expansion speed, capillary number, and vessel branching order inside and outside the metastatic tumor. 2D and 3D mathematical models of tumor antiangiogenesis predict similar morphological behavior, such as vessels' length, branching patterns, anastomosis density, or geometric distribution, for metastatic tumor angiogenesis under the inhibitory efficiency of angiostatin. However, capillary number and microvascular density due to space growth of vessel networks are increased in the 3D model. Furthermore, the simulations reflect the influences of heterogeneous blood perfusion, widespread interstitial hypertension, and low convection within the 3D metastatic tumor by carrying out a comparative study relating to the inhibitory effect of angiostatin. We find that 2D antiangiogenesis model may be well suited to studying morphological behavior of vessel networks in the metastatic tumor, but 3D antiangiogenesis model can better analyze blood perfusion, interstitial fluid flow, or oxygen and nutrient transport within the metastatic tumor microenvironment based on its more realistic 3D microvascular networks. Although 3D simulation results are consistent with the experimental observed facts and can provide more detailed space information, however, angiogenesis and hemodynamics in the metastatic tumor by the antiangiogenic therapy are very complex. To further research tumor angiogenic mechanisms and help to improve antiangiogenic cancer therapy, more realistic features and complex biology factors need to be incorporated within the 3D model, such as the anatomy and physiology of the metastatic tumor, drug delivery of antiangiogenic therapy, behaviors of cells adhesion and interaction and coupled with the various factors or other therapy strategies.

This research is supported by the National Natural Science Foundation of China (No.

11502146), Shanghai Natural Science Foundation (No. 15ZR1429600).

**4. Conclusion**

**Acknowledgements**

**Figure 4.** Interstitial fluid pressure distributions within the metastatic tumor: (a–c) in the presence of angiostatin; (d–f) in the absence of angiostatin on the same other conditions.

with the presence of angiostatin at t = 3, 7 and 14 days, respectively, which indicated the IFP plateau is well relieved. As the growth days increase, IFP gradually elevates throughout the 3D metastatic tumor and the high pressure zone is at the center of the tumor and diminishes to the periphery and later becomes flatter. Comparing **Figure 4a**–**c** to **Figure 4d**–**f**, we come to conclude that angiostatin decreases the high IFP in the tumor, thus with the lower transvascular pressure in the 3D heterogeneous capillary networks, leading to an significantly improved situation for interstitial convection which plays a significant role in nonuniform distribution of drug delivery to the metastatic tumor. These results provide important references for cancer prevention and treatment. Furthermore, antiangiogenic therapies can normalize tumor vasculature and microenvironment, at least transiently in both preclinical and clinical settings [5].

## **4. Conclusion**

**3.2. 3D interstitial fluid flow of metastatic tumor**

74 Highlights on Hemodynamics

**Figure 4** shows the distribution of interstitial fluid pressure (IFP) within the metastatic tumor under the two mentioned situations. From the simulation results, we obtain that maximum IFP near the tumor center significantly dropped from 3.3, 11.48, and 11.53 to 0, 4.7, and 10.3 mmHg

**Figure 4.** Interstitial fluid pressure distributions within the metastatic tumor: (a–c) in the presence of angiostatin; (d–f)

in the absence of angiostatin on the same other conditions.

The inhibitory effect of angiostatin on the growth of metastatic tumor has been observed in some clinical and experimental malignancies. In this chapter, we develop three-dimensional mathematical models describing the metastatic tumor microvasculature and microenvironment to investigate the inhibitory effect of antiangiogenic factor angiostatin secreted by the primary tumor on metastatic tumor angiogenesis, blood perfusion, and interstitial fluid flow. Simulation results demonstrate that angiostatin has an obvious impact on the morphology, expansion speed, capillary number, and vessel branching order inside and outside the metastatic tumor. 2D and 3D mathematical models of tumor antiangiogenesis predict similar morphological behavior, such as vessels' length, branching patterns, anastomosis density, or geometric distribution, for metastatic tumor angiogenesis under the inhibitory efficiency of angiostatin. However, capillary number and microvascular density due to space growth of vessel networks are increased in the 3D model. Furthermore, the simulations reflect the influences of heterogeneous blood perfusion, widespread interstitial hypertension, and low convection within the 3D metastatic tumor by carrying out a comparative study relating to the inhibitory effect of angiostatin. We find that 2D antiangiogenesis model may be well suited to studying morphological behavior of vessel networks in the metastatic tumor, but 3D antiangiogenesis model can better analyze blood perfusion, interstitial fluid flow, or oxygen and nutrient transport within the metastatic tumor microenvironment based on its more realistic 3D microvascular networks. Although 3D simulation results are consistent with the experimental observed facts and can provide more detailed space information, however, angiogenesis and hemodynamics in the metastatic tumor by the antiangiogenic therapy are very complex. To further research tumor angiogenic mechanisms and help to improve antiangiogenic cancer therapy, more realistic features and complex biology factors need to be incorporated within the 3D model, such as the anatomy and physiology of the metastatic tumor, drug delivery of antiangiogenic therapy, behaviors of cells adhesion and interaction and coupled with the various factors or other therapy strategies.

### **Acknowledgements**

This research is supported by the National Natural Science Foundation of China (No. 11502146), Shanghai Natural Science Foundation (No. 15ZR1429600).
