1 Introduction

One characteristic of abnormal microenvironment in solid tumors is hypoxia^[1,2,3] .Regions far from blood vessels are usually hypoxic because of diffusion limit of oxygen through tissue. However,even the tissue nearby vessels maybe still in insufficient oxygenation due to low hematocrit ^[4] . Such phenomenon can significantly suppress sensitivity of radiotherapy and induce expression of angiogenic factors,which causes proliferation and metastasis of malignant carcinomas ^[5] .

The organization,structure and function of tumor vasculature are different from normal blood vessels. Tumor vasculature is not only a supply of nutrients to malignant tissue,but also governs phenotype of solid tumors including cell proliferation,invasion,metastasis,and responses to different clinical manipulations ^[6] . Jain proposed a hypothesis that there exists a “vasculature normalization window” during which the anti-angiogenic therapies could transiently “normalize” the abnormal structure and function of tumor vessels and the surrounding microenvironment^[7] . Further clinical practices have supported this notion.

However,our understanding of how vascular-targeted therapies affect the microenvironment surrounding the tumor is still superficial because of the inadequate imaging resolution which is incapable to reveal the fine details of tumor micro-vessels. Particularly,no reliable measurement of interstitial fluid velocity in tumors hasbeen reported due to the technical difficulties up till now ^[6] . Therefore,numerical simulation can play an important role in study. Mekheimer and Kot made a series of studies on blood flows through an artery,as well as the influence of inner tube balloon,heat,and chemical reactions on blood flow,by mathematical modeling and numerical simulation ^[8,9,10] . However,few model of tumor hemodynamics and oxygen transport during vasculature normalization has been developed so far.

The aim of this study is to investigate the effects of vascular normalization on hypoxia environment and blood perfusion in tumors. In the previous researches of tumor oxygen transport ^[11,12,13] ,since they chose to concentrate on the tumor cell growth,the vasculature distribution inside the tumor was not considered in their models. They gave the average vessel density uniformly distributed in the tumor tissue and assumed the oxygen production proportional to the vessel density. In the model of Cai ^[14] ,itwasassumedthatthemicro-vessels provided local oxygen which influences the tumor growth,and the oxygen concentration was uniform inside of the vessels. However,It has reported that there is no clear relationship between blood flow rate and oxygen tension (pO₂) of individual vessels,as revealed by high-resolution intravital microscopy and phosphorescence quenching microscopy. Surprisingly,almost no oxygen was observed in some perfused tumor vessels ^[2] .

In this study,the 3D mathematical models of tumor angiogenesis and vascular-disrupting are used to simulate the “un-normalized” and “normalized” vasculatures. The structure,density, maturity,and mechanical properties of the vessels become different during the normalization period. A new model which combines the tumor hemodynamics and oxygen transport is developed. In this model,the intravascular-transvascular-interstitial flow with blood rheology is tightly coupled,and we put emphasis on the red blood cells (RBCs) delivery through the microvasculature,with consideration of vasculature normalization and the consequent change of blood perfusion. Finally,the oxygen transport inside of tumor tissue is calculated,in which the oxygen resource is produced bythe heterogeneous distribution of hematocrit,instead of by the vascular average density artificially given or the vessel nodes distribution in the previous models. Figure 1 describes the major structure of this research.

2 Materials and methods 2.1 Tumor vascular normalization

It has been proven that tumor vascular system could be disrupted or pruned effectively by some vascular-disrupting agents. Trifurcation vessels which are twisted and tangled with each other are believed to be two typical network structure abnormalities of tumor vessels,which greatly influence blood perfusion in tumors ^[15,16] . In addition,immature vessels in tumors will not only help local spread of cancer cells,but also promote cancer cell infiltration through its own protease partial role. It has been reportedthat,the treatments efficacy will be improved if the high proportion of tumor immature vessels is damaged by the drugs ^[17] .Also,theresearches have predicated that anti-tumor drugs can destroy some abnormal blood vessels with lower blood flow rate ^[18] . Therefore,in this study,three vascular-disrupting approaches are adopted,corresponding to the abnormal characteristics of tumor vasculature described above.

The vascular networks are generated from our models of “tumor angiogenesis” ^[19] and “tumor vascular-disrupting”^[20] ,as shown in Fig. 2. Figure 2(a) is the tumor vasculature with “un-normalization” (Network 1(U),the control one),Figs. 2(b),2(c),and 2(d) are the normalized vasculatures by the three disrupting schemes,corresponding to the network structure, vessel maturity,and blood flow rate,respectively. Network 2(S) is obtained by disrupting the trifurcation and twisted vessels from Network 1(U); Network 3(M) is by trimming the low mature vessels with a growing time less than 40% of the whole growing time of Network 1(U); Network 4(F) is by cutting off the vessels with blood flow rate less than 40% average blood flow rate of Network 1(U).

The simulation is carried out on a 3D domain of 2 mm×2mm×2 mm. An arteriole and a venule are located on the right top and right bottom boundaries,respectively,and a half tumor with a radius (R_t) of 1 mm is on the left and surrounded by normal tissues of the host. The detailed description of the angiogeneses model can be found in Ref. ^[19]. 2.2 Tumor blood perfusion and RBCs delivery within tumor

A coupled flow model of tumor hemodynamics has been developed in our pervious work, which incorporates the intravascular blood flow,the transvascular leakiness,and the interstitial fluid movement ^[21,22] . In this study,we put emphasis on the RBCs delivery through the microvasculature,to investigate the hematocrit distribution within the tumors which provides the oxygen resource to the tumor tissue. The major governing equations are as follows.

At each node,the flux conservation is

where the index k refers to adjacent nodes. Q_V is the intravascular flow rate without fluid leakage,described by local Poiseuille’s law,i.e., Q_t is the transvascular flow rate,given by Starling’s law,i.e., where P_V is the intravascular pressure,P_i is the interstitial fluid pressure,μ is the blood viscosity,Δ_l is the vessel length,R is the vessel radius,L_pV is the hydraulic permeability of blood vessel wall,σ_T is the average osmotic reflection coefficient for plasma proteins,and π_V and π_i are the colloid osmotic pressure of plasma and interstitial fluid. The interstitial fluid flow is governed by Darcy’s law,i.e., where U_i is the interstitial fluid velocity,and Kis the hydraulic conductivity coefficient of the interstitium.

The continuity equation is given by

whereSV/V is the surface area of blood vessel wall per unit volume of tissue,and P_eV = P_V−σ_T(π_V−π_i) is the effective pressure inside blood vessels.

The delivery of RBCs at microvascular bifurcations follows the empirical formula proposed by Pries and Secomb ^[23] . The fractional flow of RBC into one daughter branch (F_QE)iscalculated from the respective fractional blood flow (F_QB) as follows:

where logit x=ln(x/(1−x)). Parameter Φ₁ represents the difference between the relations derived for the two daughter branches,Φ₂ denotes the nonlinearity of the relation between F_QE and F_QB,and Φ₃ defines the minimal fractional blood flow required to draw RBCs into the daughter branch,d_α,d_β,and d_m are the diameters of the daughter branches and the mother vessel,H_m is the discharge hematocrit in the mother vessels. The formula describing the tumor vessel compliance and blood viscosity can be found in Ref. ^[21]. 2.3 Tumor oxygen transport

Tumor cells need oxygen to grow and invade. The fundamental role of RBCs is to carry oxygen to tissues because the amount of dissolved oxygen in RBC is about 70 times greater than that of plasma. In the present model,oxygen production is assumed to be proportional to the hematocrit value in the vessels. The oxygen concentration cis governed by diffusion, production,consumption by the tumor cells,anddecay of itself. The oxygen equation therefore has the form as follows:

where D_c is the oxygen diffusion coefficient,h_RBC=H_nRBC represents the hematocrit within the vessel present (CA model),H is the hematocrit. The value of h_RBC is either H if a vessel is present (n_RBC= 1) or 0 if it is not (n_RBC= 0),which combines the information of the network structure and the RBCs distribution. nTrepresents the tumor cell (CA model); if there is a tumor cell n_RBC = 1,ifnot n_RBC = 0.β,γ,andαare positive constants.

In the previous studies ^{[11,12,13,14]} ,since they chose to concentrate on the tumor cell proliferation, the vasculature and blood perfusion inside the tumor were not included in their models. It was assumed that the oxygen resource was proportional to the vessel density through the tumor tissue and the oxygen concentration was uniform inside the vessels,and hence no RBC delivery and heterogeneous hematocrit distribution through the vessels were calculated. In the present model,the oxygen is produced by the heterogeneous distribution of RBCs through the vasculature,represented by the spatial variable h_RBC. Its value is obtained from the above flow simulation,which synthesizes the information of vasculature structure and hematocrit distribution. Therefore,both of the vasculature normalization and the consequent change of blood perfusion with RBCs delivery are taken into account. 2.4 Simulation methods

In the simulations,the baseline parameter values are listed in Table 1. As for the “vascular normalization” cases,the values of several parameters describing microenvironment become close to normal tissues ^[20,21] . In order to solve the coupling flows,we establish a specific numerical computational procedure,based on the iterative simulation methods,including local iterations at individual parameter level and one global loop to provide coupling and control of the simulation convergence.

3 Results and discussion 3.1 Tumor blood perfusion and RBCs distribution during vascular normalization

In the following figures,each line which provides a general trend of the flows is the average flow value based on 30 simulated networks of the corresponding disrupting approach.

Changes induced by abnormalities of both vasculature and viscosity in tumor exaggerate the resistance to blood flow ^[27] . The overall perfusion rates in malignant tissues are much lower than those in healthy tissues. The average blood velocity in tumor vessels can be an order of magnitude lower than that in normal conditions ^[28] . Figure 3 shows the distribution of average blood flow velocity along the radial-direction,before and after the vascular normalization,respectively. (In the figures,x-axis represents the normalized distance from the tumor center to the parent vessels,whereRTis the tumor radius,r/R_T=0.0,1.0 correspondent to the tumor center and boundary,respectively.) As seen from Fig. 3,as for the un-normalized vasculature Network 1(U),the blood flows extremely slowly especially in the tumor interior. After the normalization treatments,all the tumor blood perfusions are increasing. Some experimental researches have reported that vascular-disrupting treatments could increase blood perfusion in tumors, e.g.,the blood perfusion within the breast tumor accelerated following vascular-disrupting by anti-VEGF TKRi (provided by Dr. A. R. Padhani,Paul Strickland Scanner Centre,U. K.). Increase of blood perfusion is helpful to drug delivery,and therefore may enhance the efficacy of therapeutic agents to tumors.

The distribution of hematocrit is shown in Fig. 4. The value of hematocrit in the parental vessels is 0.45. In the un-normalized case,the hematocrit reduces rapidly inside the tumor tissue,because the majority of RBCs are distributed into the vessels near the parental vessels caused by the chaotic vascular structure and the lower blood velocity in the tumor. Especially,Fig. 4 demonstrates that the hematocrit within the interior of the tumor (except for the periphery of the tumor) approaches zero in the most area. As shown in Fig. 3,although blood perfusion can be observed in the interior region of tumor in certain cases,no RBC is present in those perfusion (see Fig. 4). Such a phenomenon that no clear relationship exists between blood flow rate and oxygen tension (pO₂) in individual tumor vessels has been revealed by the highresolution intravital microscopy and phosphorescence quenching microscopy,and unexpectedly, there is no oxygen observed in some perfused tumor vessels ^[2] . Our result shown in Fig. 4 could well explain this experimental discovery in Ref. ^[2]. The fundamental role of RBCs is to carry oxygen to tissues because the amount of dissolved oxygen in RBC is about 70 times greater than that of plasma. Thus,devoid of RBCs in the interior region of tumor results in hypoxia and the formation of necrosis. Our result is consistent with this significant phenomenon. During the vascular normalization,the blood hematocrit is increasing greatly,which indicates more RBCs delivery to the tumor interior after the treatments,as shown in Fig. 4. Accordingly,the tumor hypoxia environment may be greatly improved,as discussed below.

3.2 Tumor hypoxia environment during vascular normalization

The oxygen concentration c^* along the radial-direction is shown in Fig. 5. The oxygen concentration is the lowest in the central region of the tissue and rises towards the periphery of the tumor. The change trend of oxygen concentration (see Fig. 5) is similar with that of RBC hematocrit (see Fig. 4). Unlike the hematocrit,the oxygen in the tumor center on the un-normalized vasculature is not close to zero,due to the diffusion effect of oxygen. From the calculation,the total oxygen contents inside the tumor tissue on Network 2(S),Network 3(M),Network 4(F) increase 67% ,51%,95%,respectively,compared with that of the vasculature Network 1(U). The result indicates that the tumor hypoxic environment,especially in the center of tumor,has been improved greatly afterthe vascular normalization,in all the three vascular-disrupting approaches,especially for the one according to the flow rate Network 4(F).

A special subset of tumor cells which are more malignant,aggressive,and genetically unstable and less susceptible to apoptosis are selected and rendered for resistance to various therapies ^[29,30,31] . Hypoxia can promote growth and metastasis of tumors via the induction of angiogenic factors ^[5] . Therefore,the elevation of oxygen concentration in tumors could improve its metabolic environment,and consequently reduce malignancy of tumor cells.

Hypoxia is not only associated with resistance to some chemotherapeutics but also reduces the radiation sensitivity in solid tumors because oxygen is an essential component of radiation therapy ^[32] . In hypoxic and/or acidic circumstances,immune system is not fully functional and allow tumor cells to evade host immune response and attack of certain cell-based therapies. Therefore,the vascular normalization could also enhance the radiation and chemotherapeutics to tumors.

Furthermore,since the proliferation rate of tumor cells is greater than the angiogenic capacity,the distance between tumor cells and vessels will increase. Once this distance exceeds the diffusion distance of oxygen and nutrients,necrosis of tumor cells will occur. Generally, the distances between tumor cells and vessels are divided into three groups ^[33] : >150μm, 100-150μm,<100μm. The farther away from the vessels,the lower pO₂ is,and the more easily to induce the necrosis of tumor. Tumor cells in>150μm group are confronted with the greatest environmental stress. In such a circumstance,adaptive changes in the proteome and genome of neoplastic cells occur and result in the emergence of more malignant clones that are able to overcome nutrient deprivation or to escape their hostile microenvironment. At the same time,this is the very region where drug delivery is particularly ineffective. Adaptive changes include accelerated erythropoiesis and glycolysis,promotion of cell survival and further angiogenesis,inhibition of apoptosis and cell differentiation. Simultaneously,there are sufficient oxygen and/or nutrients for tumor cell growth in the periphery of tumor tissues because of the abundant blood perfusion with adequate hematocrit (see Figs. 4 and 5). In reality,the two physiological processes of tumor proliferation and angiogenesis are coupled,although the relationship has not been considered in our model yet. 4 Conclusions

In this study,we investigate the blood perfusion,especially the RBCs delivery through the microvasculature,and the improvement of hypoxia microenvironment in tumors,during the tumor vascular normalization. The 3D mathematical models of tumor angiogenesis and vascular-disrupting are used to simulate the “un-normalized” and “normalized” vasculatures. A new model which combines the tumor hemodynamics and oxygen transport is developed,in which the oxygen resource is produced by the heterogeneous distribution of the RBC hematocrit from the flow simulation,instead of by the ones artificially given or being proportional to the uniform vessel density through the tumor tissue in the previous models.

Our result could well explain the experimental discovery about the unclear relationship between blood perfusion and oxygen concentration in the tumor vessels. Additionally,the results show that,both of the tumor blood perfusion and hematocrit through the microvasculature are increased to some extent,and the hypoxia microenvironment in the tumor center is greatly improved,during the vascular normalization. The total oxygen contents inside the tumor tissue increase about 67%,51%,and 95% for three approaches of normalization,respectively. The elevation of oxygen concentration in tumors could improve its metabolic environment, and consequently reduce malignancy of tumor cells. Also,it could enhance the radiation and chemotherapeutics to tumors.