Computational Multi-Scale Modeling of Drug Delivery into an Anti-Angiogenic Therapy-Treated Tumor

1. Introduction

Investigation of different methodologies to improve the quality of drug delivery and the effectiveness of chemotherapy is of great interest among researchers [1,2]. Abnormally tortuous tumor microvasculature generated via the angiogenesis process is one of the main reasons for unsuccessful drug delivery [3,4]. Therefore, anti-angiogenesis, an adjuvant treatment strategy [5], needs to be studied, especially by using mathematical modeling [6]. The concept of anti-angiogenesis was introduced by Folkman [7] regarding the prevention of capillaries’ sprouting into the tumor site. It has been reported in clinical studies [8,9,10] and review papers [11,12,13] that anti-angiogenic drug administration improves chemotherapeutic drug delivery, its efficiency, and its penetration depth.

The modeling of solid tumors involves multiple spatial and temporal scales of complexity [14,15]. The formation of a tumor-induced capillary network in nanometers, intravascular blood flow in the micrometer dimension, blood flow distribution in the capillary network in millimeters, and fluid flow and solute transport in tumor and normal tissues on the scale of centimeters are all examples of multi-scale modeling of cancer-related studies [16].

Mathematical and computational studies have made great strides in cancer modeling to provide qualitative and quantitative comprehension of the complex dynamics of cancer. Hadjicharalambous et al. [17] conducted a review of in silico studies that assessed tumor perfusion, angiogenesis, drug delivery, and investigations leveraging clinical data. Jain and his colleagues [18,19,20] conducted basic studies on drug delivery into solid tumors. They considered tumor tissue as a porous medium and introduced high interstitial fluid pressure (IFP) as one of the main barriers to effective drug delivery. In 2007, Jain et al. [21] studied vascular normalization by modeling the interstitial fluid flow in a macroscopic model. They showed that IFP decreased after vascular normalization. This model was improved to consider solute transport analysis in a single tumor nodule [22] and a non-homogeneous macroscopic model [23,24]. Time course of perfusion was introduced as a controlling factor of normalization efficiency, which depends on tumor size, normalization intensity, and concurrent therapeutic agents [23].

Mathematical modeling of angiogenesis should be considered for extracting the capillary network to develop a microscopic analysis. Anderson and Chaplain [25,26] developed a mathematical study on continuous and discrete models of angiogenesis, which is the base of different study [27,28,29]. In our recent study [30,31], the mathematical model of angiogenesis was modified to consider the effect of proliferation and death of endothelial cells and matrix-degrading enzymes. The morphology of a tumor-induced microvascular network with two parent vessels was simulated under the inhibitory effect of an anti-angiogenic agent, angiostatin.

Though the macroscopic view of the tumor microenvironment could provide a qualitative description of different phenomena of cancer, considering the microscopic capillary network of the tumor is an important factor in achieving a more realistic illustration. Stéphanou et al. [32] and McDougall et al. [33] investigated tumor-induced angiogenesis via a mathematical model that simultaneously simulated blood flow and dynamic capillary network progression by considering the non-Newtonian blood behavior and non-uniform distribution of the hematocrit in the bifurcations. The objective of their research was to develop a model to make tumor-induced angiogenesis more precise. Accordingly, they did not consider transvascular flow and interstitial fluid flow. Alamer and Xu [34] studied the effect of microvasculature on interstitial fluid flow and investigated vascular normalization via capillary pruning. Their model had the limitation of not considering non-Newtonian blood behavior and the adaptation of micro-vessel diameter in response to transmural pressure and metabolic stimuli. Soltani and Chen [35] investigated interstitial fluid flow distribution in relation to blood flow in a dynamic tumor-induced microvasculature from one parent vessel by considering the non-continuous behavior of blood. Sefidgar et al. [16] further developed the model of Soltani and Chen [35] to include solute transport analysis to reflect drug distribution in the tumor site. Wu et al. [27] used a mathematical model of angiogenesis, which produces capillary networks originating from two distinct vessels of arteriole and venule, to study blood flow and interstitial flow distribution by decreasing the capillary network’s density to mimic the anti-angiogenic effect. In another study by this group [36], the anti-angiogenic effect of angiostatin and endostatin in combination with intravascular and interstitial flow is considered. The blood vessels in this study were assumed to be dynamic based on the compliance method of Netti in the tumor tissue. In another study [37], the anti-angiogenic effect of angiostatin on interstitial fluid flow behavior was investigated in a rigid capillary network without considering the non-continuous behavior of blood.

Ozturk et al. [38] investigated the effect of vascular normalization on liposome delivery into a homogeneous solid tumor based on a study by Jain et al. [21] They found that the efficiency of normalization in improving liposome delivery is a function of tumor size. Stylianopoulos and Jain [39] investigated the effect of vascular normalization by assuming a decrease in micro-vessels’ diameter and their pruning. They concluded that normalization is more effective in microvasculature with more permeability and less compressibility characteristics.

Steuperaert et al. [40] investigated intraperitoneal drug delivery using a macroscopic model based on an actual image extracted via magnetic resonance imaging. They also took into account interstitial transport properties, considering a non-uniform distribution. Image processing techniques were used to develop a macroscopic model [41] of a brain tumor to investigate the effect of administration of bevacizumab on drug delivery. Sweeney et al. [42] studied anti-angiogenesis by applying the modifications to transport properties in solid tumors with a microvasculature based on the real image.

To develop a comprehensive numerical microscopic model that simulates the effect of anti-angiogenic adjuvant therapy on the quality of drug delivery, one should consider multi-physics in a multi-scale context. This model should describe (a) the generation of tumor-induced angiogenesis under the inhibitory effect of the anti-angiogenic agent, angiostatin in this study; (b) intravascular blood flow in connection with interstitial fluid flow in a dynamic microvasculature, taking into account the non-Newtonian behavior of blood and adapting the micro-vessels’ diameter in response to hemodynamic and metabolic stimuli; and (c) spatiotemporal solute transport into the tumor tissue. All of these aspects are addressed in the present paper for the first time. Consideration of different tumor sizes and vascular normalization approaches are other contributions of this paper. Three approaches are observed for investigating anti-angiogenic therapy. In the first approach, the microvasculature is updated under the influence of an anti-angiogenic agent without considering modifications in transport properties. In the second approach, modifications in transport properties experimentally induced via anti-angiogenesis are considered. The third approach applies the model by combining the former two.

2. Materials and Methods

2.1. Computational Model Geometry

In this study, a circular tumor (with different radius sizes of $R=0.8\hspace{0.17em}\mathrm{cm}$, $R=0.6\hspace{0.17em}\mathrm{cm}$, $\mathrm{R}=0.4\hspace{0.17em}\mathrm{cm}$, and $\mathrm{R}=0.2\hspace{0.17em}\mathrm{cm}$) surrounded by normal tissue (a rectangular domain of $2\times 4\hspace{0.17em}\mathrm{cm}$) is considered. A parent vessel with ten sprouts is defined on the left side of the tumor, and one parent vessel with five sprouts is located on the right side of the tumor. Figure 1 shows an illustration of the computational field, which is rendered in a two-dimensional domain.

Figure 1. Schematic view of the computational domain, coordinate origin, parent vessels, cut lines, and boundaries.

2.2. Governing Equations

In this multi-scale study, angiogenesis and network formation under the influence of an anti-angiogenic agent, blood flow distribution in microvascular networks, and interstitial fluid flow and solute transport in tumor and normal tissues are mathematically modeled. The following sections describe the equations that govern the physics of each phenomenon.

2.3. Numerical Simulation Explanation

To clarify the numerical model applied in the present study, boundary and initial conditions, numerical modeling process, grid-independent solution, and parameter values are described in the following sections.

3. Validation

As the present study involves complex multi-scale and multi-physics modeling, it is not possible to validate it experimentally. Moreover, the present study addresses various complications, which contribute to its progressive nature. Therefore, to verify the present model, various phenomena from the case studies in the literature [16,28,66] are duplicated in this study. In the first stage, angiogenesis is modeled under the influence of endostatin, another anti-angiogenic agent, based on the study by Tee and DiStefano III [28], by administering a continuous injection with a dose of 20 mg/kg/day. As reported in our previous study [30], capillary growth toward the tumor was halted in this case study, which is consistent with the literature [28]. Experimental research has demonstrated a reduction in tumor microvascular density following anti-angiogenic therapy. For instance, Soto-Pantoja et al. [67] conducted an in vivo study, revealing a 50% decrease in vessel density in human A549 lung tumor xenografts subcutaneously implanted in mice following angiostatin injection. Similarly, a decrease in blood vessel density was observed in established ovarian cancer in mice after angiostatin treatment [68]. To verify the interstitial fluid flow behavior, the physical conditions of the experimental work by Boucher et al. [66] are imposed on the present study’s model, and the results show good agreement between the two research studies, as shown in Figure 6a. In the third stage, the average drug exposure is determined in $R=0.4\hspace{0.17em}\mathrm{c}\mathrm{m}$, considering the same methodology as described in the previous study [16], while taking into account the microvasculature of the present research. The comparison is shown in Figure 6b. Upon comparing the results, it is evident that the solute accumulation process over time aligns well with the literature [16]. However, a discrepancy arises due to the variation in computational domains. Sefidgar et al. [16] considered a single parent vessel sprouting toward half of a circular tumor, leading to a different capillary network structure.

Figure 6. Comparison between results of the present model and literature [16,66]. (a) A comparison between the results of the interstitial fluid pressure distribution in the present study and the work of Boucher et al. [66]. (b) A comparison between the results of the average solute exposure in the present study and the work of Sefidgar et al. [16].

4. Results and Discussion

Drug delivery into solid tumors with different sizes and remodeled dynamic networks are investigated numerically in this study. The effect of the anti-angiogenic adjuvant treatment on the quality of drug delivery is studied here. Three cases are considered to study this effect. In case one, modification in response to the inhibitory effect of angiostatin and consequently updated tumor-induced microvascular network is considered without any change in the interstitial transport properties. In the second case, only modifications in transport properties are considered. In the third case, both modifications in the microvascular network and transport properties in response to the anti-angiogenic therapy are considered.

Interstitial fluid flow is carried out to find out the IFP and IFV distributions. Solute transport analysis is considered to evaluate the concentration of the therapeutic agent delivered into the tumor site. Two parameters of non-dimensional average solute exposure (NDASE) and non-dimensional average solute distribution non-uniformity (NDASDNU) are introduced based on our previous research [23] as indicators of the quality of drug delivery into the tumor.

As different tumor sizes are considered, the final time for each geometry is defined such that the average amount of solute in the tumor site reaches one percent of its maximum value after considering vascular normalization modification (case 3). This time is equal to 820,210 s, 709,150 s, 511,750 s, and 646,160 s in tumors with $R=0.8\hspace{0.17em}\mathrm{c}\mathrm{m}$, $R=0.6\hspace{0.17em}\mathrm{c}\mathrm{m}$, $R=0.4\hspace{0.17em}\mathrm{c}\mathrm{m}$, and $R=0.2\hspace{0.17em}\mathrm{c}\mathrm{m}$, respectively.

4.1. Fluid Flow Analysis

Investigating the interstitial fluid flow is significant because high IFP in the tumor site, its sudden decrease, and consequently the sudden increase of IFV at the tumor margin were introduced as a main barrier for qualified drug delivery in the literature [18,19,20,21,69]. Therefore, interstitial fluid flow analysis in connection with intravascular blood flow is performed. Figure 7, Figure 8 and Figure 9 demonstrate the non-dimensional distribution of IBP, IFP, and IFV (relative to their maximum value) in different case studies of the present research.

Figure 7. Non-dimensional IBP contour in different tumor sizes and states.

Figure 8. Non-dimensional IFP contour in different tumor sizes and states.

Figure 9. Non-dimensional IFV contour in different tumor sizes and states.

Figure 7 shows the non-dimensional IBP contour in different tumor sizes and states. It can be seen that the distribution of IBP is dependent on the microvascular network morphology in the first order and on the transport properties next.

Figure 8 and Figure 10 show the non-dimensional contour of IFP and IFP distribution along cut lines shown in Figure 1. According to these figures, the IFP in the tumor site is more than the surrounding normal tissue. There are three main reasons for this phenomenon: higher density of the microvascular network, more leakage of the capillaries, and lack of an efficient lymphatic system in the tumor site. It is obvious that IFP does not have a uniform distribution in the tumor site as what is in the macroscopic analysis. Because there is non-uniform distribution of the blood vessels and consequently non-uniform fluid flow sources (${\mathsf{\varphi }}_{\mathrm{B}}$ in Equation (15)). These results improve the visualization of the tumor microenvironment behavior and bring a more realistic view of that.

The value of the IFP in the tumor area before anti-angiogenic therapy in different tumor sizes studied in this research is in accordance with an experimental study by Butcher and Jain [70], who reported the tumor pressure range from 586 to 4200 Pa. In comparison to our previous study [23], the IFP has a higher value in the tumor site before considering the anti-angiogenic therapy. This result is in accordance with our previous research [16].

By comparing the IFP distribution before considering anti-angiogenesis and considering it by case 1 in Figure 8 and Figure 10, the significant effect of micro-vessels feeding the tumor is demonstrated. IFP distribution in $R=0.8\hspace{0.17em}cm$ and $R=0.6\hspace{0.17em}cm$ in Figure 8 and Figure 10 shows that even though the density of micro-vessels is reduced by 24% and 13%, respectively [30], in response to the inhibitory effect of angiostatin, the maximum amount of IFP occurs in approach 1 of considering anti-angiogenesis. This result shows that the interstitial fluid flow behavior is more dependent on the structure of the microvascular network than on its density. This behavior changes in $R=0.2\hspace{0.17em}cm$ by pruning severely micro-vessels, such that IFP has its maximum value in the case without considering the anti-angiogenic therapy.

It has been demonstrated preclinically [71] that anti-angiogenic therapy leads to the establishment of a pressure gradient across vasculature, as depicted in Figure 8 and Figure 10. It is evident that various approaches to modeling anti-angiogenesis result in the development of a pressure gradient between the microvasculature wall and its periphery. Another observation in the present study is the reduction of IFP induced by anti-angiogenic therapy, which has been reported in different experimental trials [71,72,73,74]. The IFP in tumors implanted subcutaneously in mice with an approximatively equivalent radius of 4 mm decreases ~50% after treatment with the anti-angiogenic agent EGCG [75]. The reduction in the average amount of IFP in the tumor with R = 0.4 cm is close to this value (~45%). However, this close agreement may be interpreted qualitatively rather than quantitatively, as different algorithms were utilized between the two studies.

Figure 9 and Figure 11 show the contour of IFV and its distribution along cut lines. In dissimilarity with macroscopic studies, IFV has a non-uniform distribution in tumor tissue. IFV has a non-zero value in some parts of tumor tissue, as IFV is proportional to the IFP gradient (Darcy’s law). Non-uniform IFP distribution, and consequently the existence of pressure gradient in the tumor site, results in non-zero IFV inside the tumor.

In Figure 11, in all tumor sizes, IFV along the horizontal line has its non-zero maximum value at the tumor margin before considering the anti-angiogenic treatment because of the almost uniform distribution of IFP along this line. However, IFV has a non-zero value not only in the tumor margin but also outside the dense region of micro-vessels along the vertical line.

In addition to the discussion made about the effect of anti-angiogenesis case 1 on interstitial fluid flow behavior, it is shown that other than R = 0.2 cm, in which anti-angiogenesis induced via angiostatin application causes a 55% decrease in microvascular network density [30], this approach (case 1) causes a pressure gradient in the inner areas only along the vertical direction. The modification in IFV distribution in $R=0.2\hspace{0.17em}cm$ is obvious. An intensive decrease in capillary density limits the source of flow, which is responsible for the decrease in IFP and non-zero IFV inside the tumor along both horizontal and vertical directions. Figure 10 shows that the second approach of anti-angiogenic treatment causes a decrease in IFP in all tumor sizes. Moreover, this approach modifies the steep pressure gradient in the tumor margin and shifts the pressure gradient to the inner areas. In accordance with the modification in IFP with the second approach, IFV behavior is also modified. The effect of normalization induced by the third approach on IFP in $R=0.4\hspace{0.17em}cm$ is shown in Figure 10 just by causing the IFP gradient, which causes non-zero IFV in inner areas and a decrease in IFV at the tumor margin. By decreasing the tumor size, the IFP decrease induced by the third approach of vascular normalization is increased.

4.2. Solute Transport Analysis

A single bolus injection, whose equation is described in Section 2.3.1, is considered in this study, and the convection-diffusion equation is solved numerically to find the concentration distribution of the therapeutic agent.

Figure 12 shows the non-dimensional solute concentration contour in $R=0.8\hspace{0.17em}cm$ at different post-injection times. This figure is dimensionless relative to its maximum value. The non-uniform distribution of the solute is obvious, which is the result of the tortuous structure of microvasculature. As shown in Figure 12, vascular normalization modifies the drug wash-out phenomenon in the tumor periphery. This is evident from the modification of the spatial distribution range toward the inner parts of the tumor (case 1), and moreover a reduction in the amount of drug wash-out (cases 2 and 3) compared to the pre-anti-angiogenic therapy.

By considering the modifications in transport properties (cases 2 and 3), while the maximum amount of solute reaching the tumor site is decreased, the tumor is exposed to the drug for a longer period of time. This behavior causes more uniformity of solute distribution because the mechanism of diffusion in the interstitium has more time for carrying solute to the sites with less or no density of micro-vessels. It is observed in Figure 12 that the uniformity in the solute distribution in the entire tumor region at 72 h post-injection is higher in cases 2 and 3 compared to the case without considering anti-angiogenic therapy.

Figure 13 shows the distribution of solute along horizontal and vertical cut lines in $R=0.8\hspace{0.17em}cm$ at different post-injection times. It is obvious that the solute has a heterogeneous distribution in the tumor region because of the heterogeneity in micro-vessels distribution as the channels for transporting the therapeutic agents into the tumor. By comparing the present study’s results with previous ones, which assumed a uniform distribution of blood vessels in vital regions of tumors, it is obvious that considering a dynamic microvascular network based on real phenomena is essential to have a more realistic view.

The solute concentration has its jumped wash-outed maximum value in the tumor margin along the horizontal line before considering anti-angiogenesis treatment because of a sudden increase in IFV profile in the margin (Figure 13a). The first approach of anti-angiogenesis treatment cannot modify this behavior because it cannot modify the IFP and IFV distribution in $R=0.8\hspace{0.17em}cm$ along the horizontal line in Figure 10 and Figure 11. The solute concentration along the vertical line has its maximum value in the inner areas of the tumor at early injection (1 h). Over time, the solute concentration starts to wash out from the tumor boundaries because of the out-flow convective mechanism. Vascular normalization induced via anti-angiogenesis modifies this behavior at each post-injection time (Figure 13b–d, vertical line).

The solute concentration increases fast before anti-angiogenic therapy. Then, the concentration starts to decrease (as seen in Figure 13c) because of plasma clearance. Vascular normalization, especially by the second and third approaches, makes a difference in the timing of drug delivery by modifying the transport properties (which is discussed in detail in our previous study [23]) or modifying both transport properties and micro-vessels distribution.

It is shown in Figure 13 that at 24 h post-injection, drug exposure and uniformity are improved by vascular normalization, especially by the second and third approaches. This phenomenon is more obvious at 72 h post-injection. This behavior is related to (a) the trade-off between $\mathrm{P}$, $\frac{\mathrm{S}}{\mathrm{V}}$, and ${\mathrm{C}}_{\mathrm{P}}$, which control the trans-vascular diffusion as the dominant transport mechanism in areas with dens micro-vessels, and (b) modifications in trans-vascular convection and interstitium convection transport mechanisms (via the modifications in IFP, IFV, ${\mathrm{L}}_{\mathrm{p}}$, and $\frac{\mathrm{S}}{\mathrm{V}}$), which play critical roles in drug delivery in areas with lower micro-vessels density and tumor margins. In other words, by modifying transport properties (case 2) and also the microvascular network morphology (case 3) due to vascular normalization, the tumor exposure to the therapeutic agent occurs slowly, and subsequently, plasma clearance does not occur fast like the untreated tumor. Therefore, the tumor is exposed to the drug for a longer time (it is shown in Figure 14 that at 120 h and 216 h post-injection, in contrast to the untreated tumor and the first approach used for anti-angiogenesis, the second and third approaches of vascular normalization cause the presence of solute inside the tumor region long time after a single injection).

Based on the aforementioned discussion, the type of therapeutic agent used in either chemotherapy alone or in combination with anti-angiogenic treatment is another crucial factor that governs the effectiveness of drug delivery and vascular normalization. In further detail, the type of therapeutic agent through its molecular weight is a determinant of the plasma clearance (which its rate is expressed by drug half-life [76]), osmotic filtration reflection coefficient, effective diffusion coefficient, and micro-vessel permeability coefficient [18]. For example, a therapeutic agent with a rapid plasma clearance half-life results in a faster decrease in concentration within the tumor interstitium. Consequently, the interstitium has a shorter window of exposure to it. Vascular normalization induced via anti-angiogenic therapy reduces the excessive leakiness of the microvascular network [77,78] and may improve the quality of drug delivery. As outlined in our prior study [23], vascular normalization has the potential to improve clearance behavior by directly modifying the parameters governing trans-vascular diffusion, trans-vascular convection, and interstitial convection and indirectly influencing the interstitial diffusion transport mechanism. In other words, modification of transport properties and microvascular network induced via vascular normalization can cause an improvement in solute distribution via a regulated process of drug entry into the interstitium and subsequent return to the plasma. Thus, during a specific perfusion time frame, precise intensities of vascular normalization have the potential to enhance the delivery of a specific therapeutic agent to a solid tumor with predefined properties. More discussion can be found in [23]. Thus, it is important to study effective parameters in drug delivery, with the specific plasma clearance half-life of the therapeutic agent being one of them.

It is observed that the first approach considered for anti-angiogenesis can decrease drug wash-out (along the vertical axis) due to the modification in the distribution of blood micro-vessels caused by the inhibitory effect of angiostatin. However, this approach cannot improve the distribution of solute.

The values of two parameters, NDASE and NDASDNU, are determined for different tumor sizes and under various vascular normalization approaches at different final time durations. The results are reported in . In terms of both average drug exposure and uniformity of drug exposure, the first approach considered to mimic anti-angiogenesis does not yield a positive effect. Since the blood micro-vessels transfer drugs to the tumor site, updating the micro-vessel distribution due to the inhibitory effect of angiostatin not only leads to a reduction in microvascular density but also concentrates them in central areas of the tumor. This, in turn, increases non-uniformity in drug distribution without significantly altering exposure. The non-uniformity increases as the tumor size decreases because of the increase in micro-vessel pruning due to the inhibitory effect of angiostatin. This demonstrates that the heterogeneous distribution of micro-vessels results in a non-uniform distribution of therapeutic agents at the tumor site, transported via trans-vascular diffusion and convection mechanisms via blood vessel sources. On the contrary, it has been demonstrated that a more uniform distribution of the microvascular network leads to less non-uniformity in solute distribution, as demonstrated in $R=0.2\hspace{0.17em}cm$, even before considering anti-angiogenesis in the present study. Clinical studies demonstrate that tumor vessels exhibit significant heterogeneity in their distribution, diameter, density, and serpentine shape. This leads to low and heterogeneous blood flow in tumor tissue. The primary factors responsible are the mechanical forces generated via fluid and solid stress, along with an excess of vessel permeability [79]. Thus, reengineering the abnormal and heterogeneous microenvironment of solid tumors via approaches such as normalizing tumor blood vessels and the extracellular matrix, and alleviating vessel compression is conducted to overcome the challenges associated with cancer heterogeneity [39,77,78]. From another point of view, this behavior shows the determinative effect of the architecture of the microvascular network, which can be affected by the anti-angiogenic agent. So, the type of anti-angiogenic agent, as a controlling factor of pruning the micro-vessels and consequently the uniformity of their distribution, is an important factor in reengineering the microvascular distribution.

In addition to the microvasculature update, considering the modifications in transport properties (case 3) modifies the non-uniformity caused by case 1. This is due to the earlier discussion regarding prolonged tumor exposure to the therapeutic agent and improvements in IFP and IFV induced by the second approach of anti-angiogenesis. In tumor size $R=0.8\hspace{0.17em}cm$, where microvascular network density is decreased by 13% [30], and, of even greater importance, micro-vessel morphology is modified to a more uniform distribution, anti-angiogenesis via the third approach increases drug uniformity by 7%.

The results indicate that the second approach of anti-angiogenesis improves drug distribution by 19%, 17.3%, 14.1%, and 38.7% in $R=0.8\hspace{0.17em}cm$, $R=0.6\hspace{0.17em}cm$, $R=0.4\hspace{0.17em}cm$, and $R=0.2\hspace{0.17em}cm$, respectively, without causing a significant difference in average drug exposure compared to before anti-angiogenesis. In this case, the microvascular network suppression is not considered, instead transvascular diffusion and convection mechanisms are improved in areas with a blood source due to the modification of IFP and IFV, as demonstrated in Figure 10 and Figure 11. Furthermore, the modification of IFV due to the establishment of an IFP gradient in the inner areas of the tumor, achieved via the second approach of anti-angiogenesis, improves the interstitial convection mechanism. All of these factors contribute to a greater uniformity in the distribution of the therapeutic agent. The average drug exposure does not change significantly because, although applying anti-angiogenesis initially decreases drug exposure, it increases afterward. The experimental evidence supports the notion that when used as an adjuvant treatment alongside basic therapies such as chemotherapy and radiotherapy, antiangiogenic treatment can lead to a more uniform distribution of therapeutic agents within the tumor [80,81].

The greatest increase in uniformity occurs in $R=0.2\hspace{0.17em}cm$. This is primarily because the entire tumor site is supplied by blood micro-vessel sources, resulting in uniform delivery via transvascular mechanisms. Furthermore, modifying the interstitial fluid flow improves the role of interstitial convection in this size. This result highlights the strong dependency of drug delivery on the microvascular network structure, or to be more precise, on the distribution and density of micro-vessels.

The modification of solute distribution in different time windows varies depending on the specific cases considered for anti-angiogenesis simulation in this study. However, based on the results of NDASE and NDASDNU across different tumor sizes, the second approach to anti-angiogenic therapy can be interpreted as the most effective method for enhancing the quality of drug delivery in this current research. This illustrates the significant impact of modifying transport properties via vascular normalization on the response to the therapeutic agent. However, the decision regarding the effectiveness of the adjuvant anti-angiogenic treatment strategy cannot be accurate without considering the importance of the uniformity of the microvascular structure at the tumor site. This is highly dependent on the type of anti-angiogenic agent. Therefore, in order to gain a comprehensive understanding of the effectiveness of anti-angiogenic therapy, factors beyond those previously discussed, such as tumor size, concurrent therapeutic agent type, and time course of perfusion [23], the type of anti-angiogenic agent and its role in normalizing the microenvironment are crucial.

Based on the results, it should be emphasized that different parameters of anti-angiogenic agent type, concurrent therapeutic agent type, and tumor geometric and physical characteristics are determinative factors that specify the efficiency of anti-angiogenic adjuvant therapy. Therefore, quantitative results obtained from the present study, such as the percentage of improvement in drug delivery induced via anti-angiogenic therapy and the proposal of the most effective approach to vascular normalization, are dependent on the aforementioned factors. However, Qualitative findings from the present study emphasize the crucial influence of microvascular network structure. This underscores the importance of selecting the most appropriate anti-angiogenic strategy, one that optimizes the distribution and density of the microvascular network and response of interstitial fluid flow and solute transport properties to anti-angiogenic therapy, rather than merely suppressing micro-vessels. These insights can potentially be applied to other types of tumors, where drug delivery can be simulated using the numerical model employed in this study. In other words, the findings of the current research can open a new horizon on the dual function of the tumor microvascular network and how to take advantage of anti-angiogenic adjuvant treatment in improving the quality of drug delivery into the tumor.

It is important to mention that this study was not experimentally validated due to constraints related to available laboratory resources. To rely more quantitatively on the results of the present study, the numerical model could be integrated into experimental studies. There exists experimental research that assesses the combination therapy of chemotherapy and anti-angiogenic therapy [8,9,10,82,83,84]. However, developing an experimental study to investigate the details of tumor microvasculature considered in the current numerical model could be achieved by examining the effect of anti-angiogenic therapy on drug delivery to tumor fragments or cells implanted in the rat cornea [85,86,87]. Furthermore, the impact of anti-angiogenic agents on the quality of chemotherapy could be assessed in vivo using a zebrafish model [88,89]. Another potential platform for examining the tumor, its microvasculature, and methodologies for its therapy is via development in vitro using microfluidic systems [90,91].

5. Conclusions

In this study, a multi-scale numerical model, ranging from cell to tissue level, is developed to explore the impact of vascular normalization on drug delivery within a dynamic solid tumor microvasculature. This paper integrates mathematical models of intravascular blood flow and interstitial fluid flow to compute IFP, IFV, and IBP. These values are then applied to the convection-diffusion equation to simulate the solute distribution in different tumor sizes while considering various vascular normalization approaches.

It is shown that the interstitial fluid and solute are distributed heterogeneously, a result of the heterogeneous distribution of micro-vessels. The results demonstrate a high dependency of IFP, IFV, and solute distributions on the microvasculature structure (distribution and density). Consequently, the impact of vascular normalization on drug delivery is greatly influenced by the microvascular structure.

The results demonstrate the effectiveness of all vascular normalization approaches in correcting drug wash-out from the tumor margins. This is achieved by modifying convection as the dominant transport mechanism of drug delivery in the tumor margin.

The results of the first approach to vascular normalization show that the inhibitory effect of angiostatin in suppressing the microvasculature and reducing its density cannot lead to an improvement in drug delivery to the tumor site in terms of both average drug exposure and its uniformity. In other words, this paper illustrates the function of the tumor capillary network as a double-edged sword. On one hand, it disrupts drug delivery; on the other hand, it aids in delivering the drug to the tumor site. This is demonstrated using a model that incorporates more realistic considerations in the present study for the first time. Modifying the transport properties accompanied by the microvasculature reform caused by vascular normalization in the third approach results in improving the outcomes of the first approach by improving the drug delivery schedule via modifications in solute transport mechanisms.

According to the results, it can be concluded that the effect of the anti-angiogenic agent is vital in influencing drug delivery in combination therapy. This is because the distribution and density of the microvascular network are highly dependent on the mechanism of action of the anti-angiogenic agent. Therefore, choosing the most appropriate strategy for deriving benefits from anti-angiogenic therapy is one of the most important factors in combination therapy with chemotherapy and anti-angiogenic therapy.

The second approach to anti-angiogenic therapy results in an improvement in drug distribution uniformity. This improvement is contingent on the microvascular distribution and density within a specific tumor size, indicated as the uniformity of capillary distribution. In other words, the benefit of drug delivery via anti-angiogenic adjuvant therapy, resulting from modifications in transport properties that extend the tumor’s exposure to the drug, will be most pronounced when the tumor is supplied by a more uniformly distributed capillary network. This scenario arises in $R=0.2\hspace{0.17em}\mathrm{c}\mathrm{m}$, as observed in the second approach of anti-angiogenic therapy, considering the physics and conditions outlined in this paper, resulting in a 39% increase in drug exposure uniformity.

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