On the other hand oxygen concentration in tumor

On the other hand, oxygen concentration in tumor tissue also depends on its consumption rate by tumor cells, which decrease will also lead to alleviation of hypoxia. This fact is not taken into account in the majority of studies on the topic, although it is well-known that antiangiogenic therapy does result in alterations of tumor metabolism (Curtarello et al., 2015). Of especial interest is the fact that inflows of two major nutrients, i.e., oxygen and glucose, change differently during the process of microvessels normalization. Transvascular transport of both these small molecules is governed by the process of diffusion, but oxygen is able to diffuse directly through the capillary wall, so its transport remains practically unaltered, while diffusion of glucose (as well as of all other nutrients) strongly depends on the number and sizes of pores (Levick, 2013), thus its inflow significantly decreases due to normalization of capillaries’ structure, which in turn limits cell proliferation, since this process requires glucose as main energetic nutrient and major substrate for biosynthesis. This is well-illustrated by the fact that overexpression of angiopoietin-1, which leads to increased vascular maturation status without major changes in vascular density and VEGF expression, results in severely impaired tumor growth (Hawighorst et al., 2002). The mathematical modeling of tumor growth and therapy with account of angiogenesis is connected with certain difficulties. Since the growth of tumor is an incredibly complex process, one has to thoroughly select what features of it should be regarded in the model and how they should be described mathematically in order to objectively capture all the necessary details, and not to overcomplicate the model. At that, the major challenge is the necessity to account for objects and processes on different size scales. Indeed, the typical capillary diameter is the typical cell size is the typical distance between cyproheptadine hcl synthesis is the average length of a capillary is 0.5–1 mm, and the tumor size can reach several centimeters, which results in the difference in scales of more than three orders of magnitude. There are several common types of tumor models. The first type is non-spatially-distributed models, which bypass the mentioned difficulty by using a set of ordinary differential equations with phenomenological dependencies (Benzekry, Chapuisat, Ciccolini, Erlinger, Hubert, 2012, Letellier, Sasmal, Draghi, Denis, Ghosh, 2017), which makes them relatively simple but also heavily restricts their applicability. The more sophisticated approach is using of cellular automata or other agent-based techniques to account for separate tumor cells and vessels, while dynamics of substances is usually described via reaction–diffusion equations (Stéphanou, Lesart, Deverchère, Juhem, Popov, Estève, 2017, Wu, Frieboes, McDougall, Chaplain, Cristini, Lowengrub, 2013). These models are able to produce great illustrative result but generally require large computational resources so the simulations are usually carried out in small regions, thus reproducing only initial stage of tumor growth. This obstacle is overcome in models, governed by system of partial differential equations, in which not only substances, but also tumor cells and microvasculature are modeled via spatially-distributed variables (Alfonso, Köhn-Luque, Stylianopoulos, Feuerhake, Deutsch, Hatzikirou, 2016, Swanson, Rockne, Claridge, Chaplain, Alvord, Anderson, 2011, Szomolay, Eubank, Roberts, Marsh, Friedman, 2012). This approach is linked with certain problems with correct description of microvasculature, however, it allows to consider growth of large neoplasms at relatively low computational costs. The model presented herein is of reaction–diffusion–convection type and is developed from our previous works in this field (Kolobov, Kuznetsov, 2015, Kuznetsov, Kolobov, 2017). Its main distinguishing feature is simultaneous account of two basic nutrients, i.e., glucose and oxygen, with detailed consideration of their inflow and consumption, which is crucial for investigating the considered phenomenon.