Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3779_Библиотеки_им_академика_М_И_Перельмана

This reasoning leads to an important physiological conclusion that changes in microvessel density during angiogenesis and antiangiogenic therapy should serve only as a preliminary indicator of changes in oxygen supply. Although these changes should have the same sign, their values are not necessarily proportional, especially in very dense capillary networks. Thus, several-fold increase in capillary density leads to the only conclusion that the increase in oxygen supply is less. Note that sufficiently active proangiogenic activity also results in considerable increase in interstitial pressure, which leads to compression of capillaries and thus slows down the blood flow in them. This effect is considered further in

Section 8.4.2. However, it should be noted that, for other major metabolites, the situation

should be more straightforward. As we discussed in Section 8.1, this happens because metabolities local inflow, which is diffusion limited rather than flow limited, is almost proportional to the capillaries surface area density.

Such kind of investigation makes it possible to obtain a fitting function, which provides a link between local capillary network density and local oxygen inflow. This result can be useful for spatially distributed mathematical models of complex physiological processes, e.g., tumor growth in tissue, an example of which is presented in Chapter 10. At the moment, a common approach in such models is the description of the microcirculatory network via spatially distributed variables. This approach is fairly coarse but unavoidable simplification when dealing with equations of reaction-diffusion type. Inability of such models to adequatelydfrom a physiological point of viewddescribe inflow of oxygen in the tissue is one of the major drawbacks of such approach. In the work [380], a new approach to modeling of oxygen inflow has been presented, which relies on the preliminary results, obtained in such investigation. It also relies on some other general physiological assumptions, discussed in detail further in Section 8.4.3. The results from [401] will be further discussed in Section 8.4.2.

As discussed in the introduction to this chapter, an excessive production of proangiogenic factors by tumor leads to the formation of new capillaries with altered properties including increased permeability, which leads to an increased influx of nutrients and blood plasma into the tissue and a subsequent local increase in interstitial pressure (further IP for short), and a lack of supporting cells, in particular pericytes, which reduces elasticity of capillary wall. As a result of this alteration, due to an increased IP, the lumina of capillary wall may

justified by the fact that the pressure gradient in a tumor is usually small [387]. At the same time, alterations in the resistances of arterioles and venules are not taken into account, since it is known that in response to changes in transmural pressure (i.e., the pressure difference between the inner and outer side of the microvessel), they are able to actively change the size of their lumina. This size adjustment is possible due to the presence of smooth muscle layer in their walls. It helps to maintain constancy of blood flow in the capillary network and so to meet the metabolic needs of the perfused tissue. This phenomenon is well studied and is called autoregulation of blood flow [32]. Due to this reason, the input and output pressures are also considered to be unchanged during AAT, which typically does not affect blood pressure in practice. However, it should be noted that according to clinical studies, the chance of hypertension due to VEGF inhibitors is approximately 20% [388].

Of course, a sufficiently large increase in IP will result in compression of the microvessels, but this does not occur within the range of typical values of IP in tumors, which does not rise higher than the average pressure in the capillaries [389]. Moreover, such values of pressure do not lead to the collapse of capillaries (not to mention larger vessels), which can happen inside a tumor for another reason, i.e., due to the pressure caused by proliferation of cells [390].

The graphs in Fig. 8.4 show the values of blood flow density in the simulation domain, depending on the number of capillaries in the network and the fraction of angiogenic capillaries for three values of k

estimations [386,503].

microvascular network remodeling. A completely integrated model may become an extremely valuable tool for research and development strategies for antiangiogenic therapy. The steps made in this direction and the prototype of the coupled model are discussed further.

Note that simultaneous simulations of microvasculature network remodeling along with tumor growth become a rather popular field of mathematical modeling. Early modeling setups included a consideration of tumor growth with fixed capillary network [391,392] and, vice versa, evolving vasculature with unchangeable tumor which play a role of a source of proangiogenic factors [393,394]. The majority of papers accounting for the dynamics of both tumor and microvasculature network are based on discrete methods, such as cellular automata approach, where capillaries are built from a portion lattice elements, while other elements represent tumor cells [394,395]. For other approaches see [396,397]. These models produce fascinating illustrative results but, due to heavy computational cost, may be suitable only for simulation of the initial stage of tumor growth. In the work [347], an approach is presented, similar to the one presented here in the sense that tumor is modeled via continuum variable, which allows to run simulations for a tumor up to 5 mm in diameter. However, on a more detailed level, the methods discussed here differ from those in [365] in many aspects, including algorithms of generation of microvasculature.