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**Mathematical modeling of tumor-induced angiogenesis.**
*(English)*
Zbl 1109.92020

From the introduction: Cancer is a collection of diseases with the common feature of uncontrolled cellular growth. Most tissues in the body can give rise to cancer, some even yield several types, and each cancer has unique features. The salient feature of cancer cells is that the mechanisms that control growth, proliferation, and death of cells in a multicellular organism are disrupted, often as a result of mutations. The \(\sim 10^{13}\) cells in the human body are subject to numerous checks and balances that, to varying degrees, are absent, ignored, or affirmatively avoided during cancer development. In effect, cancer cells escape the usual controls on cell proliferation and proliferate excessively to form a neoplastic growth or tumor.

Initially, solid tumors are avascular, i.e., they do not have their own blood supply, and rely on diffusion from nearby vessels to supply oxygen and nutrients and to remove waste products. As the tumor grows nutrient demand increases until the flux of nutrients through the surface of the tumor is too small to supply the entire mass of cells. A necrotic core of dead cells develops at the center and eventually the tumor stops growing and reaches a steady state size of \(\sim 1-3\) mm, in which the number of dying cells counterbalances the number of proliferating cells. Growth can resume only if the tumor becomes vascularized, i.e., if it becomes permeated with a network of capillaries. An early response of tumor cells lo hypoxia (oxygen deprivation) is the expression of genes that code for signaling molecules (growth factors, primarily vascular endothelial growth factors (VEGF), and basic fibroblast growth factor (bFGF; also called FGF-2)) that are used to induce a nearby vessel to grow new capillaries to vascularize the tumor through a process called angiogenesis. These growth factors diffuse from tumor cells to the nearby primary vessels, and initiate a cascade of processes, including the activation of endothelial cells (ECs) that line the blood vessel walls, inducing them to proliferate and migrate chemotactically towards the tumor. This results in the creation of a new capillary network that extends from a primary vessel into the growth-factor-secreting tumor, thereby bringing essential nutrients to the tumor and providing a shorter route for the spread of cancer cells to other parts of the body. Thus angiogenesis is the sine qua non for cancer invasion, and understanding the mechanisms that control it will provide the basis for rational therapeutic intervention. Angiogenesis involves many intermediate steps, and to give the reader a brief overview of the entire process, we summarize some of the main steps involved. Each of these steps, will be discussed in greater detail in this paper.

The aim of this review is to identify aspects of the angiogenic process for which mathematical models can provide significant new insights. Models for the growth of the tumor itself have been addressed by J. A. Adam and N. Bellomo [A survey of models for tumor-immune system dynamics. (1997; Zbl 0874.92020)]. To give a flavor of what has been done in regard to tumor-induced angiogenesis we describe some of the models briefly; details are given later. Much of the mathematical modeling has focused on the way in which tumor angiogenic factors initiate and coordinate capillary growth. For example, M. E. Orme and M. A. J. Chaplain [IMA J. Math. Appl. Med. Biol. 13, No. 2, 73–98 (1996; Zbl 0849.92009)] and H. A. Levine el al. [Math. Biosci. 168, No. 1, 77–115 (2000; Zbl 0986.92016)] have developed continuum models for initiation and outgrowth of buds from a primary vessel. These models postulate that the vascular endothelial cells that form capillary tips migrate up a gradient of angiogenic factor released by the tumor. The density of new capillaries as well as the concentrations of angiogenic factors evolve according to coupled nonlinear partial differential equations. Others have used continuum models to study the interactions between endothelial cells and the extracellular matrix during angiogenesis in order to understand how cells respond to not only chemical signals via chemotaxis, but also to mechanical signals via haptotaxis.

Discrete models that treat cells as individual units have also been developed to model angiogenesis. In contrast with the continuum models, discrete models can track individual cells and can incorporate more details about cell movement and interaction with the tissue. The results of the discrete model agree with the predictions of the continuum model and in addition are able to produce capillary networks with structure and morphology similar to those observed in vitro.

Initially, solid tumors are avascular, i.e., they do not have their own blood supply, and rely on diffusion from nearby vessels to supply oxygen and nutrients and to remove waste products. As the tumor grows nutrient demand increases until the flux of nutrients through the surface of the tumor is too small to supply the entire mass of cells. A necrotic core of dead cells develops at the center and eventually the tumor stops growing and reaches a steady state size of \(\sim 1-3\) mm, in which the number of dying cells counterbalances the number of proliferating cells. Growth can resume only if the tumor becomes vascularized, i.e., if it becomes permeated with a network of capillaries. An early response of tumor cells lo hypoxia (oxygen deprivation) is the expression of genes that code for signaling molecules (growth factors, primarily vascular endothelial growth factors (VEGF), and basic fibroblast growth factor (bFGF; also called FGF-2)) that are used to induce a nearby vessel to grow new capillaries to vascularize the tumor through a process called angiogenesis. These growth factors diffuse from tumor cells to the nearby primary vessels, and initiate a cascade of processes, including the activation of endothelial cells (ECs) that line the blood vessel walls, inducing them to proliferate and migrate chemotactically towards the tumor. This results in the creation of a new capillary network that extends from a primary vessel into the growth-factor-secreting tumor, thereby bringing essential nutrients to the tumor and providing a shorter route for the spread of cancer cells to other parts of the body. Thus angiogenesis is the sine qua non for cancer invasion, and understanding the mechanisms that control it will provide the basis for rational therapeutic intervention. Angiogenesis involves many intermediate steps, and to give the reader a brief overview of the entire process, we summarize some of the main steps involved. Each of these steps, will be discussed in greater detail in this paper.

The aim of this review is to identify aspects of the angiogenic process for which mathematical models can provide significant new insights. Models for the growth of the tumor itself have been addressed by J. A. Adam and N. Bellomo [A survey of models for tumor-immune system dynamics. (1997; Zbl 0874.92020)]. To give a flavor of what has been done in regard to tumor-induced angiogenesis we describe some of the models briefly; details are given later. Much of the mathematical modeling has focused on the way in which tumor angiogenic factors initiate and coordinate capillary growth. For example, M. E. Orme and M. A. J. Chaplain [IMA J. Math. Appl. Med. Biol. 13, No. 2, 73–98 (1996; Zbl 0849.92009)] and H. A. Levine el al. [Math. Biosci. 168, No. 1, 77–115 (2000; Zbl 0986.92016)] have developed continuum models for initiation and outgrowth of buds from a primary vessel. These models postulate that the vascular endothelial cells that form capillary tips migrate up a gradient of angiogenic factor released by the tumor. The density of new capillaries as well as the concentrations of angiogenic factors evolve according to coupled nonlinear partial differential equations. Others have used continuum models to study the interactions between endothelial cells and the extracellular matrix during angiogenesis in order to understand how cells respond to not only chemical signals via chemotaxis, but also to mechanical signals via haptotaxis.

Discrete models that treat cells as individual units have also been developed to model angiogenesis. In contrast with the continuum models, discrete models can track individual cells and can incorporate more details about cell movement and interaction with the tissue. The results of the discrete model agree with the predictions of the continuum model and in addition are able to produce capillary networks with structure and morphology similar to those observed in vitro.

### MSC:

92C50 | Medical applications (general) |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |