Biomechanical Models for Predicting Behavior of Growing Tumors

Abstract

Solid tumors are characterized by high mechanical stresses, interstitial fluid pressure (IFP), and low oxygen levels. Tumor growth and treatment effectiveness are significantly affected by biological, chemical, and mechanical relevant factors. For example, solid stress may lower tumor cell growth rate, stimulate invasiveness, increase metastatic possibility, and induce apoptosis. While the cancerous cells' response to treatments like radiation is considerably influenced by the oxygen level in the lesion. Even though tumors develop in a heterogeneous microenvironment, the majority of current mathematical frameworks assume homogeneous growth and uniform mechanical characteristics of the tissue. For instance, the brain is composed of three substances: white matter, gray matter, and Cerebrospinal fluid (CSF). Therefore, tissue heterogeneity should be considered in the modeling.

Here, we established two theoretical models for investigating the growth of solid tumors. The first model is based on continuum mechanics principles, considering the tissue as a poroelastic material. The second model is based on convection-diffusion equations and continuum mechanics. The Gompertz law and the convection-diffusion equation were used to describe the oxygen concentration and cancer cell proliferation, respectively. This model explicitly describes the influence of mechanical factors, such as solid stresses and IFP, chemical factors, such as oxygen concentration, and biological factors, such as cancer cell concentration, on growing solid tumors. On a macroscopic level, it also considers fluid-solid interaction.

The proposed models may provide an understanding of the mechanisms behind tumor development and growth. We also solved the second model for a brain tumor located within the white matter, taking into consideration the geometrical inhomogeneity of the brain such that the realistic geometry of the brain was constructed based on an axial MRI.

This research may have a clinical value in the diagnosis and prognosis of cancer by estimating the stress, IFP, cell concentration, and oxygen concentration in the tissue and predicting the time at which the tumor can be detectable by CT scan and the time at which the tumor can be fatal.