Abstract
A continuous mathematical model of non-invasive avascular tumor growth in tissue is presented. The model considers tissue as a biphasic material, comprised of a solid matrix and interstitial fluid. The convective motion of tissue elements happens due to the gradients of stress, which change as a result of tumor cells proliferation and death. The model accounts for glucose as the crucial nutrient, supplied from the normal tissue, and can reproduce both diffusion-limited and stress-limited tumor growth. Approximate tumor growth curves are obtained semi-analytically in the limit of infinite tissue hydraulic conductivity, which implies instantaneous equalization of arising stress gradients. These growth curves correspond well to the numerical solutions and represent classical sigmoidal curves with a short initial exponential phase, subsequent almost linear growth phase and a phase with growth deceleration, in which tumor tends to reach its maximum volume. The influence of two model parameters on tumor growth curves is investigated: tissue hydraulic conductivity, which links the values of stress gradient and convective velocity of tissue phases, and tumor nutrient supply level, which corresponds to different permeability and surface area density of capillaries in the normal tissue that surrounds the tumor. In particular, it is demonstrated, that sufficiently low tissue hydraulic conductivity (intrinsic, e.g., to tumors arising from connective tissue) and sufficiently high nutrient supply can lead to formation of giant benign tumors, reaching tens of centimeters in diameter, which are indeed observed clinically.
Highlights
All cancers, except for blood cancers, form solid tumors
P was varied to study the influence of nutrient supply level on tumor growth, since it varies in different normal tissues depending on their metabolic demand
≈0.3% under P = 16 and by ≈0.5% under P = 64. The smallness of this difference was due to the fact that such high value of hydraulic conductivity led to sufficiently fast smoothing of the solid stress gradients, caused by spatial variations of cell fractions in tissue
Summary
Except for blood cancers, form solid tumors. Under favorable conditions, cancer cells should divide virtually indefinitely [1], resulting in exponential growth of tumor volume. Transition from avascular to vascular tumor leads to increase of its growth speed at a linear stage and to increase of its maximum volume [6] Another mechanism is invasion of cells into surrounding tissue, which allows them to move away from nutrient-deficient regions and changes tumor growth pattern at a qualitative level. Models, based on approaches from solid mechanics, can yield more realistic reproduction of solid stress distribution within the tissue [23] and can even provide quantitative predictions that are consistent with experimental results [26] Their solution is associated with much greater computational costs and they are not amenable to analytical investigation, like simpler approaches [20].
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