A multiphase model for three-dimensional tumor growth

Abstract

Several mathematical formulations have analyzed the time-dependent behaviour of a tumor mass. However, most of these propose simplifications that compromise the physical soundness of the model. Here, multiphase porous media mechanics is extended to model tumor evolution, using governing equations obtained via the Thermodynamically Constrained Averaging Theory (TCAT). A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TC), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HC); and an interstitial fluid (IF) for the transport of nutrients. The equations are solved by a Finite Element method to predict the growth rate of the tumor mass as a function of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion and geometry. Results are shown for three cases of practical biological interest such as multicellular tumor spheroids (MTS) and tumor cords. First, the model is validated by experimental data for time-dependent growth of an MTS in a culture medium. The tumor growth pattern follows a biphasic behaviour: initially, the rapidly growing tumor cells tend to saturate the volume available without any significant increase in overall tumor size; then, a classical Gompertzian pattern is observed for the MTS radius variation with time. A core with necrotic cells appears for tumor sizes larger than 150 μm, surrounded by a shell of viable tumor cells whose thickness stays almost constant with time. A formula to estimate the size of the necrotic core is proposed. In the second case, the MTS is confined within a healthy tissue. The growth rate is reduced, as compared to the first case - mostly due to the relative adhesion of the tumor and healthy cells to the ECM, and the less favourable transport of nutrients. In particular, for tumor cells adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas tumor cell infiltration is predicted for the opposite condition. Interestingly, the infiltration potential of the tumor mass is mostly driven by the relative cell adhesion to the ECM. In the third case, a tumor cord model is analyzed where the malignant cells grow around microvessels in a 3D geometry. It is shown that tumor cells tend to migrate among adjacent vessels seeking new oxygen and nutrient. This model can predict and optimize the efficacy of anticancer therapeutic strategies. It can be further developed to answer questions on tumor biophysics, related to the effects of ECM stiffness and cell adhesion on tumor cell proliferation.

Keywords: Finite Elements; cell adhesion; multiphase systems; porous mechanics; tumor cord; tumor growth; tumor spheroid.

Figure 1

The multiphase system within a representative elementary volume (REV).

Figure 2

Pressure difference - saturation relationship.

Figure 3

Geometry and boundary conditions for a multicellular tumor spheroid (red) in a medium (not to scale).

Figure 4

(a) Volume fraction of the tumor cells (total and living) during 360h. (b) Volume fraction of the tumor cells phase over 120h; lines drawn at every 10h of simulations. (c) Numerical results compared with different in vitro experiments. The symbols are data obtained in the following in vitro cultures: squares = FSA cells (methylcholantrene-transformed mouse fibroblasts, Yuhas et al., 1977); diamonds = MCF7 cells (human breast carcinoma, Chignola et al., 1995); circles = 9L cells (rat glioblastoma, Chignola et al., 2000). (d) Numerical results (points) for spheroid and necrotic core radii, and their interpolations (solid lines).

Figure 5

(a) Apparent volume of the tumor spheroid, effective volume of the tumor cells, and the effective volume of the living tumor cells, over time. (b) Mass fraction of oxygen over 360h. (c) Pressure in the tumor cells phase over 360h. (d) Numerical prediction of the interstitial fluid pressure over 180h; Lines drawn at every 10h of simulations.

Figure 6

Geometry and boundary conditions for a multicellular tumor spheroid growing within a healthy tissue. (not to scale)

Figure 7

(a,b)Numerical prediction of the volume fractions of the living tumor cells (LTC), the necrotic tumor cells (NTC) and the host cells (HC), at different times (from up to down: 1h, 180h, and 360h). The left column (a) is for a_h = a_t, while the right column (b) is for a_h = 1.5·a_t. (c) Evolution of the effective volume of the tumor cells, and the effective volume of the living tumor cells. The black lines refer to the case (a_h = a_t), while the grey lines refer to the case (a_h = 1.5a_t). (d) Scaled effective volume of tumor (normalized by initial value) after 360 hours for different radii of the computational domain.

Figure 8

(a) Initial conditions of the third case. Yellow shows the axes of the two capillary vessels. (b) Geometry and boundary conditions. (c) Volume fractions of the living tumor cells (first column) of the healthy cells (second column) and mass fraction of oxygen (third column) for the case S1.

Figure 9

(a) Mass fraction of oxygen along the line joining points A and B for S2. (b) Volume fractions of the LTC and oxygen mass fraction for S2 at 15 days. (c) Volume fractions of the LTC for S2 at 20 days. “N” indicates the necrotic areas. (d) Volume of the tissue invaded by the tumor.