G. Sciumè

A multiphase model for three-dimensional tumor growth

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.

A tumor growth model with deformable ECM

Existing tumor growth models based on fluid analogy for the cells do not generally include the extracellular matrix (ECM), or if present, take it as rigid. The three-fluid model originally proposed by the authors and comprising tumor cells (TC), host cells (HC), interstitial fluid (IF) and an ECM, considered up to now only a rigid ECM in the applications. This limitation is here relaxed and the deformability of the ECM is investigated in detail. The ECM is modeled as a porous solid matrix with Green-elastic and elasto-visco-plastic material behavior within a large strain approach. Jauman and Truesdell objective stress measures are adopted together with the deformation rate tensor. Numerical results are first compared with those of a reference experiment of a multicellular tumor spheroid (MTS) growing in vitro, then three different tumor cases are studied: growth of an MTS in a decellularized ECM, growth of a spheroid in the presence of host cells and growth of a melanoma. The influence of the stiffness of the ECM is evidenced and comparison with the case of a rigid ECM is made. The processes in a deformable ECM are more rapid than in a rigid ECM and the obtained growth pattern differs. The reasons for this are due to the changes in porosity induced by the tumor growth. These changes are inhibited in a rigid ECM. This enhanced computational model emphasizes the importance of properly characterizing the biomechanical behavior of the malignant mass in all its components to correctly predict its temporal and spatial pattern evolution.

A two-phase model of plantar tissue: a step toward prediction of diabetic foot ulceration

A new computational model, based on the thermodynamically constrained averaging theory, has been recently proposed to predict tumor initiation and proliferation. A similar mathematical approach is proposed here as an aid in diabetic ulcer prevention. The common aspects at the continuum level are the macroscopic balance equations governing the flow of the fluid phase, diffusion of chemical species, tissue mechanics, and some of the constitutive equations. The soft plantar tissue is modeled as a two-phase system: a solid phase consisting of the tissue cells and their extracellular matrix, and a fluid one (interstitial fluid and dissolved chemical species). The solid phase may become necrotic depending on the stress level and on the oxygen availability in the tissue. Actually, in diabetic patients, peripheral vascular disease impacts tissue necrosis; this is considered in the model via the introduction of an effective diffusion coefficient that governs transport of nutrients within the microvasculature. The governing equations of the mathematical model are discretized in space by the finite element method and in time domain using the θ-Wilson Method. While the full mathematical model is developed in this paper, the example is limited to the simulation of several gait cycles of a healthy foot.