Effective Macroscopic Equations for Biological Fluid and Nutrients' Transport in Vascularized Tumors Growing Via Proliferation and Chemotaxis

This paper presents a system of partial differential equations designed to model fluid and nutrient transport within the growing tumor microenvironment. The fluid phase, representing both cells and extracellular fluids flowing within the interstitial space, is assumed to be intrinsically incompressible, so that growth can be modeled as a source, generally defined by volumetric growth terms and nonconvective mass fluxes. Specifically, we consider the volumetric growth term proportional to the nutrient concentration and the nonconvective mass flux driven by the nutrient gradient. By exploiting the scale separation between the microscopic vascular structures and the larger tumor tissue, we employ asymptotic homogenization to derive effective macroscopic equations that integrate detailed microscale characteristics. The resulting model operates as a double porous medium framework, where fluid dynamics are driven by both pressure and concentration gradient, which reduces to a more standard Darcy's law when microscale variations of the convective mass flux are neglected. Nutrient transport is captured through a coupled advection-diffusion-reaction system. Permeability and diffusivity tensors, which encapsulate the influence of microvascular geometry, are computed via cell-problem analysis to accurately reflect the microscale structure within the macroscopic model. Given that the tissue model includes a fluid phase that continually exchanges with the surrounding vasculature, along with nutrients, the Kedem-Katchalsky formulation is employed to represent fluid and nutrient transport across the capillary walls. This approach provides valuable insights into the interactions between vascular architecture and tumor growth. Although certain limitations remain, such as the static tumor domain and assumptions regarding cell proliferation, the framework offers a foundation for further development. It is adaptable for numerical simulations based on real tumor geometries, with promising potential to inform and improve anticancer treatment strategies through the integration of patient-specific clinical data.