Interstitial Pressure Dynamics Due to Bacterial Infection

Abstract

Edema is often one of the symptoms found in infections along with skin warm to touch, shivering, aching muscles, pain, redness, swelling and so on. Edema is a consequence of interstitial fluid dynamics and their interactions with the immune system. When pathogens enter into the body, the natural consequence is an immunological reaction triggered by the production of cytokines by macrophages. This immunological response recruits other immune cells, the neutrophils, responsible for seeking and destroying the pathogens. This physiological process can be mathematically modeled by a nonlinear system of partial differential equations (PDE) based on porous media approach. In order to simplify the model, just neutrophils and pathogens (an unspecified bacteria) are considered. The equations that represent these populations are coupled with a pressure equation that is dynamically influenced by them, as well as by the lymphatic system. The pressure equation is modeled using the Starling hypothesis of fluid exchange and lymph flow. This coupling is a challenge due to the boundary conditions that are necessary to model the influence of the lymphatic system. The model is presented in its general three-dimensional form but is numerically solved in an one dimensional domain to make easier the comprehension of the results. The numerical method used to approach the solution of the PDE system is the finite volume method (FVM) with a first order upwind (FOU) scheme to ensure a stable solution of the advection term. Therefore, this study describes the mathematical and numerical tools used to model and solve the resulting PDE system.