In silico estimates of the free energy rates in growing tumor spheroids

The physics of solid tumor growth can be considered at three distinct size scales: the tumor scale, the cell-extracellular matrix (ECM) scale and the sub-cellular scale. In this paper we consider the tumor scale in the interest of eventually developing a system-level understanding of the progression of cancer. At this scale, cell populations and chemical species are best treated as concentration fields that vary with time and space. The cells have chemo-mechanical interactions with each other and with the ECM, consume glucose and oxygen that are transported through the tumor, and create chemical byproducts. We present a continuum mathematical model for the biochemical dynamics and mechanics that govern tumor growth. The biochemical dynamics and mechanics also engender free energy changes that serve as universal measures for comparison of these processes. Within our mathematical framework we therefore consider the free energy inequality, which arises from the first and second laws of thermodynamics. With the model we compute preliminary estimates of the free energy rates of a growing tumor in its pre-vascular stage by using currently available data from single cells and multicellular tumor spheroids.

💡 Research Summary

The paper presents a continuum‐scale mathematical framework for describing the growth of avascular tumor spheroids, with the explicit aim of quantifying the rates at which free energy is produced, stored, and dissipated during the process. The authors begin by situating tumor growth within three hierarchical size scales—tumor (macroscopic), cell‑extracellular matrix (mesoscopic), and sub‑cellular (microscopic)—and then focus on the tumor scale where cell populations and chemical species can be treated as spatially varying concentration fields.

The core of the model consists of coupled partial differential equations. Cell density (c(\mathbf{x},t)) obeys a conservation law that includes advective transport by a velocity field (\mathbf{v}) (representing chemotactic and mechanical migration) and a source term (\Gamma) that captures proliferation and apoptosis. Nutrient transport is described by diffusion‑consumption equations for glucose (g) and oxygen (o): \