In silicoestimates of the free energy rates in growing tumor spheroids

Abstract

Abstract. 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. BackgroundThe progression of a tumor involves (a) cell proliferation, (b) cell motility, (c) metabolism by which the cells consume glucose and oxygen and create byproducts, (d) mechanical interactions between cancer cells, the ECM and surrounding tissues, and (e) mass transport of chemical species to and through the tumor. Each of these processes has a physically-distinct contribution to the free energy rate in the developing tumor. Complex biophysical interactions between these processes are more broadly observable at the tumor scale than in single-cell studies. Additionally, as we demonstrate in this communication, tumor scale studies have the potential of identifying the relevant questions regarding energy rates that must be considered at the lower, cell-ECM and sub-cellular scales. Using the tumor scale studies, it is of interest to track the free energy rates and thereby gain a system-level understanding of the processes listed above in developing tumors. More broadly, we argue in the Discussion of this paper that there is an interest in combining free energy studies at the tumor, cell-ECM and sub-cellular scales. This will reveal how the energetics change with time and state of the tumor between these different scales and between the processes underlying tumor growth to affect the progression of the cancer.

Perspectives on biological growth and remodeling

“…The formal expertise developed in continuum thermodynamics can thus make fundamental contributions to understanding the energetic basis of growth in normal and pathological states. Such an approach was recently adopted by Narayanan et al (2010) in an in silico study of the free energy changes during growth of solid tumors. The authors grew tumor spheroids of a human colon cancer adenocarcinoma cell line and characterized the growth and mechanical properties of these tumor spheroids.…”

“…(f) The rate of change of free energy density due to glucose consumption. See Narayanan et al, (2010).…”

A tumor growth model with deformable ECM

“…This model does not need interface tracking; they arise naturally from the solution of an initial-boundary value problem that must be comprised of the mass balance equations of all phases involved [5]. Another model without interface tracking is that of Narayanan et al [18], where the free energy rates associated with biochemical dynamics and mechanics of tumors are investigated. The model is derived within the theory of mixtures involving coupled reaction–transport equations for the concentration of cells, of the ECM, of oxygen and glucose, and a quasi-static balance of momentum equation that governs the mechanics of the tumor.…”

Mass transport in morphogenetic processes: A second gradient theory for volumetric growth and material remodeling

“…From experimental measurements, the elastic modulus for a tumor aggregate is c 1 ρ g = 6 KPa (Suresh, 2007), the diffusion coefficient of glucose is D g = 4.22 10 −11 m 2 s −1 , and the cell doubling time τ c ranges between (in vitro) 14-28 hours and (in-situ) 20 days (Narayanan et al, 2010).…”