We consider the mechanisms by which folds, or sulci (troughs) and gyri (crests), develop in the brain. This feature, common to many gyrencephalic species including humans, has attracted recent attention from soft matter physicists. It occurs due to inhomogeneous, and predominantly tangential, growth of the cortex, which causes circumferential compression, leading to a bifurcation of the solution path to a folded configuration. The problem can be framed as one of buckling in the regime of linearized elasticity. However, the brain is a very soft solid, which is subject to large strains due to inhomogeneous growth. As a consequence, the morphomechanics of the developing brain demonstrates an extensive post-bifurcation regime. Nonlinear elasticity studies of growth-driven brain folding have established the conditions necessary for the onset of folding, and for its progression to configurations broadly resembling gyrencephalic brains. The reference, unfolded, configurations in these treatments have a high degree of symmetry-typically, ellipsoidal. Depending on the boundary conditions, the folded configurations have symmetric or anti-symmetric patterns. However, these configurations do not approximate the actual morphology of, e.g., human brains, which display unsymmetric folding. More importantly, from a neurodevelopmental standpoint, many of the unsymmetric sulci and gyri are notably robust in their locations. Here, we initiate studies on the physical mechanisms and geometry that control the development of primary sulci and gyri. In this preliminary communication we carry out computations with idealized geometries,

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.