I. Representation of Tumor Tissue Heterogeneity

Tumors are typically heterogeneous tissues comprised of multiple cell species in addition to extra-cellular matrix (ECM) and water fluid. It is difficult to model these components at the tissue (10–10m) scale, where individual cells cannot be represented without prohibitive computational burden. Assuming that same-kind components tend to cluster together, a multiphase approach can be applied to represent heterogeneous tumor tissue at this larger physical scale. This method enables simulating mixture of elements within tissues, e.g., geno-/phenotypic heterogeneity underlying mutationor microenvironment-driven tumor progression. Further, by not explicitly tracking interfaces, this methodology facilitates realistic modeling of tissue in 3-D. I. REPRESENTATION OF TUMOR TISSUE HETEROGENEITY With a multiphase approach, a solid tumor is modeled as a saturated medium consisting of at least one solid phase (cells, ECM, etc.) and one liquid phase (fluid) [1]. The approach can be generalized to represent additional phases to describe multiple tissue elements. The volume fractions or mass at a given location describe the relative amounts of the different elements. The system is governed by mass and momentum balance equations for each phase, interphase mass and momentum exchange, along with appropriate constitutive laws to close the equations [1]. The approach does not require enforcing complicated boundary conditions between elements. This method stands in contrast to modeling using sharp boundaries, in which tumor elements at the tissue scale are delineated from each other without mixing, hence potentially representing a challenge to properly simulate the tumor biological complexity. Many multiphase mixture models have been developed to account for cell type and mechanical response heterogeneity of the solid and liquid tumor phases. Work includes [1-39] in the past decade and associated earlier-date references. II. ILLUSTRATIVE RESULTS OF APPLICATION OF METHOD Multiphase tumor models in 3-D [39] have been combined with models of tumor-induced angiogenesis [4041] to fully couple heterogeneous tumor growth with vascularization [37-38, 42-44]. The nonlinear effects of cellto-cell adhesion as well as taxis-inducing chemical/molecular species are implemented by using an approach based on energy variation [1]. This means that the energy within the tumor system accounts for all the processes to be modeled. In particular, adhesion is represented through an interaction energy that leads to narrow transition layers originating from differential adhesive forces among the cell species [1]. The resulting diffuse-interface mixture equations for the cellspecies volume fractions are a coupled system which includes Research supported by National Cancer Institute. H. B. Frieboes is with the Dept. of Bioengineering, University of Louisville, Louisville, KY 40292 USA (phone: 502-852-3302; fax: 502852-6806; e-mail: hbfrie01@ louisville.edu. fourth-order nonlinear advection–reaction–diffusion equations of Cahn–Hilliard-type [45]. Cell substrate elements (e.g., oxygen) are included via reaction–diffusion equations. This framework allows for a detailed description of solid tumor growth without the need to track sharp boundaries between the tumor constituent elements. It also firmly links the dependence of cell–cell and cell–matrix adhesion on cell genoand phenotype and on local microenvironmental conditions (such as levels of oxygen) [1]. A. Simulation of vascularized tumor growth Fig. 1(left) simulates growth of a vascularized tumor [42]. Hypoxia leads to necrosis in the interior (darker color), with tumor angiogenic factors (TAF) diffusing from the interior to the surroundings to stimulate new capillaries (small lines) from pre-existing vasculature (not shown). Endothelial cells proliferate up the TAF gradient, forming branches and then loops to conduct blood (darker lines). Vessels are uniformly distributed around and inside the lesion. In time, the tumor becomes asymmetric, influenced by heterogeneity in cell proliferation and death, which is in turn based on availability of cell substrates in the microenvironment as a function of the vasculature. Fig. 1(right) shows that viable tissue cuffs around the vessels, as observed experimentally [46] and clinically [37-38], with areas distal from conducting vessels undergoing necrosis. Figure 1. Simulation of vascularized tumor with one viable cell species [42]. Left: Viable tumor tissue (orange/ red) is shown in 3-D contours representing density values of 0.1, 0.2, and 0.6 (min.:0.0; max.:1.0). Conducting vessels: blue; non-conducting: gray. Time unit = 1 day; grid length = 200μm. Right: Slice of tumor (plane x=10) shows that viable tissue cuffs around the vessel locations (darker areas). Color coding: density of viable tissue; highest=1.0; unit length = 100μm. B. Simulation of multiple viable tumor species Fig. 2 shows a tumor that originally started as a small, round avascular tumor nodule with one viable species [42]. As in the previous example, hypoxia leads to necrosis in the interior, with TAF diffusing from the interior to the surroundings to stimulate new capillaries that eventually conduct blood (darker lines). A mutated species (red) with equal oxygen uptake as the original species (gray) is made to appear randomly, expressing a phenotype which upregulates Overview: Modeling Heterogeneous Tumor Tissue as a Multiphase Material Hermann B. FrieboesUSC PSOC Member . CC-BY-NC-ND 4.0 International license peer-reviewed) is the author/funder. It is made available under a The copyright holder for this preprint (which was not . http://dx.doi.org/10.1101/031534 doi: bioRxiv preprint first posted online Nov. 12, 2015;