Cell-Kinetics Based Calibration of a Multiscale Model of Structured Cell Populations in Ovarian Follicles

Abstract

Abstract. In this paper, we present a strategy for tuning the parameters of a multiscale model of structured cell populations in which physiological mechanisms are embedded into the cell scale. This strategy allows one to cope with the technical difficulties raised by such models, that arise from their anchorage in cell biology concepts: localized mitosis, progression within and out of the cell cycle driven by time-and possibly unknown-dependent, and nonsmooth velocity coefficients. We compute different mesoscopic and macroscopic quantities from the microscopic unknowns (cell densities) and relate them to experimental cell kinetic indexes. We study the expression of reaching times corresponding to characteristic cellular transitions in a particle-like reduction of the original model. We make use of this framework to obtain an appropriate initial guess for the parameters and then perform a sequence of optimization steps subject to quantitative specifications. We finally illustrate realistic simulations of the cell populations in cohorts of interacting ovarian follicles.Key words. transport equations, parameter calibration, structured cell populations, cell kinetics AMS subject classifications. 92B05,35Q92Introduction. In this paper, we deal with the question of the numerical calibration of an existing multiscale model of cell-structured populations in the physiological context of ovulation. This model was formulated as a system of weakly coupled, non conservative transport equations with controlled velocities and sink terms, where the unknowns are the cell densities in each follicle [9, 8]. A number of theoretical studies have established the well-posedness of the model [19], examined optimal control problems related to the ovulatory trajectories in the framework of hybrid optimal control theory [6], and studied the reachability of final states corresponding to either ovulatory or atretic cases in the framework of backwards reachable sets [8]. Implementation of the model in an efficient and reliable computing environment has involved the design of a finite-volume scheme dealing with the discontinuous coefficients [3], embedding this scheme within a dedicated adaptive mesh based on a multi-resolution approach [4], and implementing it on parallel architecture [2]. This has left the question of model calibration to biological specifications to be resolved. We have to face a generic, yet unsolved issue in parameter fitting for physiologicallyoriented multiscale mathematical models: although mechanistic knowledge in molecular and cell biology is available on the lower scales, quantitative experimental data are rather available on the higher scales. In our case, the question is how to infer the parameters entering the microscopic functions (on the level of the follicular cells) from mesoscopic (on the level of the individual follicles, i.e. the number of follicular cells) or macroscopic (on the level of the populations of follicles) information. In addition, even on the macroscopic level, data remain rather scarce and ...

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“…This means connecting fine-grain models designed a t each level of the HPG. Some steps forward have already been made through the design and study of spatio-temporal multi-scale models for structured cell populations [117,118] Note that these pathways are neither linear chain nor independent of each other, as multiple cross-talks and retroaction loops exist. LHCGR and EGFR are also represented at the cell membrane, to highlight possible cross-talks and receptor trans-activation (LHCGR is known to activate Gq, Gi, Gs, β-arrestin and Src pathways; EGFR activates PI3K and ERK).…”

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“…The average cell maturity, displayed in the bottom left panel, shows that the first follicle (blue line) is from this time on right in the middle of the dangerous area [y − s , y + s ] centered on y s = 0.5, where it remains more or less trapped, encountering massive cell death, while the second follicle (red line), rapidly crosses the dangerous zone, and stabilizes with a cell number sufficient to ensure ovulation. We refer the reader to [3] for a thorough discussion and biological interpretation of such a simulation. What interests us here is that the complex and highly nonlinear competition phenomenon has been correctly captured by the model reduction whose numerical solution (solid lines) exhibits virtually the same features as the original 2D model solution (dashed lines).…”

“…To investigate further the issue of parameter calibration already addressed in [3], we plan to perform intensive numerical simulations to identify various parametric configurations corresponding to specific physiological (e.g. mono-ovulation versus poly-ovulation) or pathological situations (e.g.…”