Inria, Université Paris-Saclay, France.
LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, France.
Math Biosci Eng. 2019 Apr 10;16(4):3018-3046. doi: 10.3934/mbe.2019150.
In this work, we study a multiscale inverse problem associated with a multi-type model for age structured cell populations. In the single type case, the model is a McKendrick-VonFoerster like equation with a mitosis-dependent death rate and potential migration at birth. In the multi-type case, the migration term results in an unidirectional motion from one type to the next, so that the boundary condition at age 0 contains an additional extrinsic contribution from the previous type. We consider the inverse problem of retrieving microscopic information (the division rates and migration proportions) from the knowledge of macroscopic information (total number of cells per layer), given the initial condition. We first show the well-posedness of the inverse problem in the single type case using a Fredholm integral equation derived from the characteristic curves, and we use a constructive approach to obtain the lattice division rate, considering either a synchronized or non-synchronized initial condition. We take advantage of the unidirectional motion to decompose the whole model into nested submodels corresponding to self-renewal equations with an additional extrinstic contribution. We again derive a Fredholm integral equation for each submodel and deduce the well-posedness of the multi-type inverse problem. In each situation, we illustrate numerically our theoretical results.
在这项工作中，我们研究了与年龄结构细胞群体多类型模型相关的多尺度反问题。在单类型情况下，模型是一个具有有丝分裂依赖性死亡率和潜在出生时迁移的 McKendrick-VonFoerster 型方程。在多类型情况下，迁移项导致从一种类型到另一种类型的单向运动，因此在年龄 0 处的边界条件包含来自前一种类型的额外外在贡献。我们考虑从宏观信息（每个层的细胞总数）中检索微观信息（分裂率和迁移比例）的反问题，给定初始条件。我们首先使用从特征曲线导出的 Fredholm 积分方程证明了单类型情况下反问题的适定性，并使用构造方法来获得格子分裂率，考虑同步或非同步初始条件。我们利用单向运动将整个模型分解为嵌套子模型，这些子模型对应于具有额外外在贡献的自我更新方程。我们再次为每个子模型推导一个 Fredholm 积分方程，并推导出多类型反问题的适定性。在每种情况下，我们都通过数值说明来展示我们的理论结果。
