Mathematical Models of Organoid Cultures

Abstract

Organoids are engineered three-dimensional tissue cultures derived from stem cells and capable of self-renewal and self-organization into a variety of progenitors and differentiated cell types. An organoid resembles the cellular structure of an organ and retains some of its functionality, while still being amenable to in vitro experimental study. Compared with two-dimensional cultures, the three-dimensional structure of organoids provides a more realistic environment and structural organization of in vivo organs. Similarly, organoids are better suited to reproduce signaling pathway dynamics in vitro, due to a more realistic physiological environment. As such, organoids are a valuable tool to explore the dynamics of organogenesis and offer routes to personalized preclinical trials of cancer progression, invasion, and drug response. Complementary to experiments, mathematical and computational models are valuable instruments in the description of spatiotemporal dynamics of organoids. Simulations of mathematical models allow the study of multiscale dynamics of organoids, at both the intracellular and intercellular levels. Mathematical models also enable us to understand the underlying mechanisms responsible for phenotypic variation and the response to external stimulation in a cost- and time-effective manner. Many recent studies have developed laboratory protocols to grow organoids resembling different organs such as the intestine, brain, liver, pancreas, and mammary glands. However, the development of mathematical models specific to organoids remains comparatively underdeveloped. Here, we review the mathematical and computational approaches proposed so far to describe and predict organoid dynamics, reporting the simulation frameworks used and the models' strengths and limitations.

Keywords: 3D tissue; agent-based models; computational modeling; differential equations; mathematical modeling; organoids.

Figure 1

Graphical representation of computational models developed to understand the intrinsic dynamics of organoid cultures. (A–C) Agent-based models, (D–H) equation-based models, color coded as per description in each panel. (A) A 3D model of intestinal organoids developed to investigate the distribution of cell populations and growth patterns provoked by Wnt and Notch signaling dynamics (Buske et al., 2012; Thalheim et al., 2018). (B) A 2D model of the cross section of a confluent intestinal epithelial layer, designed to study the biomechanical interactions between cells to produce crypt fission (Langlands et al., 2016; Almet et al., 2018). (C) Representation of a simulated optic-cup organoid (Okuda et al., 2018a). (D) Computational model of colon organoids created to study the effect of exogenous substances in the growth pattern and spatial distributions, to compare them with cancer phenotypes (Yan et al., 2018). (E) Diffusion model of a spheroid that simulates the consumption of nutrients in cerebral organoids to predict growth patterns (McMurtrey, 2016). (F) Model of oxygen consumption by a midbrain organoid grown in a millifluidic chamber to compare it with the oxygen consumption that occurred in the common well (Berger et al., 2018). (G, H) Equation-based reaction-diffusion models of gene networks used to simulate and predict fate patterning expression in gastruloids. The patterning expression and positional information paradigm plots show the signaling expressions of each primary germ layer observed experimentally (Etoc et al., 2016; Tewary et al., 2017).