Mathematical Modelling and Avascular Tumour Growth: Interdisciplinary Research

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling

In vitro tumour spheroids have been used to study avascular tumour growth and drug design for over 50 years. Tumour spheroids exhibit heterogeneity within the growing population that is thought to be related to spatial and temporal differences in nutrient availability. The recent development of real-time fluorescent cell cycle imaging allows us to identify the position and cell cycle status of individual cells within the growing spheroid, giving rise to the notion of a four-dimensional (4D) tumour spheroid. We develop the first stochastic individual-based model (IBM) of a 4D tumour spheroid and show that IBM simulation data compares well with experimental data using a primary human melanoma cell line. The IBM provides quantitative information about nutrient availability within the spheroid, which is important because it is difficult to measure these data experimentally.

Designing and interpreting 4D tumour spheroid experiments

Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal sampling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.

Quantitative analysis of tumour spheroid structure

Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.

Model of Drug Delivery to Populations Composed of Two Cell Types

The rate of drug delivery to cells and the subsequent rate of drug metabolism are dependent on the cell membrane permeability to the drug. In some cases, tissue may be composed of different types of cells that exhibit order of magnitude differences in their membrane permeabilities. This paper presents a brief review of the components of the tissue scale three-compartment pharmacokinetic model of drug delivery to single-cell-type populations. The existing model is extended to consider tissue composed of two different cell types. A case study is presented of infusion mediated delivery of doxorubicin to a tumor that is composed a drug reactive cell type and of a drug resistive cell type. The membrane permeabilities of the two cell types differ by an order of magnitude. A parametric investigation of the population composition is conducted and it is shown that the drug metabolism of the low permeability cells are negatively influenced by the fraction of the tissue composed of the permeable drug reactive cells. This is because when the population is composed mostly of drug permeable cells, the extracellular space is rapidly depleted of the drug. This has two compounding effects: (i) locally there is simply less drug available to the neighboring drug resistant cells, and (ii) the depletion of the drug from the extracellular space near the vessel-tissue interface leaves less drug to be transported to booth cell types farther away from the vessel.

Mathematical Model of Tumour Spheroid Experiments with Real-Time Cell Cycle Imaging

Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential cancer drug therapies. Standard experiments involve growing and imaging spheroids to explore how different conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI) has enabled real-time in situ visualisation of the cell cycle progression. Observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work, we adapt the mathematical framework originally proposed by Ward and King (Math Med Biol 14:39-69, 1997. https://doi.org/10.1093/imammb/14.1.39 ) to produce a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red; (ii) living cells in S/G2/M phase that fluoresce green; and (iii) dead cells that are not fluorescent. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can qualitatively capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to qualitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software programs required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub. at https://github.com/wang-jin-mathbio/Jin2021.