J. E. F. Green

Distinguishing between mechanisms of cell aggregation using pair-correlation functions

Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.

Non-local models for the formation of hepatocyte-stellate cell aggregates

Liver cell aggregates may be grown in vitro by co-culturing hepatocytes with stellate cells. This method results in more rapid aggregation than hepatocyte-only culture, and appears to enhance cell viability and the expression of markers of liver-specific functions. We consider the early stages of aggregate formation, and develop a new mathematical model to investigate two alternative hypotheses (based on evidence in the experimental literature) for the role of stellate cells in promoting aggregate formation. Under Hypothesis 1, each population produces a chemical signal which affects the other, and enhanced aggregation is due to chemotaxis. Hypothesis 2 asserts that the interaction between the two cell types is by direct physical contact: the stellates extend long cellular processes which pull the hepatocytes into the aggregates. Under both hypotheses, hepatocytes are attracted to a chemical they themselves produce, and the cells can experience repulsive forces due to overcrowding. We formulate non-local (integro-partial differential) equations to describe the densities of cells, which are coupled to reaction-diffusion equations for the chemical concentrations. The behaviour of the model under each hypothesis is studied using a combination of linear stability analysis and numerical simulations. Our results show how the initial rate of aggregation depends upon the cell seeding ratio, and how the distribution of cells within aggregates depends on the relative strengths of attraction and repulsion between the cell types. Guided by our results, we suggest experiments which could be performed to distinguish between the two hypotheses.

A Mathematical Model of Liver Cell Aggregation In Vitro

The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work.

Modeling oxygen transport in surgical tissue transfer

Reconstructive microsurgery is a clinical technique used to transfer large amounts of a patients tissue from one location used to another in order to restore physical deformities caused by trauma, tumors, or congenital abnormalities. The trend in this field is to transfer tissue using increasingly smaller blood vessels, which decreases problems associated with tissue harvest but increases the possibility that blood supply to the transferred tissue may not be adequate for healing. It would thus be helpful to surgeons to understand the relationship between the tissue volume and blood vessel diameter to ensure success in these operations. As a first step towards addressing this question, we present a simple mathematical model that might be used to predict successful tissue transfer based on blood vessel diameter, tissue volume, and oxygen delivery.

A mechanistic study on tumour spheroid formation in thermosensitive hydrogels: experiments and mathematical modelling

A tumour is a complex, growing tissue with a dynamic microenvironment. Its shape and size affect mass transport and thus the ability of drugs to penetrate into the tumour. Three-dimensional (3D) tumour spheroid culture has thus been recognised as an advanced tool for anti-cancer drug screening. However, the use of tumour spheroids has been hampered by the large variations in spheroid size, their irregular shape and the labour-intensive culture process. We explore thermosensitive hydrogels, P(NIPAM-AA), for culturing tumour spheroids and compare our approach with a traditional suspension culture method (non-adhesive surface) in terms of cell proliferation, tumour spheroid size distribution and spheroid morphology. Spheroids cultured in the microgel network show a narrower size distribution and a more spherical shape. We hypothesised that these observations could be explained by the fact that cells are homogeneously retained inside the hydrogels, cell aggregate formation is much slower due to network resistance and the cell death rate is smaller in comparison with the suspension culture. We developed a cellular automata (CA) model to validate these hypotheses. Spheroid formation with different parameter values, representing culture in suspension and in microgels, is simulated. Our results are consistent with the hypothesis that the microgel culture produces a more uniform size distribution of spheroids as a result of reduced cell death and the gel network resistance.

An investigation of the influence of extracellular matrix anisotropy and cell–matrix interactions on tissue architecture

Mechanical interactions between cells and the fibrous extracellular matrix (ECM) in which they reside play a key role in tissue development. Mechanical cues from the environment (such as stress, strain and fibre orientation) regulate a range of cell behaviours, including proliferation, differentiation and motility. In turn, the ECM structure is affected by cells exerting forces on the matrix which result in deformation and fibre realignment. In this paper we develop a mathematical model to investigate this mechanical feedback between cells and the ECM. We consider a three-phase mixture of collagen, culture medium and cells, and formulate a system of partial differential equations which represents conservation of mass and momentum for each phase. This modelling framework takes into account the anisotropic mechanical properties of the collagen gel arising from its fibrous microstructure. We also propose a cell–collagen interaction force which depends upon fibre orientation and collagen density. We use a combination of numerical and analytical techniques to study the influence of cell–ECM interactions on pattern formation in tissues. Our results illustrate the wide range of structures which may be formed, and how those that emerge depend upon the importance of cell–ECM interactions.

Identifying the necrotic zone boundary in tumour spheroids with pair-correlation functions

Automatic identification of the necrotic zone boundary is important in the assessment of treatments on in vitro tumour spheroids. This has been difficult especially when the difference in cell density between the necrotic and viable zones of a tumour spheroid is small. To help overcome this problem, we developed novel one-dimensional pair-correlation functions (PCFs) to provide quantitative estimates of the radial distance of the necrotic zone boundary from the centre of a tumour spheroid. We validate our approach on synthetic tumour spheroids in which the position of the necrotic zone boundary is known a priori. It is then applied to nine real tumour spheroids imaged with light sheet-based fluorescence microscopy. PCF estimates of the necrotic zone boundary are compared with those of a human expert and an existing standard computational method.

A MATHEMATICAL MODEL FOR CELL-INDUCED GEL COMPACTION IN VITRO

We present a mathematical model for cell-induced gel contraction in vitro. The core of the model consists of conservation equations for the mass of the gel and the number of cells, coupled to a force balance on the gel. On the basis of previously reported experimental findings for collagen gels, which are frequently used experimentally, the gel is treated as a compressible viscous fluid while inertial effects are neglected. The flow is assumed to be isothermal, and a linear pressure–density relation is adopted. The force exerted on the gel by cells is assumed to depend upon the local environment surrounding the cell: influences can include the cell and extracellular matrix density, and the concentration of a diffusible chemical produced by the cells. We consider the simple, but experimentally relevant, case of spherically symmetric gels. For cell-free gels, we show how simple experiments might be used to determine the parameters in the model. When the cell-derived forces are given by a prescribed function of position, we are able to obtain the early time and steady-state behavior of the solution analytically. We perform numerical simulations which generate predictions of how the gel density evolves during compaction under differing assumptions concerning the factors influencing the force exerted by the cells. These results are compared with some previous observations reported in the literature.

A structure-function analysis of the left ventricle.

This study presents a structure-function analysis of the mammalian left ventricle and examines the performance of the cardiac capillary network, mitochondria, and myofibrils at rest and during simulated heavy exercise. Left ventricular external mechanical work rate was calculated from cardiac output and systemic mean arterial blood pressure in resting sheep (Ovis aries; n = 4) and goats (Capra hircus; n = 4) under mild sedation, followed by perfusion-fixation of the left ventricle and quantification of the cardiac capillary-tissue geometry and cardiomyocyte ultrastructure. The investigation was then extended to heavy exercise by increasing cardiac work according to published hemodynamics of sheep and goats performing sustained treadmill exercise. Left ventricular work rate averaged 0.017 W/cm3 of tissue at rest and was estimated to increase to ∼0.060 W/cm3 during heavy exercise. According to an oxygen transport model we applied to the left ventricular tissue, we predicted that oxygen consumption increases from 195 nmol O2·s-1·cm-3 of tissue at rest to ∼600 nmol O2·s-1·cm-3 during heavy exercise, which is within 90% of the oxygen demand rate and consistent with work remaining predominantly aerobic. Mitochondria represent 21-22% of cardiomyocyte volume and consume oxygen at a rate of 1,150 nmol O2·s-1·cm-3 of mitochondria at rest and ∼3,600 nmol O2·s-1·cm-3 during heavy exercise, which is within 80% of maximum in vitro rates and consistent with mitochondria operating near their functional limits. Myofibrils represent 65-66% of cardiomyocyte volume, and according to a Laplacian model of the left ventricular chamber, generate peak fiber tensions in the range of 50 to 70 kPa at rest and during heavy exercise, which is less than maximum tension of isolated cardiac tissue (120-140 kPa) and is explained by an apparent reserve capacity for tension development built into the left ventricle.

On the derivation of approximations to cellular automata models and the assumption of independence

Cellular automata are discrete agent-based models, generally used in cell-based applications. There is much interest in obtaining continuum models that describe the mean behaviour of the agents in these models. Previously, continuum models have been derived for agents undergoing motility and proliferation processes, however, these models only hold under restricted conditions. In order to narrow down the reason for these restrictions, we explore three possible sources of error in deriving the model. These sources are the choice of limiting arguments, the use of a discrete-time model as opposed to a continuous-time model and the assumption of independence between the state of sites. We present a rigorous analysis in order to gain a greater understanding of the significance of these three issues. By finding a limiting regime that accurately approximates the conservation equation for the cellular automata, we are able to conclude that the inaccuracy between our approximation and the cellular automata is completely based on the assumption of independence.