John C. Dallon

Fibroblast migration and collagen deposition during dermal wound healing : mathematical modelling and clinical implications

The extent to which collagen alignment occurs during dermal wound healing determines the severity of scar tissue formation. We have modelled this using a multiscale approach, in which extracellular materials, for example collagen and fibrin, are modelled as continua, while fibroblasts are considered as discrete units. Within this model framework, we have explored the effects that different parameters have on the alignment process, and we have used the model to investigate how manipulation of transforming growth factor-β levels can reduce scar tissue formation. We briefly review this body of work, then extend the modelling framework to investigate the role played by leucocyte signalling in wound repair. To this end, fibroblast migration and collagen deposition within both the wound region and healthy peripheral tissue are considered. Trajectories of individual fibroblasts are determined as they migrate towards the wound region under the combined influence of collagen/fibrin alignment and gradients in a paracrine chemoattractant produced by leucocytes. The effects of a number of different physiological and cellular parameters upon the collagen alignment and repair integrity are assessed. These parameters include fibroblast concentration, cellular speed, fibroblast sensitivity to chemoattractant concentration and chemoattractant diffusion coefficient. Our results show that chemoattractant gradients lead to increased collagen alignment at the interface between the wound and the healthy tissue. Results show that there is a trade-off between wound integrity and the degree of scarring. The former is found to be optimized under conditions of a large chemoattractant diffusion coefficient, while the latter can be minimized when repair takes place in the presence of a competitive inhibitor to chemoattractants.

A review of fibroblast-populated collagen lattices.

Bells introduction of the fibroblast‐populated collagen lattice (FPCL) has facilitated the study of collagen–cell interactions. As a result of the numerous modifications of the casting of FPCLs, the in vivo applications of these in vitro findings have been confusing. Here experimental FPCL contraction findings are viewed in regard to three proposed mechanisms responsible for lattice contraction. The cellular mechanisms responsible for generating FPCL contraction are cell contraction, cell tractional forces related to cell locomotion, and initial cell elongation and spreading.

Theoretical models of wound healing: past successes and future challenges

The complex biology of wound healing is an area in which theoretical modelling has already made a significant impact. In this review article, the authors describe the key features of wound healing biology, divided into four components: epidermal wound healing, remodelling of the dermal extracellular matrix, wound contraction, and angiogenesis. Within each of these categories, previous modelling work is described, and the authors identify what they regard as the main challenges for future theoretical work.

Mathematical modelling of anisotropy in fibrous connective tissue

We present two modelling frameworks for studying dynamic anistropy in connective tissue, motivated by the problem of fibre alignment in wound healing. The first model is a system of partial differential equations operating on a macroscopic scale. We show that a model consisting of a single extracellular matrix material aligned by fibroblasts via flux and stress exhibits behaviour that is incompatible with experimental observations. We extend the model to two matrix types and show that the results of this extended model are robust and consistent with experiment. The second model represents cells as discrete objects in a continuum of ECM. We show that this model predicts patterns of alignment on macroscopic length scales that are lost in a continuum model of the cell population.

Modeling the Effects of Transforming Growth Factor-beta on Extracellular Matrix Alignment in Dermal Wound Repair

We present a novel mathematical model for collagen deposition and alignment during dermal wound healing, focusing on the regulatory effects of transforming growth factor‐β (TGFβ.) Our work extends a previously developed model which considers the interactions between fibroblasts and an extracellular matrix composed of collagen and a fibrin based blood clot, by allowing fibroblasts to orient the collagen matrix, and produce and degrade the extracellular matrix, while the matrix directs the fibroblasts and control their speed. Here we extend the model by allowing a time varying concentration of TGFβ to alter the properties of the fibroblasts. Thus we are able to simulate experiments which alter the TGFβ profile. Within this model framework we find that most of the known effects of TGFβ, i.e., changes in cell motility, cell proliferation and collagen production, are of minor importance to matrix alignment and cannot explain the anti‐scarring properties of TGFβ. However, we find that by changing fibroblast reorientation rates, consistent with experimental evidence, the alignment of the regenerated tissue can be significantly altered. These data provide an explanation for the experimentally observed influence of TGFβ on scarring.

Numerical aspects of discrete and continuum hybrid models in cell biology

In this paper we introduce a method of modeling which mixes continuum and discrete variables, and explain two models in cell biology that use this method. The first application deals with wound healing, more specifically the collagen alignment in scar tissue formation and the second models early aggregation in the cellular slime mold Dictyostelium discoideum. We solve these models using numerical techniques similar to the particle-in-cell method which requires that the discrete and continuum variables are interpolated one to the other. The implementational and numerical details are discussed in an informal and practical manner with particular attention given to the problem of interpolation.

A MATHEMATICAL MODEL FOR SPATIALLY VARYING EXTRACELLULAR MATRIX ALIGNMENT

Orientation of extracellular matrix fibers in the skin is a keyingredient of tissue appearance and function, and differences in fiber alignment are one of the main distinctions between scar tissue and normal skin. In this paper, the authors develop a mathematical model for alignment of collagen fibers and the fibroblast cells that remodel them; the model extends previous work in which spatial variation was excluded. Numerical simulations of the model are presented, which show spatial variations in alignment over long transients, but with spatiallyuniform behavior in the long term. This is investigated further via asymptotic analysis, using the angular diffusion coefficient as a small parameter. This method enables calculation of the form of the steadystate orientation peaks observed numerically; by considering behavior at large times, the rate of approach to these peaks is shown to be exponential. Extension of this analysis to the spatially varying model confirms that long-time behavior will be spatiallyuniform except in one special, and biologicallyunrealistic, case. The authors conclude that the spatiallyvary ing alignment patterns observed in skin are in fact slow transients, and biological implications of the modeling are discussed.

Differences in the mechanism of collagen lattice contraction by myofibroblasts and smooth muscle cells.

Both rat derived vascular smooth muscle cells (SMC) and human myofibroblasts contain α smooth muscle actin (SMA), but they utilize different mechanisms to contract populated collagen lattices (PCLs). The difference is in how the cells generate the force that contracts the lattices. Human dermal fibroblasts transform into myofibroblasts, expressing α‐SMA within stress fibers, when cultured in lattices that remain attached to the surface of a tissue culture dish. When attached lattices are populated with rat derived vascular SMC, the cells retain their vascular SMC phenotype. Comparing the contraction of attached PCLs when they are released from the culture dish on day 4 shows that lattices populated with rat vascular SMC contract less than those populated with human myofibroblast. PCL contraction was evaluated in the presence of vanadate and genistein, which modify protein tyrosine phosphorylation, and ML‐7 and Y‐27632, which modify myosin ATPase activity. Genistein and ML‐7 had no affect upon either myofibroblast or vascular SMC‐PCL contraction, demonstrating that neither protein tyrosine kinase nor myosin light chain kinase was involved. Vanadate inhibited myofibroblast‐PCL contraction, consistent with a role for protein tyrosine phosphatase activity with myofibroblast‐generated forces. Y‐27632 inhibited both SMC and myofibroblast PCL contraction, consistent with a central role of myosin light chain phosphatase. J. Cell. Biochem. 111: 362–369, 2010.

Models with Lattice-free Center-based Cells Interacting with Continuum Environment Variables

In this chapter we describe a discrete continuum hybrid method applied to two biological systems. The cells are modeled as discrete objects which are free to move in space (lattice-free), the forces which act on the cells are applied to their center of mass (center-based), and the cells interact with something represented as a continuum variable. Dictyostelium discoideum is the first system modeled by the method. The cells move and communicate with each other through a diffusible chemical. In the second system, scar tissue formation, the cells interact with the extracellular matrix which is represented as a continuous vector field.

Models of Dictyostelium discoideum Aggregation

Since its discovery in the 1940’s, the life cycle of the cellular slime mould Dictyostelium discoideum has attracted the interest of developmental biologists. It involves a relatively simple transition from unicellular to multicellular organization. Briefly, amoebae feed on bacteria in the soil and divide. Exhaustion of the food supply triggers a developmental sequence which leads, via cell aggregation, to the formation of a migrating slug-like “organism”. The slug eventually culminates into a fruiting body, aiding the dispersal of spores from which, under favourable conditions, new amoebae develop. To date a variety of species in different taxonomic groups are known whose life cycles follow a similar pattern (Margulis & Schwartz 1988). Over the past fifty years, many of the molecular and cellular mechanisms which are involved in cell aggregation, collective movement and differentiation have been identified, and much work is devoted to the understanding of the interaction of these mechanisms in shaping Dictyostelium development. Mathematical modelling has proved a useful tool with which to study these interactions on a quantitative basis.