Computational approaches can simulate new or incompletely understood treatment strate- gies in a non-invasive and inexpensive way. For example, a novel technique for treating severe burns involves collecting and culturing keratinocytes from a patient’s healthy skin, which can then be sprayed onto the affected area to promote or accelerate healing. Although initial in vitro and in vivo studies demonstrated accelerated re-epithelialisation after treat- ment, the optimal cell distribution and concentration for spraying had not been established experimentally. A reaction-diffusion PDE system, based on the descriptive studies of Sher- ratt and Murray [87], was used to model growth patterns of cell colonies following aerosol application to predict changes in epidermal coverage over time [185].
Rather than assuming a homogenous population of keratinocytes, the model considered different levels of mitotic activity in separate equations: epidermal stem cells, transit am- plifying cells recently progressed from stem cells, transit amplifying cells and quiescent cells were each represented by a separate variable. Parameters were experimentally determined, although using data from a different system to burn wounds. It was found that choosing initial positions of cell colonies from an approximately uniform distribution (which repre- sents keratinocytes being sprayed from a distance) resulted in faster healing than when initial positions were chosen randomly from a normal distribution (which would occur when
so the findings are less clinically relevant. Aspects for further exploration, for example the frequency of application, were also identified. This study has been highlighted as one of the first directly translational mathematical models of epidermal healing [220]; however, a period of examination and assessment will be required to determine how and where these advances in understanding may be applied.
Another clinically motivated study considered the process of surgical debridement: a pro- cedure that is widely used to stimulate or expedite healing despite the limited clinical data available to support its benefits. A geometric model was formulated to fit ODEs to data on changes to wound size over time, but without incorporating any of the cellular mechanisms underpinning the process of healing [188]. Proportions of wound to slough area were pre- dicted under a variety of conditions, including the timing of debridement and the amount of slough removed. The model identified timepoints at which debridement is most effective in forcing a non-healing wound to resume repair, although careful parameterisation would be required to move from the theoretical framework to a clinical recommendation. Questions that could be addressed by extending the model were also proposed, including identification of debridement strategies that would require the fewest number of surgical interventions, or would minimise the time required to complete healing. Despite neglecting to incorporate underlying biological processes, the study illustrates how simple mathematical models may describe clinically relevant scenarios and produce novel insights into potential interventions.
Another non-mechanistic approach was used in an early investigation of leg ulcer healing, which combined theoretical studies with clinical assessment to predict healing outcomes at early timepoints [196]. Instead of including mechanistic descriptions of the processes involved, the model predicted the evolution of an ulcer given its initial shape, producing a binary output designating the wound as healing or chronic. The model produced good spatial agreement between theoretical predictions and clinical observation, but gave limited information on the temporal dynamics of healing. If time evolution could be incorporated this may provide a valuable clinical tool to make early predictions of which wounds are likely to become chronic.
Models that are primarily designed to study biological hypotheses may be further exploited to investigate clinical scenarios. For example, the previously described hybrid model de- veloped by Sun et al. (2009) [192] was also used to investigate healing in a large wound. Model output predicted that wounds above a certain size wound not heal regardless of the length of the time period simulated, which is a situation directly comparable to chronic wounds in the clinic. Model analysis suggested that auto-regulation of keratinocyte colony
formation promoted migrating keratinocytes to differentiate into committed cells and cor- neocytes before re-epithelialisation was complete, preventing the wound from fully healing: a novel and clinically applicable conclusion. However, to save computation time only small areas of epidermis were modelled, so a comparatively large wound in the simulation was not equivalent to a large wound in clinical practice. Thus the model provided some mechanistic evidence for the causes of non-healing wounds but the authors noted that it would require modification, in particular increasing the dimensions of the area of skin simulated, to draw clinically applicable conclusions [192].
Predicting outcomes and personalising treatment decisions based on characteristics of a patient and their wound would constitute an important improvement in the care of chronic wounds. A recent study used statistical methods to predict delayed wound healing [167]. Whilst not a mathematical model in the traditional sense of those described previously, the study applied computational approaches to a large dataset of information collected during routine wound care. 865 possible predictors of delayed healing were identified and a subset of data was used to develop a model of the predictors and their relevant importances. The model was then applied to the remaining data, and could predict which wounds would become chronic with reasonable reliability. Predictors with high importance could also be identified: these included patient age, wound dimensions and changes in wound size observed between assessments. Wound type and location had relatively little importance, suggesting that careful quantification and monitoring of changes to wound size over time should be prioritised soon after injury to identify indications of pathological healing. Results such as these could be used to recommend areas of clinical need, for example the development and evaluation of novel technologies to measure wounds accurately and non-invasively.
Diabetic wounds are known to exhibit non-healing pathologies, but our understanding of which cellular processes contribute to delayed repair is limited. A study by Bowden et al. (2014) [184] used an ODE model to describe healing of full-thickness wounds, incorporating equations for epidermal tissue undergoing cell migration and proliferation, dermal tissue undergoing contraction and growth, and the forces acting on the wound due to attachments between healthy tissue, granulation tissue and sub-dermal tissue. Model output suggested that growth in the dermis dominates healing and, when fitted to experimental data, identi- fied a reduction in the rate of dermal tissue growth (as opposed to decreased levels of dermal contraction) to be the key difference between normal and diabetic healing. The ability of
One commonly proposed limitation of studies using mathematical models to inform treat- ment strategies is their failure to include every component and process of epidermal repair. Although it may be possible to formulate models that attempt to encompass the entire healing process, problems would occur due to the complexity of cellular processes and sig- nals involved. Such models would suffer from reduced utility due to an increased number of unknown parameters, and coupling several subsystems can obscure the analysis of main healing processes. The studies described in this section show that, whilst mathematical methods are yet to unearth a novel strategy for the treatment of chronic wounds, they may be used to refine nascent procedures. One reason for the success of models investigating a particular treatment is that they focus on a specific subset of processes in the wound healing cascade, which may be studied in detail in the context of an intervention.
3.7
A marriage between experiments and theory: collabora-
tion not competition
Computational models of healing cannot make significant advances in isolation from experi- mental studies, but together the two techniques represent a powerful approach for addressing open questions in epidermal repair. Models are often more successful in generating hypothe- ses than drawing firm conclusions; these hypotheses may be investigated experimentally and their findings then used to further update models in an iterative process that is more likely to yield novel and relevant results than experiments or modelling alone.
Another important link between theoretical studies and experiments is the reliance of com- putational models on parameter estimates derived from in vivo and in vitro work. Unreliable parameter estimation or misinterpretation of the underlying biology can be hugely detrimen- tal, so models should be formulated in close collaboration with biologists and clinicians to identify uncertainties in processes and parameters from the outset. To facilitate this, when assessing the expense of conducting a computational study the additional costs associated with robust experimental derivation of parameter values should not be overlooked. Compu- tational studies could be improved by noting the level of uncertainty associated with each experimentally determined parameter value: none of the studies described here list the con- fidence intervals of their parameters and, whilst some do carry out sensitivity analyses, a more explicit discussion of the potential estimation errors and their effects on model output would be a welcome addition. For example, Bayesian sensitivity analysis is a technique that has been implemented in models of electrical activation and recovery in cardiac cells, and
could also be applied to models of the healing epidermis [221].
When using models to investigate competing biological hypotheses, it is important that scientists and clinicians clearly communicate open questions that are not amenable to ex- perimentation, and that theoreticians correctly interpret the associated hypotheses. For example, Sun et al. (2009) [192] claimed that experimental studies had purported increased mitotic activity in adjacent tissue as a possible stimulus for the onset of keratinocyte mi- gration at the wound edge. However, the studies they cited in fact proposed that additional keratinocyte proliferation is not strictly required for migration, but increases the population of cells available to restore epidermal integrity [99]. Therefore, whilst the model output provided further evidence for this assertion it did not substantially increase understanding of the migratory stimulus. Closer collaboration between experimental and computational biologists will help ensure that theoretical work addresses relevant and open questions, thus increasing the value of modelling studies.
Conversely, clinicians and experimentalists should be open to the use of modelling as one of many important techniques for investigating biomedical systems. Wider understanding of the benefits and limitations of mathematical modelling, along with acknowledgement that all methodologies in vitro, in vivo, clinical and computational involve some degree of ab- straction would help to achieve this. In particular, it should be recognised that models describing only a subset of the processes involved in healing may be equally if not more informative than those attempting to exhaustively investigate the entire system. Making model simplifications is a powerful tool that may enable a reduction of uncertainty in the model’s processes and parameters, and therefore limit the noise associated with solution behaviour: balancing complexity and utility is of paramount importance. It is also impor- tant that clinicians and scientists engage with theoreticians at all stages of the modelling process to produce models that further the understanding of epidermal healing, and to en- sure that theoretical results are presented in a form that is understandable to those without an extensive background in mathematical modelling. In particular model output should be considered carefully: quantifying the degree of injury and repair is of greatest relevance to healthcare practitioners but determining how best to extrapolate this from models on a cellular or mechanical level is non-trivial. Enabling scientists and clinicians without a math- ematical background to evaluate models including making judgements on their confidence in the underlying biological assumptions, the suitability of the model formulation, and the
imental and clinical studies, but rather complementary techniques that play an important role in answering complex questions. Just as quantitative and qualitative insights from the wet-lab inform model development, the output of theoretical studies can be exploited to generate hypotheses that are testable using experimental techniques. Thus computational modelling can be used to organise complex biological data, connecting experimental results to fundamental biological principles in an iterative manner. Mathematical modelling of wound healing is an established field, but the scope for increased interaction with experi- ments and patients provides an exciting prospect for addressing biological unknowns and supporting personalised therapeutic decision making in epidermal repair.