Biological events rarely occur with linear kinetics and attempting to describe wound repair with any single simple formula is problematic, since the healing process is a complex combination of events such as re-epithelialization and contraction. Further, wounds present in a wide range of sizes, and "50 percent healing" of a large wound represents much more extensive cellular proliferation and migration than a similar degree of linear healing in a tiny lesion.

Several studies have shown that wound contraction can be described by an exponential curve. In a classic study on full-thickness rabbit wounds, the reduction in the logarithm of wound area over time corresponded to a series of linear vectors, indicating a logarithmic process of contraction [1]. In a subsequent investigation, the closure of excised wounds in rats was shown to follow an exponential decrease in wound area during the contraction phase, supporting the notion of a contraction decay constant could describe the rate of wound closure, independent of lesion size and geometry [2]. Porcine full thickness wounds have been shown to correlate to an exponential decay of area over time [3, 4]. A mechanistic wound contraction model confirmed the exponential relationship of wound contraction versus time, and the size-and-shape independence of the contraction decay constant [5]. Numerical solutions of a wound contraction mathematical model yielded comparable results to the excised rat wounds [6]. An analysis of contraction inhibition in hairless mice also indicated an exponential decrease in wound area during the first half of the study period [7].

Research on epithelialization and keratinocyte proliferation models has produced evidence of both linear and nonlinear patterns of wound healing. One might expect, a priori, that the exponential nature of cell division might lead to ever accelerating epithelial growth and wound closure. However, both cell-to-cell contact inhibition and the limited ability of keratinocytes to divide before they reach senescence limit this acceleration and prevent theoretical prediction about the kinetics of re-epithelialization. Further, the keratinocytes on the healing edge of a wound are restricted to proliferation and migration towards the open wound area, in contrast to the centrifugal growth of a cell colony observed in vitro. Models based on the relationship of cell proliferation to the rate of increase in cell density and chemical concentration produced numerical solutions exhibiting a lag and linear phase in reduction of wound radius over time [8]. Cell fronts in assays have been shown to travel with constant speeds under assumption of a logistic cell proliferation rate [9]. Another research group assumed an exponential growth pattern for the proliferation rate parameter of clonal subtypes in a study on aerosolized skin graft modeling [10]. Elsewhere, a delayed exponential model was used to predict time to full re-epithelialization in chronic wounds treated with electrical stimulation [11]. Exponential patterns of epithelialization have also been observed in corneal defects [12]. A significant limitation of in vitro and animal studies is that they typically study the response to a one-time injury rather than mimicking the repeated damaging effects of venous hypertension, matrix metalloproteinases, trauma, etc. that a real-life human venous leg ulcer experiences.

In previous work, we reported on the statistical reliability of standard early wound healing rate measurements for predicting ultimate closure [13]. While there is significant clinical value for the use of linear healing rates and wound trajectories, neither method is an approximation to a standard mathematical function [14, 15]. Regression fitting of data points to a function potentially solves this issue by estimating values for the healing rate constant (or contraction constant) independent of initial wound size and geometry. Prediction intervals, goodness-of-fit statistics, and residual error plots may lend more analytical information to the nature of the wound healing process than summary statistics of standard healing rates. Nonlinear curve fitting algorithms, primarily using a Gompertz function, have been applied to serial wound measurements with positive results [16–19]. Regression curve fitting may be a powerful method to mathematically describe contraction and epithelialization rates in a clinical context.

The results of our study strongly suggest that wound surface area reduction for healing VLUs, particularly early in the treatment phase, follows an exponential decay curve. It is inaccurate to claim, though, that wound healing per se follows an exponential decay/growth process. In our study, for example, a few wounds that worsened with time projected with significance to an exponential growth model, which can not be sustained over time due to anatomical limits. It is also not clear whether chronic wounds that heal over a longer time frame (i.e. 6–9 months) follow a similar exponential decay pattern. Further, we have not stratified the data to investigate the influence of patient age, wound duration and various comorbidities and medications on adherence to the model.

There is also additional debate as to whether wound surface area is the best metric for wound size quantification. Wounds that present with significant depth are not adequately quantified by the 2-dimensional surface area measurements used in our regression tests; this may corrupt some outcome predictions. Similarly, wounds of equivalent surface area but of varying geometrical shapes may be best represented by linear healing parameter, wound perimeter, or perimeter/area ratio measurements.

Another important flaw in the application of a generalized exponential model to wound healing data is that the model only truly fits some contraction and epithelialization mechanics. While many VLUs follow an exponential healing pattern, in general these lesions are relatively superficial and in some wounds contraction may be less important than granulation and epithelialization; in addition the mode of healing may be influenced by various therapeutic modalities. An exponential decay function does not intersect at zero; the limit of the function is zero only as *b* (healing rate) or *x* (time) gets very large in scale. This further supports the notion of a wound contraction rate via an exponential decay model, followed by or partially combined with an unmodeled proliferation rate of the wound margins during epithelialization. Often times it was observed in this study that regression residual variation was negative in trend as the wound reached closure. This suggests that the process of epithelialization during the final stages of wound healing may have not fit the model accurately.