Mathematical model for wound healing following autologous keratinocyte transplantation

Abstract

In times of increasing economical pressure on the health care systems, it is important to optimise the outpatient treatment of chronic wounds. Another aim of wound healing research is to discover agents to accelerate healing. Wound healing trajectories or healing velocities can provide information to demonstrate the endpoints for wound healings. A great problem in clinical trials is to specify these parameters. Therefore, we developed a mathematical model for more transparency. In this initial project, we observed 19 wounds to construct the wound healing trajectories after transplantation of autologous keratinocytes, and the results are so encouraging that investigation in this area will continue. The developed mathematical model describes the clinical observed healing process. It was possible to find parameters to distinguish between old and young patients, retrospectively or prospectively calculate the healing rates and to determine exactly the endpoint of healing. Therefore, our model might be very useful in practices or for studies.

Introduction

Because of the increasing life expectancy of the population, chronic venous insufficiency and its sequel chronic venous ulceration are increasing. Median prevalence of chronic venous insufficiency is about 20–25% in females and 10–15% in males, and prevalence of chronic wounds is estimated to range around 5% in patients older than 80 years. In 20% of these patients, the leg ulcer exists over 10 years 1, 2, 3, 4

Speeding up the healing process reduces the time during which the patient needs high attention from nurses or other health care professionals, thus reducing costs for the health care system. Additionally, wound healing also helps to increase quality of life for the individual.

Patients and methods

We retrospectively analysed 15 patients within total 19 transplantations of autologous keratinocytes. There was no treatment of more than one ulcer at one time, but some of the patients underwent transplantation of the same ulcer twice.

Autologous grafts were cultured out of follicular outer root sheath cells to a full‐thickness, layered skin sheet graft (EpiDex™) according to current good manufacturing practice. These sheets are put on the wound bed.

The examined group was composed of 11 female patients and 4 male patients and included only patients with venous leg ulcers, confirmed by duplex ultrasound imaging. It was not differentiated between postthrombotic and primary epifascial venous insufficiencies. Peripheral arterial disease was excluded by ankle brachial pressure index measurement. The mean age of the patients was 69 years, with a minimum of 48 years and a maximum of 82 years. Detailed additional data are shown in Table 1.

Observation time was at least 12 weeks after transplantation. Patients were observed twice a week in the first 2 weeks, then once weekly for 4 weeks, then every 4 weeks regularly. Additional visits of the patient depended on the current wound condition.

Four of the observed patients showed an abnormal healing process: in the beginning, normal growth, adhesion of the grafts and epithelisation of the ulceration were observed, but after about 40–50 days, wound area increased to primary level or beyond it. We interpreted this because of an insufficient nutriment of the keratinocytes and/or clinical obvious signs of a beginning infection. These four patients were not included in the following mathematical analysis.

Potential risk factors like thrombosis, diabetes mellitus, history of previous surgery, lymphoedema and polyneuropathy could not be used statistically because the number of patients was too low.

All patients received common medical device like compression stockings or compression bandages for optimal therapy of the venous insufficiency. All of them had been treated with optimal wound management.

The wound area usually had the form of an ellipse with the diameters D and d and was calculated using the equation

Mathematical model

Our mathematical model is based on simple generally accepted presuppositions like growth of a bacterial burden or tumour growth (medical biology). It is well known that healing of open wounds follows an exponential course 5, 6, with the changing rate of wound area progressively decreasing as the residual wound area approaches total closure. This is typical for the so‐called logistic growth.

Our model was not intended to correct for different confounding factors. The parameters of our equations are adapted to our observed measurements.

We observed that most of the wounds showed epithelisation from the edge of the wound. Most of the implanted cells in the middle of the wound died. Cell support, for example growth factors or oxygen, must be carried out through diffusion in the first time after transplantation 7, 8, so growth rate will stagnate after a while, if distance of diffusion gets too long. The cell population reached a maximum value in our model symbolised by the parameter K.

Unlimited cell growth can be described by dA(t)/dt = rA(t), where the factor r describes the initial growth rate and A(t) describes the area of the new cells in total at the time t.

If the support of nutriments is more and more reduced, A(t) will reach the limit K and epithelisation will stop.

Mathematical experience shows that in this case, we can modify this formula for the growth rate to dA(t)/dt = λA(t)(K − A(t)), where λis a proportional factor.

The integration of the equation above leads to A(t) = K/(1 +((K/A ₀) − 1)e^−rt) where A ₀ corresponds to the area of implanted keratinocytes that are capable of growing.

If the substitution A ₀ = pK is applied, and we look at the relative change of the area instead of the absolute area increase, we generated the growth function we have used:

This equation describes every individual wound healing trajectories of our patients.

Collective wound healing with total closure of the ulcers in a population N ₀ describes our second model. In this setting, we suppose that patients are in a fictive population at the same time. Some patients whose wounds healed 100% in the time dt will leave the population. This process can be described with a differential equation of the type dN(t)/dt = −v(t)N(t), where v(t) determines the healing rate of the patients.

The chance of total wound closure in a collective of patients diminishes in accordance with the time the group persists. New problems that prolong wound healing causes other problems, which decrease the possibility of wound closure.

Our clinical experience therefore confirms the following approach for the healing rate:

An elementary integration leads to v(t) = λe^−kt.

The reaction of our group of patients can now characterised by the following equation:

This is a differential equation according to the type Gompertz with the following solution: . To simplify the formula, we substitute λ/k = α.

Remember, N ₀ describes all patients in the group at the time t = 0 with ulcerations. We receive the ones with healed wounds in their chronological order through the following equation:

where αand k are two parameters to fit our observed to the calculated values with α= 0.55 and k = 0.2.

The relative part of patients with healed wounds (100% wound closure) is shown in Figure 1.

Figure 1.

Relative number of patients with wound closure in the group N ₀.

Results and discussion

In 11 of 19 applications, we noticed the so‐called ‘edge effect’, which means that there was a boosting effect of epithelisation in the wound margin area.

Minimal wound area was reached after 41 days (median), with a minimum of 10 days and a maximum of 125 days.

Also some patients underwent transplantation twice, for example in our statistical analysis of age, they are only mentioned once. So, there are slightly different numbers of patients in our graphics.

The categorisation of the patients was carried out in dependence of the data.

Wound healing trajectories were constructed for patients with different ages who achieved total healing and those who did not.

In addition, the percentage of patients who achieved total healing was plotted versus time of wound treatment to versify our second model.

By a suitable choice of p and r, this function is adapted to the measurements in the best possible way as shown in Figure 2. It shows two of the observed and calculated growth functions, with one example of very good epithelisation (red line) and one example of medium epithelisation (blue line). In this diagram, which is representative of the other patients, A(t) = A _initial is marked on the y axis as percentage of increase in epithelisation over the time.

Figure 2.

Two of the growth functions with different growth rates adapted to the observed.

As shown in Figure 2, great values of r cause very fast a maximum covering degree of the wound. Small values indicate poor growth conditions. Values for p are usually at a range between 10⁻³ and 10⁻⁴ and values for r seem to be strongly dependent on the age of the patient and the duration of the ulcer.

Many processes of growth or decay can be described by exponential functions. Therefore, we postulate that our parameter r might also be described by an exponential function. In the following figures, we illustrate this part. So we can also introduce the descriptive term of half‐life time.

3, 4 show the growth factor r and its dependence on the patients’ ages and the duration of the ulcer. The values of r vary between 0·2 and 0·85 depending on the age. The marked trend lines are exponential and lead, dependent on the patients’ ages, to a half‐life time of the growth factor r of approximately 100 years if the ulceration exists less than 1 year. The half‐life time of r in reference to the age of the ulcer is approximately 20 years.

Figure 3.

Correlation between growth factor r and age of the patients.

Figure 4.

Correlation between growth factor r and ulcer duration.

We have shown that growth capability of keratinocytes in the wound area decreases with the patients’ ages as well as the duration of the ulcer. It is supposed that the cells on the edges of a wound die if their minimal supply is not longer supported. The ulcer then enlarges. This is shown in Figure 5. The wound area doubles every 1·75 years within the limited period of 10 years.

Figure 5.

Wound area dependent on ulcer duration.

It is a known fact that numbers of patients with chronic wounds are increasing, depending on the patients’ age. This was confirmed in our test population (Figure 6). Because the growth factor r halves within 100 years (Figure 3) for patients with duration of their ulcer less than 1 year, the possibility of healing is good if the ulcer is treated right from the start. However, if treatment of the ulceration is delayed, chances of healing are getting worse rapidly because the growth factor r halves in about 20 years under the conditions of chronic ulceration (Figure 4). In addition, wound areas will double every 1·75 years, which has also a negative effect.

Figure 6.

Number of patients with leg ulcers dependent on age of the patients.

Therefore, therapeutic strategies to prevent ulceration are useful because the expected number of patients with ulcers is doubled every 15 years (Figure 6). Because of the increasing life expectancy of the population, this could have an enormous impact on our health system.

Duration of the growth phase to reach the minimal wound area W _min depends on many factors. Figure 7 describes the growth rates of the keratinocytes of four different patients. The change in per cent represents the relative change according to the initial wound area, for example 40% means that 40% of the initial wound area is newly epithelised. The numbers represent the age of the patients.

Figure 7.

W _min is dependent, additionally, on the beginning of growth period, which is also depending on the patient’ s age.

• (i) (51) and (79) have roughly equal values for their ulcerated areas and their growth factor r. W _min is nevertheless very different [e.g. W _min(51) ≈12 days and W _min(79) ≈22 days]. We suggest that the reason might be the covered area of the ulcer with keratinocytes. This area is five times bigger in (51) than in (79) corresponding to factor p. Remember, growth area also depends on the available viable keratinocytes at the beginning. If many keratinocytes survive at the beginning, reduction of the wound area is faster than with a low amount of viable keratinocytes. The phase of strong growth therefore starts much sooner at (51) than at (79): W _min(51) < W _min(79).
• (ii) A long‐lasting ulcer and being among the older observed patients lead to a very small r value of 0·2 at (65). This small growth factor leads to the observed W _min≈45 days.
• (iii) (76/1) and (76/2) is the same female patient. She was treated twice. The initial ulcer surface was 10 cm². It was epithelised with new cells to a degree of 42%. The second treatment led to an increase of the covered area up to 60% in reference to the treated ulcer area. In addition, the second treatment led to an increase of the factor p. The factor r decreased by 30%. This might be the reason why the two factors of W _min differ very much, W _min(76/1) ≈20 days and W _min(76/2) ≈15 days. The original ulcerated area decreased from 10 cm² to approximately 2·5 cm² after the second treatment. We expected that one more treatment will close the wound.

Minimal wound area W _min depends on ulcer duration: if there has been a long duration of the treated ulcer, the growth period of the transplanted autologous keratinocytes is reduced and stops earlier. Growth factor r is reduced, too. These findings in our hypothetical model correspond with the clinical impression of decreasing capability for wound healing in the elderly and – as shown – the tendency of the wound area to increase in long‐lasting ulcers that correlates with decreasing value of r in progressive ulcer duration. This supports the notion that frequent autologous keratinocyte transplantation at an early stage could have much more effect and better results. So far, in Germany only, patients who had failed all other therapeutic options in local therapy and surgical therapy (ulcer shaving, vein stripping and skin transplantation) get an approval from their health insurance company for autologous keratinocyte transplantation (negative selection).

We observed two different responders to therapy: in most cases (group a), minimal wound area was reached within the first 14–56 days (mean 35·8 days). In group b (three applications), we observed an initiation of wound healing, but minimal wound area was reached in 119–149 days. In most of the observed wounds, particularly, autologous grafts in the centre of the wound area were rejected, but they induced a new wound milieu presumably by releasing different cytokines and growth factors, favouring granulation and epithelisation. This may be a possible explanation of the edge effect. In addition, it is likely that keratinocytes at the edge are getting better nutriment including growth factors than those in the centre of the ulceration supposing a transport through diffusion.

Sabolinski et al. (9) also reported about three different reactive patterns following transplantation of a bilayered cultured allogenous keratinocytes, which are similar to our findings: (a) promotion of healing by secondary intention, (b) persistent closure of the wound with the transplanted cells with ‘partial’ take and healing occurring underneath the graft and (c) healing by graft take of the cultured skin equivalent with remodelling of the tissue over time (9).

In other studies, wound healing varied between 26 and 39·9 days after transplantation of allogenic cryopreserved cultured keratinocytes. The size of the ulcer was mainly reduced rapidly within the first to fourth week 10, 11, 12, 13, 14. The authors believe that this is not only because of the release of different cytokines and growth factors by the transplanted keratinocytes but probably also because of deposition of extracellular matrix and basement membrane components.

Overall in our patients, we noticed an average decrease in the wound area by 55%. This implies that approximately three to four applications are needed for complete wound closure if the ulcerations react to the treatment and forces healing. With multiple applications of keratinocytes, wound area could be epithelised completely. In our study, we applied keratinocytes only once or twice, implying that our results could have been better with more repeated applications 15, 16.

It makes sense to control the ulceration following autologous keratinocyte transplantation at the beginning of the treatment between fifth and tenth day to evaluate if keratinocyte transplantation will be successful and after reaching the maximum growth period (median time between the days 25 and 30). Assuming this medium growth period of 25 days, complete wound closure can be achieved in about 100 days, including a consolidation phase between the single treatments. Three to four applications will be necessary for complete wound closure if the ulceration react to the treatment.

Minimal wound area following autologous keratinocyte transplantation depends also on ulcer duration: if there has been a long duration of the treated ulcer, the growth period of the transplanted autologous keratinocytes is reduced and stops earlier. This supports the notion that frequent autologous keratinocyte transplantation at an early stage could have much more effect and better results.

Conclusions

Our mathematical model describes very well wound healing following transplantation of autologous keratinocytes, which is dependent on time. We think our model is enormous useful because it provides an equation for wound healing trajectories, and so it is possible to retrospectively or prospectively determine exactly the moment of wound closure or the healing rate. These interesting details we found in this initial project are worth to be investigated further in new studies.