Letters to the Editor July 2003

Dear Editor: Regarding the article, Healing Rate as a Prognostic Indicator of Complete Healing: A Reappraisal, in the March, 2003, issue of WOUNDS, may I say that a rather simple mathematical approach I published in Dermatologic Surgery [1997;23:1219–25] entails Gilman’s and other workers’ empirical results from a theoretical approach. Wounds heal by secondary intention from the edges, and so irrespective of the shape of the wound the rate of change in the area of the wound is proprotional to the area, since there are more cells involved in healing in larger wounds. Thus, we may write: dA/dT = kA, where A = area, t is the time, and k is a healing constant. Solution of this differential equation is easy. Rearrange the equation to read dA/A = kdt and integrate both sides of the equation giving us (1) lnA = kt, where ‘ln’ is the log to the base e=2.71828. This says that the total time for healing is proportional to the log of the area, a well-known empirical rule. The constant in this proportion is the healing constant, k. Taking the antilog of both sides of equation ‘1’ gives us (2) At = A0 e -kt, where At is the area of the wound at some time ‘t,’ and A0 is the area of the wound at the outset. Solving this equation for ‘k,’ the constant of healing, we obtain, (3) [ln (A0/At)]/[t] = k. Equation ‘3’ tells us to: 1) measure the area of the wound at the outset, then at some time ‘t’ in the future, and then take the log (ln) of this ratio; and 2) divide this value by the time over which the measurement is made. The result is healing constant, k, a measure of the overall propensity of the wound to heal. Over shorter healing periods ‘k’ can be regarded as a prognostic indicator for the total healing time of the wound in question. Now the area of the largest circle, Ac, within an irregularly shaped wound determines the outside boundary conditions for healing—the longest overall time for the wound to heal. Thus, we can measure the area of the wound at different times by using Ac measured at different times. The area can, therefore, be measured by simple means in a clinical setting. All of this reduces to the following steps: 1. Measure Ac at two times, say four weeks apart. 2. Take the log to the base ‘e’ of the ratio of the earlier area divided by the later area. 3. Divide this number by the total time (four weeks). 4. The result is the value for the healing constant, ‘k.’ The higher the value of ‘k’ the faster the wound is healing. If ‘k’ is calculated over a relatively short period of time, one can determine the prognosis for healing early in the treatment. Repeated measurements of ‘k’ can also be averaged and used directly to gauge the success of any treatment regimen. Marshall P. Goldberg, MD Laguna Niguel, California Dear Editor: I would like to offer some comments and corrections relative to a recent article entitled "Healing Rate as a Prognostic Indicator of Complete Healing: A Reappraisal", published in WOUNDS in March, 2003 [2003; 15(3):71-76].[1] This article has several errors and misstatements that I believe are not helpful to readers who are trying to understand the best way to measure healing progress. The most important errors in this article, I believe, are misstatements of equations for healing progress in terms of wound measurements. The article, in Table 1, mistakenly presents “Gilman’s transformed equation:” as: {(Aa-Ab)/[(Pa-Pb)/2]} / (b-a). The subtraction of the two perimeters should actually be a sum. The correct equation of this form is: {(Aa-Ab)/[(Pa+Pb)/2]} / (b-a). Similarly, the authors misquote an equation from a previous publication.[3] They say that Bulstrode, et al., “To assess epithelial migration, …divided the change in wound area by the change in wound perimeter…dA/dP.”[1] In fact, Bulstrode used the perimeter sum in the denominator, and probably used “…the mean perimeter…”3 as denominator. This is an important point, because any parameter that has the perimeter difference in the denominator will be smaller when the perimeter difference is larger. So, if the perimeter is reduced a great deal due to a great deal of wound healing progress, this erroneous parameter will actually be small, rather than large. It will not reflect wound progress, but just the opposite. The authors go on to say that this dA/dP parameter was used in another previously published analysis of clinical data.[4] They then discuss the value of DA/DP versus “Gilman’s transformed equation.” However, the other previously published article makes no mention of using dA/dP.[4] This article is unclear as to how healing was measured, as it states that an area difference was used, but the units are linear, not area. A minor error in the present review article is a repeat of an error first published by Tallman, et. al.,[5] the idea that “Gilman’s equation” measures the healing progress at each visit in comparison to the initial ulcer size on Day 0. In fact, my publication did not specify any particular time period over which progress should be assessed, but did encourage investigators to use the shortest possible time span when calculating the linear parameter.[2] I disagree with the conclusion of the Donohue and Falanga article that there are several different methods for assessing epithelial migration which are of comparable value. I believe there is great value in standardizing on the linear parameter first used by Hopkins and Jamieson[6] and proven by me to be a valid measure of wound progress toward the wound center, for wounds of any size or shape.[2] This parameter, used with the proper measure of wound size, should be able to make predictive statements about time to complete healing for individual wounds.[7] I do respect and appreciate the contributions Dr. Falanga and his co-authors have made to this area. I regret that this paper contains these errors and would welcome the authors’ response. Thomas Gilman, PhD Principal Scientist Hollister, Inc. Libertyville, Illinois References 1. Donohue K, Falanga V. Healing rate as a prognostic indicator of complete healing: A reappraisal. Wounds 2003;15:71–6. 2. Gilman T. Parameter for measurement of wound closure. Wounds 1990;2:95–101. 3. Bulstrode CJ, Goode AW, Scott PJ. Measurement and prediction of progress in delayed wound healing. J R Soc Med 1987;80:210–2. 4. Falanga V, Sabolinski M. Prognostic factors for healing of venous and diabetic ulcers. Wounds 2000;12:42A–6A. 5. Tallman P, Muscare E, Carson P, et al. Initial rate of healing predicts complete healing of venous ulcers. Arch Dermatol 1997;133:1231–4. 6. Hopkins NF, Jamieson CW. Diffusion barriers in venous ulceration. J R Soc Med 1985;78:355–7. 7. Gilman TH. Letter to Editor. Br J Dermatol 2003;In press. Authors’ Response: Dear Editor: We are pleased to see the renewed interest this review has generated. Obviously, we regret the typos in the “Gilman transformed equation” and in describing the method from the Bulstrode article, which bypassed our proofreading of the manuscript; we thank Dr. Gilman for pointing out these errors. Certainly, however, we expect that investigators wishing to utilize these equations would refer to the original publications. Determination of the use of dA/dP in the Falanga and Sabolinski article was based on an abstract of theirs that detailed a portion of this work.[1] In reviewing again the article in light of the points made by the letters to the editor, we still feel correct about our conclusions, and in particular that at this point no method can definitely be said to be superior. Of course, one can get very much involved with the mathematics of these descriptions. However, it must be kept in mind that we are dealing with a complex biological system. Moreover, to properly examine the value of these and other methods to predict wound closure, they should be employed on a prospective basis. This has not been done on a large scale. We do have data developed on a prospective basis, but it has not been published yet. Kevin G. Donohue, MD Clinical Research Fellow Department of Dermatology and Skin Surgery Roger Williams Medical Center Providence, Rhode Island Vincent Falanga, MD Professor of Dermatology and Biochemistry Boston University Chairman of Dermatology Roger Williams Medical Center Providence, Rhode Island References 1. Sabolinski M, Falanga V, Giovino K, Toole T. Mean healing rates at four weeks can predict complete wound closure of diabetic foot ulcers. J Invest Dermatol 2001;117:544.