The Impact of the Wound Shape on Wound Healing Dynamics: Is it

Time to Revisit Wound Healing Measures?

Gennadi Saiko

1,2 a

Swift Medical Inc., 1 Richmond St. W., Toronto, Canada

Ryerson University, Toronto, Canada

Keywords: Wound, Wound Healing. Epithelisation, Wound Measurements, Planimetry.

Abstract: Introduction: Wound healing is a multifaceted process, which can be impacted by many endogenous (e.g.,

compromised immune system, limited blood supply) or exogenous (e.g., dressing, presence of infection)

factors. An essential step in wound management is to track wound healing progress. It typically includes

tracking the wound size (length, width, and area). The wound area is the most often used indicator in wound

management. In particular, wound closure is the single parameter used by the FDA to measure wound

therapeutics' efficiency. Here, we present some arguments on why the wound area alone is insufficient to

predict wound healing progress. Methods: We have developed an analytical approach to characterize an

epithelization process based on the wound's area and perimeter. Results: We have obtained the explicit results

for wound healing trajectory for several wound shapes: round (2D), elongated cut (1D), and rectangular. The

results can be extended to complex shapes. Conclusions: From geometrical considerations, the wound healing

time is determined by the shortest dimension (the width) of the wound. However, within that time, the wound

healing trajectory can be different. Our calculations show that the shape of the wound may have significant

implications on a wound healing trajectory. For example, in the middle of the wound healing process (t/T=0.5),

the 1D wound model predicts 50% closure, while the 2D model predicts 75% closure (25% remaining). These

considerations can be helpful while analyzing clinical data or designing clinical or pre-clinical experiments.

1 INTRODUCTION

Wound healing is a multifaceted process, which can

be impacted by many endogenous (e.g., compromised

immune system, limited blood supply) or exogenous

(e.g., dressing, presence of infection) factors.

Successful acute wound healing depends on

orderly progression through four phases: hemostasis,

inflammation, proliferation, and remodeling or

maturation. During hemostasis and the early

inflammatory phase, platelets and inflammatory cells

migrate into the wound bed. During the inflammatory

phase, neutrophils enter the wound (Diegelmann,

2004), followed by macrophages that are responsible

for neutrophil and damaged matrix removal

(Meszaros, 2000). During the proliferative phase, the

migration of fibroblasts and keratinocytes into the

wound occurs. Keratinocytes cause re-epithelization

of the wound. Fibroblasts produce collagen and other

extracellular matrix (ECM) proteins necessary for

granulation tissue formation. During the final phase

https://orcid.org/0000-0002-5697-7609

of wound remodeling (which takes months and

years), collagen deposition continues, and collagen

III is gradually replaced with collagen I (Xue, 2015).

The emphasis of the current work is the

proliferation phase. The sign of the successful

proliferation is the re-epithelization of the tissue,

which occurs due to keratinocytes' migration.

An important aspect of wound management is to

track wound healing progress. Geometrical wound

measurements (length, width, and depth) are essential

tools in wound care armamentarium. In particular,

wound closure is the single parameter used by the

FDA to measure wound therapeutics' efficiency.

The geometrical wound measurements typically

are being performed manually, using a ruler. There

are two primary methods used for wound

measurements (Swezey, 2014):

- “Greatest length and width method: In this

method, the greatest length and the greatest width of

the wound are measured across the wound's diameter,

from wound edge to the opposite wound edge.

182

Saiko, G.

The Impact of the Wound Shape on Wound Healing Dynamics: Is it Time to Revisit Wound Healing Measures?.

DOI: 10.5220/0010337601820187

In Proceedings of the 14th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2021) - Volume 2: BIOIMAGING, pages 182-187

ISBN: 978-989-758-490-9

- Clock method: In this method, the face of a clock

is used to guide measurement. The 12:00 reference

position is towards the head of the body, and

measurements are obtained from 12:00 to 6:00

(length) and from 9:00 to 3:00 (width).” Notice that

the width can be larger than the length in this case.

However, only the length, L, and width, W, can

be determined using these methods. The surface area

of the wound, S in this case, can be estimated as

S=LxW, which is a very rough approximation that

does not take into account the shape of the wound.

This type of calculation has been shown to

overestimate the wound area by 10% to 44%

(Goldman, 2002), with accuracy decreasing as wound

size increases (Majeske, 1992). Thus, tracking the

wound healing progress in time manually will be

relatively imprecise. The accuracy of wound

measurements can be increased with digital

photography. In this case, the precise wound area can

be calculated in addition to the more accurate length

and width measurements. Note that it can also be

done with wound tracing on a sterile sheet or film

(Langemo, 1998). However, it is a very labor-

intensive process. In particular, digital technology

may lead to a 10x increase in the accuracy of wound

measurements. However, the initial implementation

of such techniques using DSL cameras got limited

clinical traction, primarily due to the significant extra

time required to take pictures using specialized

equipment. Thus, this process did not fit well in a

busy clinical workflow. With the advent of

smartphones, wound management was

revolutionized. The ability to capture an image and

annotate the wound using the same tool significantly

improved the overall wound management workflow.

Many wound healing measures are proposed (a

brief overview can be found in (Cukjati, 2001)).

However, most commonly, a measure of the change

in wound area is used and is expressed either as a raw

number (cm

) or as a percentage of the initial wound

area. The wound area, S, is an important clinical

indicator of the wound status and can be used to

predict wound healing progress and clinical

outcomes. In particular, S is a part of several wound

indices (e.g., PUSH score for pressure injuries).

However, it is known that the wound healing rate

expressed as absolute area healed per day tends to

exaggerate larger wounds' healing rates. Similarly,

the healing rate expressed as a percentage of initial

area healed per day tends to exaggerate smaller

wounds' healing rates (Cukjati, 2001). Thus, more

objective methods need to be adopted into clinical

use.

Here we present some arguments why wound area

alone is not sufficient to predict wound healing

progress. The shape of the wound (e.g., round,

complex, or elongated) also play an essential role in

wound healing dynamics. In particular, we argue that

wound healing for quasi-2D (round shape) wounds

will have significantly different dynamics from quasi-

1D wounds (e.g., cuts). Thus, objective wound

healing measures need to account for the wound

shape.

2 METHODS

Let's consider a wound with an arbitrary shape. It can

be characterized by the area S and the perimeter P. If

the epithelization starts from the wound margin, and

keratinocytes propagate on the distance dx, then in the

first approximation, we can write

𝑑𝑆  𝑃𝑑𝑥

(1)

Now, let's consider this process in more detail. In

particular, we will be interested in how the shape of

the wound affects wound healing dynamics.

2.1 Lattice Model

To elucidate the details, we will consider a lattice

model of the wound. The epithelized tissue bounds

the wound. During each time interval



t, the wound

wall propagates inwards by



x=v



t, where v is the

wound closure speed. For practical reasons, the time

interval



t can be set to one day.

If we consider the wound of an arbitrary shape,

then the wall propagation dynamics will be different

in different parts of the curve. For the square lattice,

there are three cases to consider: a convex angle

(Figure 1), a concave (or reflex) angle (Figure 2), and

a flat segment (Figure 3).

Figure 1: Healing dynamics in the case of a convex

segment.

The Impact of the Wound Shape on Wound Healing Dynamics: Is it Time to Revisit Wound Healing Measures?

183

Figure 2: Healing dynamics in the case of a concave

segment.

Suppose we calculate the perimeter at each step of the

wall propagation. In that case, one can see that the

perimeter (and according to Eq.1, the rate of wound

closure) increases at reflex segments (Figure 2),

decreases at convex segments (Figure 1), and stays

constant for the flat segments (Figure 3).

Figure 3: Healing dynamics in the case of two parallel walls.

Thus, intuitively, the wound healing rate behaves

very differently at various points of the closed curve

(wound margin).

2.2 Wound Closure Rate

Let's discuss what the wound closure speed, v is. The

naïve idea would be to link this speed with the

migration rate of keratinocytes. According to Tao et

al. (Tao, 2007), the keratinocytes migration rate is 10-

20 m/h in vitro. However, to have a viable epithelial

layer, it needs to be supplied with oxygen and

nutrients. In particular, to have adequate oxygen

supply, the oxygen diffusion length should not exceed

200m. Thus, the tissue underneath a new epithelium

needs to be vascularized, which may include

angiogenesis and collagen matrix deposition by

fibroblasts. Without an appropriate collagen scaffold,

the vascular network will collapse.

Consequently, we can consider the re-

epithelization process as three processes run in

parallel: 1) keratinocytes migration, 2)

revascularization via angiogenesis beneath a newly

epithelized surface, and 3) fibroblast migration and

collagen deposition. Based on this model, we can

estimate the wound closing speed, v. Based on the

maximal oxygen diffusion length (L

=200m), we

can expect that the epithelial layer cannot overtake

the new vasculature by much more than L

. Thus, we

can expect that the wound closing rate, v, will be

equal to the smallest of three values: keratinocytes

migration rate, angiogenesis rate, and fibroblast

migration rate.

Epithelization happens through two primary

mechanisms: crawling of cells on the substratum and

constriction of a supracellular actomyosin cable at the

edge of the gap in a purse-string-like mechanism

(Klarlund, 2012). In the presence of ECM, crawling

is a predominant mechanism. However, in the general

case, wound closure is an interplay of these two

mechanisms (Ravasio, 2015). In (Ravasio, 2015), it

was also found that the shape of the wound affects the

epithelization rate. These two mechanisms act in the

same direction near a convex surface; thus, the

closure rate is higher (up to 75 m/h for MDCK

(Madin-Darby canine kidney) cells). Near a concave

surface, these two mechanisms act in opposite

directions. Thus the closure speed is lower (close to 0

m/h for MDCK cells).

Angiogenesis is a multistage process by itself.

According to Felmeden et al. (Felmeden, 2003),

angiogenesis consists of 7 distinct phases (endothelial

cell and pericyte activation, degradation of the

basement membrane, migration of endothelial cells

(e.g., sprouting), proliferation of endothelial cells,

differentiation of endothelial cells, and reconstitution

of basement membrane). The primary factors, which

drive the angiogenesis and the morphology of the

newly developed vascular network are proliferation

rate (PR) and migration rate (MR) of endothelial

cells. The overall growth of vasculature is a result of

both proliferation and migration. They were

researched intensively in (Norton, 2016) numerically,

where the authors found that to get normally

vascularized tissue, these parameters must be close to

PR=0.025 1/h and MR=10-16 m/h. These values

agree with experimental observations that doubling

times for microvascular endothelial cells range from

about 12 hours to 4 days (Anagnostou, 1990). Based

on these values, we can conclude that assuming a

sufficient supply of endothelial cells, the endothelial

cell migration rate will limit the vascularization rate.

Subsequently, the wound healing rate, v, will be

limited by a smaller value (vascularisation rate in this

case), which is 10-16 m/h=240-380m/d. Note that

it is hard to expect that the vascularization rate will be

affected by the wound curvature.

This estimation is very close to experimental data

on humans and animal models. For example, the

wound closure rate can be estimated as 0.25mm/d in

the rat model (Rong, 2019).

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3 RESULTS

3.1 2D Case

Let's consider the 2D case: a round wound with the

radius r. In this case, the perimeter 𝑃2𝜋𝑟

2𝜋

/

𝑆

/

. If we substitute this expression in Eq.1

and notice that dx=vdt, we can write

𝑑𝑆  2𝜋

/

𝑆

/

𝑣𝑑𝑡

(2)

We can divide both sides of the equation by 2𝑆

/

and integrate them



𝑆





√

𝑆 𝜋

/

𝑣𝑡

(3)

Here, S

is the wound area at the moment t=0.

From Eq.3, we can find the explicit expression for the

wound area, S

𝑆



𝑆





√

𝜋𝑣𝑡



(4)

The wound healing time will be 𝑇



𝑆

/𝜋

/2𝑣 

𝑊/2𝑣

3.2 1D Case

Let's consider the 1D case: a long rectangular cut,

where the length of the cut L is much bigger than its

width, W. In this case, the perimeter P=2L+2W

≅

2L.

The wound area is S=LW. From this expression, one

may deduce that P=2S/W. However, W is not

constant. It decreases over time. Thus, this expression

is not useful for calculations. Instead, we may notice

that the perimeter P is approximately constant during

wound healing. Therefore, we can write P=2L instead

(note that P~S

). If we substitute this expression in

Eq.1 and notice that dx=vdt, we can write

𝑑𝑆  2𝐿𝑣𝑑𝑡

(5)

If we integrate both parts and reshuffle terms, we

will get

𝑆  𝑆



2𝐿𝑣𝑡

(6)

The wound healing time will be 𝑇

𝑆

/2𝐿𝑣 

𝑊/2𝑣

3.3 The Generalization to Simple

Shapes

For a rectangular wound, we can find an exact

solution to the wound closure problem. If we consider

the rectangle with the length L and width W (𝐿𝑊)

and the wall is moving with the speed v, then the

unclosed area of the wound at time t will be

𝑆



𝐿2𝑣𝑡



𝑊2𝑣𝑡





𝑊





𝐿/𝑊  2𝑣𝑡/𝑊



12𝑣𝑡/𝑊



(7)

for 0𝑡𝑊/2𝑣.

If we introduce the wound healing time 𝑇𝑊/2𝑣

and normalized time 𝜏𝑡/𝑇 then we can obtain an

expression for the normalized surface area s:

𝑠

𝑆







1𝜏𝑊/𝐿



1𝜏



(8)

Figure 4: Wound healing trajectories for a rectangular

wound for various values of W/L ratios: 0 (red curve), 0.3

(green curve), 0.6 (purple curve), and 1 (blue curve).

In Figure 4, one can see several wound healing

scenarios for various W/L ratios.

The case W/L->0 corresponds to the 1D case

considered above. If W/L=1, then it is a square

wound, which is very similar to the round wound

considered in the 2D section.

3.4 The Generalization to Complex

Shapes

2D and 1D cases can be generalized differently if we

consider the wound margin as a fractal curve with a

dimension d

, where 1𝑑



2. One can see that the

perimeter P can be expressed through wound area S

𝑃𝑎𝑆









(9)

where a is a constant. In particular, this expression

holds in the case of d=2 and d=1. This expression can

be used to calculate d

for a specific curve. We need

to visualize the curve at several pixel sizes, calculate

P and S for each of them, and plot these values against

each other (P vs. S). Then, the parameter a and the

fractal dimension d

can be found numerically using

curve fitting.

If we substitute Eq.9 in Eq.1, divide both parts on

𝑆





/

and integrate them, we will get

𝑆









𝑆









3𝑑



𝑎𝑣𝑡

(10)

Thus, we can obtain an explicit expression for the

area S at any given time t

The Impact of the Wound Shape on Wound Healing Dynamics: Is it Time to Revisit Wound Healing Measures?

185

𝑆𝑆











3𝑑



𝑎𝑣𝑡







(11)

This expression holds for d=2 (see Eq.4) and d=1

(see Eq.6). Similarly, we can get an expression for the

normalized area s:

Figure 5: Wound healing scenarios for various fractal

dimensions d

: 1 (red curve), 1.3 (green curve), 1.6 (purple

curve), and 2 (blue curve).

𝑠

𝑆



1𝜏







(12)

In Figure 5, one can see several wound healing

scenarios for various fractal dimensions d

. Note that

the wound healing time, in this case, will be

𝑇2𝑆









/𝑎𝑣3  𝑑

(13)

4 DISCUSSION

From simple geometrical considerations, one can see

that the wound healing time is determined by the

shortest dimension (the width) of the wound. That is

why the surgical closure of the wound (wherever

possible) is the best strategy. However, within that

time, the wound healing trajectory can be different.

Our simple calculations show that the wound's shape

may have significant implications on a wound healing

trajectory. For example, in the middle of the wound

healing process (t/T=0.5), the 1D wound model

predicts 50% closure, while the 2D model predicts

75% closure (25% remaining).

We have proposed two approaches to account for

the wound shape: a) based on the W/L ratio and b)

based on the fractal dimension. Both methods do not

require any clinical workflow changes if the wound

were measured using digital photography.

The rectangular wound model can be easily

extended to any elliptical shape. In this case, the

wound trajectory (Eq.8) will be the same. The fractal

model can be extended to more complex shapes,

which include concave segments. Thus, the models

are complementary, and the combination of these two

models covers the broad range of wound shapes.

We also found that the wound perimeter is a fairly

important factor in the wound healing process. For

example, the perimeter can be linked with the wound

area through a fractal dimension of the shape. Note

that the "fractal dimension" term is used quite loosely

here. We don't expect wound shape similarity

extended through multiple scales.

The proposed approach also helps identify which

wound healing rate used in clinics is the most

informative. As we mentioned in the introduction, the

current wound healing tracking methods based on

wound area S are imprecise. Partially it can be

explained by the fact that they do not account for the

wound shape. The more relevant metric could be a

linear healing rate D proposed in (Gilman, 1990),

which can be calculated as D=∆S/P from the change

in the area



S and mean perimeter P. It can be seen

that D, which was originally termed as “the advance

of the wound margin toward the wound centre,” is an

experimental measure for the wound closure rate v.

In particular, a study on 49 wounds found (Gorin,

1996) that the linear healing rate does not correlate

with the initial wound area, perimeter, and W/L ratio.

Thus the linear healing rate is independent of the

wound shape. These results were confirmed in

(Cukjati, 2001) on 300 wounds. Therefore, these

results can be considered as experimental

confirmation of the validity of the proposed model.

To validate these theoretical predictions further,

experimental verification is required, particularly for

complex shapes.

There are certain limitations to the proposed

model. Firstly, it was derived under the assumption

that wound epithelization occurs for all wound

perimeter points. From a physiological point of view,

it means that wound healing is not impaired. While it

can be the case for an acute wound, it is not apparent

for chronic wounds. For example, the

revascularization of the ischemic wound can be

impossible without revascularization surgery.

Secondly, the wound healing rate may depend on the

wound thickness. In the case of superficial wounds,

the vascular network may stay almost intact. Thus

only re-epithelization is required. Therefore, the rate

of wound closure is limited by keratinocytes

migration only (up to 20 m/h) and can be higher than

for partial-thickness wounds (10-16 m/h). For the

full-thickness wounds, which require collagen

deposition, the wound closure rate will be even

slower.

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The linear healing rate was assessed in several

studies. Pecoraro et al. (Pecoraro, 1991) found 0.064

mm/day on diabetic foot patients. Margolis et al.

(Margolis, 1993) found 0.093 mm/day on venous

ulcers. Gorin et al. (Gorin, 1996) found a similar

result of 0.11 mm/day on venous ulcers. Cukjati et al.

(Cukjati, 2001) found 0.068 mm/day for the wound

of unknown etiology. All these values are 2-4 times

lower than the angiogenesis-limited healing rate.

Thus, one can expect that these rates are limited to

slower collagen-deposition processes or presence

areas with impaired healing.

These considerations can be helpful while

analyzing clinical data or designing clinical or pre-

clinical experiments.

5 CONCLUSIONS

Wound shape has significant implications on a wound

healing trajectory, which is not taken into account by

metrics currently used for wound healing progress

tracking. Wound area (closure) and wound area

(closure) as a percentage of the initial wound area are

important clinical endpoints. However, they do not

account for wound shape and do not allow an accurate

comparison of different wounds and treatment

methods. With the ubiquity of smartphones and

digital wound measurements, it is time to start

developing more accurate wound healing metrics.

The smallest size of the wound (width) and a linear

wound healing rate can be the basis for such metrics.

REFERENCES

Diegelmann, R.F., Evans, M.C., 2004, Wound Healing: An

Overview of Acute, Fibrotic and Delayed Healing.

Frontiers in Bioscience, 9: 283-289. doi:10.2741/1184

Meszaros, A.J., Reichner, J.S., Albina, J.E., 2000,

Macrophage-induced neutrophil apoptosis. J Immunol.

165(1):435-41. doi: 10.4049/jimmunol.165.1.435.

Xue, M, Jackson, C.J., 2015, Extracellular Matrix

Reorganization During Wound Healing and Its Impact

on Abnormal Scarring. Adv Wound Care. 4(3):119-136.

doi:10.1089/wound.2013.0485

Swezey, L., 2014, Methods and Strategies for Accurate

Wound Measurement. https://www.woundsource.com/

blog/5-techniques-accurate-wound-measurements

Goldman, R.J, Salcido, R., 2002, More than one way to

measure a wound: an overview of tools and techniques.

Adv Skin Wound Care. 15(5):236-43.

Majeske, C., 1992, Reliability of wound surface area

measurements. Phys Ther. 72(2):138-41.

Langemo, D.K, Melland, H., Hanson, D., et al., 1998, Two-

dimensional wound measurement: comparison of 4

techniques. Adv Wound Care 11(7): 337-43.

Cukjati, D., Reberšek, S., Miklavčič, D., 2001, A reliable

method of determining wound healing rate. Med. Biol.

Eng. Comput. 39: 263–271. doi: 10.1007/BF02344811

Tao, H, Berno, A.J, Cox, D.R, et al., 2007, In Vitro Human

Keratinocyte Migration Rates Are Associated with

SNPs in the KRT1 Interval. PLoS ONE 2(8): e697. doi:

10.1371/journal.pone.0000697

Klarlund, J. K., 2012, Dual modes of motility at the leading

edge of migrating epithelial cell sheets. Proc. Natl

Acad. Sci. USA 109: 15799–15804.

Ravasio, A., Cheddadi, I., Chen, T. et al., 2015, Gap

geometry dictates epithelial closure efficiency. Nat

Commun 6:7683.doi: 10.1038/ncomms8683

Felmeden, D.C, Blann, A.D, Lip, G.Y.H, 2003,

Angiogenesis: basic pathophysiology and implications

for disease, European Heart Journal, 24(1): 586–603,

doi: 0.1016/S0195-668X(02)00635-8

Norton, K., Popel, A., 2016, Effects of endothelial cell

proliferation and migration rates in a computational

model of sprouting angiogenesis. Sci Rep 6, 36992. doi:

10.1038/srep36992

Anagnostou, A., Lee, E. S., Kessimian, N., et al. 1990,

Erythropoietin has a mitogenic and positive

chemotactic effect on endothelial cells. Proceedings of

the National Academy of Sciences of the United States

of America 87: 5978–5982.

Rong, X., Chu, W., Zhang, H., et al. 2019, Antler stem cell-

conditioned medium stimulates regenerative wound

healing in rats. Stem Cell Res Ther 10: 326.

doi:10.1186/s13287-019-1457-9

Gilman, T., 1990, Parameter for measurement of wound

closure. Wounds; 2:95-101.

Gorin, D.R, Cordts, P.R, LaMorte, W.W, et al. 1996, The

influence of wound geometry on the measurement of

wound healing rates in clinical trials. J Vasc Surg.

23(3):524-8. doi: 10.1016/s0741-5214(96)80021-8.

Pecoraro, R.E, Ahroni, J.H, Boyko, E.J, et al., 1991,

Chronology and determinants of tissue repair in

diabetic lower extremity ulcers. Diabetes;40:1305-13.

Margolis, D.J., Gross, E.A., Wood, C.R., et al. 1993,

Planimetric rate of healing in venous ulcers of the leg

treated with pressure bandage and hydrocolloid

dressing. J Am Acad Dermatol; 28:418-21.

The Impact of the Wound Shape on Wound Healing Dynamics: Is it Time to Revisit Wound Healing Measures?

187