ORIGINAL ARTICLE

A Mathematical Model for Decreasing the Surface Area ofSurgical Excisions

SAMANTHA DAVIDSON, MBBS, AND KARYN LUN, MBBS, FACD, MACMS*

BACKGROUND Repair of lower extremity excision defects poses a surgical challenge, and as a result,split-thickness skin grafting is often used to close large defects. By minimizing the size of the defect, asmaller graft can be used, which may translate into improvements in wound healing and the aesthetic out-come.

OBJECTIVE To demonstrate, using a mathematical model, how to decrease the surface area of excisionson lower extremities requiring split-thickness skin grafting.

METHODS Four patients had cutaneous neoplasms excised from their lower legs. The resulting defectsunderwent partial primary closure with removal of Burrow’s triangle. The new dimensions of the defectwere recorded, and the surface area of the pre- and postprimary closure was calculated.

RESULTS Modest decreases in the dimensions of the ovoid–ellipsoid defect translated to large decreasesin the surface area requiring split-thickness skin graft repair.

CONCLUSION Using a mathematical model, we quantified how it is possible to decrease the size of anexcision site. This reduction in surface area may translate to benefits in a postoperative outcomes.

The authors have indicated no significant interest with commercial supporters.

Excisions on the lower extremities commonly

present a surgical challenge. Defects too large

to repair primarily are often closed using a split-

thickness skin graft (STSG). STSGs provide good

coverage of a wound that lacks an adequately vas-

cularized base, as is commonly seen on the leg.1

Although this method is reasonable, it may result

in a longer healing time and a less-appealing aes-

thetic appearance than direct closure. Healing time

depends on a multitude of factors, including vascu-

lar supply, surrounding skin integrity, medical

comorbidities, and most importantly, the size of

the defect.2

To excise a lesion completely and minimize the

defect area on the lower leg, we have demonstrated

a method combining partial primary closure and

STSG. A mathematical model that shows how

modest decreases in the length and width of the

defect dramatically alter the surface area of the

defect and subsequently the area requiring grafting

supports this technique.

Methods

Four patients with nonmelanoma skin cancer on

the lower leg underwent excision and repair

under local anesthesia. Lesions were basal cell

carcinoma (two patients) and squamous cell carci-

noma (two patients). Each lesion was excised

with a margin of 4–6 mm, with resultant ellip-

soid defects. The lesions chosen for excision and

grafting were considered to be too large for pri-

mary closure, and the site and size of the lesions

made repair using a full-thickness skin graft

unsuitable.

*Both authors are affiliated with the Queensland Institute of Dermatology, Greenslopes Private Hospital,Brisbane, Qld, Australia

© 2012 by the American Society for Dermatologic Surgery, Inc. � Published by Wiley Periodicals, Inc. �ISSN: 1076-0512 � Dermatol Surg 2012;1–5 � DOI: 10.1111/j.1524-4725.2012.02363.x

1

After removal of the lesions, the length and width

of the defect were measured in millimeters. A small

amount of undermining was performed, and each

pole of the ellipse was sutured, creating a dog ear,3

which was excised and closed using subcutaneous

absorbable and external nonabsorbable sutures.

Dimensions were recorded after each dog-ear

repair (Table 1).

For each patient, a rectangular graft site was pre-

pared on the upper thigh of the ipsilateral leg. The

length and width of the site was planned to be

10 mm greater than that of the recipient site. The

STSG was harvested using a sterile technique using

a 0.010-G Weck knife, sutured into the recipient

site using silk sutures, and secured using a tie

over bolster dressing. Sutures were removed after

10–14 days. The donor site was also dressed using

an absorbent, nonadhesive dressing.

Using a mathematical model and integration tech-

niques, we calculated the surface area of a true

ellipse, which closely corresponded to the shapes of

the defects.

Results

When excising skin lesions, the resulting defect is

often an ellipsoid or ovoid shape because of the

presence of relaxed skin tension lines. Although

surgical defects are not an exact geometric shape,

it is reasonable to use the equation for calculating

the surface area of the ellipse to estimate the size

of the defect.4 This can be seen graphically in

Figures 1A–C. The equation for plotting of an

ellipse (Figure 2) is

x2

a2þ y2

b2¼ 1;

with a and b being the horizontal and vertical

axes, respectively. Solving for y and integrating this

function gives the area under the curve in one

quadrant (as highlighted in Figure 2), and multi-

plying by four gives the total area of an ellipse.5

Solving for y

y ¼ � b

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � x2

p

Integrating using a trigonometric substitution

(x = asinh)

Za

0

b

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � x2

pdx ¼ abp

4

Multiplying by 4, because this is only representa-

tive of one-quarter of the ellipse, gives

Area ¼ pab

Major and minor radii (a and b) are equal to half

the major and minor diameters (length, L and

width, W), so

TABLE 1. Dimensions and Surface Area for Four Patients

Patient

Length

(mm)

Width

(mm)

Area

(mm2)

Decrease

in Length

(9100 = %)

Decrease

in Width

(9100 = %)

Decrease

in Area

(9100 = %)

1 43 31 1046.94 0.12 0.23 0.32

38 24 716.28 0.05 0.29 0.33

36 17 480.66 0.16 0.45 0.54

2 36 35 989.60 0.06 0.17 0.22

34 29 774.40 0.06 0.28 0.32

32 21 527.79 0.11 0.40 0.47

3 38 28 835.66 0.11 0.21 0.30

34 22 587.48 0.03 0.32 0.34

33 15 388.77 0.13 0.46 0.53

4 35 30 824.67 0.06 0.17 0.21

33 25 647.95 0.09 0.24 0.31

30 19 447.68 0.14 0.37 0.46

SURFACE AREA OF SURGICAL EXCIS IONS

DERMATOLOGIC SURGERY2

pab ¼ pL

2�W

2:

Figures 1A–C demonstrate patient 3’s excisions

with the plotted ellipses overlying them. The

photographs have been scaled to real dimensions,

and each gridline represents 5 mm. It can be seen

in the diagrams that the ellipses drawn using the

equation

x2

a2þ y2

b2¼ 1

are representative of the sizes of the excisions in

our patients.

Table 1 displays all of the dimensions of the

lesions, including their length and width, and the

surface area as calculated using the formula for an

ellipse

SA ¼ pL

2�W

2

(SA = surface area in mm2, L = length and

W = width, both in mm).

The absolute and percentage change in length,

width, and surface area have also been calculated.

Because both dimensions (length and width) of an

ellipse may be altered independently of the other,

their relationship with surface area needs to be

plotted on a three-dimensional graph. Table 2

shows the surface area, which corresponds to vari-

ous measurements typical of excisions on a lower

extremity, and Figure 3 demonstrates the relation-

ship between length, width, and surface area.

The measurements and calculations taken from our

four patients show that we were able to decrease

the length of the defects by an average of 13.7%

(A)

(B)

(C)

Figure 1. (A) Initial excision with graph of ellipse. (B) Initial“dog ear” or “Burrows” trianglewith overlyinggraph. (C) Sec-ond “dog ear” or “Burrows” triangle with overlying graph.

Figure 2. Plot of an ellipse.

DAVIDSON AND LUN

2012 3

and the width by an average of 42.1%. These

changes equate to a mean surface area reduction of

50.0%.

The amount that the length or width can be reduced

is partly operator dependent and also depends on

the tissue movement available and the strength of

the dermis. In patient 3, we were able to reduce the

length 13% and the width 46%, correlating to a

surface area reduction of 53%, although in patient

2, we were able to decrease the length dimension by

only 11% and the width by 40%, corresponding to

a surface area reduction of 47%.

Discussion

For large surgical defects on the lower limb, which

often have poor vascular supply, impaired sur-

rounding skin integrity, and minimal tissue move-

ment, STSGs provide rapid wound coverage.1

STSGs also have better survival characteristics than

full thickness-skin grafts because they do not con-

tain a full-thickness dermis and related adnexal

structures.4 Although the STSG has many benefits,

it is often considered as a last resort in wound

repair because it is the least durable form of

wound closure, and contraction and dyspigmenta-

tion commonly occur at the donor and recipient

sites.1 In addition, the donor site creates a signifi-

cant wound, requiring postoperative care.

The benefits of primary closure, as opposed to any

type of grafting, include shorter healing time,

better cosmesis, less risk of infection, and superior

scar strength.

Healing time of ulcers has been shown to be

related to size,2 and this principle may be trans-

TABLE 2. Surface Area (mm2) Calculated for Variable Lengths (mm) and Widths (mm)

Length

Width

15 20 25 30 35 40 45 50

15 176.71 235.62 294.52 353.43 412.33 471.24 530.14 589.05

20 235.62 314.16 392.70 471.24 549.78 628.32 706.86 785.40

25 294.52 392.70 490.87 589.05 687.22 785.40 883.57 981.75

30 353.43 471.24 589.05 706.86 824.67 942.48 1060.29 1178.10

35 412.33 549.78 687.22 824.67 962.11 1099.56 1237.00 1374.45

40 471.24 628.32 785.40 942.48 1099.56 1256.64 1413.72 1570.80

45 530.14 706.86 883.57 1060.29 1237.00 1413.72 1590.43 1767.15

50 589.05 785.40 981.75 1178.10 1374.45 1570.80 1767.15 1963.50

Figure 3. Surface area of an ellipse.

SURFACE AREA OF SURGICAL EXCIS IONS

DERMATOLOGIC SURGERY4

lated to surgical defects. By decreasing the surface

area of the defect with primary repair, we effec-

tively minimize the area that requires grafting and

thus improve postoperative outcomes.

We have quantified how altering the dimensions of

an ellipsoid excision equates to a change in surface

area. The results (Table 1) of our four patients

demonstrate how decreasing the dimensions by

only small increments corresponds to large percent-

age changes in area. The results obtained in this

study have shown surface area reductions of

greater than 53% in two of the cases and a mean

reduction of 50.0% overall (Table 3).

This simple technique, which uses partial primary

closure to minimize the size of the recipient graft

site, is easy to use as an adjunct to STSGs and to

reduce the size of a defect on a lower limb and

improve postsurgical outcomes. All of the patients

in this study had STSGs after partial primary clo-

sure of a defect on the lower leg, although this

technique can be used on other areas of the body

in conjunction with other closure methods.

Excision of cutaneous lesions results in an ellipsoid

or ovoid shape because of the radial growth pattern

of cutaneous neoplasms and the presence of relaxed

skin tension lines. Although the resultant defect is

not an exact geometric shape, it can be seen from

our figures with overlying graphs that the excisions

follow the plot of an ellipse closely. Furthermore,

by overlaying the graph on the photograph of our

excisions, we can demonstrate graphically and

mathematically the accuracy of our calculations.

The aim of this study was not to identify the bene-

fits of decreasing the size of a surgical defect but to

quantify by how much this can be done. Much

research has gone into the effects of wound size

and other comorbidities and local factors on heal-

ing time. We have quantified how much we are

able to reduce the surface area of a defect site

using a simple mathematical model and have been

able to demonstrate graphically the accuracy of

this method. It would be useful in future to evalu-

ate to what extent this benefits patients’ after care

in terms of healing time, infection rates, and cos-

mesis.

References

1. Adams DC, Ramsey ML. Grafts in dermatologic surgery: review

and update on full- and split-thickness skin grafts, free cartilage

grafts, and composite grafts. Dermatol Surg 2005;31:1055–67.

2. Moffatt CJ, Doherty DC, Smithdalea R, Franks PJ. Clinical

predictors of leg ulcer healing. Brit J Dermatol 2010;1:51–8.

3. De Giorgi V, Mannone F, Quercioli E, Giannotti V, et al. Dog-

ears: a useful artifice in the closure of extensive wounds. J Eur

Acad Dermatol Venereol 2003;17(5):572–4.

4. Qian W, Mei C, Yu-Le W, Guo-Cheng Z. Mathematical guide to

minimize donor size in full-thickness skin grafting. Dermatol

Surg 2009;35(9):1364–7.

5. Garner W. Area of an Ellipse; 2008. Available from: http://math.

ucsd.edu/~wgarner/math10b/area_ellipse.htm. Accessed

November 12, 2011.

Address correspondence and reprint requests to:Samantha Davidson, MBBS, Queensland Institute ofDermatology, Greenslopes Private Hospital, Brisbane,Qld, Australia, or e-mail: samjdavidson@gmail.com

TABLE 3. Average Change in Dimensions for Four

Patients

Dimension Average Change (%)

Length (cm) 13.7

Width (cm) 42.1

Area (cm2) 50.0

DAVIDSON AND LUN

2012 5