The Mathematics of Hair Restoration Surgery

Mathematics, in many ways, is the quintessence of life itself. Whether it be the interrelationship of the keys of the piano, the frequencies of the sounds around us, or the speed at which light reaches us from the most distant stars-mathematics seems to find its way into every crevice of life.

On the other hand, it would appear at first thought that the simplistic field of hair restoration surgery could somehow avoid this ever pervasive arithmetical order. But as one enters into this arena, it becomes evident that mathematics controls almost every facet of this procedure, just as it does with all earthly existence.

Be it the donor site, the recipient site, or the vectors of the advancements of scalp reductions and flaps, it becomes obvious relatively quickly that mathematics does, indeed, pervade almost every aspect of hair restoration surgery. To elucidate this realization, let us go through a few of the various subdivisions of surgical hair restoration and explore some of these mathematical relationships.

THE MATHEMATICS OF THE DONOR SITE

This, however, changed with the introduction of strip harvesting.^1-3 Now the surgeon needed to calculate the width and length of the strips to be able to meet the requirements at the recipient site. When initially delving into this approach, I did, as many others: I used intuition to estimate the length and width of the strips. This approach, however, often left the patient with either too many or too few grafts for the given procedure. This rough technique also made it difficult to teach the procedure to new practitioners.

To accurately measure a strip donor harvest, one must first be able to calculate the density of the donor hair in question. This can be accomplished with the help of an otoscope, trichoscope,⁴ or densitometer.⁵ To do this with an otoscope, a 4.0-mm speculum is placed over four shaved areas at the donor site. The number of hairs viewed per each 4.0-mm circle is then recorded, added together, and divided by four. This number represents the donor density (hairs per 4.0-mm circle) in the area to be harvested.

A = TTr²
A = 3.14 x2.0 x2.0 mm
A = 12.56 mm²
A = 13 mm² when it is rounded off

To demonstrate the concept of calculating area based on the hair density, let us take who averages 20 hairs per 13 mm² in the harvested. Let's say the doctor desires:

200 eight-haired grafts (200 X 8) = 1600 hairs
and
100 one-haired grafts (100 X 1) = 100 hairs

This means that a total of 1700 hairs will be needed to accomplish the task at hand. To find how many 13-mm² circles are needed to get the desired number of hairs, we must divide 1700 by the number of hairs per one 13-mm² circle.

1700 / 20 = 85

Eighty-five represents the number of 13-mm² circles needed. In the days of conventional punch-grafting, this would have equaled 85 (4.0-mm) plugs.

Now the total area that is required to be harvested needs to be calculated. We do this by multiplying 85 X 13 mm².

85 X 13 mm² = 1105 mm² = 11.05 cm²*

If the harvest is decided to be 1 cm wide, the harvest will be then be approximately 11 X 1 cm.

If, on the other hand, a quadruple bladed knife is utilized with 3 mm spacers, the total width will be 9 mm or 0.9 cm (3 X .3 cm). The length would then be calculated by dividing 11.05 cm² by 0.9 cm.

11.05 cm² - 0.9 cm² = 12.20 cm²*

The harvest would therefore be approximately 12.2 X 0.9 cm.

MATHEMATICS OF THE RECIPIENT SITE

Figure 1. An acetate grid that is divided into square inches helps the surgeon determine the number of hair grafts required.

*It is usually necessary to add a centimeter or two to the length to account for hairs that may be mutilated by the graft cutters, the graft inserters, or the multiblades themselves.

Figure 2. The above stencil has 20 slits per square inch and utilizes six-haired grafts.	Figure 3. The below stencil has 40 holes per square inch and utilizes three-haired grafts.

Another tool that helps us with our preoperative calculations is a stencil system (Figures 2 and 3),⁶ that allows the surgeon to know exactly how many grafts (and hairs) are going to be transferred into each square inch. This system consists of two stencils that are based on a systematic three-step approach that was described by Brandy⁷ and involve two basic methodologies: one using 20 six-haired grafts/in² (Figure 4), and the other using 40 three-haired grafts/in² (Figure 5). As can be observed, the number of hairs transferred per square inch is the same with each approach:

6 hairs per graft x 20 grafts/in² = 120 hairs/in² 3 hairs per graft X 40 grafts/in² = 120 hairs/in²

Figure 4. A schematic of the systematic three step approach using 20 six-haired grafts per square inch. The number of hairs transferred per square inch is 120 hairs (6 x 20 = 120).

Figure 5. A schematic of the systematic three-step approach using 40 three-haired grafts per square inch. The number of hairs transferred per square inch is 120 hairs (3 x 40 = 120).

If we know the number of grafts (and hairs) that will be transplanted into each square inch, and we also know the number of square inches that need transplanted, we should be able to easily calculate the number of grafts (and hairs) that will be transferred during each session (Figure 6).

Figure 6. The patient with the appropriate areas stenciled immediately before the surgery.

Once we know the number of hairs that are required, we can determine the needed size of the donor site from the donor density calculated with an otoscope or trichoscope. With this added donor site information we can also now convey to the patient the approximate recipient density achievable relative to the donor density (ie, 10%, 40%, 50%, etc).

Once again, let us look at an example to illustrate this concept. A patient presents as a type VI male pattern baldness (Figure 7A). When the donor site is evaluated, he has approximately 700 hairs per square inch. He desires to have the frontal one-half of his baldness ameliorated and would like to achieve approximately 50% of the donor hair density (ie, 350 hairs per square inch). When this area is measured with an acetate grid, the area desired to be corrected calculates out to be 14 square inches. The patient also desires a 0.5-inch hairline zone of one- to three-haired grafts and also refinement of the posterior hairline zone. The frontal hairline measures 8 inches in length.

Figure 7. (A)A patient with type IV alopecia. He has approximately 700 hairs per square inch in the donor site, and desires 50% of the donor density (ie, 350 hairs er square inch). (B)After the first session, the patient has approximately 130 hairs per square inch. (C)After the second session the patient has approximately 260 hairs per square inch. (D)After the third session, the patient has approximately 390 hairs per square inch.

In this case, we must first calculate the area of the 0.5-inch-wide feathered zone. If we are using the micropattern stencil for feathering, there will be 40 threehaired grafts per square inch in that area. We should thus be able to calculate the number of grafts (and hairs) in that feathered area.

8 in long X 0.5 in wide = 4 in² of feathered zone 40 three-haired grafts (120 hairs)/in² X 4 in² = 160 three-haired grafts = 480 hairs (Figure 8, pink zone)

The remaining area is now 10 in² where we are using the six-haired graft stencil.

20 six-haired grafts (120 hairs)/in² X 10 in² = 200 six-haired grafts = 1,200 hairs (Figure 8, blue zone)

Up to this point there are to be 360 grafts and 1,680 hairs utilized. We then recommend 75 random one-haired grafts to the frontal hairline and 75 random one-haired grafts to the posterior hairline, a total of 150 one-hair micrografts to the hairline edges (Figure 8, yellow zone). This brings the total number of grafts to 510 and the total number of hairs to 1,830. Let us review the calculations:

Figure 8. The pink area (the feathered zone) is 4 in2 accepting 160 three-haired grafts per session (4 x 40 = 160). The blue area is 10 in2 and will accept 200 six-haired grafts (10 X 20 = 200). The yellow area will accept approximately 150 one-haired grafts (75 to the frontal hairline and 75 to the posterior hairline). At the posterior hairline the one-haired grafts are spread out over a wider distance.

Figure 9. Schematic helping to demonstrate the equation for calculating the extender length: W - D = R. W = the width (cm) of the bald area after the first reduction. D = the desired amount of alopecia removal (cm) from the second reduction. This is also the distance that the extender is stretched. R = the resting length of the extender.

MATHEMATICS OF MEGASESSIONS

This will dictate how large the harvest should be based on the short- and long-term goals for the patient.

For example, a patient with Type VI baldness can easily have 35 in² or approximately 225 cm² of recipient area that needs to be treated. If the harvest was 22.5 cm² for example and the surgeon was attempting to perform a megasession throughout a 225-cm² area, the best possible density would be 10% of the original recipient density based on the donor density. The issue of how many hairs and how many incisions can be safely made into each square centimeter during each session to allow for consistently good yield is a discussion in and of itself and will not be addressed in this article.

MATHEMATICS OF SCALP EXTENSION

Calculating the Amount of Extension To be able to calculate the degree of extension, one must first have an equation to use. After using this modality for approximately 2 years and collating the data from the cases performed, it became evident that the amount of extra tissue developed was directly related to the distance that the extender was stretched. The equation to explain this is:

where L = length (cm) that the extender is stretched beyond its resting state and also the amount (cm) that will be removed from the second procedure; E = expected removal (cm) from the second procedure. This number is the expected gain based on the first procedure minus the expected stretch-back; and I = improved alopecia removal (cm) from extender usage over the first two procedures. For example, if patient A had a circumferential scalp reduction¹¹ with 4 cm of width removed, and we desire to remove 5 cm during the next procedure, it will be necessary to stretch the silicone sheeting 5 cm beyond its resting state. The numbers will fit into the equation as follows:

Calculating the Length of the Extender
The length of the extender to be used can also be calculated with an equation that I developed, which reads:

where W = the width (cm) of the bald area after the first reduction; D = the desired amount of alopecia removal (cm) from the second reduction. This is also the distance that the extender will be stretched from the resting position; and R = the resting length (cm) of the scalp extender.

Once again, let us take the example of the patient who has a 12-cm width of male pattern baldness. If a circumferential scalp reduction is performed on this patient and the total width of the excision is 4 cm, the remaining width of the bald area will be 8 cm (W = 8 cm).

If the surgeon desires to remove 5 cm during the next procedure, he/she will be required to stretch the silicone extender 5 cm. This reality is due to the fact that the width of the alopecia excision is limited by the length that the strip is tensed past its resting length. Once the extender has shrunk back to its resting state, no further traction is exerted on the scalp. Therefore, no significant improvement (past 5 cm of width removal) will occur during a 1-month period (D = 5 cm).

If the equation W - D is then used, the resting length of the extender should be 3 cm (8 cm - 5 cm = 3 cm). When this equation is put to the test in other similar situations, one will notice that lengths of 1, 2, 3, 4, and 5 cm will be the usual sizes utilized. The width of the strip should be between 4 and 6 cm.

Figure 10. When a resting extender length of 3 cm is utilized, the thinner thickness of the sheeting allows one to create much more extension without causing excessive discomfort secondary to the grams force being generated.

Figure 11. As in Figure 10, a 4-cm extender exhibits the same inverse relationship between thickness of the sheeting and grams force being generated. There is no reason to believe that this inverse relationship exists for all lengths of extenders.

Figures 10 and 11 demonstrate another interesting mathematical relationship in scalp extension. When a 4-cm-wide extender is continuously stretched, there is a linear correlation between the amount of stretch and the grams of force generated. It is also interesting to note that the thicker the silicone sheeting, the higher the linear equation is shifted on the graph. This mathematical relationship has caused us to change to a 0.05-inthick silicone sheet because greater extension can be generated with less discomfort. With the conventional 0.1-in-thick sheeting, 200% of the original length was the general limit of the extension. With 0.05-in-thick sheeting, the limit is around 300%.

MATHEMATICS OF SCALP REDUCCION

There have been studies analyzing this phenomenon performed by Nordstrom,¹⁴ Unger,¹⁵ Hitzig,¹⁶ and Brandy,¹⁷ with the common denominator being that the more tension applied to the incision, the more stretchback that will ensue. Additionally, when the bald skin is not lifted off the undersurface as with curvilinear scalp reduction,¹⁶ circumferential scalp reduction,¹¹ and scalp-lifting,¹⁸ little to no stretch-back occurs postoperatively because the central bald skin is essentially plastered down to the undersurface and cannot move. It also has been observed by Brandy and Nordstrom¹⁴ that the greater the tension applied to the wound, the greater the stretch-back, but even greater is the net gain.

Table 1

To demonstrate this concept, Table 1 was created. These relationships between gross gain and stretchback are theoretical, since the same patient cannot have two different operations performed at the same time. The below linear correlation, however, was frequently observed in my patients with similar skin properties when the alopecia removals were within the range of 1.5-4.5 cm (the usual amount of removal with midline procedures):

Figure 12. The relationship between the gross gain and the net gain after stretch-back with midline scalp reductions is essentially linear in the range of normal alopecia excisions, 1.5-4.5 cm of width removal. In the graph, X = net gain, Y = gross gain, the equation is Y = 5X - 5.5.

As can be noted, when these numbers are plugged into a graph (Figure 12) there is a linear relationship (Y = 5X - 5.5) between stretch-back and the amount of bald skin removed when removing between 1.5 and 4.5 cm (the usual amount removed with these procedures). Once one gets beyond 4.5 cm or below 1.5 cm, the linear relationship starts to dissipate and the equation does not fit reality. As stated, however, most removals are usually between 1.5 and 4.5 cm.

The basic precept to be gained from this demonstration is that when a greater amount of baldness is removed, a greater force is generated, which results in a greater amount of new skin production (stretch-back). However, there is a greater net gain when a greater amount of tension is applied.

Conversely, one must weigh the consequences of applying great force to the wound-the scar can widen, the patient has more discomfort, the wound may dehisce, temporary or permanent hairloss may occur adjacent to the incision line, etc. Therefore, each surgeon must decide on the degree of tension that they feel comfortable with exerting on the wound edges. If a physician wants maximum removal, he/she will be forced to deal with some of the problems related with high tension. On the contrary, if the surgeon wants minimal problems, but is willing to accept less alopecia removal, that is a choice that he/she is free to make.

SUMMARY

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