An e-mail survey was conducted utilizing several dermatology e-mail lists including RxDERM-L at ucdavis and the Academic Dermatologic Surgeons listserve. 28 Mohs surgeons responded, and were asked the questions seen in [Table 1].

To best appreciate the following mathematical model, it is crucial for one to be familiar with the processing of tissue in Mohs surgery. For those not involved with Mohs surgery on a daily basis, this can be challenging to visualize. As such, a clay animation of ideal Mohs tissue processing is provided to clarify the geometry of expected tissue movement during processing [see Additional file 1].

Using this clay model, one can begin to imagine where errors may occur during tissue processing. The first example of processing error can we call "Edge Lift Roll". An animation of this potential processing error can be seen at [see Additional file 2]. In this case, when the tissue is processed, asymmetrical compression is applied to the tissue (Figure 7). This results in a rolling of the specimen while processing, and may result in one edge lifting from the plane of sectioning (Figure 8, 9). This can potentially result in a false negative, and a future recurrence of tumor. One can also easily imagine that asymetircial compression could also lead to an edge folding into the plane of sectioning thereby causing a false positive.

"The Squash" [see Additional file 3] can occur if a block is thick with dermis or fat bellowing from the midsection (Figure 10). If redundancy from the core of the block slides into the plane of sectioning (Figure 11, 12) – one may observe a false positive, resulting in additional and unnecessary layer harvesting.

"Tip Lift" [see Additional file 4] may occur if during the attempt to flatten the outer edge of a block (Figure 13), the inner tip lifts from the plane of sectioning (Figure 14, 15). This may result in a false negative, and potentiates future recurrence.

"Thin Section Collapse" [see Additional file 5] error is based on an exaggerated model where a layer is cut into thin slivers (Figure 16). We mention it here as a subtle variation may occur in clinical practice. In this case, when an attempt is made to flatten the epidermis, the tissue collapses and rolls to one side (Figure 17, 18). In this example, the tumor that was reaching the base of the specimen is lifted away from the base and is removed from the plane of sectioning. This can potentially result in either a false positive or a false negative – as a tumor can be lifted away (as demonstrated here), or brought into the plane of sectioning.

Experience ranged from 2 to 29 years, with a mean of 12 years. 46% of the time, the Mohs Surgeon reported cutting the excised layer into blocks (see Figure 1). As expected, the average number of blocks needed for a given layer increased from 1 or 2 blocks for a 10 mm specimen to 6 blocks for a 40 mm specimen (see Figure 2). However, when evaluating the range of this data, one sees that there is great variability. To make the sizes referred to less abstract, consider (figure 3). Here we see that a dime is about 15 mm, a nickel is 20 mm, half dollar 30 mm, and portrait of George Washington 40 mm. Some surgeons reported processing a dime size layer whole, while others reported cutting it into three blocks. Similarly, some would process George Washington's portrait (40 mm) into 2 blocks – while others would process the same layer into 6 blocks (see figure 4).

Regarding the thickness of Mohs layer's, similar variability was reported. Though the average depth showed progressive thickening, as may have been expected (see figure 5), analysis of the range reveals that some surgeons tend take thin layers while others tend to cut to subcutis – regardless of specimen size (see figure 6).

The question remains... does it matter? A mathematical model was created to assess the importance of these Mohs technique variables.

Careful analysis of tissue movement in "ideal" processing, and the errors that may occur when tissue is processed allows one to derive a mathematical expression. This is a useful exercise because analysis of the expression can allow one to draw conclusions related to the specific aspects of the technique that contribute most to potentiating error.

It is clear that for any layer, there is an "ideal area" (Figure 19) that represents the perfect footprint of the tissue, allowing for 100% visualization of the surgical margins. Errors in tissue processing will result in either a loss of ideal area (false negative), or gain in area (false positive). Review of the models above identifies that the loss or gain in processed tissue is related to the area of the sidewalls of the block. (Figure 20) This is a part of the tissue often overlooked – as it has no significance if the tissue is processed correctly. To explain the errors identified here, it is important to precisely calculate the area of these sidewalls. While tissue has dynamic properties categorized as stress relaxation and creep, we have ignored these in this model, as their impact is minimal on the analysis we present.

Step 1: Calculation of ideal area (Figure 19)

A_base = Πr²

Step 2: Calculation of the area of the side wall of one block (Figure 22)

A_side = (d)(r₁) + 1/2(d)²

Step 3: Calculation of the total area of the side walls (Figure 21)

Total area of the side walls = (N) × (A_side)

Step 4: A false negative is the ideal area (A_base) minus a percentage of A_side. And a false positive is the idea area plus a percentage of A_side.

Let k = the percentage roll, falling between 0 and 1.

Substituting what we know, and performing some simple trigonometry, we continue our derivation as shown in [Table 2]

We must solve for r₁ (See Figure 22) Note: although r1 and r2 are not collinear, they become collinear when the Mohs tissue is processed (see Additional file 1 for review of movement during processing)

r = r₁ + r₂

r₁ = r - r₂

Sin (45) = d/r₂

r₂ = d/Sin (45) = d/0.851

r₁ = (r - d/0.851)

Step 5: We can place the expression of error over the ideal area, to create a mathematical formula that predicts error. This formula will produce the value that is equal to the percentage of tissue that is lost from the ideal preparation of a specimen. If we assume only a 5% roll (k = 0.05), we have the following expression (see Figure 23) (Note: for simplicity, let us assume that k is the same on each side)

Alternatively, we could calculate the percentage of the tissue that is viewed on a prepared histological preparation of a Mohs slide. The percentage of viewable surface area would be calculated by subtracting the result of Figure 23 from 1, as shown in Figure 33.

The final mathematical expression derived from the proof above can be seen in Figure 25. Of note, N (the number of blocks), and d (depth of layer) are directly related to the degree of processing error anticipated. This final expression is demonstrated with real numbers to illustrate its importance.

To illustrate variability of thickness of specimens, Figures 24, 25, 26, 27 are shown. All of these figures assume a 5% roll (k = 0.05). Several conclusions can be made by looking at this series of figures. First, it is clear that the greater the number of blocks (N), the higher the predicted error. Looking across the figures, one sees how anticipated error grows with thicker layers.

A 5% roll is an estimation used, but is not based on any known data. Figures 28, 29, 30, 31 demonstrate changes in error that can be anticipated if there is greater than a 5% roll, with Figure 28 illustrating a 5% roll, and Figure 31 a 25% roll.

Figure 32 demonstrates a proposed clinically relevant set of parameters. Illustrated is anticipated error for a 10 mm layer, with a 5% roll during processing. One sees as the thickness increases from 2 to 15 mm, anticipated error grows. It is also apparent that as N grows, so does the anticipated error. Error rates reported in this graph are between 1 and 7%, consistent with reported rates of recurrence and far below the recurrence rates seen with standard excision and breadloaf sectioning.