Mohs math – where the error hides
 Jeffrey I Ellis^{1}Email author,
 Tatiana Khrom^{1},
 Anthony Wong^{1},
 Mario O Gentile^{2} and
 Daniel M Siegel^{1}
DOI: 10.1186/14715945610
© Ellis et al; licensee BioMed Central Ltd. 2006
Received: 18 August 2006
Accepted: 06 December 2006
Published: 06 December 2006
Abstract
Background
Mohs surgical technique allows a full view of surgical margins and has a reported cure rate approaching 100%.
Method
A survey amongst Mohs surgeons was performed to assess operator technique. In addition, an animated clay model was constructed to identify and quantify tissue movement seen during the processing of Mohs surgical specimens.
Results
There is variability in technique used in Mohs surgery in regards to the thickness of layers, and the number of blocks layers are cut into. A mathematical model is described which assesses the clinical impact of this variability.
Conclusion
Our mathematical model identifies key aspects of technique that may contribute to error. To keep the inherent error rate at a minimum, we advocate minimal division and minimal physical thickness of Mohs specimens.
Background
Over the past sixty years, Mohs micrographic surgery has become the standard of care in the management and treatment of many skin cancers. Unlike standard vertical sectioning, the horizontal sectioning utilized by Mohs technique allows a full view of surgical margins [1] and has a reported cure rate between 88 to 100% [2–7]. Differences in operator technique are already known [8, 9], however their impact into the ability to fully view the surgical margins have not been defined. This paper is divided into two parts; Part I: A survey of the techniques of practicing Mohs Surgeons. Part II: A mathematical model is described which assesses the clinical impact of technique variability.
Methods
Survey methods
Survey Questions
1. How many years have you been performing Mohs Surgery? 

2. Who cuts your excised layer into blocks? 
a. You 
b. Fellow 
c. Tech 
3. For specimens ranging from 1–4 cm, on average 
a. How many blocks is the excised layer cut into when processing? 
b. What is the thickness (depth) of your first Mohs layer? 
Mathematical model methods
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].
Results
Survey results
The question remains... does it matter? A mathematical model was created to assess the importance of these Mohs technique variables.
Mathematical model results, derivation of a mathematical proof
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.
Step 1: Calculation of ideal area (Figure 19)
A_{base} = Πr^{2}
A_{side} = (d)(r_{1}) + 1/2(d)^{2}
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.
Mathematical proof
False ()  Ideal  False (+) 

A_{base}  k(A_{side})(N)  Abase  A_{base} + k(A_{side})(N) 
Πr^{2}  k(A_{side})(N)  Πr^{2}  Πr^{2} + k(A_{side})(N) 
Πr^{2}  k((d)(r_{1}) + 1/2(d)^{2})(N)  Πr^{2}  Πr^{2} + k((d)(r_{1}) + 1/2(d)^{2})(N) 
Πr^{2}  k((d)((r  d/0.851)) + 1/2(d)^{2})(N)  Πr^{2}  Πr^{2} + k((d)((r  d/0.851)) + 1/2(d)^{2})(N) 
r = r_{1} + r_{2}
r_{1} = r  r_{2}
Sin (45) = d/r_{2}
r_{2} = d/Sin (45) = d/0.851
r_{1} = (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)
Discussion
Is recurrence of a tumor after Mohs surgery always a result of error? Persistent tumor may be related to "difficulties of anatomic site[10], tumor size and histological subtype[11], as well as observer error in histological interpretation and potential tumor multifocality[12]" [13]. There are also many processing errors that may occur including inaccurate mapping, tissue staining, and tissue preparation for sectioning. It is clear that in order to maximize the value of the technique, processing of tissue must be as ideal as possible.
The importance of processing tissue in an 'ideal way' is not a new one. The benefits of processing a layer as one block have been previously described [14]. In addition, several authors have suggested techniques to facilitate obtaining quality and complete horizontal sections [15–17].
It seems prudent to anticipate some questions that this paper may raise, and provide answers at this time. One frequently asked question is "Wouldn't you notice missing tissue (i.e.: edge role)" The answer is simply no. Remember that the clay models show an exaggerated event to help illustrate a potential event. If only 5% of the tissue rolled, this would unlikely be perceivable. Even if it were perceived that this tissue seemed "smaller", it would be easy to disregard this fact as anticipated tissue shrinkage [18].
Another question often asked relates to tissue dyes. In the models presented, the clay was not marked with an orientation dye. If the edge lifted, wouldn't the marked edge be lost? The answer is that it depends. As we know, the orientation dye we use is far from precise, and often "bleeds" slightly. It is easy to imagine tissue could be removed form the plane of section, while some orientation dye remains. One must remember that the only absolute edge is an epidermal edge; it is the nonepithelial edges that are subject to the errors we have demonstrated. As tissue dyes do "bleed", they cannot be considered absolute boundary markers.
Finally, curetting or debulking a tumor may have additional benefit related to processing. Though this is controversial amongst Mohs surgeons, removing the bulk of a tumor will serve to significantly decrease the thickness of a Mohs layer. In doing so, it may serve to decrease the likelihood of the processing errors described here.
The model presented in this paper could be adapted to any layer of Mohs surgery, with or without debulking. The conclusions will always be the same. A variety of processing errors can be significantly reduced by taking thin layers, and processing tissue in the least number of blocks possible.
Conclusion
As previously described, variability exists in the technique of Mohs Surgery. This paper represents the first known attempt to quantitate in a mathematical way the consequence of some components of this variation. Evidence is provided which suggests that minimizing the number of blocks an excised layer is cut into when processing, and minimizing the thickness or depth of an excised layer can dramatically improve the cure rate of Mohs Surgery.
Abbreviations
 Aside:

Area Side
 r = r_{1} + r_{2} :

radius of Abase
 r_{1} :

length of base
 r_{2} :

length of side wall
 N:

Number of blocks
 d:

depth (thickness)
Declarations
Authors’ Affiliations
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