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Anatomic Pathology / SPITZ NEVUS AND PATIENT AGE

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Am J Clin Pathol 2004;121:872-877

DOI: 10.1309/E14CJ6KRD092DP3M

© American Society for Clinical Pathology

Patient Age in Spitz Nevus and Malignant Melanoma

Implication of Bayes Rule for Differential Diagnosis

Robin T. Vollmer, MD

Key Words: Spitz nevus; Age; Bayes theorem; Probability; Melanoma

DOI: 10.1309/E14CJ6KRD092DP3M

A b s t r a c t

In the differential diagnosis of Spitz nevus vs

malignant melanoma, patient age provides a critical

piece of clinical information, because Spitz nevi occur

mostly in children and melanomas occur mostly in

adults. Nevertheless, there is overlap in the age

distributions of Spitz nevus and melanoma. The issue to

consider is how these age distributions and their

governing probability densities can impact the a priori

probability that a lesion is a Spitz nevus vs a melanoma.

Herein I introduce a quantitative approach that uses

Bayes rule together with previous published data on the

age distributions in Spitz nevi and melanoma. The

resulting algorithm yields plots and a table of predictive

a priori probabilities of Spitz nevus, given patient age

occurring within narrow intervals, and I believe these

provide useful guidelines for using age in the

differential diagnosis of Spitz nevus and malignant

melanoma.

One of the most vexing issues in the pathology of cuta-

neous pigmented lesions is the choice between Spitz nevus and

malignant melanoma.1-9Although many have assembled

useful lists of histologic features to differentiate these lesions

from one another,2-6the final diagnosis often implies a degree

of uncertainty.3,5Of course, clinical features also are helpful,

and the one most often considered is patient age. Clearly, Spitz

nevi tend to occur in children, and melanomas tend to occur in

adults. Thus, when a patient is a child, we favor Spitz nevus if

the histologic features are consistent with Spitz nevus, and

when the patient is an adult, we favor melanoma if the histo-

logic features are consistent with melanoma. Yet, we know that

melanomas occur in children, that Spitz nevi occur in adults,

and that the age distributions for Spitz nevus and melanoma

overlap. The issue to consider is how the numeric nature of age

and the probability densities of its distributions in these

diseases impact the predictive probability that any given lesion

is a Spitz nevus. Knowledge, then, of this predictive probability

for a particular age interval could provide us with quantitative

information that we then could combine with the histologic

features to reach a more informed diagnosis. Bayes theorem is

ideal for this task, and herein it is applied to the problem of age

in the differential diagnosis of Spitz nevus vs melanoma.

Materials and Methods

Bayes Rule

What we require is a formula for the probability that a

lesion is a Spitz nevus vs a malignant melanoma, given that

the patient’s age falls into a certain interval, symbolized as

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Iage. Throughout this article, Iage is defined so that age is

greater than some number “a” and less than or equal to a

larger number “b,” so that:

❚Equation 1❚

Iage = {a < age ≤ b}

Using probability notation, a formula is sought for the

conditional probability P(Spitz | Iage) with Iage symbolizing

the age interval. Bayes rule provides such a formula and is

written as:

❚Equation 2❚

P(Iage | Spitz) × P(S)

P(Spitz |Iage) =

P(Iage | Spitz) × P(S) + P(Iage | Mel) × (1 – P(S))

Bayes rule says that the desired probability P(Spitz |

Iage) can be calculated if the values of the 3 probabilities

appearing on the right side of Equation 2 are known. These 3

are: (1) P(Iage | Spitz), the probability of age in the interval

Iage given that the patient has a Spitz nevus; (2) P(Iage |

Mel), the probability of age in the interval Iage given that the

patient has a melanoma; and (3) P(S), the a priori probability

that the lesion is a Spitz nevus vs melanoma.

Use of Age Distribution Functions

The medical literature often provides tables or plots of

frequency distribution of ages. For example, ❚Figure 1A❚

shows a consolidated frequency distribution of age from 2

large studies of Spitz nevi.10,11These 2 studies included a

total of 449 patients, and the frequency on the vertical axis is

expressed as a fraction of that total. The frequency distribu-

tion of a continuous variable like age is an estimate of the

probability density function, f(x). By contrast, ❚Figure 1B❚

shows an estimate of the distribution function of age, which

was estimated from the cumulative sum of patient numbers

for each age and then divided by the total. The distribution

function often is symbolized as F(x), and for a continuous

variable like age, f(x) and F(x) are related to one another

through calculus in the following equation.12

❚Equation 3❚

F(a) = ∫f(x) dx

with the integration limits between 0 and a, because negative

age is not defined. Thus, frequency distributions of age and

distribution functions of age are directly related to one

another, so that one can be used to estimate the other.

The probability that age is less than some value, a, is

defined as equal to F(a). In other words,

❚Equation 4❚

P(age ≤ a) = F(a)

The probability of observing an age in the interval {a <

age ≤ b}, therefore, can be obtained from the distribution

function of age as follows:

❚Equation 5❚

P(a < age ≤ b) = F(b) – F(a)

This equation provides the method to be used to esti-

mate the probability of age occurring in some interval, Iage.

10 20 304050 60 70 8090

0.5

0.4

0.3

0.2

0.1

0.0

f(age)

Age in Years

1020 30 4050 6070 8090

1.0

0.8

0.6

0.4

0.2

0.0

F(age)

Age in Years

A

B

❚Figure 1❚ A, Frequency distribution of ages in 2 consolidated studies of Spitz nevus.10,11B, Cumulative distribution function

corresponding to the frequency distribution in A. f, probability density function; F , distribution function.

Page 3

❚Figure 2❚ A plot of P(Age > x | Spitz) vs patient age. The

smooth line comes from the exponential probability

distribution function, ie, exp(–α × x), and the value used for α

is a weighted mean of the studies in Table 1. The points on

the plot come from the raw data of the studies in Table 1.

The values for F from the medical literature were obtained

as follows.

Modeling the Distribution Function of Age in Spitz Nevi

To discover the distribution function appropriate for age

in Spitz nevi, I searched the literature for articles that

provided histograms or frequency tables of age in cases of

Spitz nevi. I identified 7 such articles,2,3,10,11,13-15

comprising a total of 1,085 patients with Spitz nevus.

Because several of these studies shared cut points in age, I

was able to partially combine patient counts, and this

consolidation yielded 3 cumulative distribution functions,

one of which appears in Figure 1. The frequency distribu-

tions of these 3 also seemed to follow an exponential func-

tion (eg, Figure 1A). Consequently, I chose the exponential

distribution function to match these published data. The

exponential function implies that the probability that age

exceeds a value, a, given that the lesion is a Spitz nevus can

be written as:

❚Equation 6❚

P(Age > a | Spitz) = 1 – F(a) = exp(–α × a)

where α is the parameter for the distribution function. I used

the least squares fitting technique to estimate values of α

from the data for each of 3 cumulative distributions ❚Table

1❚, and the weighted mean of these 3 estimates was 0.0754.

❚Figure 2❚ shows a plot of P(Age > x | Spitz) vs patient age

for the 3 observed distributions (points on the plot), and the

smooth line demonstrates the fit obtained with the exponen-

tial distribution function using the weighted mean for α.

Finally, I used the weighted mean of α to calculate the prob-

ability of age falling in a specified age interval, Iage, as:

❚Equation 7❚

P(Age in Iage | Spitz) = exp(–α × a) – exp(–α × b)

Modeling the Distribution Function of Age in Melanoma

To discover the probability density function, f(x), appro-

priate for age in melanoma, I used the frequencies of ages for

melanoma in the National Cancer Data Base (NCDB) as

reported by Urist and Karnell,16and I consolidated their data

for the years 1985, 1988, and 1990 to achieve a single

frequency distribution of age for the total population of 20,165

patients. Because the mean of this composite distri-bution

appeared less than the mode, I used the beta probability density

function to model these data after dividing age by 100. In equa-

tion form, the beta probability density function is written as:

❚Equation 8❚

Γ(β + γ)

f(x) = × x(β – 1)× (1 – x) (γ – 1)

Γ(β) × Γ(γ)

Here, x symbolizes age divided by 100, and Γ

symbolizes the gamma function. I used the weighted

mean and apparent variance of Urist and Karnell’s16

composite data to estimate the values of β and γ, respec-

tively, as 4.0 and 3.17, and I used SPLUS software (Math-

Soft, Seattle, WA) to calculate the beta distribution func-

tion for these values. Henceforth, the beta distribution

function, which is the integral of Equation 8, is symbol-

ized as Fβ(x). ❚Figure 3❚ shows a plot of the probability

that age exceeds a value, x, given that the lesion is a

melanoma, ie, P(Age > x | Mel) or 1 – Fβ(x). The smooth

line provides the result using the beta distribution func-

tion, and the points on the plot are from the raw NCDB

data. The proximity of points to the line demonstrates that

the beta distribution function provides a good model for

these data. Finally, I calculated the probability of age

falling in a specified age interval, I, between 2 ages, a and

b, given that the lesion is a melanoma as:

❚Equation 9❚

P(Age in Iage | Mel) = Fβ(b) – Fβ(a)

❚Table 1❚

Studies of Age Distributions in Spitz Nevi*

ReferenceN Estimate of α α

Coskey and Mehregan10

and Dal Pozzo et al11

Weedon,2Chung et al,14

and Herreid and Shapiro15

Paniago-Pereira et al3

and Peters and Goellner13

4490.0831

4160.0785

2200.0538

*N is the combined number of patients from the studies cited; the combined number

was used in the present study; α is the parameter for the exponential probability

density and was estimated from the published data by using the least squares

technique (see the “Materials and Methods” section).

020 40 6080

1.0

0.8

0.6

0.4

0.2

0.0

P(Age > x | Spitz)

Age in Years

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Choice of P(S)

The choices for P(S) were obtained as follows: First, in

this binary differential diagnosis, there are only 2 possibili-

ties—Spitz nevus and melanoma. Consequently, the a priori

probability that the patient has a melanoma is simply 1 –

P(S). For Bayes rule to work, P(S) should reflect the local

prevalence of Spitz nevus vs melanoma. Thus, if a search of

coded diagnoses for 1 year yielded 5 Spitz nevi and 100

primary melanomas, then P(S) would be estimated as 5/105

or approximately 0.048. Because some have suggested that

Spitz nevi represent fewer than 1% of all nevi3and because

in most pathology practices melanomas are less common

than nevi, P(S) should be higher than 0.01. A recent search

of the pathology files at Duke University Medical Center,

Durham, NC, which has a relatively large number of referred

melanomas, produced a value of 0.03 for P(S). Weedon2esti-

mated P(S) to be approximately 0.05. Finally, the report by

Herreid and Shapiro15of 177 cases of Spitz nevus during an

interval when 625 melanomas were observed yields a value

for P(S) of 0.22. Consequently, to illustrate Bayes rule, I

chose 4 values for P(S): 0.01, 0.03, 0.05, and 0.25.

Results

❚Figure 4❚ shows 3 plots of the conditional probability

that a lesion is a Spitz nevus given that patient age exceeds a

threshold, x, ie, P(Spitz | Age > x). The separate lines are for

3 values of P(S), namely 0.25 (curve A), 0.10 (curve B), and

0.01 (curve C), and all 3 were calculated using Equation 2.

These curves demonstrate how the probability of Spitz nevus

falls with increasing age, as expected, and they also demon-

strate that the value of P(Spitz | Age > x) depends strongly

on the choice of a priori probability of Spitz nevus (ie, P(S)).

The main results are given in ❚Table 2❚, which lists the

calculated conditional probabilities of Spitz nevus given that

patient age falls into specified intervals and using Equation

2. The entries demonstrate several aspects of Bayes rule in

the differential diagnosis. When the patient is younger than 4

years, the probability of a Spitz nevus is much higher than

melanoma, regardless of P(S). However, for patients older

than 8 years, the probability of Spitz nevus depends on P(S).

For example, when P(S) is approximately 0.03, then the

probability of Spitz nevus becomes less than that of

020 40 6080

1.0

0.8

0.6

0.4

0.2

0.0

P(Age > x | Mel)

Age in Years

0 204060

0.25

0.20

0.15

0.10

0.05

0.00

P(Spitz | Age > x)

Age in Years

A

B

C

❚Figure 3❚ A plot of P(Age > x | Mel) vs patient age. The

smooth line comes from the beta probability distribution

function with β = 4.0 and γ = 3.17; SPLUS software

(MathSoft, Seattle, WA) was used to calculate the function.

The points on the plot come from National Cancer Data Base

data on 20,165 patients with melanoma (Mel).

❚Figure 4❚ Three plots of the conditional probability that a

lesion is a Spitz nevus given that patient age exceeds a

threshold, x, ie, P(Spitz | Age > x). The separate lines are for

3 values of P(S), namely 0.25 (curve A), 0.10 (curve B), and

0.01 (curve C), and all 3 were calculated using Equation 2.

❚Table 2❚

Effect of Patient Age on Predictive Probabilities of Spitz Nevus*

P(S)

Age Interval (y) 0.010.030.05 0.25

0-4

4-8

8-12

12-20

20-30

30-40

40-60

0.98

0.77

0.38

0.10

0.02

0.005

0.001

0.99

0.91

0.66

0.26

0.06

0.01

0.003

1.00

0.95

0.76

0.37

0.09

0.02

0.005

1.00

0.99

0.95

0.79

0.40

0.13

0.03

*Entries in the table are values of the conditional probability P(Spitz | Iage), which

was obtained from Bayes rule and Equation 2. P(S) is the a priori probability that

the lesion is a Spitz nevus in the local practice, and it can be estimated by

searching local databases for the relative frequencies of Spitz nevus and melanoma.

The probabilities in Table 2 are for the differential that comprises just 2 possible

diagnoses, ie, Spitz nevus and melanoma. Thus, for each entry in Table 2, the

estimated probability of melanoma [ie, P(Mel | Iage)] can be obtained by

subtracting the entry from 1.0.

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melanoma for ages older than 12 years, and between the ages

of 12 and 20 years, approximately 75% of lesions should be

expected to be melanoma. By contrast in a referral practice

with a higher frequency of Spitz nevi as, for example, with a

P(S) of 0.25, Spitz nevus should be expected to be more

likely than melanoma until patient age exceeds 20 years.

The entries in Table 2 also can be used as guidelines for

the overall probability of Spitz nevus given that the patient’s

age falls into certain groups. For example, for a pigmented

lesion in a 25-year-old patient in a practice where P(S) is

less than 0.06, Table 2 shows that Spitz nevus is less likely

than melanoma, with probabilities ranging from 0.02 to

0.09. If the diagnosis of Spitz nevus is chosen, Table 2 indi-

cates that it is a diagnosis of low overall probability. By

contrast if the diagnosis of melanoma is chosen for a 6-year-

old child, Table 2 once again indicates that the diagnosis is

unlikely, with a probability of occurring from 0.01 to 0.23.

Thus, Table 2 provides feedback on the likelihood of the

chosen diagnosis occurring in this age group. If an unlikely

diagnosis has been chosen, further consultation or even

staining for Ki-67 to evaluate the tumor’s proliferation rate

might be considered.

Discussion

The purposes of the present study were to inject a

degree of quantitative objectivity into the consideration of

age in the differential diagnosis between Spitz nevus and

melanoma and to take advantage of previous experience with

age in these lesions. Because age is a numeric phenomenon,

the problem is numeric, and Bayes rule provides an ideal

solution. Its result, Equation 2, flows from pure mathemat-

ical logic. What this article does is connect that logical result

to a consolidation of a large amount of data regarding age

distributions in Spitz nevus and melanoma—more data than

any one medical center or observer is likely to obtain. Thus,

Table 2 provides guidelines that flow from mathematical

logic plus a large amount of previous data, and I hope that

this result will efficiently summarize the impact of patient

age on the differential diagnosis of Spitz nevus vs malignant

melanoma.

I have found that the exponential probability density

function works well for the distribution of ages in Spitz nevi

and that the beta probability density function works well for

the distribution of ages in melanoma. The combination of

these 2 distributions and their respective parameters of α, β,

and γ provide a concise summary of the collected data from

more than 21,000 patients, so that these distributions and

their parameters make these data portable to practices with

far less experience with either Spitz nevus or melanoma. Use

of these 2 distributions in Bayes rule permits us to move

from subjective considerations of the likelihood of Spitz

nevus or melanoma in certain age groups to a more objective

and quantitative evaluation of age in the probability of either

Spitz nevus or melanoma. However, the Bayes algorithm

also requires us to estimate the a priori probabilities of Spitz

nevus and melanoma. For this binary differential diagnosis,

P(S) is simply 1 – P(Mel), the probability of melanoma.

Thus, to estimate P(S), all one must do is search the local

coded diagnoses for Spitz nevus and melanomas, and P(S)

would be estimated as the ratio of the number of Spitz nevi

to the total numbers of cases found in the search.

The use of Bayes rule could be expanded to include

additional types of nevi, but such expansion would require a

change in the denominator of Equation 2 to:

P(Iage | Spitz) × P(S) + P(Iage | Mel) × P(Mel)

+ P(Iage | Nevus) × P(Other Nevus)

Furthermore, this expansion of Bayes rule also would require

new data, including the distribution of age in nevi other than

Spitz nevi, and estimates of the 3 separate probabilities of

P(S), P(Mel), and P(Other Nevus) from a pool of cases

comprising all types of pigmented lesions.

The influence of P(S) on P(Spitz | Iage) via Bayes rule is

one of the more important results of this study, and it is illus-

trated in Figure 4 and Table 2. Local practice patterns can

influence P(S) dramatically and even alter the locally

observed age distributions for Spitz nevus and melanoma.

For example, an institution like Duke treats a large number

of referred patients with melanoma, whereas children with

Spitz nevi might be treated at community hospitals or clinics.

Such factors can reduce the observed P(S). Furthermore,

adults with Spitz nevi initially misdiagnosed as melanoma

might increase the local age distribution for Spitz nevi in a

referral center, and children with melanoma sent to the

referral center might lower the age distribution for

melanoma. In fact, the scatter of points in Figure 2 and the

variability of estimated values of α in Table 1 might be due

to referral patterns of patients with Spitz nevus. Whereas

such concerns do not weaken the validity of Bayes rule, they

do emphasize the importance of the data and resulting distri-

bution functions used in Equation 2. For these reasons, I

caution that the probabilities listed in Table 2 are intended as

guidelines derived from a large amount of historic data.

The entries in Table 2 also do not suggest definitive cut

points for the diagnosis of Spitz nevus or melanoma. That

there should be residual uncertainty in probabilities,

however, is no more than the expected nature of probability,

because for any given case with a probability of Spitz nevus,

there always is a corresponding chance of melanoma. In fact,

the entries in Table 2 demonstrate that Spitz nevi might occur

at any of the listed age intervals. Such considerations simply

emphasize the probabilistic nature of the diagnosis of Spitz

nevus as suggested by Piepkorn5and others.3Finally, I

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emphasize that the diagnosis of Spitz nevus or melanoma is

a histologic exercise using an adequate biopsy sample. In

other words, routine histologic findings trump all of the

aforementioned probabilistic considerations, and it remains

our responsibility to recognize these lesions even when they

seem improbable.

From the Department of Pathology, Veterans Affairs, and

Department of Pathology, Duke University Medical Centers,

Durham, NC.

Address reprint requests to Dr Vollmer: Laboratory

Medicine 113, VA Medical Center, 508 Fulton St, Durham, NC

27705.

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