ANALYZING THE SPATIAL RANDOMNESS IN THE DISTRIBUTION OF ACQUIRED MELANOCYTIC NEOPLASMS

Abstract

Based on the clinical impression and current knowledge, acquired melanocytic nevi and melanomas may not occur in random localizations. The goal of this study was to identify whether their distribution on the back is random and the location of melanoma correlates with its adjacent lesions. Therefore, patient-level and lesion-level spatial analyses were performed using the Clark-Evans test for complete spatial randomness (CSR). 311 patients with 3D total body photography (average age 40.08 [30–49]; male/female ratio: 128/183) with 5108 eligible lesions in total were included in the study (mean sum of eligible lesions per patient 16.42 [3–199]). The patient-level analysis revealed that the distributions of acquired melanocytic neoplasms were more likely to deviate towards clustering than dispersion (average z-score −0.55 [95% CI −0.69 to −0.41; P<.001]). The lesion-level analysis indicated a higher portion of melanomas (79.2% [n=57/72, 95% CI 69.4% to 88.9%]) appearing in proximity to neighboring melanocytic neoplasms compared to nevi (45.3% [n=2281/5036, 95% CI 43.9% to 46.7%]). In conclusion, the nevi and melanomas’ distribution on the back tends towards clustering as opposed to dispersion. Furthermore, melanomas are more likely to appear proximally to their neighboring neoplasms than nevi. These findings may justify various oncogenic theories and improve diagnostic methodology.

INTRODUCTION

Spatial analysis of the distribution of melanocytic neoplasms has been a matter of research, considering it could provide insights about nevogenesis and perhaps clues regarding melanomagenesis. It has been shown that large congenital melanocytic nevi manifest distinct patterns of anatomical distribution (Martins da Silva VP et al., 2017), which is believed to be related to developmental pathways during embryogenesis (de Wijn et al., 2010; Martins da Silva et al., 2019). In addition, based on the genetic analysis of tissue samples obtained from patients with multiple congenital nevi, Kinsler et al. (2013) found that congenital nevi may derive from a single, post-zygotic NRAS mutation that follows patterns of cutaneous mosaicism in the context of mosaic RASopathies.

Similarly, for the acquired melanocytic nevi (hereafter ‘nevi’), the empirical visual assessment based on the experience from examining patients with multiple nevi suggests that their distribution may also be not random; nevi are often found aggregated in some anatomical areas while sparse in others. Various efforts to address nevi distribution have measured overall nevus density, defined as the number of nevi per unit skin surface area, and total nevus count, defined as the absolute number of nevi per patient regardless of body surface size, as it correlates with melanoma risk (Welkovich et al., 1989; Zattra et al., 2009). Although local nevus density is associated with increased melanoma risk (Vredenborg et al., 2014), it remains unclear if the location of melanomas correlates with the location containing the highest density of nevi (Juhl et al., 2009; Rieger et al., 1995). It has been demonstrated that a significant number of nevi are genetically determined (Bataille et al., 2000) and that body site-specific nevus counts in women, particularly for the lower limbs, are associated with specific genetic alterations (Visconti et al., 2020). The observations that many nevi harbor BRAF-V600E mutations and the detection of incipient nevomelanocytic nests in clinically normal-appearing skin (Dadzie et al., 2008; Roh et el., 2015; Scope et al., 2009) raise the possibility that the presence of nevi may be determined in utero. If this is true, nevi may represent tardive congenital nevi, and their eventual growth and clinical visibility are potentially dependent on factors such as UV exposure, immunosuppression, hormones, among others (Bauer et al., 2005; Driscoll and Grant-Kels, 2007; Zalaudek et al., 2007).

The primary aim of this study is to investigate whether the distribution of acquired melanocytic neoplasms (hereafter ‘neoplasms’) over the surface of the back of high-risk patients is random. The secondary aim of this study is to determine whether the location of melanoma is associated with the nevi distribution pattern among patients with melanoma in the defined area of the back.

RESULTS

The initial dataset consisted of 422 patients between the ages of 30–49 with 3D total body photography acquired between March 2017 and February 2020. Two patients were excluded due to technical issues that precluded us from accessing their 3D body maps, 14 patients due to difficulty in visualizing their nevi and differentiating their nevi from seborrheic keratosis, and 95 patients were excluded since they had less than three eligible neoplasms on the back.

The final dataset consisted of 311 patients. The demographic characteristics are presented in detail in Table 1. There were 152 (48.9%) patients with a history of melanoma, with 65 (20.9%) of these patients having at least one melanoma located on the back. In particular, 59 (19.0%) patients were diagnosed with a single melanoma on the back, 5 (1.6%) patients were diagnosed with two, and 1 (0.3%) patient had three. In total, 5108 eligible lesions were collected from the 311 patients, of which 5036 (98.6%) were classified as nevi and 72 (1.4%) as melanomas. The median sum of eligible lesions per patient was 12 (Q25=6, Q75=20).

Table 1.

Demographic summary of sample characteristics

	N (%)
Total patients	311 (100%)
Male	128 (41.2%)
Female	183 (58.8%)
History of melanoma
True	152 (48.9%)
False	159 (51.1%)
Melanomas on back
0	246 (79.1%)
1	59 (19%)
2	5 (1.6%)
3	1 (0.3%)
Atypical Mole Syndrome
True	257 (82.6%)
False	54 (17.4%)
	Median (Q25, Q75)
Age	40 (35, 44)
Moles on back	12 (6, 20)
Melanoma	0 (0, 0)
Nevi	12 (6, 20)

The patient-level analysis reported that the Clark-Evans z-scores of the patient-sample follow a gaussian distribution, as presented in Figure 1. Overall, the spatial distribution of the patient’s neoplasms did not exhibit CSR but rather deviated towards clustering with an average Clark-Evans z-score of −0.55, which was statistically significantly less than 0 (95% CI −0.69 to −0.41; P<.001). Each of the various patient groups investigated in the study exhibited a deviation towards clustering at a statistically significant degree. More specifically, mean z-scores were lower among patients with a history of melanoma on the back versus those without (P=.008), whereas for the group of patients with AMS versus those without there was no statistically significant difference between their mean z-scores. The results of the patient-level analysis with the average z-scores of the sample are presented thoroughly in Table 2 and visual examples of the three types of distribution for patients with high and low counts of eligible lesions in Figure 2.

Figure 1. Allocation of the distributions of melanocytic lesions according to their classification for spatial randomness, categorized in various patient subgroups.

Y axis refers to the patient-sample subgroups, based on their eligible lesion counts (A) or history of melanoma on the back and presence of AMS (B). X axis refers to the z-scores calculated with the Clark-Evan test. 0 represents complete spatial randomness (CSR), while negative and positive z-scores represent clustering and dispersion, respectively. The orange lines correspond to the critical z-scores +/−1.96, based on the standard normal distribution levels of significance: (95% confidence level). The boundaries of each boxplot represent the interquartile range Q25 to Q75 and the blue circles represent the average values. The figure demonstrates that the Clark-Evans z-values of the various patient-sample subgroups follow a gaussian distribution.

Table 2.

Average Clark-Evans scores according to the patient-level spatial distribution analysis

Patient	N	%	Clark-Evans Score		p-value1	p-value2
			Average	95% CI
All Patients	311	100.0%	−0.55	(−0.693 – −0.415)	<0.001
Sex
Male	128	41.2%	−0.48	(−0.676 – −0.29)	<0.001	0.385
Female	183	58.8%	−0.60	(−0.8 – −0.408)	<0.001
History of melanoma
True	152	48.9%	−0.54	(−0.766 – 0.317)	<0.001	0.862
False	159	51.1%	−0.57	(−0.737 – −0.395)	<0.001
Melanoma on the back
True	65	20.9%	−0.93	(−1.234 – −0.617)	<0.001	0.008
False	246	79.1%	−0.46	(−0.611 – −0.301)	<0.001
Atypical mole syndrome
True	257	82.6%	−0.59	(−0.738 – −0.437)	<0.001	0.342
False	54	17.4%	−0.40	(−0.768 – −0.024)	0.038

¹One sample two-sided t-test

²Welch two sample two-sided t-test

Figure 2. Visual examples of distributions for patients with high and low counts of eligible lesions.

The images demonstrate models of backs of patients included in the study, which were created after 3D TBP. The skin layer was removed and a green circle was placed in the center of each eligible lesion included in the study. Each eligible lesion is tagged with the letter ‘l’ followed by an ascending number. The top row contains patients with high counts of eligible lesions whereas the bottom row contains patients with low ones. Based on the Clark-Evans test, the first column contains patients with distributions classified as significantly clustered (Z < −1.96), the second column contains patients with random distributions (Z close to 0) and the third one includes patients with significantly dispersed distributions (Z > 1.96).

Although most patients revealed a distribution pattern that achieved z-scores between −1.96 and 1.96, 11.3% (n=35, 95% CI 7.7% to 14.8%) of the patients exhibited significant clustering of their melanocytic neoplasms with a z-score lower than −1.96 compared to 1.9% (n=6, 95% CI 0.6% to 3.5%) of the patients who manifested a significantly dispersed distribution with a z-score higher than 1.96. Furthermore, 20.0% (n=13, 95% CI 10.8% to 30.8%) of the patients with melanoma on the back demonstrated a significantly clustered distribution, whereas none of them had a significantly dispersed distribution (95% CI 0% to 23.1%). All the patient-level comparisons of deviation from randomness are summarized in Table 3.

Table 3.

Frequencies of significant clustering and dispersion based on the Clark-Evans scores of the patient-level spatial distribution analysis with a 95% level of confidence

Patient	N	%	Clustered (Z < −1.96)			Dispersed (Z > 1.96)
			N	%	(95% CI)	N	%	(95% CI)
All Patients	311	100.0%	35	11.3%	(8.0% – 14.8%)	6	1.9%	(0.6% – 3.5%)
Sex
Male	128	41.2%	10	7.8%	(3.2% – 12.4%)	1	0.8%	(0% – 2.3%)
Female	183	58.8%	25	13.7%	(8.7% −18.6%)	5	2.7%	(0.6% – 5.5%)
History of melanoma
True	152	48.9%	19	12.5%	(7.9% −17.8%)	4	2.6%	(0.7% – 5.3%)
False	159	51.1%	16	10.1%	(5.7% −15.1%)	2	1.3%	(0% – 3.1%)
Melanoma on the back
True	65	20.9%	13	20.0%	(10.8% – 30.8%)	0	0.0%	(0% – 4.6%)1
False	246	79.1%	22	8.9%	(5.7% −12.6%)	6	2.4%	(0.8% – 4.5%)
Atypical mole syndrome
True	257	82.6%	28	10.9%	(7.4% −14.8%)	4	1.6%	(0.4% – 3.1%)
False	54	17.4%	7	13.0%	(5.6% – 22.2%)	2	3.7%	(0% – 9.3%)

¹95% CI calculated by the rule of three

The results of the lesion-level analysis are presented in Table 4. 45.8% (n=2339, 95% CI 44.4% to 47.2%) of all neoplasms arose in proximity to neighboring neoplasms and 25.1% (n=1282, 95% CI 23.9% to 26.3%) arose in distance from them. Examining only nevi (n=5036), 45.3% (n=2281, 95% CI 43.9% to 46.7%) appeared in proximity to other neoplasms, while 25.2% (n=1269, 95% CI 24.0% to 26.3%) appeared at a distance from neighboring neoplasms.

Table 4.

Lesion Nearest Neighbor Analysis

		Proximal to Neighbor (${\text{D}}_{\text{r}}<{\text{D}}_{\text{C}}$)		Distant from Neighbor (${\text{D}}_{\text{r}}>{\text{D}}_{\text{D}}$)
Lesion	n (%)	n (%)	(95% Cl)	n (%)	(95% Cl)
All melanocytic neoplasms	5,108 (100.0)	2,339 (45.8)	(0.44–0.47)	1,282 (25.1)	(0.24–0.26)
Melanomas	72 (1.4)	57 (79.2)	(0.69–0.89)	13 (18.1)	(0.10–0.28)
On patients with a clustered distribution	13 (18.1)	13 (100.0)	(0.77–1.00)†	0 (0.0)	(0.00–0.23)†
On patients with dispersed distribution	0 (0.0)	0 (0.0)	NA	0 (0.0)	NA
On all other patients	59 (81.9)	44 (74.6)	(0.63–0.85)	13 (22.0)	(0.12–0.32)
Acquired melanocytic nevi	5,036 (98.6)	2,281 (45.3)	(0.44–0.47)	1,269(25.2)	(0.24–0.26)
On patients with history of melanoma	2,509 (49.8)	1,129 (45.0)	(0.43–0.47)	650 (25.9)	(0.23–0.28)
On patients w/o history of melanoma	2,527 (50.2)	1,155 (45.7)	(0.44–0.48)	617 (24.4)	(0.23–0.26)
On patients with melanoma on back	1,205 (23.9)	565 (46.9)	(0.44–0.50)	319 (26.5)	(0.24–0.29)
On patients w/o melanoma on back	3,831 (76.1)	1,716 (44.8)	(0.43–0.46)	946 (24.7)	(0.23–0.26)
On patients with atypical mole syndrome	4,709	2,182 (46.3)	(0.45–0.48)	1,214 (25.8)	(0.25–0.27)
On patients w/o atypical mole syndrome	327	100 (30.6)	(0.26–0.36)	53 (16.2)	(0.12–0.20)

Abbreviations: Cl, confidence interval; NA, not applicable; w/o, without.

^†95% Cls calculated by the rule of three.

Among the melanoma lesions (n=72), the percentage developing in proximity to their nearest neighbor was higher than that of nevi, with 79.2% (n=57, 95% CI 69.4% to 88.9%) of melanomas occurring in proximity to adjacent neoplasms. 18.1% (n=13, 95% CI 9.7% to 27.8%) of melanomas were located in distance from neoplasms. All melanomas among patients with a significantly clustered distribution (n=13, 95% CI 76.9% to 100%), classified as described above, appeared in proximity to surrounding nevi. None of the melanoma lesions were present on backs of patients with a dispersed distribution of nevi.

DISCUSSION

This study was designed to investigate the randomness of the spatial distribution of neoplasms on a defined surface area of the back in high-risk populations that receive TBP and dermoscopy as part of the regular monitoring for melanocytic skin cancer. We hypothesized that the pattern of distribution of neoplasms, being a manifestation of the pathways involved in melanocytic oncogenesis, may not be random. An additional goal was to specify whether melanoma is spatially related to the adjacent neoplasms, which we hypothesized may be the case since superficial spreading melanoma often shares common oncogenic pathways (i.e., BRAF) with nevi (Duffy et al., 2018).

The non-random distribution of neoplasms can manifest in a clustered or dispersed pattern. According to the patient-level analysis, on average, patients’ neoplasms spatial distributions deviated towards clustering as opposed to dispersion. This tendency was seen in the overall patient sample but was more evident among those with a history of melanoma on the back. However, only a small portion of them were classified as significantly clustered. The lesion-level analysis revealed that the majority of melanomas arose in proximity to neighboring neoplasms, and this observation was more prominent among patients with a distribution pattern of neoplasms classified as clustered. Interestingly, melanoma appeared more frequently in the vicinity of other neoplasms compared to nevi.

Evidence to date regarding the arrangement of melanocytes across the epidermis is ambiguous, with previous studies identifying either high variability in the degree of randomness in the distribution of fetal melanocytes (Holbrook et al., 1989), uniform distribution of melanocytes among adult population (Sun et al., 2021) or marked increase in the melanocytic density in skin tissues surrounding melanoma sites (Barlow et al., 2007). Furthermore, while nevus counts across body sites has occasionally been attributed to certain genetic effects (Visconti et al., 2020), data concerning the association of the melanocyte density with the anatomical site are also contradictory (Hirobe and Enami, 2019; Sun et al., 2021). Nevertheless, little is known about the distribution of melanocytes that share a mutational profile correlating with elevated neoplasia risk within a certain anatomical area.

The finding of the non-random distribution of neoplasms and, therefore, the non-random distribution of neoplasia-related melanocytes could be attributed to a variety of possible causes, ranging from early genetic events, resulting in genetically distinct melanocytic subpopulations (i.e., mosaicism), to their migration, maturation and survival pathways. Taking into the account the speculations that at least some nevi are genetically determined in utero (Bataille et al., 2000; Happle, 1995; Sarma, 2012; Torrelo, 2010), the distribution of subgroups of melanocytes or incipient nevomelanocytic nests, sharing a variable predisposition for generating melanocytic neoplasia, across certain skin domains may be influenced by embryogenetic mechanisms. Skin mosaicism underlines not only the potential genetic variability within the same skin cell type population but also the fact that the various cellular subpopulations are distributed in a non-random fashion throughout the skin surface. Numerous pigmentary skin conditions have been described as appearing with a mosaic phenotype, with the resulting distribution patterns of the skin manifestations being attributed to the type, time-point and affected cell type of the causal genetic alteration during embryogenesis (Kinsler et al., 2020). Moreover, various genetic mutations and rearrangements associated with nevogenesis and melanomagenesis are shown to be present in mosaics, such as HRAS, NRAS and V600E BRAF among others (Groesser et al., 2012; Gerami and Paller, 2013; Levinsohn et al., 2015; Lindhurst et al., 2011; Maitra et al., 2002; Rübben et al., 2002; Žilina et al., 2015; Alomari et al., 2020; Bastian et al., 2003; Piotrowski et al., 2008). It would be interesting to investigate the mutational profile of different nevi in the same patient, as findings of an identical profile would advocate for a common origin of nevi. In addition, differences in migration and maturation of melanoblasts to their final destination (Gerami and Paller, 2013) and subsequent uneven deposition of incipient nests during embryogenesis may also play a role in explaining the study findings. If the hypotheses mentioned above prove true, then shared exposure of skin surface areas to exogenous factors could have a disproportionate impact on subgroups of melanocytes. Alternatively, the variability in the melanocytic microenvironment (stroma), ranging from immune system responsiveness (Poźniak et al., 2019) and intercellular interactions between the diverse skin cell types (Eliades and Tsao, 2016), might also be expressed in the non-random spatial occurrence of melanocytic neoplasia. Further studies would be essential to determine the influence of these scenarios on the distribution patterns of neoplasms, further contributing to solving the complex enigma of melanocytic oncogenesis.

The occurrence of melanomas in proximity to other neoplasms observed in this study, considering also that the likelihood of having a proximal nearest neighbor is higher for melanomas than nevi, hints that not all melanocytes in the study area are at equal risk for developing into a malignancy, while the nevus-melanoma spatial relationship enhances the presumption of the existence of cutaneous micro-territories that are at increased potential for hosting melanoma development. This site-specific variability of proneness to the development of melanocytic neoplasia could lead to discrimination of areas with stratified risk for melanocytic neoplasia with direct implications on the clinical practice. Areas of elevated melanoma risk would necessitate heightened vigilance from the clinician for new or changing lesions. Moreover, evaluating the localization of a suspicious melanocytic lesion with respect to its neighboring neoplasms during melanoma screening examination could add to the established methods of seeking the morphological outlier lesion and the comparative approach. Lastly, it could be additionally integrated into an artificial intelligence model for enriching the diagnostic algorithms from a topographical standpoint.

This study focused on investigating the distribution of neoplasms on the back. However, since nevogenesis pathways differ across the body sites (Olsen et al., 2009; Randi et al., 2016), other studies need to determine if our findings are also seen on anatomical areas, such as the extremities. Moreover, additional studies are necessary to determine the impact of cumulative UV exposure or sunburns on the spatial manifestation of neoplasms. Another limitation of our study is that we only included highly selected patients in terms of age and epidemiological profile, as they represent high-risk populations undergoing increased surveillance for melanoma. Even though large differences in the total nevus count are not expected after this age range, our results might correspond to the behavior of melanomas arising at a younger age. Furthermore, further studies involving random samples of individuals would be necessary in order to prove the generalizability of our results. We also limited our lesion sample to nevi larger than 5mm in maximum diameter; thus, it is possible that the distribution of large nevi may differ from smaller nevi. However, differential diagnosis of pigmented lesions smaller than 5mm would be highly challenging and it would therefore require collecting dermoscopic evaluation for each lesion, which surpasses clinical practice. Lastly, even though the model for calculating spatial randomness in this study did not consider border coordinates, therefore ignoring edge effects and underestimating the degree of clustering, this is not a limitation that biases the results towards our conclusions.

In conclusion, the findings of this study suggest that the location of acquired melanocytic neoplasms across the skin of the back may not be of random occurrence, as nevi and melanomas are more likely to be distributed in a clustered than a dispersed pattern and melanomas are more likely to appear closer to a preexisting melanocytic lesion than nevi. Expanding the spatial analysis of melanocytic neoplasms with integration of clinical, dermoscopic, histopathologic and molecular data could amplify our knowledge of oncogenic pathways, predict domains of increased probability for melanocytic neoplasia, optimize our screening methodology and eventually improve earlier melanoma detection and management.

MATERIALS & METHODS

Data Collection

This retrospective study was conducted after approval by the International Review Board of Memorial Sloan Kettering Cancer Center. We selected the back as the area of study because it constitutes a hotspot for the development of nevi and melanoma and its geometry permitted easier measurements of spatial relationships. Data were analyzed between February and June 2020.

The sample was collected from the archives of the Pigment Lesions clinic, Dermatology Service, MSKCC. The patient population of the clinic that receives TBP is generally comprised of individuals at a high risk for melanocytic neoplasia due to either personal/family history or phenotype, for whom monitoring with TBP and dermoscopy is indicated. All patients with baseline 3D TBP acquired between March 2017, and February 2020 that met the inclusion criteria were included in the study. Inclusion criteria for the study were: (i) age between 30 and 49 years at the time of imaging and (ii) manifestation of at least three melanocytic neoplasms on the back. The age range was decided based upon the fact that the many nevi tend to become clinically manifest and stop growing by the age of 30–50 (Randi et al., 2006) and in an effort to eliminate confusion with age-related, pigmented non-melanocytic lesions, such as seborrheic keratosis. The 3D TBP maps of the patients were reviewed for empirical assessment of AMS, which, due to lack of strict criteria, was defined as the presence of more than 50 nevi with at least one larger than 5mm and at least one that was clinically dysmorphic. In addition, the following variables were also collected: age, sex, and personal melanoma history at the time of image capture.

Nevi larger than 5 mm in greatest diameter and melanomas of any type on the back were included in our analysis. We utilized the filtering tools within the VECTRA (Canfield Scientific, Inc) software to select all nevi larger than 5 mm in diameter. A reviewer (EC) manually confirmed the eligibility of every lesion by evaluating the recorded clinical images of each lesion and dermoscopic images, when available, to confirm that the lesions were, in fact, nevi. None of the patients had medium to large congenital melanocytic nevi. We did not attempt to differentiate small congenital nevi from acquired nevi based on the low prevalence of the former (Ingordo et al., 2007) and our inability to conclusively differentiate small congenital nevi from acquired nevi based on clinical morphology alone (Stefanaki et al., 2018). Melanoma selection was performed by identifying the corresponding scar based on the patient’s records. Scars associated with nevi removal were assessed based on the dermoscopic image of the preexisting lesion and the available pathology and clinician’s notes. Scars not associated with nevus or melanoma excision were excluded from the data analysis. The center of each eligible lesion was marked on the patient’s body map on the software, and their location 3D coordinates were recorded. Variations in the Z dimension, which reflects the curvature of the surface, were negligibly minimal.

We outlined the anatomical boundaries of the back of the patients and calculated its surface area using a specialized plugin of the VECTRA software called VAM (Canfield Scientific, Inc). On a standardized view of every patient’s 3D TBP map, we selected the right acromion, left acromion, 7^th cervical vertebra, right iliac crest, left iliac crest, and median sacral crest as anchor points. Through its lasso tool, VAM fits a closed convex boundary that circumscribes the area of interest. The surface area of the circumscribed region was then calculated using VAM, also taking into the account the curvature of the surface.

Evaluation of the pattern of distribution

Once the area on the back was identified, we set out to quantify the randomness in the spatial distribution of nevi using a model called CSR (Zalaudek et al., 2011; Maimon and Rokach, 2010) widely used in geographic information systems. CSR indicates if samples (e.g., lesions) over a target area (e.g., the surface of the back) are spread in a random or organized fashion. The spread of points over a target area can be summarized using the Clark-Evans (Clark and Evans, 1954) hypothesis test, which quantifies the extent of deviation from CSR. Let’s assume that patient $\text{k}$ has ${\text{M}}_{\text{k}}$ lesions, whose distances to their nearest neighbor (in our case the nearest eligible nevus or melanoma) are ${\text{D}}_{\text{r,k}}$, $\text{r}=1,\dots ,{\text{M}}_{\text{k}}$. The Clark-Evans test assumes that the patient’s average nearest neighbor over sufficiently large ${\text{M}}_{\text{k}}$ lesions follows a normal distribution defined by a mean and standard deviation, which under CSR (i.e., null distribution) are defined as

$${\text{E}}_{\text{k}}(\text{D})=\frac{1}{2\sqrt{{\lambda }_{\text{k}}}},{\text{std}}_{\text{k}}(\text{D})=\frac{4-\pi }{{\text{M}}_{\text{k}}4\pi {\lambda }_{\text{k}}},$$

where ${\lambda }_{\text{k}}$ is the point density defined as $\frac{{\text{M}}_{\text{k}}}{{\text{SA}}_{\text{k}}}$ and ${\text{SA}}_{\text{k}}$ is the surface area of the bounded region on patient $\text{k}$ (Zalaudek et al., 2011). The patient’s average nearest neighbor distance is defined as ${\overline{\text{D}}}_{\text{k}}=\frac{1}{\text{M}}\sum _{\text{r}=1}^{\text{M}}{\text{D}}_{\text{r},\text{k}}$ and it is converted to a standard normal ${\text{z}}_{\text{k}}$ score as:

$${\text{z}}_{\text{k}}=\frac{{\overline{\text{D}}}_{\text{k}}-{\text{E}}_{\text{k}}(\text{D})}{{\text{std}}_{\text{k}}(\text{D})}.$$

According to a specified level of significance, the set of lesions may deviate far enough from CSR and be considered clustered or dispersed. Clustering is defined as lesions are located in close proximity to one another. Similarly, dispersion is defined as the lesions being spread far apart over the defined surface. We will define clustering and dispersion using ${\text{z}}_{\text{k}}$ as described in the next section.

Statistical methods

We conducted both a patient-level and a lesion-level analysis. In the patient-level analysis, we classified patients with two binary variables – clustered and dispersed – defined by their ${\text{z}}_{\text{k}}$ values. ${\text{z}}_{\text{k}}$ values larger than 1.96 were considered significantly dispersed, and values smaller than −1.96 were considered significantly clustered (Smith, 2020). These correspond to the critical values on the standard normal distribution for two-sided hypothesis tests. We aggregated the binarized classifications at various patient-level factors such as the history of melanoma and AMS. The average ${\text{z}}_{\text{k}}$-scores’ deviation from zero were analyzed with two-sided t-tests. Welch’s t-tests compared average ${\text{z}}_{\text{k}}$-score differences between subsets split into binary factors.

In the lesion-level analysis, we set out to explore the spatial distribution of each lesion with respect to the other lesions over the area of interest. Given ${\text{E}}_{\text{k}}$ (D) and ${\text{std}}_{\text{k}}$ (D) for each patient, we calculated the average nearest neighbor distance ${\text{D}}_{\text{C}}$ and ${\text{D}}_{\text{D}}$ that would have resulted in a Clark-Evans statistic of ${\text{z}}_{\text{k}}=-1.96$ (clustering) and ${\text{z}}_{\text{k}}=1.96$ (dispersion), respectively. We then classified each lesion as proximal $\left({\text{D}}_{\text{r}}<{\text{D}}_{\text{C}}\right)$, distant $\left({\text{D}}_{\text{r}}>{\text{D}}_{\text{D}}\right)$, or neither $\left({\text{D}}_{\text{C}}<{\text{D}}_{\text{r}}<{\text{D}}_{\text{D}}\right)$, according to its actual nearest neighbor distance (${\text{D}}_{\text{r}}$) relative to these derived critical distances. We estimated for various lesion subsets the proportion that were clustered and dispersed according to this definition.

All confidence intervals for proportions were derived through Monte Carlo bootstrap resampling and, when necessary, the rule of three (Hanley and Lippman-Hand, 1983). The significance level chosen for all inferential estimates and hypothesis tests was $\sigma =0.05$. All analyses were performed using R Statistical Software (v4.0.3; R Core Team 2020).

Abbreviations:

TBP

total body photography

AMS

atypical mole syndrome

CSR

complete spatial randomness

DATA AVAILABILITY STATEMENT

No datasets were generated or analyzed during the current study.