Pairwise interaction Markov model for 3D epidermal nerve fibre endings

• Published in: Journal of microscopy (Oxford) Vol. 288; no. 1; pp. 54 - 67
• Main Authors: Konstantinou, Konstantinos, Särkkä, Aila
• Format: Journal Article
• Language: English
• Published: England Wiley Subscription Services, Inc 01.10.2022; John Wiley and Sons Inc
• Subjects: Anisotropy; Attraction; Confocal microscopy; cylindrical K function; Diabetes; Diabetes mellitus; Diabetic Neuropathies; Diabetic neuropathy; Epidermis; Humans; Markov chain Monte Carlo; Markov chains; Markov random field; Microscopy, Confocal; Nerve Fibers - chemistry; Nerve Fibers - physiology; Nerves; Original; point process; pseudo-likelihood; Three dimensional models; cylindrical K function; Markov random field; point process; pseudo-likelihood; Markov chain Monte Carlo; anisotropy
• ISSN: 0022-2720; 1365-2818; 1365-2818
• DOI: 10.1111/jmi.13142

In this paper, the spatial arrangement and possible interactions between epidermal nerve fibre endings are investigated and modelled by using confocal microscopy data. We are especially interested in possible differences between patterns from healthy volunteers and patients suffering from mild diabetic neuropathy. The locations of the points, where nerves enter the epidermis, the first branching points and the points where the nerve fibres terminate, are regarded as realizations of spatial point processes. We propose an anisotropic point process model for the locations of the nerve fibre endings in three dimensions, where the points interact in cylindrical regions. First, the locations of end points in R2$\mathbb {R}^2$ are modelled as clusters around the branching points and then, the model is extended to three dimensions using a pairwise interaction Markov field model with cylindrical neighbourhood for the z‐coordinates conditioned on the planar locations of the points. We fit the model to samples taken from healthy subjects and subjects suffering from diabetic neuropathy. In both groups, after a hardcore radius, there is some attraction between the end points. However, the range and strength of attraction are not the same in the two groups. Performance of the model is evaluated by using a cylindrical version of Ripley's K function due to the anisotropic nature of the data. Our findings suggest that the proposed model is able to capture the 3D spatial structure of the end points. Lay description Diabetic neuropathy is a serious complication which affects a lot of people suffering from diabetes. It causes nerve damage, resulting in a variety of unpleasant symptoms such as pain and loss of sensation. As there is no treatment able to completely cure the diabetic neuropathy, diagnosis at the earliest possible stage is important to slow down its progression. To develop better diagnostic tools, skin samples from patients suffering from diabetic neuropathy and healthy controls have been collected. From these samples, the locations, where nerves enter the epidermis, branch, and terminate, can be identified using confocal microscopy imaging and image analysis procedures. In this work, we focus on the changes in the spatial pattern of the nerve endings at the early stages of the neuropathy. As data, we have skin samples from the feet of healthy subjects and subjects with mild diabetic neuropathy. To investigate potential differences between the nerve end patterns between healthy subjects and patients suffering from diabetic neuropathy, we developed spatial point process models for the three‐dimensional locations of the nerve endings given the locations of the points, where nerves branch for the first time, and the locations of the entry points. These models were fitted to the available data for both groups and the estimated model parameters were compared. The model was able to describe the arrangement of the nerve endings quite well. Our results indicate that the nerves grow and interact in a different way in diabetic neuropathy and healthy samples. In particular, the clusters of the nerve endings in the diabetic neuropathy samples are tighter across the skin but further apart in the vertical direction compared to the clusters in the healthy patterns.