A model-based hierarchical Bayesian approach to Sholl analysis

Abstract

Motivation: Due to the link between microglial morphology and function, morphological changes in microglia are frequently used to identify pathological immune responses in the central nervous system. In the absence of pathology, microglia are responsible for maintaining homeostasis, and their morphology can be indicative of how the healthy brain behaves in the presence of external stimuli and genetic differences. Despite recent interest in high throughput methods for morphological analysis, Sholl analysis is still widely used for quantifying microglia morphology via imaging data. Often, the raw data are naturally hierarchical, minimally including many cells per image and many images per animal. However, existing methods for performing downstream inference on Sholl data rely on truncating this hierarchy so rudimentary statistical testing procedures can be used.

Results: To fill this longstanding gap, we introduce a parametric hierarchical Bayesian model-based approach for analyzing Sholl data, so that inference can be performed without aggressive reduction of otherwise very rich data. We apply our model to real data and perform simulation studies comparing the proposed method with a popular alternative.

Availability and implementation: Software to reproduce the results presented in this article is available at: https://github.com/vonkaenelerik/hierarchical_sholl. An R package implementing the proposed models is available at: https://github.com/vonkaenelerik/ShollBayes.

Figure 1.

The mean model induced by Equation (1) as each parameter varies. (A) The growth parameter ${\alpha }_1$ controls the behavior of the curve before the change-point. (B) The decay parameter ${\alpha }_2$ controls the behavior of the curve after the change-point. (D) The parameter $\tau$ controls the branch maximum of the fitted curve via $e^{\tau }$. (C) The parameter $\gamma$ controls the critical value, i.e. the change-point, of the fitted curve.

Figure 2.

Hierarchical structure for a common experimental design with two binary factors (ND/MD and I/C) and an interaction. We assume parameters at any level are randomly sampled from the corresponding distribution in the next highest level. Here, for some combination of groups $\ast$, $\varphi (\ast )$ denotes the Gaussian distribution, ${\zeta }^{\ast }$ denotes group-level parameters, ${\Omega }^{\ast }=({\omega }_1^{\ast },\dots ,{\omega }_L^{\ast })$ denotes animal-level parameters, and $Y^{\ast }=(y_{1lm},\dots ,y_{\mathit{Nlm}})$ denotes Sholl curve process crossings. Additionally, group combination is indexed by m and we model group level effects as additive terms $b^{\ast }$ on the mean parameter for group level distributions. For a given parameter subscript, $\Sigma$ denotes the corresponding variance parameters for the Gaussian distribution for that parameter. Gaussian priors are truncated via A and B to enforce the parameter constraints of Equation (1).

Figure 3.

Group-level fitted curves obtained by fitting model 2 to the MD/ND dataset. (A) Fitted curves faceted by group, superimposed over animal level Sholl curves. One microglia from each group is superimposed over the corresponding panel. (B) All four facets from panel (A), superimposed to better show hyper-ramification of the MD/Contra group.

Figure 4.

95% credible intervals for each effect in the MD/ND dataset, computed as the highest density posterior interval. Credible intervals are superimposed over the approximate posterior distributions obtained via MCMC. Estimated posterior means are represented by black dots with point estimates displayed above. The dotted vertical line is fixed at zero.