An automatic counting algorithm for the quantification and uncertainty analysis of the number of microglial cells trainable in small and heterogeneous datasets

TL;DR

This work tackles the problem of counting microglial cells in lumbar spinal cord cross-sections of rats by omitting cell detection and focusing only on the counting task, and designs an automatic kernel counter that is a non-parametric and non-linear method.

Abstract

Counting immunopositive cells on biological tissues generally requires either manual annotation or (when available) automatic rough systems, for scanning signal surface and intensity in whole slide imaging. In this work, we tackle the problem of counting microglial cells in lumbar spinal cord cross-sections of rats by omitting cell detection and focusing only on the counting task. Manual cell counting is, however, a time-consuming task and additionally entails extensive personnel training. The classic automatic color-based methods roughly inform about the total labeled area and intensity (protein quantification) but do not specifically provide information on cell number. Since the images to be analyzed have a high resolution but a huge amount of pixels contain just noise or artifacts, we first perform a pre-processing generating several filtered images {(providing a tailored, efficient feature extraction)}. Then, we design an automatic kernel counter that is a non-parametric and non-linear method. The proposed scheme can be easily trained in small datasets since, in its basic version, it relies only on one hyper-parameter. However, being non-parametric and non-linear, the proposed algorithm is flexible enough to express all the information contained in rich and heterogeneous datasets as well (providing the maximum overfit if required). Furthermore, the proposed kernel counter also provides uncertainty estimation of the given prediction, and can directly tackle the case of receiving several expert opinions over the same image. Different numerical experiments with artificial and real datasets show very promising results. Related Matlab code is also provided.

Figures (9)

• Figure 1: Example of complete image (spinal cord cross-section). The zoom frame indicates part of the ipsilateral dorsal horn with microglial cells (scale = 200$\mu m$).
• Figure 2: Examples of microglial cells (up: ramified; down: amoeboid). Microglia is always represented by a brownish tinction, whereas blue colored cells usually correspond to the nuclei of neurons.
• Figure 3: Graphical representation of the analysis performed for the feature extraction in each image. In each image filtered by ${\bf t}^{(k)}$, the total number of objects $r_{kd}$ is obtained by clustering.
• Figure 4: Illustrative example of a generic $d$-th image and $T=4$ corresponding filtered images, with different threshold vectors ${\bf t}^{(k)}$. In this graphical example, we have $N_d=4$ objects of interest (i.e., in our application, microglial cells). The rest of the $6$ objects in the image play the role of irrelevant artifacts. In each filtered image, the total number of objects $r_{kd}$ is given.
• Figure 5: (a) The two functions $v_i=v(\alpha_i)$ and $s_i=s(\alpha_i)$ in Eqs.Ê\ref{['Vieq']} and \ref{['Sieq']}. Recall that $\min F_i=v_i$ and $\max F_i=s_i$. ( b) The mean square error (MSE) in Eq. \ref{['MSEEq']} as function of the number of threshold vectors used, i.e., $T$.