Introduction

Segmentation is crucial in brain imaging analysis to determine volumes of tissue compartments (TCs). Changes in tissue volume are often investigated in studies concerning healthy development, or pathological changes e.g. neurodegeneration. Segmentation software that produces probability maps of TCs is commonly used. However, this requires the selection of a probability threshold (T_p) to produce binary maps to segment TCs and calculate their volumes. For a given TC, every voxel with probability above T_p is included, and every voxel with probability below T_p is excluded. An issue with this approach is the lack of consensus on the choice of optimal T_p^1-5, although previous studies agree that the choice of T_p has large effect on binary segmentation results^1,2. We present a way to minimise those effects by quantifying uncertainty using Shannon entropy. Entropy measures the average uncertainty in a random variables^6,7, and was used previously in segmentation to assess the uncertainty using machine learning⁸. To our knowledge, it has never been utilised in probabilistic segmentation (using e.g. SPM12⁹) of healthy brain MRI. We investigate to what degree Shannon entropy can help with minimising underestimation or overestimation of brain TCs volume measurements by providing an uncertainty statement on the results of segmentation.

Methods

We used T1-weighted scans of 5 healthy adult brains, acquired on a Siemens Prisma 3T scanner. The results are presented for a representative example. The data were not further corrected to avoid contaminating results with effects of pre-processing on uncertainties. Probability maps of grey matter (GM), white matter (WM) and cerebrospinal fluid (CSF) were derived using SMP12 software⁹ with default settings. Uncertainties were derived as entropy map using Shannon entropy equation⁶:

$$H(p)=\sum_{i=1}^np(x_i)log_2(p(x_i))\quad(1)$$

where: p(x_i) is the probability of a voxel belonging to TC class i, with i = 1,2,…n, where n is a number of TCs. A bespoke algorithm was used to capture uncertainty in the high probability class (consisting of voxels with probability > T_p) by adding a new class – uncertain class. The algorithm utilises entropy map and probability maps, with corresponding thresholds. Each voxel can be in one or more certain or uncertain classes. If the entropy of a given voxel is below entropy threshold (T_H), we assume the voxel belongs to one tissue only, and place it in the certain class for which probability > T_p. Otherwise the voxel is placed in all uncertain classes of TCs for which probability > T_p. Two assumptions are made: 1) For a given voxel with entropy < T_H, only one TC have probability > T_p; 2) If a voxel has entropy < T_H and probability > T_p for one TC, it cannot belong to uncertain class of other TCs. We also investigated how volumes of certain and uncertain classes for each TC vary with fixed T_H and T_p ranging from 0.0 to 1.0 (step size 0.05). Empirical choice of T_H = 0.4 was made so the volumes of the certain class are not particularly sensitive to the choice of T_p.

Results

Entropy map (Fig. 1) informs about uncertainties in each individual voxel. Higher entropy means higher uncertainty. We observe that higher uncertainty is present specifically at the edges between TCs. Fig. 2 shows each segmented TC at low T_p (0.2) with estimated uncertainty (red regions). Fig. 3 shows that at a fixed T_H, for T_p between ≈ 0.2 and 0.8, there is a plateau in the variation of the certain class. Table 1 summarises the calculated volumes of each TC with percentages indicating the portion of total volume belonging to uncertain class. Fig. 4 presents all segmented tissues with uncertainties.

Discussion

Brain tissue volume estimation is a key indicator for healthy development as well as disease progression, but it is highly dependent on segmentation. Previous studies^1,2 established that the choice of threshold on probability maps highly influences resulting volumes. However, there is little consensus on optimal threshold choice ^1-5. Including more volume solves possible underestimation. Including uncertainty (estimated with Shannon entropy) solves possible overestimation by informing which regions may still belong to a given tissue but with lower certainty. In Fig. 2, the choice of low T_p was motivated by inspecting results from Fig 3, and that minimises the effect of choice of T_p on the resulting segmentation. The uncertainty maps (Fig. 1 and Fig. 2) highlights tissue boundary regions as the most uncertain, which is expected due to partial volume effects. In these regions the choice of T_p can be particularly important and inconsistencies can easily mask important changes. Estimating and visualising uncertainties is a crucial step in making measurement of brain tissue volume a technically complete measurement. Furthermore, together Fig. 2 and Fig. 4 could present additional information that could be utilized in e.g. surgical planning, visualising uncertainties at tissue boundaries.

Conclusions

We showed that Shannon entropy can inform about uncertainty in MRI scan segmentation, which could be a useful information for e.g., studies investigating longitudinal volume changes. There is more uncertainty at the edges between tissues and the choice of T_p has a large effect on binary maps. Including uncertainty in volume measurement can minimise the effects of choice of T_p on the results.

Acknowledgements

This project is funded by UCL and the National Physical Laboratory as part of the EPSRC iCASE 2021-22 programme. The NPL contribution was funded by the Department of Business, Engineering and Industrial Strategy (BEIS) through the National Measurement Strategy under the Data Science Modelling & Analytics Applications theme.