A General Bayesian Functional Spatial Partitioning Method for Multiple Region Discovery Applied to Prostate Cancer MRI

Abstract

Current protocols to estimate the number, size, and location of cancerous lesions in the prostate using multiparametric magnetic resonance imaging (mpMRI) are highly dependent on reader experience and expertise. Automatic voxel-wise cancer classifiers do not directly provide estimates of number, location, and size of cancerous lesions that are clinically important. Existing spatial partitioning methods estimate linear or piecewise-linear boundaries separating regions of local stationarity in spatially registered data and are inadequate for the application of lesion detection. Frequentist segmentation and clustering methods often require pre-specification of the number of clusters and do not quantify uncertainty. Previously, we developed a novel Bayesian functional spatial partitioning method to estimate the boundary surrounding a single cancerous lesion using data derived from mpMRI. We propose a Bayesian functional spatial partitioning method for multiple lesion detection with an unknown number of lesions. Our method utilizes functional estimation to model the smooth boundary curves surrounding each cancerous lesion. In a Reversible Jump Markov Chain Monte Carlo (RJ-MCMC) framework, we develop novel jump steps to jointly estimate and quantify uncertainty in the number of lesions, their boundaries, and the spatial parameters in each lesion. Through simulation we show that our method is robust to the shape of the lesions, number of lesions, and region-specific spatial processes. We illustrate our method through the detection of prostate cancer lesions using MRI.

Keywords: Biomedical imaging; Functional estimation; Reversible Jump MCMC; Spatial partitioning; Spatial statistics.

Figure 1:

An example of how spatial locations are assigned given multiple boundaries. Boundaries $B_1$ and $B_2$ are shown in black. $s=\left(x,y\right)$ is a spatial location, $c_1$ is the centroid of $B_1$ at the current iteration, ${\theta }_1$ is the angle between the horizontal and the line from $s$ and $c_1$, and $f\left({\theta }_1\right)$ is the magnitude of the boundary at ${\theta }_1$. In this example, $s$ is categorized as within region $D_2$ but not $D_1$ : the distance from $s$ to $c_1$ is greater than $f\left({\theta }_1\right)$.

Figure 2:

Estimated partitioning boundary (black) and 95% credible bands (grey) for one simulated dataset. Color represents voxel intensity. The procedure for simulating this data is outlined in Section 5.

Figure 3:

Distributions of sensitivity, specificity, and Dice coefficient of BFSP-M, KM, Two-Stage CART, BayesImageS, and BFSP-1 over 400 simulated data sets.

Figure 4:

From left to right: average partitioning results of the BFSP-M, KM, Two-Stage CAFT, BayesImageS, BFSP-1 methods in the simulation study. Each image represents 100 simulations of the first simulation setting. The color represents the proportion of time the method classified each spatial location as within a target region with red close to 1 and blue close to 0. The black outline shows the true boundaries.

Figure 5:

From left to right: Map of voxelwise predicted cancer probabilities from five slices where color indicates probability of cancer with red close to 1 and blue close to 0 (Jin et al., 2018), true cancer status, partitioning results from BFSP-M, Two-Stage CART, KM, BayesImageS, and BFSP-1.

Figure 6:

From left to right: Map of voxelwise predicted cancer probabilities derived from mpMRI data from 2 slices containing multiple lesions, where color indicates probability of cancer with red close to 1 and blue close to 0 (Jin et al., 2018), true cancer status, lesion uncertainty of BFSP-M where color indicates the proportion of time that a lesion was included in the MCMC posterior draws post burn-in.