The Annals of Applied Statistics

Standardization of multivariate Gaussian mixture models and background adjustment of PET images in brain oncology

Meng Li and Armin Schwartzman

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In brain oncology, it is routine to evaluate the progress or remission of the disease based on the differences between a pre-treatment and a post-treatment Positron Emission Tomography (PET) scan. Background adjustment is necessary to reduce confounding by tissue-dependent changes not related to the disease. When modeling the voxel intensities for the two scans as a bivariate Gaussian mixture, background adjustment translates into standardizing the mixture at each voxel, while tumor lesions present themselves as outliers to be detected. In this paper, we address the question of how to standardize the mixture to a standard multivariate normal distribution, so that the outliers (i.e., tumor lesions) can be detected using a statistical test. We show theoretically and numerically that the tail distribution of the standardized scores is favorably close to standard normal in a wide range of scenarios while being conservative at the tails, validating voxelwise hypothesis testing based on standardized scores. To address standardization in spatially heterogeneous image data, we propose a spatial and robust multivariate expectation-maximization (EM) algorithm, where prior class membership probabilities are provided by transformation of spatial probability template maps and the estimation of the class mean and covariances are robust to outliers. Simulations in both univariate and bivariate cases suggest that standardized scores with soft assignment have tail probabilities that are either very close to or more conservative than standard normal. The proposed methods are applied to a real data set from a PET phantom experiment, yet they are generic and can be used in other contexts.

Article information

Ann. Appl. Stat., Volume 12, Number 4 (2018), 2197-2227.

Received: November 2016
Revised: January 2018
First available in Project Euclid: 13 November 2018

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Background adjustment PET images tumor detection multivariate Gaussian mixture model outlier detection robust EM algorithm spatial modeling standardized scores tail distributions voxelwise inference


Li, Meng; Schwartzman, Armin. Standardization of multivariate Gaussian mixture models and background adjustment of PET images in brain oncology. Ann. Appl. Stat. 12 (2018), no. 4, 2197--2227. doi:10.1214/18-AOAS1149.

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Supplemental materials

  • Supplement: Additional material. The Supplementary material contains: (A) proofs of all theorems and lemmas in the main paper; (B) a simulation study to compare the proposed robust EM with the multivariate $t$ mixtures method [Peel and McLachlan (2000)] in a nonspatial setting; (C) additional simulation studies of the proposed RB-SGMM approach when lesions have smaller sizes and are noncircular.