Estimating Variability in Functional Images Using a ... - Semantic Scholar

Estimating Variability in Functional Images Using a Synthetic Resampling Approach




Ranjan Maitra and Finbarr O’Sullivan



Statistics and Data Analysis Research Group, Bellcore, Morristown, NJ 07960-6438, USA  Department of Statistics, University of Washington, Seattle WA 98195, USA

Abstract Functional imaging of biologic parameters like in vivo tissue metabolism is made possible by Positron Emission Tomography (PET). Many techniques, such as mixture analysis, have been suggested for extracting such images from dynamic sequences of reconstructed PET scans. Methods for assessing the variability in these functional images are of scientific interest. The nonlinearity of the methods used in the mixture analysis approach makes analytic formulae for estimating variability intractable. The usual resampling approach is infeasible because of the prohibitive computational effort in simulating a number of sinogram datasets, applying image reconstruction, and generating parametric images for each replication. Here we introduce an approach that approximates the distribution of the reconstructed PET images by a Gaussian random field and generates synthetic realizations in the imaging domain. This eliminates the reconstruction steps in generating each simulated functional image and is therefore practical. Results of experiments done to evaluate the approach on a model one-dimensional problem are very encouraging. Post-processing of the estimated variances is seen to improve the accuracy of the estimation method. Mixture analysis is used to estimate functional images; however, the suggested approach is general enough to extend to other parametric imaging methods.

I. I NTRODUCTION The potential to quantitate tissue metabolism from a sequence of PET scans is one of the most powerful features of this imaging modality. In this context, the protocol typically consists of injecting a patient with a radio-tracer and recording the emissions at discrete time-points. From emissions recorded at each time-point, the tissue isotope concentration or source distribution is estimated, giving us a time-course sequence of reconstructed PET scans. These scans form the input for algorithms that output pixel-wise estimates of biologic parameters like metabolic rate, phosphorylation ratio, etc. A number of techniques have been developed to generate these images; we have been using a mixture analysis technique (O’Sullivan [12]) in our experiments. Assessing variability in these estimated functional images is of scientific importance because these can potentially be used to develop inference tools like calculating significance levels of tests of hypothesis on biologic activities in different regions in single-patient studies. The problem of developing

practical variability measures for reconstructed PET scans at fixed time-points has been studied extensively ([3],[7],[8]). Blomqvist et al. [2] noted the desirability of extending these results to functional images. Unfortunately, the nonlinear formulations used in constructing the biologic parameter estimates make analytic variance formulae intractable. Extending the resampling approach of Haynor and Woods [7] would involve simulating a number of sinogram data sets, applying image reconstruction, mixture analysis and generating parametric images. This approach is impractical because of the excessive computational effort required to replicate dynamic reconstructed PET sequences. In this paper, we suggest a simulation approach via the parametric bootstrap [5] executed in the imaging domain. We use the result in Maitra [11] that with increased count rate, each reconstructed PET scan has an approximate multivariate Gaussian distribution. The mean is estimated by the reconstructed image. Computationally feasible and accurate dispersion estimates are suggested. This model is used to simulate dynamic PET sequences, from each of which biologic parameters are extracted. This yields a bootstrap sample of the functional images, which can be used to assess variability. The advantage of this synthetic approach over the usual one is that it eliminates the computationally expensive step of reconstructing time-course sequences after simulating from the observation process. In the sequel, Section 2 formulates the problem and outlines the theory and develops the methodology behind our approach. Section 3 details the experimental evaluations that were carried out to examine the performance of our suggested approach. Since it is not possible to validate our methods in a two-dimensional PET setup, the suggestions are evaluated on experiments performed in a model one-dimensional deconvolution problem with reconstruction characteristics similar to PET. The results are presented in Section 4. Finally, Section 5 summarizes the contributions of this paper and poses questions for future research.

II. THEORY

AND

M ETHODS

A. Problem Formulation 1) Image Reconstruction The standard reconstruction methodology for PET is an algorithm known as filtered backprojection (FBP). In convolution form, this method involves filtering of the data from each projection angle followed by back-projection. The

equation for the  ’th reconstructed pixel value

   

        !"$#&%('



is, (1)

&#*),+-%/.10 32 65

Here, % denotes angle, 0 distance, 42 is the corrected sinogram data and   78 is the convolution filter with resolution size (FWHM) 9 . In matrix notation, the reconstruction equation can be written in terms of the expression,  

:  =?;@BA



(2)

;>=C2

where 2 is the vector of corrected projection data, ; is the discretized version of the Radon transform, and :  represents the smoothing operation of FWHM 9 that is applied to the raw reconstructions in order to obtain acceptable solutions [13]. 2) Functional Imaging via Mixture Analysis Local tissue metabolism has usually been assessed from dynamic PET scans by modeling locally averaged time-course measurements [15]. Functional imaging techniques, like mixture analysis [12], generate more comprehensive pixel-wise representations.  Let 3D represent the true source distribution in the D ’th time-bin at the   ’th pixel in the PET5 imaging domain. The  vector 78  4 ED F*D HG IJKKLK*M is called the true timeactivity curve (TAC) at the  ’th pixel. A ; -component mixture model represents the  ’th pixel TAC as a weighted average of ; underlying curves (sub-TACs), NLOJP QG BI LKKLK*; . R 

3D 

(3)

3ONLO3D  OTS $U

R 

3O  O S [U 6

W X OTZ

W

(4)

 W  X O6Z 3 O  O S  U T



3) Assessing Variability

(6)

W

Analytic expressions for Var( ) are intractable because of the nonlinear methods used in the extraction. The prohibitive cost of generating time-course sequences from realizations in the observation domain makes the usual resampling approach impractical. This suggests the need for development of variance estimation strategies.

B. A Synthetic Variability Estimation Strategy 1) Approximate Distribution of

 

Maitra [11] shows that under idealized projection conditions of no detector effects such as scatter, attenuation, etc., the   at any fixed distribution of the reconstructed PET scan time-point can be approximately and adequately specified by a multivariate Gaussian distribution. The mean of this distribution is :  while From (2), the dispersion matrix b of   is given by,

5

where the mixing proportions 4 3O FP  lie G IJKKLK*; U in the ; -dimensional simplex. The physical basis for such a representation is that the sub-TACs (N ’s) correspond to the different tissue types represented in the image and the underlying