2010 Walker-Samuel Et Al. Magnetic Resonance in Medicine

Bayesian Estimation of Changes in Transverse Relaxation Rates Simon Walker-Samuel, 1 * y Matthew Orton, 1 Lesley D. McPhail, 1 Jessica K. R. Boult, 1 Gary Box, 2 Suzanne A. Eccles, 2 and Simon P. Robinson 1 Although the biasing of R* 2 estimates by assuming magnitude MR dat a to be nor mal ly dist ributed has been descri bed, the effect on changes in R* 2 ( D R* 2 ), such as induced by a paramagnetic contrast agent, has not been reported. In this study, two versions of a novel Bayesian maximum a posteriori approach for estima ting D R* 2 ar e descri bed and evaluated: one tha t assumes normally distributed data and the other, Rice-distributed data. The approach enables the robust, voxelwise deter- minati on of the uncertain ty in D R* 2 estimates and provides a useful statistical framework for quanti fyi ng the probabilit y that a pixel has been significa ntly enhanced. This technique was evalu ated in vivo, using ultra small superpara magne tic iron oxide particles in orthotopic murine prostate tumors. It is shown that assuming magnitude data to be normally distributed causes D R* 2 to be under estima ted when signal-to-n oise rat io is modest . Howeve r , the biasin g eff ect is less than is found in R* 2 estimates, implying that the simplifying assumption of nor mal ly distri but ed noi se is mor e justif iable whe n evaluating D R* 2 compa red with when evalua ting precontras t R* 2 values . Magn Res on Med 64:9 14–9 21, 20 10. V C 2010 Wiley- Liss, Inc. Key words : Bayes ian; infer ence; transvers e; relax ation; iron oxide; susceptibility; Rician; Rice; orthotopic; prostate; tumor; likelihood; maximum a posteriori A range of applications in MRI relies on the estimation of changes in the transverse relaxation rate R* 2 (¼ 1/T * 2 ) fr om si gnal ma gn it ud e da ta . Su ch change s can be brought about by the administration of paramagnetic contra st agen ts suc h as gadoli nium che late s (1) , ult ra- sma ll superparamag net ic iron oxide (USPIO) par tic les (2, 3), or inhala tion of hig h oxy gen con tent gas es (4, 5). Suc h tech niq ues can allow blo od vol ume , flow, blood oxyge nation , and/o r vesse l size to be estimated (4,6–9). To estimate R* 2 , mag nit ude image dat a are typ ica lly fi tte d to a si mple exponential decay wi th echo time, from which R* 2 is the associated rate cons tan t of the decay. Whenever a parameter value such as this is estimated from data corrupted by even moderate noise, the point estimate has an associated uncertainty. By robu stly modeling a given set of data, it is generally straightfor- war d to derive estimates of these unc ert ainties, whi ch can then be propagated into any further calculation. For such estimates to be reliable, it is required that an appro- pr iate mo de l of the nois e be inco rporat ed into the analysis. The inf lue nce of the Ric e dis tri but ion of mag nit ude MR data on par ameter est ima tes, whe n ass umi ng normally distrib uted data, has been studi ed exten sively in the literature (10–14). It has been shown that, when signal -to-noise ratio (SNR) is mod est or low, bia s can be int rod uce d int o est ima tes of expone ntial dec ay con- stants (11–13,15–17). Strategies have been developed to acco unt for thi s eff ect; for exa mpl e, in app arent dif fu- sion coefficient estimates in brain diffusion tensor studies (18–22) and in heterogeneous solid tumors (23). Simila rly, bias intro duced into R* 2 estimates by assuming normally distributed data was reported in functional brain MRI studies (24) and in R 2 measurements of iron- loaded liver (25). Anot her st udy expl icit ly recom- mended the use of a rob ust analysis in the estimatio n of R* 2 (17). However, when estimating changes in relaxation rates, it has not been reported whether the full representation of the Rician data distribution is necessary. Although it mi ght be assumed that bi as evident in R* 2 estimates wou ld be ref lec ted in DR* 2 estimates (i.e., estimate s of the chang e in R* 2 ), an ass ess ment of the magni tude of this effect is useful; if it is minimal, then a more compu- tationally efficient algorithm based on the assumption of normally distributed data can be used for parameter estimation . Furthermore, when utilizing negative contr ast mechanisms such as USPIO nanoparticles, which inher- ently reduce signal to noise, a robust analysis approach is required that can take this into account. As such, a Baye si an maxi mu m a po st erio ri (MAP) mode l is des cri bed in thi s stu dy tha t, thr oug h its genera tio n of marginal DR* 2 parameter distr ibutio ns, prov ides a thor- ough treatment of point estimates and uncertainties and can estimate the probability that DR* 2 is greater than or less than zero, on a pixel-by-pixel basis, in vivo. THEORY Parameter Optimization R* 2 is typically estimated from magnitude MR data, using a multigradient echo (MGE) sequence with multiple echo time acquisitions. Here, the signal magnitude is modeled 1 Cance r Resea rch UK & EPSR C Cance r Imag ing Centre, The Insti tute of Cancer Research, Sutton, Surrey, United Kingdom. 2 Can cer Research UK Centre for Cancer The rapeutics, The Ins tit ute of Cancer Research, Sutton, Surrey, United Kingdom. Grant spo nsor: MRC and Dep art men t of Health (En gland; Institu te of Cancer Researc h CR-UK and EPSRC Cancer Imagin g Centre and King’ s Colleg e Lond on and UCL Compr ehens ive Cance r Imag ing Centre ); Gran t number : C10 60/ A1033 4 and C1519/A1 033 1; Gra nt spo nsor: CR-UK Project; Grant number: C16412/A6269. y Present address: Centre for Advanced Biomedical Imaging, Lower Ground Floor , Paul O’Gorman Building , Unive rsity Colleg e Lond on, 72 Huntle y Street, London WC1E 6DD, UK. *Corre spond ence to: Simo n Walker-Samuel , Ph.D ., Centr e for Adva nced Bio medical Ima gin g, Lower Gro und Flo or , Pau l O’Gorman Bui ldi ng, University College London, 72 Huntley Stree t, London WC1E 6DD, UK. E-mail: [email protected] Receiv ed 27 November 2009; revised 3 March 2010; accepte d 5 March 2010. DOI 10.1002/mrm.2 2478 Published online 4 May 2010 in Wiley Online Library (wileyonlinelibrary.com). Magnetic Resonance in Medicine 64:914–921 (2010) V C 2010 Wiley-Liss, Inc. 914

2010 Walker-Samuel Et Al. Magnetic Resonance in Medicine

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