This work is motivated by a quantitative Magnetic Resonance Imaging study

This work is motivated by a quantitative Magnetic Resonance Imaging study of the differential tumor/healthy tissue change in contrast uptake induced by radiation. summarize the tumor contrast uptake relative to healthy tissue. In the second stage, we fit a population model to the U estimate and statistic when it achieves its maximum. Our initial findings suggest that the maximal contrast uptake of the tumor core relative to healthy MG-132 tissue peaks around three weeks after initiation of radiotherapy, though this warrants MG-132 further investigation. index the visits at time (= 1, 2, ? , at time and pixel at time are the same pixel). We denote the observed contrast uptake of pixel at time by in vector form as y= (images as We decompose the observed contrast uptake into a true contrast uptake and a measurement error, i.e. = + where denotes the unobserved true contrast uptake of pixel at time = (We specify flat (improper) prior distribution for the standard deviation MG-132 of the measurement error, i.e. as the true contrast uptake from the Neurod1 right time > 1, this is formalized as = is the structural change (since is the evolution error of pixel at time for all and as the true change in contrast uptake with change in quotes for simplicity. We write in vector form = (, and C 0. Specifically, let (= 1, 2, ? , possible values, (= (possible values: ~ Uniform( assume one of possible values at time for all 1 = 2.5 and = 5 , such that the prior mode of is 0.3. In Section 4 Later.1, {we investigate the sensitivity of the results to other choices of on the state space = {1,|we investigate the sensitivity of the total results to other choices of on the state space = 1, ? , index the value of and out of the choices, i.e. = and when = ~ and exp{= = (= = controls the strength of the spatial correlation of the labels, e.g. large encourages smoother configurations. A priori, we assume is uniform on the integers from ~ Uniform(0, = (= 0, 0.1, ? , 3.0 and = 2, 3, ? , 20 (Appendix 1, Zhang et al. 2008b). Its value on non-grid points is computed via linear interpolation. Furthermore, our model induces the following covariance structure at each pixel over time. The covariance of the observed contrast uptake at the same pixel at any two visits (= {: = at time have closed form due to conjugacy. By Bayes Theorem the conditional posterior distributions of Zand satisfy = 1, ? , Following Zhang et al. (2008b), we characterize the contrast uptake by its posterior mean, i.e. and when ) is greater than a random drawn value from healthy tissue (> from > > = > + > + percentile of the posterior mean (baseline contrast uptake) image as the threshold between tumor core and annulus. In principle, the threshold of tumor core versus annulus MG-132 should be high (e.g. 95-th percentile of healthy tissue) in order to protect the bulk of healthy brain tissue. We are not aware of systematic or optimal choice of such a value. Therefore, we conduct sensitivity analysis on our choice in Section 4.4.2. 2.3 (1 at time (1 (gray profiles in Figures 5a and 5c). Since the U statistic is bounded between 0 and 1, we apply a logit transformation to before fitting the population model, i.e. logit(C log(1 C ~ N(0, ~ N(0, ~ N(0, ~ Uniform(0, 183) and an improper flat prior density for = (= = = 1, = = 0.1, and = 0.5 such that when the visit of subject is left out and use a checking function is the posterior predictive mean of (Gelfand et al. 1992). A straightforward approach is to fit the population model multiple times with one observation left out each time. Instead, the method is used by us described in Chen, Shao and Ibrahim (2000) to simplify the posterior predictive check. We denote the posterior distribution by P(O ) and the distribution of the data by P(O O A) P(O A\O O A) not P(O Avisits of subject sequentially. At time have closed form and are updated via standard Gibbs sampling steps. The spatial.

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