There is an abundance of prior information available for image-guided radiotherapy, making it ideally suited for Bayesian techniques. I will demonstrate some results from applying the method of Teo, Sapiro & Wandell (1997) to cone-beam computed tomography (CT). A previous CT scan of the same object forms the prior expectation. The posterior probabilities of class membership are smoothed by diffusion, before labeling each pixel according to the maximum a posteriori (MAP) estimate. The effect of the prior and of the smoothing is discussed and some potential extensions to this method are proposed.
Informative Priors for Segmentation of Medical Images
Motivation Method 1 Method 2 Extensions Conclusion Informative Priors for Segmentation of Medical Images Matt Moores1,2 , Cathy Hargrave3 , Fiona Harden2 & Kerrie Mengersen1 1 Discipline of Mathematical Sciences, Queensland University of Technology 2 Discipline of Medical Radiation Sciences, Queensland University of Technology 3 Radiation Oncology Mater Centre, Queensland Health Bayes on the Beach, 2011
Motivation Method 1 Method 2 Extensions ConclusionitkBayesianClassifierImageFilter 1 estimate µ using k-means 2 estimate σ 2 for each cluster (mixing proportions are assumed equal) 3 create a matrix y∗ : for each pixel yi and each cluster Ck ∼ N(µk , σk ), yik = p(yi |µk , σk ) 6 5 4 classify each pixel yi according to the largest value of yik
Motivation Method 1 Method 2 Extensions ConclusionitkBayesianClassifierImageFilter 1 estimate µ using k-means 1 select initial values for µ 2 assign each pixel y to the nearest µk 3 recalculate each µk by averaging over the members of k 4 repeat steps 2 & 3 until none of the pixel assignments change 2 estimate σ 2 for each cluster (mixing proportions are assumed equal) 3 create a matrix y∗ : for each pixel yi and each cluster Ck ∼ N(µk , σk ), yik = p(yi |µk , σk ) 6 5 4 classify each pixel yi according to the largest value of yik
Motivation Method 1 Method 2 Extensions ConclusionPrior 4 matrix pik representing the prior probability of pixel i belonging to cluster k then pixel classiﬁcation is based on the posterior pik × yik but: this has no eﬀect on the number of clusters, nor on their parameters µk and σk can’t use the posterior from one classiﬁcation as the prior for another, unless the clusters are the same
Motivation Method 1 Method 2 Extensions Conclusionhidden Markov random ﬁeld Joint distribution of observed intensities y and unobserved labels z: p(y, z|µ, τ ) ∝ p(y|µ, τ , z)p(z) (1) 1 yi |µj , τj , zi = j ∼ N µj , (2) τj N p(z) = C(β)−1 exp αi (zi ) + β wij f (zi , zj ) (3) i=1 i∼j simple Potts model (without external ﬁeld): p(z) = C(β)−1 exp β I(zi = zj ) (4) i∼j
Motivation Method 1 Method 2 Extensions Conclusioninformative prior for µ and τ 200 200 0 0 −200 −200 Hounsfield unit pixel intensity −400 −400 −600 −600 −800 −800 −1000 −1000 0 1 2 3 4 0 1 2 3 4 Electron Density Electron Density (a) Fan-Beam CT (b) Cone-Beam CT
Motivation Method 1 Method 2 Extensions Conclusionexternal ﬁeld N In equation (3) earlier, the term exp i=1 αi (zi ) deﬁnes an external ﬁeld. Figure: manual contours of the organs of interest.
Motivation Method 1 Method 2 Extensions Conclusionexternal ﬁeld II Prior probabilities αi (zi ) for each pixel can be generated by simulation, based on: geometry of each organ, from the treatment plan variability in size and position, from published studies Axis prostate seminal vesicles Ant-Post x = 0.1, sd = 4.1 mm x = 1.2, sd = 7.3 mm Sup-Inf x = −0.5, sd = 2.9 mm x = −0.7, sd = 4.5 mm Left-Right x = 0.2, sd = 0.9 mm x = −0.9, sd = 1.9 mm Table: Mean x and standard deviation sd of observed  variability in position, along three axes: anteroposterior (Ant-Post); superoinferior (Sup-Inf); & lateral (Left-Right) relative to the patient.
Motivation Method 1 Method 2 Extensions Conclusionhybrid model Chen & Metaxas [6, 7] deﬁne the object boundary implicitly as the zero level set of a cost function: ∂φi φi φi = λ1 M i + λ 2 Pi · − (λ2 Pi + λ3 ) · ∂t φi φi (5) where: Mi is the inﬂation force (total gradient magnitude) Pi is the local image force at each pixel (probability of pixel j belonging to object i) non-overlapping constraint φi · φi is the local curvature (surface smoothness constraint)
Motivation Method 1 Method 2 Extensions ConclusionSummary Two Bayesian approaches to medical image segmentation: k-means with posterior diﬀusion (itkBayesianClassifierImageFilter) hidden Markov random ﬁeld (PyMCMC) Potential extensions to Potts MRF: external ﬁeld deﬁned by size and position of objects hybrid Level Set model
Motivation Method 1 Method 2 Extensions ConclusionReferences I P. Teo, G. Sapiro and B. Wandell (1997) Creating connected representations of cortical gray matter for functional MRI visualization. IEEE Trans. Med. Imag. 16: 852-863. J. Melonakos, K. Krishnan and A. Tannenbaum (2006) An ITK Filter for Bayesian Segmentation: itkBayesianClassifierImageFilter The Insight Journal http://hdl.handle.net/1926/160 Strickland, C. M., Denham, R. J., Alston, C. L., & Mengersen, K. L. (2011) PyMCMC : a Python package for Bayesian Estimation using Markov chain Monte Carlo. Journal of Statistical Software (In Press) C. Alston, K. Mengersen, C. Robert, J. Thompson, P. Littleﬁeld, D. Perry and A. Ball (2007) Bayesian mixture models in a longitudinal setting for analysing sheep CAT scan images. Computational Statistics & Data Analysis 51(9): 4282-4296.
Motivation Method 1 Method 2 Extensions ConclusionReferences II S.J. Frank, L. Dong, R. J. Kudchadker, R. De Crevoisier, A. K. Lee, R. Cheung, S. Choi, J. O’Daniel, S. L. Tucker, H. Wang, et al. (2008) Quantiﬁcation of Prostate and Seminal Vesicle Interfraction Variation During IMRT. International Journal of Radiation Oncology*Biology*Physics 71(3): 813-820. T. Chen and D. Metaxas (2005) A hybrid framework for 3D medical image segmentation. Medical Image Analysis 9(6): 547-565. T. Chen, S. Kim, J. Zhou, D. Metaxas, G. Rajagopal & N. Yue (2009) 3D Meshless Prostate Segmentation and Registration in Image Guided Radiotherapy. In Proceedings of MICCAI 43-50. P. Th´venaz, T. Blu & M. Unser (2000) Interpolation Revisited. e IEEE Trans. Medical Imaging 19(7): 739–758.
Motivation Method 1 Method 2 Extensions ConclusionAcknowledgements Bayesian Research & Applications Group at QUT Radiation Oncology Mater Centre: Emmanuel Baveas Rebecca Owen Timothy Deegan Steven Sylvander John Baines Dr. Michael Poulsen