Improved Inference in Bayesian Segmentation Using Monte Carlo Sampling: Application to Hippocampal Subfield Volumetry

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1 Improved Inference in Bayeian Segmentation Uing Monte Carlo Sampling: Application to Hippocampal Subfield Volumetry Juan Eugenio Igleia a, Mert Rory Sabuncu a, Koen Van Leemput a,b,c, for the Alzheimer Dieae Neuroimaging Initiative a Martino Center for Biomedical Imaging, Maachuett General Hopital and Harvard Medical School, USA b Department of Applied Mathematic and Computer Science, Technical Univerity of Denmark c Department of Information and Computer Science and of Biomedical Engineering and Computational Science, Aalto Univerity, Finland Abtract Many egmentation algorithm in medical image analyi ue Bayeian modeling to augment local image appearance with prior anatomical knowledge. Such method often contain a large number of free parameter that are firt etimated and then kept fixed during the actual egmentation proce. However, a faithful Bayeian analyi would marginalize over uch parameter, accounting for their uncertainty by conidering all poible value they may take. Here we propoe to incorporate thi uncertainty into Bayeian egmentation method in order to improve the inference proce. In particular, we approximate the required marginalization over model parameter uing computationally efficient Markov chain Monte Carlo technique. We illutrate the propoed approach uing a recently developed Bayeian method for the egmentation of hippocampal ubfield in brain MRI can, howing a ignificant improvement in an Alzheimer dieae claification tak. A an additional benefit, the technique alo allow one to compute informative error bar on the volume etimate of individual tructure. Keyword: Bayeian modeling, egmentation, Monte Carlo ampling, hippocampal ubfield 1. Introduction Many medical image egmentation method perform Bayeian inference on o-called generative model to deduce egmentation label from the available image information. The employed model commonly conit of a prior decribing the patial organization of anatomical tructure in the image domain, for example via occurrence and co-occurrence tatitic. They alo contain a likelihood term that model the relationhip between anatomical label and image intenitie. Prior can take on different form. Generic prior are popular in the computer viion literature, in which domain knowledge about the image content i often limited. Markov random field model, which encourage patial regularity, are a good example of uch prior (Beag, 1986; Boykov et al., 1). In medical imaging, prior that are tailored to the anatomical tructure of interet are typically more ueful. Thee prior are often in the form of tatitical atlae (Greitz et al., 1991; Thompon et al., 1; Roland et al., 4; Johi et al., 4; Yeo et al., 8), which decribe the relative frequency of label in a reference coordinate frame, repreenting an average of the population. Image regitration technique (Brown, 199; Wet et al., 1997; Maintz and Viergever, 1998; Zitova and Fluer, Data ued in preparation of thi article were obtained from the Alzheimer Dieae Neuroimaging Initiative (ADNI) databae (adni.loni.ucla.edu). A uch, the invetigator within the ADNI contributed to the deign and implementation of ADNI and/or provided data but did not participate in analyi or writing of thi report. A complete liting of ADNI invetigator can be found at: apply/adni_acknowledgement_lit.pdf 3; Pluim et al., 3) link the reference pace to the target image pace, allowing the tranfer of the label probabilitie to coordinate of the image to be egmented. The econd component of the generative model i the likelihood term, which pecifie how the different voxel label generate the oberved image intenitie at each location. The likelihood model the image formation proce, including artifact uch a noie and MRI bia field (Well et al., 1996). Many method aume a Gauian ditribution (or a mixture thereof) for each label cla preent in the image (Van Leemput et al., 1999). If the parameter of thee ditribution are learned from the target image uing Bayeian inference, rather than predefined or learned from a training dataet, the reulting egmentation method i robut againt change in modality. Thi i in contrat with dicriminative egmentation model, which excel when the target image appearance i conitent with the training et (e.g., computerized tomography, a in Zheng et al. 7) but falter when it i not. Appearance conitency i often not the cae in MRI, due to change in acquiition hardware and pule equence. Given the prior and the likelihood term, the poterior probability of a egmentation for a given target image can be inferred uing Baye rule. The poterior i a probability ditribution over the poible labeling of the image, and the mot likely egmentation can be computed by finding it mode. Popular method uch a (Zhang et al., 1; Fichl et al., ; Van Leemput et al., 1999; Ahburner and Friton, 5; Sabuncu et al., 1) are baed on thi principle. If the ultimate goal i not to produce a egmentation, but to compute decriptor for the different tructure (e.g., volume meaurement), then the Preprint ubmitted to Medical Image Analyi April, 13

2 whole probability ditribution rather than the mode can be ued in the etimation. Both the prior and the likelihood often depend on a number of unknown model parameter. In application that ue a tatitical atla, the prior i haped by the parameter of the regitration method that i ued to deform the atla toward the target image. For example, everal tate-of-the-art egmentation method employ thouand or million of regitration parameter (Fichl et al., 4b; Ahburner and Friton, 5; Pohl et al., 6; Van Leemput et al., 9). The parametrization of the likelihood i uually more conventional. For example, if a Gauian ditribution i aumed for each label, two parameter (mean and variance) per cla are required in the model. In a truly Bayeian framework, unknown model parameter need to be integrated out in the computation of the egmentation poterior. However, all aforementioned egmentation method employ point etimate for thee parameter, thereby applying Bayeian inference only in an approximate ene. For example, the regitration parameter are either pre-computed uing metric not necearily related to the probabilitic framework (Sabuncu et al., 1), or explicitly etimated to fit the egmentation model to the target imaging data (Fichl et al., 4b; Ahburner and Friton, 5; Pohl et al., 6; Van Leemput et al., 9), but in both cae only the obtained point etimate i ued to generate the final egmentation reult. Thi may lead to biaed egmentation reult for two different reaon. Firt, many reaonable atla deformation may exit in addition to the etimated one, epecially when the boundarie between the anatomical tructure are poorly defined by image intenitie and/or when the atla deformation ha a very large number of parameter. Second, the computed point etimate may not correpond to the global optimum of the relevant objective function, ince numerical optimizer eaily get trapped in local extrema in uch high-dimenional pace. Ignoring the uncertainty in the parameter of the likelihood term, a i commonly done, may further bia the reult in a imilar way. Depite thee iue, point etimate of the model parameter are often ued in the literature due to their computational advantage: they effectively ide-tep the difficult integration over the model parameter that i required in a more faithful Bayeian analyi. In thi paper, we propoe a better approximation of the egmentation poterior that fully conider the uncertainty in the model parameter, uing a computationally feaible trategy baed on Markov chain Monte Carlo (MCMC) ampling. We intantiate the approach within a recently propoed Bayeian method aiming to egment hippocampal ubfield in brain MRI can (Van Leemput et al., 9), and how that MCMC ampling yield hippocampal ubfield volume etimate that better dicriminate control from ubject with Alzheimer dieae. Moreover, the propoed approach alo provide more realitic and ueful error bar (defined a the tandard deviation of the error in a meaurement) on the volume than thoe obtained without accounting for model parameter uncertainty. To the bet of our knowledge, integration over model parameter ha not been explored before in the medical image egmentation literature. In the context of image regitration, Simpon et al. (11) propoed an approximation of the poterior ditribution of deformation field that outperform determinitic regitration when dicriminating Alzheimer dieae patient from healthy control. Riholm et al. (1, 11) viualized regitration uncertainty and etimated it effect on the accumulated doe in radiation therapy application, wherea Allaonniére et al. (7) marginalized over deformation in the context of contructing deformable model. There have been attempt to handle uncertainty etimate of patial alignment outide the Medical Image Analyi literature, too; ee for intance Pennec and Thirion (1997); Taron et al. (9); Kybic (1), which deal with hape, et of matched point and pixel data, repectively. In Tu and Zhu (), MCMC wa ued to generate a et of ditinct olution repreentative of the entire poterior egmentation ditribution in natural image. The ret of thi paper i organized a follow. In Section, we briefly ummarize the principle behind Bayeian egmentation model and preent the baeline hippocampal ubfield egmentation framework ued in thi paper. Section 3 detail the improved inference trategy propoed in thi tudy. Section 4 decribe the experimental etup to validate the propoed approach. Section 5 preent the reult of the experiment, and Section 6 conclude the paper. An early verion of thi work appeared in a conference paper (Igleia et al., 1).. Bayeian egmentation model and baeline method In thi ection we firt ummarize the general theory behind Bayeian egmentation method (Section.1). Then, we preent the pecific hippocampal ubfield egmentation algorithm that we ue to illutrate the method propoed in thi paper (Section.)..1. General framework Let y be the voxel intenitie correponding to an image, and the underlying egmentation label we wih to etimate. Bayeian egmentation method aim at finding the mot probable egmentation given the image uing Baye rule: ŝ = argmax p( y) = argmax p()p(y ). (1) Here, p() encode prior anatomical knowledge (uually in the form of an atla), wherea the likelihood p(y ) link the underlying anatomy with the image intenitie. Both the prior and the likelihood uually depend on a number of free parameter: p( x) and p(y, θ), where x repreent the parameter related to the prior and θ thoe related to the likelihood. Both et of parameter have (hyper-)prior ditribution p(x) and p(θ) that capture any prior knowledge we may have about them, o that: p() = p( x)p(x)dx, x p(y ) = p(y, θ)p(θ)dθ. θ

3 The common practice in the literature i to firt compute the mot probable value {ˆθ, ˆx} of the model parameter in light of the image intenitie: {ˆθ, ˆx} = argmax p(θ, x y), () {θ,x} and then, rather than maximize p( y), optimize the following expreion intead: ŝ argmax p( y, ˆθ, ˆx). (3) FIMBRIA PRESUBICULUM SUBICULUM CA1 CA-3 CA4-DG HIPPO. TAIL HIPPO. FISSURE GM WM CSF Thi i only an approximation becaue a true Bayeian analyi would conider all poible value of the unknown model parameter. More pecifically, Equation 3 can be interpreted a the mode approximation for the required integral over the model parameter: p( y) = x p( y, θ, x)p(θ, x y)dθdx θ (4) p( y, ˆθ, ˆx). (5) Equation 5 will be accurate if the poterior probability of the model parameter given the image intenitie i very harp and therefore well approximated by a Dirac delta, i.e., p(θ, x y) δ(θ ˆθ, x ˆx). A large number of egmentation method fall within thi general framework, differing in the way the prior and likelihood are pecified, and in the optimization algorithm that are ued to olve Equation and 3. Example include Well et al. (1996); Guillemaud and Brady (1997); Held et al. (1997); Van Leemput et al. (1999); Zhang et al. (1); Leemput et al. (1); Marroquin et al. (); Fichl et al. (, 4a); Pratawa et al. (4); Lorenzo-Valde et al. (4); Ahburner and Friton (5); Pohl et al. (6, 7); Xue et al. (7); Menze et al. (1); Sabuncu et al. (1), among other... Baeline egmentation method To illutrate the propoed MCMC-baed method, we build on a recently developed hippocampal ubfield egmentation method (Van Leemput et al., 9), which i part of the public oftware package FreeSurfer 1 (Fichl et al., ). Automatic egmentation of the ubfield ha recently attracted the interet of the neurocience community becaue different region of the hippocampal formation are affected differently by normal aging and Alzheimer dieae (Mueller et al., 1; Yuhkevich et al., 1). Here we ummarize the baeline method of Van Leemput et al. (9) within the general Bayeian egmentation framework decribed above...1. Generative model The algorithm relie on a tatitical atla of the hippocampu, in which a total of K = 11 different label correponding to the hippocampal ubfield and urrounding brain tiue are repreented: fimbria, preubiculum, ubiculum, CA1, CA/3, Figure 1: Tetrahedral-meh-baed atla of the hippocampal ubfield (baed on the right hippocampu), howing the label probabilitie with the meh uperimpoed. From left to right, top to bottom: coronal lice, 3D rendering, axial lice, agittal lice. The color map i diplayed on the right. Note that the color of a voxel i a um of the color correponding to the label that might occur at that location, weighed by their prior probabilitie. CA4/dentate gyru (CA4/DG), hippocampal tail, hippocampal fiure, white matter (WM), gray matter (GM) and cerebropinal fluid (CSF). For the likelihood term, the model aume that the intenitie of each tiue cla follow a Gauian ditribution with parameter that are unknown a priori. The tatitical atla i a generalization of the probabilitic atlae often ued in brain MR egmentation (Ahburner and Friton, 1997; Van Leemput et al., 1999; Leemput et al., 1; Zijdenbo et al., ; Fichl et al., ; Ahburner and Friton, 5; Pratawa et al., 4; Pohl et al., 6; Awate et al., 6; Pohl et al., 7). It i automatically etimated from manual egmentation of the hippocampal formation in highreolution MRI data from ten different ubject (Van Leemput et al., 9). Rather than uing voxel-wie tatitic, the atla i repreented a a tetrahedral meh that cover a bounding box around the hippocampu (Van Leemput, 9). Each of it approximately 8, vertice ha an aociated et of label probabilitie pecifying how frequently each of the K label occur at the vertex. The meh i adaptive to the degree of complexity of the underlying hippocampal anatomy in each region, uch that uniform region are covered by larger tetrahedra. Thi yield a parer repreentation than would otherwie be attainable. The poition of the vertice in atla pace (henceforth reference poition ) i computed along with the label probabilitie in a nonlinear, group-wie regitration of the labeled training data. The atla i diplayed in it reference poition in Figure 1; note the irregularity in the hape of the tetrahedra. The meh i endowed with a deformation model that allow it vertex coordinate to change according to a Markov random field model. If the free parameter x related to the prior correpond to a vector repreenting the poition of the meh (i.e., tacked coordinate of all vertice), we have: T p(x) exp( φ(x, x re f )] = exp φ t (x, x re f ), (6) where x re f i the reference poition. The energy function φ(x, x re f ), which include a term φ t for each tetrahedron in the meh t = 1,..., T, penalize the deformation from the reference poition. t

4 Specifically, φ t (x, x re f ) follow the definition in Ahburner et al. (): 3 3 φ t (x, x re f ) = F V (t) re f 1 + ( λ t,p λ t,p + λ t,p ). (7) p=1 In Equation 7, λ t,p, p = 1,, 3, repreent the ingular value of the Jacobian matrix of the affine mapping of tetrahedron t from reference to current poition. V (t) re f i the volume of tetrahedron t in reference poition, and F > i a calar that repreent the tiffne of the meh. The function φ t goe to infinity if any of the ingular value approache zero, i.e., if the Jacobian determinant of the mapping goe to zero. Therefore, the energy function φ explicitly enforce that tetrahedra do not fold, effectively preerving the topology of the meh. In practice, it i not neceary to explicitly compute ingular value decompoition to evaluate φ t, a explained in Ahburner et al. (). A movie diplaying different ample from the reulting atla deformation prior p(x) i available a part of the upplementary material. Given the deformed meh poition x, the probability p i (k x) that label k {1,..., K} occur at a voxel i can be obtained by interpolating the probabilitie correponding to that label at the vertice of the tetrahedron containing the voxel. Thee probabilitie are aumed to be conditionally independent given x. Therefore, if i {1,..., K} i the label at voxel i, the prior probability of a labeling = ( 1,..., I ) T i given by p( x) = I i=1 p i ( i x), where I i the total number of voxel in the region of interet covered by the meh. Finally, the likelihood model connect the labeling with the oberved image intenitie y = (y 1,..., y I ) T. The intenitie of the voxel correponding to each cla are aumed to follow a Gauian ditribution parametrized by a mean and a variance aociated to that cla. The probabilitie are aumed to be conditionally independent given the label. Therefore, the likelihood term i: I p(y, θ) = p(y i i, θ) = i=1 I i=1 p=1 1 πσ i exp ( (y i µ i ) σ i where the model parameter θ related to the likelihood conit of a vector grouping all the mean and variance of thee Gauian ditribution. In practice, to reflect the fact that there i little intenity contrat between the cerebral gray matter and the hippocampal ubfield ubiculum, preubiculum, CA1, CA/3, CA4/DG and tail in our image, we conider them part of a global gray matter tiue type with a hared mean and variance. Likewie, the cerebral white matter and the fimbria are conidered part of a global white matter cla with a ingle mean and variance, and the hippocampal fiure hare Gauian parameter with the CSF. Therefore, θ i a ix-dimenional vector: θ = (µ GM, σ GM, µ WM, σ WM, µ CS F, σ CS F )T. We aume the prior ditribution on thee parameter to be uninformative, i.e., p(θ) Claical inference: egmentation and volumetry Optimizing the expreion in Equation i equivalent to a joint regitration and intenity parameter etimation proce. In ), 4 the baeline egmentation method (Van Leemput et al., 9), the variable θ and x are alternately optimized uing a coordinate acent cheme, the former with expectation maximization (Dempter et al., 1977) and the latter with the Levenberg- Marquardt algorithm (Levenberg, 1944). To reduce the computational burden of the method, the optimization of the meh deformation i retricted to a region defined by a morphologically dilated verion of the mak that the main FreeSurfer tream produce for the whole hippocampu (a in Figure 3 and 4). Once the mot likely value of the parameter have been found, the poterior ditribution of the label given the intenitie i approximately (Equation 5): p( y) p( y, ˆθ, ˆx) = I p i ( i y i, ˆθ, ˆx), (8) which factorize over voxel becaue the label are conditionally independent given x and θ. The poterior label probabilitie for each voxel are given by Baye rule: p i ( i y i, ˆθ, ˆx) = i=1 p(y i i, ˆθ)p i ( i ˆx) Kk=1 p(y i k, ˆθ)p i (k ˆx). (9) The approximate maximum-a-poteriori (MAP) egmentation (the maximizer of Equation 3) can be computed voxel by voxel: ŝ i = argmax p i ( i y i, ˆθ, ˆx). (1) i Three lice of a ample MRI can and it correponding MAP egmentation are diplayed in Figure. Finally, to infer the volume of the different hippocampal ubfield within the framework, we mut conider that they are random variable dependent on the image data y. Under the point etimate approximation for the model parameter, the expected value v k and variance γk of the poterior ditribution of the volume of the tructure correponding to label k are given by: v k = γ k = I p i (k y i, ˆθ, ˆx) (11) i=1 I p i (k y i, ˆθ, ˆx)[1 p i (k y i, ˆθ, ˆx)]. (1) i=1 3. Conidering the uncertainty in the model parameter uing MCMC ampling 3.1. Parameter uncertainty Both in egmentation and volume etimation, the framework decribed in Section.. doe not conider the uncertainty in the model parameter. In mot medical imaging application, including the hippocampal ubfield egmentation problem, thi might be a fair aumption for the likelihood parameter θ: million of voxel are typically available to etimate a low number of parameter. In other word, it i not poible to alter θ much without largely decreaing the likelihood term

5 Figure : Top row: agittal, coronal and axial lice from the left hippocampu of a ample can from the dataet. Bottom row: correponding MAP egmentation produced by the baeline Bayeian algorithm, i.e., computed by optimizing Equation 8. The color map for the ubfield i the ame a in Figure 1. of the model. However, when x i baed on a nonlinear regitration method, the number of parameter i much higher; a many a three time the number of voxel when nonparametric, voxel-wie deformation model are ued. In that cae, the mode approximation in Equation 5 may no longer be accurate. For intance, it would be relatively eay to modify the atla warp x in area of the image with low intenity contrat without changing the poterior probability of the model ubtantially. Intead of uing point etimate for x and θ, we propoe to employ a computationally more demanding but alo more accurate way of approximating the poterior than Equation 5. Rather than the mode approximation, which only conider a ingle value for the model parameter, we ue Monte Carlo ampling to account for the uncertainty in {x, θ} and obtain a better approximation of the integral in the equation. Auming that we have an efficient way of drawing N ample {θ(n), x(n)}, n = 1,..., N, from the ditribution p(θ, x y), the egmentation poterior can be approximated by: p( y) = x θ p(, θ, x y)dθdx 1 N p( x(n), θ(n), y), N (13) which i a better approximation than Equation 5, ince it conider many different value of the parameter, with more likely value occurring more frequently. The approximation can be made arbitrarily cloe to the true integral by allowing N to be large enough. Within thi framework, it can be hown (Appendix A) that the poterior mean v k of the volume correponding to cla k i: v k 1 N N v k (n), (14) where v k (n) = I i=1 p i (k θ(n), x(n), y i ) i the mean of the poterior ditribution of the volume when the model parameter are et to {x(n), θ(n)} (note the analogy with Equation 11). The expreion for the variance, alo derived in Appendix A, i: γ k 1 N ( γ N k (n) + [v k (n) v k ] ), (15) where γ k (n) = I i=1 p i (k θ(n), x(n), y i )[1 p i (k θ(n), x(n), y i )] i the variance of the poterior ditribution of the volume when the 5 model parameter are et to {x(n), θ(n)} (very imilar to Equation 1). Equation 15 ha two term: the average of the variance computed independently from each ample and the variance of the mean volume acro ample. The firt term i expected to be comparable to the etimate from conventional Bayeian method, i.e., Equation 1. However, the econd term directly reflect the uncertainty in model parameter, including the atla regitration, and can potentially be much larger. 3.. Sampling baed on Markov chain Monte Carlo (MCMC) In order to obtain the ample {θ, x} required in the propoed framework, we ue MCMC technique. Specifically, we ue a Gibb ampling cheme (Geman and Geman, 1984) that alternately draw ample of θ keeping x contant and vice vera. The ampler i initialized with the mot likely model parameter a etimated by the conventional Bayeian egmentation method (Equation and 3): x() = ˆx, θ() = ˆθ. Subequently, ample are drawn a follow: x(n + 1) p(x θ(n), y) θ(n + 1) p(θ x(n + 1), y) Since the conditional ditribution p(θ x, y) and p(x θ, y) have different nature, we ue different method to ample from each of them. We dicu thee technique below Sampling x To draw ample from p(x θ, y) we ue an efficient MCMC technique known a Hamiltonian Monte Carlo (HMC, Duane et al. 1987, alo known a hybrid Monte Carlo). HMC belong to the family of Metropoli-Hating method, in which a propoal probability ditribution that depend on the current poition of the ampler i ued to ugget a new candidate poition for the next ample. The new propoed tate i then accepted or rejected uing the Metropoli-Hating equation (Metropoli et al., 1953). In traditional Metropoli-Hating cheme, imple propoal ditribution are ued (e.g., a Gauian centered on the current ample), leading to a random walk behavior that make the exploration of the domain of the target probability ditribution very low and inefficient. In contrat, HMC i able to generate ditant propoal that are till likely to be accepted by augmenting the pace of the variable one i ampling from and taking advantage of the gradient of it probability ditribution. Specifically, HMC augment the tate pace x with a vector of momentum variable m (with the ame dimenionality a x). HMC alternate two kind of propoal: firt, randomly ampling m from a zero-mean, identity-covariance Gauian ditribution; and econd: imultaneouly updating x and m uing a imulation of Hamiltonian dynamic. The Hamiltonian i the energy of a particle with momentum m located in a potential field defined by the energy E p (x) = log p(x θ, y): H(x, m) = E p (x) + E k (m) = log p(x θ, y) + 1 mt m. where we have aumed unit ma for the particle. The two propoal are ued to generate ample from the joint probability

6 denity function: p(x, m θ, y) exp( H(x, m)) = p(x θ, y) exp( 1 mt m). Finally, we can imply dicard the momenta from the joint ample {x, m} to yield the final ample of p(x θ, y). The detail of the algorithm and of it implementation, including ome modification to improve it efficiency in our pecific application, are detailed in Appendix B. Movie illutrating atla deformation ampled from p(x θ, y) uing the propoed method are available a upplementary material. Note that, due to the information in the image intenitie y, thee ample are much more imilar to each other than thoe from the prior p(x) Sampling θ When the atla poition x i fixed, ampling from p(θ x, y) i not traightforward becaue of the unknown label, which follow the categorical ditribution in Equation 9. However, a in HMC, we can ample from the joint ditribution {θ, } p(θ, x, y) intead and imply diregard the labeling. To do o, we again ue a Gibb cheme, in which we alternately ample from and θ. Obtaining ample of p( θ, x, y) i immediate becaue it factorize over voxel, o we can ample the label i independently with Equation 9. Regarding θ, it conditional poterior ditribution p(θ, x, y) = p(θ, y) i normal-gamma (Murphy, 7). We can draw ample from thi ditribution in two tep: 1 σ k Γ( V k(), 1 V k() k ), (16) µ k N(ȳ k, σ k ), (17) V k () where Γ(α, β) i the Gamma ditribution with hape parameter α and rate parameter β, N i the Gauian ditribution, V k () i the number of voxel with label k in the current ample of, and ȳ k and k are the ample mean and variance of the intenitie correponding to uch voxel. In practice, generating ample of θ i much fater than drawing ample from p(x θ, y). Therefore, we draw many ample from θ before going back to ampling a new x. The reulting ampling cheme i ummarized in Table 1; note that we mut draw and diregard ome ample of x (burn-in period) every time we update θ and vice vera to enure that we are actually ampling from the conditional ditribution p(x θ, y) and p(θ x, y), repectively. 4. Experimental etup Ideally, we would validate the propoed method by firt having a et of brain MRI can manually labeled by an expert human rater, and then computing overlap core with the automated egmentation. However, manually labeling the ubfield in tandard brain MRI can at 1 mm reolution i extremely difficult. Therefore, we ue an indirect validation method baed 1. Initialize n = 1, {x, θ} = {ˆx, ˆθ} (optimal point etimate).. FOR t = 1 to number of ample of x to draw: A. FOR t = 1 to number of trajectorie: A-I. Sample m N(, I). A-II. Track (x, m) with Equation B.1-B.3. A-III. Accept move with probability given by Eq. B.4 END B. FOR t = 1 to number of ample of θ per ample of x: B-I. FOR t = 1 to number of ample to kip: B-Ia. Sample the image label with Equation 9. B-Ib. Sample the image intenity parameter with Equation 16 and 17. END B-II. Record ample: x(n) = x, θ(n) = θ. B-III. n=n+1. END END Table 1: Algorithm to obtain ample from p(θ, x y). on the fact that hippocampal volume are known to be a biomarker for Alzheimer dieae (AD) (Chupin et al., 9). The volume of the hippocampal ubfield, which are automatically etimated uing the Bayeian egmentation method (with and without ampling), are ued to eparate AD from control uing a imple claifier. The ability to eparate the two clae i then a urrogate for egmentation quality that we ue to ae the effect of ampling. In addition to claification accuracy, we further examine how ampling affect the error bar of the volume etimate. In thi ection, we firt decribe the MRI data ued in the tudy, then the etting of the ampler, the claification framework ued to predict AD and finally the competing approache MRI data The MRI data ued in thi tudy were obtained from the Alzheimer Dieae Neuroimaging Initiative (ADNI) databae ( The ADNI wa launched in 3 by the National Intitute on Aging, the National Intitute of Biomedical Imaging and Bioengineering, the Food and Drug Adminitration, private pharmaceutical companie and non-profit organization, a a $6 million, 5-year public-private partnerhip. The main goal of ADNI i to tet whether MRI, poitron emiion tomography (PET), other biological marker, and clinical and neuropychological aement can be combined to analyze the progreion of mild cognitive impairment (MCI) and early AD. Marker of early AD progreion can aid reearcher and clinician to develop new treatment and monitor their effectivene, a well a decreae the time and cot of clinical trial. The Principal Invetigator of thi initiative i Michael W. Weiner, MD, VA Medical Center and Univerity of California - San Francico. ADNI i a joint effort by coinvetigator from indutry and academia. Subject have been recruited from over 5 ite acro the U.S. and Canada. The initial goal of ADNI wa to recruit 8 ubject but ADNI ha been fol- 6

7 lowed by ADNI-GO and ADNI-. Thee three protocol have recruited over 1,5 adult (age 55-9) to participate in the tudy, coniting of cognitively normal older individual, people with early or late MCI, and people with early AD. The follow up duration of each group i pecified in the correponding protocol for ADNI-1, ADNI- and ADNI-GO. Subject originally recruited for ADNI-1 and ADNI-GO had the option to be followed in ADNI-. For up-to-date information, ee In thi tudy we evaluated the propoed method with 4 baeline T 1 can from elderly control (EC) and AD ubject available in the ADNI-1 tudy. The can were acquired with agittal 3D MPRAGE equence at 1 mm iotropic reolution. Since ADNI i a multi-center effort, different make and model of canner were ued to acquire the image. We refer the reader to the ADNI webite for a detailed decription of acquiition hardware and protocol. The oftware package FreeSurfer wa ued to preproce the can. FreeSurfer produce kull-tripped, bia field corrected volume along with a egmentation into ubcortical and cortical tructure, including the hippocampu. The preproceed can were then run through the hippocampal ubfield egmentation module in FreeSurfer. Thi module ue the hippocampal mak to extract the bia field corrected data within a bounding box around the hippocampu, upample thi cropped region to.5 mm iotropic reolution and give it a input to the egmentation algorithm decribed in Van Leemput et al. (9). The output from the hippocampal ubfield module wa ued to initalize the ampling algorithm, which wa run on the cropped, upampled data. Throughout the experiment we ued the default value in FreeSurfer for the tiffne parameter: F =.1. We alo recorded the intracranial volume (ICV) etimated by FreeSurfer, which are ueful to correct for brain ize in the volumetry. The FreeSurfer pipeline crahed in 17 ubject, which were removed from the final analyi. The demographic of the remaining 383 ubject are a follow: 56.% EC (age 76.1±5.6 year), 43.8% AD patient (age 75.5 ± 7.6); 53.6% male (age 76.1 ± 5.6), 46.4% female (age 75.9 ± 6.8). 4.. MCMC ampler The Monte Carlo ampler follow the algorithm in Table 1. With the help of preliminary run, we tuned the parameter value to the following value: Number of iteration: for x (tep in table), 3 ample of θ for each value of x (tep B). Thi amount to N=6 total ample. Number of trajectorie: we imulate 1 trajectorie in tep A in Table 1) before recording each ample of x. Thi burn-in period enure that we are actually ampling from the conditional ditribution p(x θ, y). In a imilar way, we kip 1 ample of θ (tep B-I in the table) for each ample that we record Claification, ROC analyi, and error bar of volume The performance when claifying AD v. EC i ued a a urrogate meaure for the quality of the hippocampal ubfield egmentation. It i thu deirable to ue a imple claifier uch that the performance i mainly determined by the input data, rather than tochatic variation in the claifier. For thi reaon, we chooe to ue a imple linear claifier (Linear Dicriminant Analyi, LDA, Fiher 1936) a the bae claifier in the experiment. LDA aume that data point (repreented by a vector z) from two different clae are ample from two Gauian ditribution with different mean but equal covariance matrice. In our cae, the clae are AD and EC, wherea z i a vector with the hippocampal ubfield volume of a ubject, leftright averaged and divided by the ICV etimated by FreeSurfer. Averaging the volume from the left and right hippocampi enhance the power of the analyi without increaing the dimenionality of the data, wherea dividing by the ICV correct for difference in whole brain ize. The mean of the the Gauian ditribution can be etimated from training data by computing the average for each cla: µ EC = (1/n EC ) j EC z j and µ AD = (1/n AD ) j AD z j, where n EC and n AD are the number of ubject in each cla in the training data. The expreion for the covariance i: j EC(z j µ EC )(z j µ EC ) T + j AD(z j µ AD )(z j µ AD ) T Σ =. n EC + n AD When a new data point i preented to the ytem, the likelihood ratio i computed and compared to a threhold to make a deciion: N(z; µ EC, Σ) N(z; µ AD, Σ) τ. It can be hown that thi tet i equivalent to: κ = (µ EC µ AD ) T Σ 1 z = w T z τ, (18) which repreent a linear deciion function in which each ubfield volume ha a different weight in w = Σ 1 (µ EC µ AD ). The threhold τ control the compromie between enitivity and pecificity. By weeping τ, we can build the receiver operating characteritic (ROC) curve, which depict the true poitive rate v. the fale poitive rate. The area under the curve (AUROC) can be ued a a meaure of the performance of the claifier acro all enitivity rate. To compare AUROC from different claifier, we ue the tatitical tet propoed by DeLong et al. (1988). In addition to the AUROC, we alo report the maximal accuracy rate of the claification acro τ. The accuracy correpond to a ingle point of the ROC and i therefore a le informative meaure, but i alo eaier to interpret. In order not to bia the reult by introducing knowledge from the tet data in the evaluation, we ue a leave one out cheme to compute the AUROC and the claification accuracy. For each can in the dataet, we build a training et T coniting of all other available can. From thi can-pecific training et, we compute w and ue it to evaluate κ for the can at hand. We alo compute the threhold ˆτ that bet eparate the 7

8 two clae (i.e., maximal claification accuracy) in T. We ue thi value to compute a claification of the tet can by teting: κ ˆτ. After repeating thee tep for all 383 can in the dataet, we compute the ROC by threholding {κ } at different value of τ, and the claification accuracy by comparing the automated claification given by κ ˆτ with the ground truth. In addition to recording the AUROC and claification accuracy, we alo evaluate the variance of the poterior ditribution of the ubfield volume with and without ampling, i.e., with Equation 1 and 15 repectively. Thi allow u to quantify to what extent the error bar on the volume are affected by the ampling Competing method We compare performance meaure of LDA claifier trained on the following volume meaurement: The whole hippocampu volume obtained by umming up the ubfield volume (computed uing Equation 11) produced by the baeline method that relie on point etimate (decribed in Section.). Thi benchmark allow u to quantify the additional information offered by ubfield volume. We denote thee meaurement a WH- PE (where WH tand for whole hippocampu and PE tand for point etimate ). The vector of ubfield volume (computed uing Equation 11) produced by the baeline hippocampal ubfield egmentation method that relie on point etimate. We abbreviate thee meaurement a SF-PE (SF tand for ubfield ). The average ubfield volume (computed uing Equation 14) obtained uing the propoed MCMC ampling cheme. We abbreviate thi method a SF-SP (where SP tand for ampling ). Finally, we ue all the ample drawn with the algorithm in Table 1 to compute the deciion boundary defined by w in Equation 18. In other word, each one of the N=6, MCMC ample computed for a ubject i treated a a eparate ubject during training. Thi contitute a richer repreentation of the data that allow the tatitical claifier to account for the uncertainty in the volume when learning the boundary between the two group. When claifying a tet ubject, we only ued the mean volume computed over all MCMC ample of that ubject (i.e., Equation 14). We denote thi method a SF-AS (where AS tand for all ample ). 5. Reult We firt preent a qualitative comparion between the propoed MCMC ampling-baed egmentation framework and the baeline method that ue point etimate of the model parameter. Both the ampling-baed and baeline method can be ued to generate ample from the poterior of the egmentation. For the baeline method, thi involve fixing the model parameter ˆθ and ˆx and drawing from an independent poterior ditribution on label (given by Equation 9) at each voxel. The amplingbaed method produce uch egmentation ample within the MCMC framework (ee tep B-Ia of Table 1). Figure 3 and 4 how example of uch egmentation for a repreentative ubject obtained uing the ampling-baed and baeline method, repectively. The two figure alo how heat map highlighting region, in which the poterior egmentation ample diagree. Specifically, we define the diagreement ς = (ς 1,..., ς I ) T a the number of pair of ample that have different label at each voxel: ς i = N 1 N n 1 =1 n =n 1 +1 δ k ( i (n 1 ) i (n )), (19) where δ k ( ) i Kronecker delta. Thi diagreement i a meaurement of how confident the method i about the egmentation at a given voxel. Table ummarize the etimated uncertainty in hippocampal ubfield volume averaged acro all 383 ubject. For the ampling method, thi i computed uing Equation For the baeline method we ued Equation To further ae the relative impact of ampling from the Gauian likelihood parameter θ, we computed the relative tandard deviation when thee are kept contant throughout the ampling (i.e., we kip tep B-I in Table 1). Figure 5 diplay the poterior ditribution of the volume meaurement for two ubfield (ubiculum and CA1) in an example can. For the propoed ampling cheme, thee ditribution were etimated uing a Parzen window etimator. For the baeline method, the poterior were approximated a Gauian with mean and variance given by Equation 11 and 1. Here, the Gauian aumption i reaonable thank to the central limit theorem, ince the total volume of a tructure i the um of the contribution from all the voxel, whoe label are aumed to be independent given the image and the model parameter (Equation 8). Figure 6 how the poterior ditribution of the Gauian likelihood parameter (the mean and tandard deviation of intenity value for each tiue type) computed uing the ampling-baed approach and Parzen window etimator on the MCMC ample. Finally, we preent a quantitative evaluation of volume meaurement obtained uing different egmentation trategie. A we decribe in Section 4.4, we ue four different type of volume meaurement to train a claifier to dicriminate AD veru EC. Figure 7 how the ROC correponding to thee different meaurement in the AD claification experiment. The correponding AUROC and accuracie are preented in Table 3. The table alo diplay pairwie p-value correponding to De- Long paired tatitical tet comparing the claification performance offered by the the different meaurement. 8

9 Subfield Fimbria Preub. Sub. CA1 CA/3 CA4/DG Tail Fiure v k (mm 3 ) γ k /v k (%) γ k / v k (%) γ k x/ vx k (%) Table : Firt row lit mean volume of the ubfield, computed with the ampling cheme, v k averaged acro all 383 ubject. The econd and third row lit the relative tandard deviation of the ubfield volume (averaged acro ubject), computed with the baeline (γ k /v k ) and ampling-baed method ( γ k / v k ). To quantify the relative effect of ampling the Gauian likelihood parameter θ (compared with the regitration parameter x), we alo include the relative tandard deviation computed baed on a ampling cheme with fixed θ. We denote thi alternative trategy via upercript x, i.e., the correponding relative tandard deviation i γ k x/ vx k. Thee value are lited in the fourth row. AXIAL CORONAL SAGITTAL AXIAL CORONAL SAGITTAL Figure 3: Sample from the poterior ditribution of the egmentation p( y) when approximated by the ampling cheme, i.e., with Equation 13. The image correpond to the right hippocampu of a ubject with Alzheimer dieae. Firt row: axial, coronal and agittal lice of the MRI data, cropped around the hippocampu a explained in Section... Row -3: correponding lice of two different ample of the egmentation; the colormap i the ame a in Figure 1. Row 4: heat map repreentation of the label diagreement defined in Equation 19 (yellow repreent mot diagreement). Thi map highlight the region in which the egmentation uncertainty i high. A oppoed to the egmentation in Figure, the ample in the econd and third row of thi figure do not maximize the approximate poterior probability of the label. 6. Dicuion Thi paper invetigated the ue of Markov chain Monte Carlo (MCMC) ampling to approximate the egmentation poterior Figure 4: Sample from the approximate poterior ditribution of the egmentation p( y) when the mode approximation i ued in the integral over model parameter (i.e., when the poterior i approximated by Equation 5 a in the baeline egmentation method). The ample can then be obtained by independently ampling the label of each voxel from the categorical ditribution in Equation 9. The image correpond to the ame ubject and lice a in Figure 3; pleae ee it caption for an explanation of the illutration. within a Bayeian framework. Thi approach i in contrat with claical Bayeian egmentation algorithm that rely on the mode approximation to integrate over the model parameter, including thoe decribing atla deformation. We ued a databae of 383 brain MRI can and an atla contructed for hippocampal ubfield egmentation to explore the difference between the MCMC approach and a claical egmen- 9

10 Normalized probability denity Ditribution of volume of ubiculum 1 Sampling No ampling Volume in mm Ditribution of volume of CA Volume in mm 3 Denity Denity 6 4 WM Mean intenity 1 5 WM Denity Denity GM Mean intenity GM Denity Denity Mean intenity CSF CSF Figure 5: Poterior ditribution of the volume of the ubiculum and CA1 for an example can, computed with the propoed approximation of p( y) uing ampling (Equation 13) and with p( ˆx, ˆθ, y), i.e., uing point etimate of the model parameter. The former i computed uing a Parzen window denity etimate baed on the ample recorded in tep B-II of Table 1 with a Gauian kernel of ize σ = 4mm 3, wherea the latter i approximated by a Gauian ditribution with mean and variance given by Equation 11 and 1. Both probability denity function have been normalized to [,1] for eaier viualization. Method Acc. AUROC p S F S P p S F PE p WH PE WH-PE 79.9% SF-PE 8.% SF-SP 83.8% SF-AS 84.3% Std. dev. of intenity Std. dev. of intenity Std. dev. of intenity Figure 6: Marginal of p(θ y), the poterior ditribution of the Gauian likelihood intenity parameter for white matter, gray matter and CSF in an example can, computed within the ampling framework. Thee ditribution were computed uing a Parzen window etimator on the ample recorded in tep B-II of Table 1 and with a Gauian kernel with σ =.1. The harpne of the ditribution (note the cale of the horizontal axi in all figure) indicate a very low uncertainty in the etimate of thee parameter. Table 3: AD/EC claification accuracy at optimal threhold ˆτ, area under the ROC curve, and p-value for one-tailed DeLong tatitical tet comparing the area under the curve for all pair of approache. The ubcript of the p indicate the method we are comparing. tation method (which we refer to a the baeline ). Firt we analyze the etimate of uncertainty computed by the MCMC method and the baeline. Theoretically, we expect the baeline to produce egmentation reult that it i very confident about, ince uncertainty in the model parameter i ignored. The MCMC reult on the other hand hould contain more uncertainty. Our experiment agreed with thi expectation: there are more red/yellow voxel in the heat map of Figure 3 compared to Figure 4. Baed on Figure 3, we oberve that egmentation uncertainty for the MCMC method i higher in voxel cloe to boundarie between tructure, and pecifically between tructure that belong to the ame tiue type (e.g., CA4-DG and CA/3, which are both gray matter tructure). Thi reflect the fact that the atla regitration i not contrained by image intenitie in thoe region. In contrat, for the baeline algorithm (Figure 4) the only ource of egmentation uncertainty i reduced atla harpne and/or partial voluming along tructure boundarie, ince uncertaintie in the atla deformation are entirely dicarded. We further explored egmentation uncertainty by computing error bar on the ubfield volume meaurement (ee Table ). In the baeline method, the relative tandard deviation (i.e., the tandard deviation divided by the volume, γ k /v k ) wa well below 1% for mot ubfield, which we deem unrealitic given the poor image contrat. The value were ignificantly larger for the MCMC method, i.e., larger than 6% for mot ub- a) b) Figure 7: Receiver operating characteritic (ROC) curve for the competing method. (a) Complete curve. (b) Detail of the top left corner (the elbow ) of the curve, which i where the operating point would typically be located. Thi region i marked with a box in (a). field. Thi difference i further highlighted in Figure 5, which diplay the poterior ditribution of two ubfield volume in an example ubject, computed uing the MCMC method and the baeline. Baed on thee plot, we make the obervation that volume etimate obtained with the baeline method can deviate ubtantially from thoe obtained with the MCMC method, i.e., the mean of the ditribution can be quite far apart. Furthermore, the uncertainty etimated by the MCMC method i dramatically higher than that computed with the baeline, which i in trong agreement with Table. To examine the influence of the uncertainty in the Gauian intenity likelihood parameter, we further computed the error bar for a modified MCMC method. In thi modified verion, we fixed the Gauian intenity likelihood parameter and only ampled over the atla deformation parameter (ee lat row of Table ). Thee relative tandard deviation value are lightly maller than thoe obtained with the full MCMC implementation, which ugget that the uncertainty in the Gauian intenity likelihood parameter ha a relatively mall contribution 1

11 11 to egmentation uncertainty. Mot of the egmentation uncertainty i due to the uncertainty in the atla deformation, which contain many more model parameter. Thi point i further reinforced with Figure 6, which reveal the harpne of the poterior ditribution of the Gauian intenity likelihood parameter. Hippocampal ubfield atrophy differentially in Alzheimer dieae (Mueller et al., 1). Thu, we utilized hippocampal ubfield volume meaurement to dicriminate AD veru control, uing claification performance to indirectly quantify the quality of different volume meaurement obtained from different egmentation cheme. Our reult (illutrated in Figure 7) revealed that ubfield volume predicted AD above and beyond the total hippocampal volume: SF-PE offered % boot in accuracy and a.1 increae in AUROC (p =.) over WH- PE. The ROC curve of thee two method might not appear very different, but there i a clear gap in the elbow (Figure 7b), which i the region with high claification accuracy, where the operating point would normally be defined. Secondly, and more importantly, ubfield volume etimate obtained uing the MCMC method were more predictive of AD than etimate computed uing the baeline (an improvement of % in accuracy,.14 in AUROC with p =.5). Thi ugget that volume meaurement extracted from MCMC egmentation can be more accurate than thoe obtained with the baeline method. Finally, when all MCMC ample were ued to train the claifier (SF-AS), a marginal improvement (though not tatitically ignificant) wa oberved. Thi ugget that the uncertainty etimate offered by MCMC method can be utilized to improve downtream analye. Overall, our empirical reult are conitent with our theoretical expectation. In addition to potentially improving egmentation accuracy, the propoed MCMC method offer a trategy to obtain a more realitic quantification of egmentation uncertainty. A we demontrated in the AD claification experiment, utilizing the uncertainty in the egmentation reult might prove ueful in variou analye. One immediate application would be to imply offer a quantification of the meaurement confidence, a thi may convey important information when thee technique are ultimately applied in clinical etting. Secondly, by examining egmentation uncertainty, one might be able to ae the effect of different parameter (e.g., the imaging modality) on egmentation quality without the neceity of ground truth. Finally, tatitical power analye commonly ued to plan population tudie might alo benefit from accurate etimate of meaurement error. For example, in a tudy deigned to examine volume difference between two group, one might be able to etimate the effect of improving egmentation quality on our ability to differentiate group. A drawback of the MCMC egmentation approach i it computational complexity. Another challenge i the fine-tuning of the ampling parameter, uch a the trajectory length, tep ize, etc. Our experience ugget that thee deign choice can have a dramatic impact on computational efficiency. The current implementation we preented in thi paper require 8 CPU hour to proce an individual MRI can, although we did not focu on optimizing run time ince our main aim wa to invetigate the empirical advantage offered by the MCMC trategy. Note, however, that the MCMC ampling tep, which i the main computational bottleneck, i amenable to dramatic parallelization, and that the number of MCMC ample required to accurately compute the mean and variance of poterior volume ditribution maybe be far le than the N = 6, ued in thi paper. The propoed MCMC ampling approach can alo be applied to other probabilitic egmentation framework, uch a Fichl et al. (4b); Ahburner and Friton (5); Pohl et al. (6); Sabuncu et al. (1). In doing o, one important conideration would be the number of free parameter in the model and identifying an efficient trategy to ample from the poterior of thee parameter. It i important to note that the baeline method we built on in thi paper utilized a pare atla repreentation with a relatively low number of free parameter, which made it poible for u to implement an MCMC ampling trategy that wa computationally practical. We are currently invetigating whether imilar technique can alo benefit whole brain egmentation, in which dozen of ubtructure are automatically egmented from brain MRI. Since the computational complexity of our atla deformation depend motly on the number of tetrahedra in the atla rather than the number of voxel of the input image, we believe uch an approach will be computationally feaible. A the model ued in Bayeian egmentation method continue to grow in complexity, with a concomitant higher number of free parameter, we expect the relevance of accurate computational approximation to the true egmentation poterior to become increaingly important in the future. Appendix A. Mean and variance of the poterior ditribution of the volume when ampling the model parameter Here we derive the expreion for the mean and the variance of the poterior ditribution of the volume correponding to label k within the ampling framework (Equation 14 and 15). Let V k () denote the volume of cla k in label map. Then, the expected value of the poterior ditribution of the volume i: v k = p( y)v k () 1 N p( y, x(n), θ(n)) N V k() = 1 N N p( y, x(n), θ(n))v k () = 1 N N v k (n), where v k (n) = I i=1 p i (k y i, θ(n), x(n)) i the expectation of the volume if the model parameter are fixed to {θ(n), x(n)}, in a imilar way a in Equation 11.