Uncertainty quantification in non-rigid image registration via stochastic gradient Markov chain Monte Carlo

We develop a new Bayesian model for non-rigid registration of three-dimensional medical images, with a focus on uncertainty quantification. Probabilistic registration of large images with calibrated uncertainty estimates is difficult for both computational and modelling reasons. To address the computational issues, we explore connections between the Markov chain Monte Carlo by backpropagation and the variational inference by backpropagation frameworks, in order to efficiently draw samples from the posterior distribution of transformation parameters. To address the modelling issues, we formulate a Bayesian model for image registration that overcomes the existing barriers when using a dense, high-dimensional, and diffeomorphic transformation parametrisation. This results in improved calibration of uncertainty estimates. We compare the model in terms of both image registration accuracy and uncertainty quantification to VoxelMorph, a state-of-the-art image registration model based on deep learning

Image registration via stochastic gradient markov chain monte carlo

We develop a fully Bayesian framework for non-rigid registration of three-dimensional medical images, with a focus on uncertainty quantification. Probabilistic registration of large images along with calibrated uncertainty estimates is difficult for both computational and modelling reasons. To address the computational issues, we explore connections between the Markov chain Monte Carlo by backprop and the variational inference by backprop frameworks in order to efficiently draw thousands of samples from the posterior distribution. Regarding the modelling issues, we carefully design a Bayesian model for registration to overcome the existing barriers when using a dense, high-dimensional, and diffeomorphic parameterisation of the transformation. This results in improved calibration of uncertainty estimates

Fast Predictive Image Registration

We present a method to predict image deformations based on patch-wise image appearance. Specifically, we design a patch-based deep encoder-decoder network which learns the pixel/voxel-wise mapping between image appearance and registration parameters. Our approach can predict general deformation parameterizations, however, we focus on the large deformation diffeomorphic metric mapping (LDDMM) registration model. By predicting the LDDMM momentum-parameterization we retain the desirable theoretical properties of LDDMM, while reducing computation time by orders of magnitude: combined with patch pruning, we achieve a 1500x/66x speed up compared to GPU-based optimization for 2D/3D image registration. Our approach has better prediction accuracy than predicting deformation or velocity fields and results in diffeomorphic transformations. Additionally, we create a Bayesian probabilistic version of our network, which allows evaluation of deformation field uncertainty through Monte Carlo sampling using dropout at test time. We show that deformation uncertainty highlights areas of ambiguous deformations. We test our method on the OASIS brain image dataset in 2D and 3D

Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach

This paper introduces Quicksilver, a fast deformable image registration method. Quicksilver registration for image-pairs works by patch-wise prediction of a deformation model based directly on image appearance. A deep encoder-decoder network is used as the prediction model. While the prediction strategy is general, we focus on predictions for the Large Deformation Diffeomorphic Metric Mapping (LDDMM) model. Specifically, we predict the momentum-parameterization of LDDMM, which facilitates a patch-wise prediction strategy while maintaining the theoretical properties of LDDMM, such as guaranteed diffeomorphic mappings for sufficiently strong regularization. We also provide a probabilistic version of our prediction network which can be sampled during the testing time to calculate uncertainties in the predicted deformations. Finally, we introduce a new correction network which greatly increases the prediction accuracy of an already existing prediction network. We show experimental results for uni-modal atlas-to-image as well as uni- / multi- modal image-to-image registrations. These experiments demonstrate that our method accurately predicts registrations obtained by numerical optimization, is very fast, achieves state-of-the-art registration results on four standard validation datasets, and can jointly learn an image similarity measure. Quicksilver is freely available as an open-source software.Comment: Add new discussion

Bayesian modelling of organ deformations in radiotherapy

Moderne strålebehandling mot kreft er skreddarsydd for å gje ein høg stråledose tilpassa svulsten (målvolumet), mens så lite dose som mogleg vert gitt til det friske vevet omkring. Den totale dosen vert levert over nokre veker i daglege "fraksjonar", noko som reduserer biverknader. Under og mellom desse fraksjonane rører dei indre organa på seg heile tida på grunn av pust, fylling av blæra, tarmar si rørsle og ekstern påverknad. Likevel vert posisjonen til målvolumet og relevante risikoorgan bestemt på grunnlag av eit statisk 3D-skann som er tatt før behandlinga startar. Den vanlege måten å sikre seg mot konsekvensar av denne rørsla er å legge til marginar rundt svulsten. Slik sikrar ein å treffe målvolumet, men til gjengjeld får det friske vevet meir dose. Marginane sin storleik er fastsett ved hjelp av statistikk over tidlegare behandla pasienter. Dei statistiske metodane som vert brukte er ofte enkle, og tek berre omsyn til rigid rørsle, altså at heile kroppen rører seg i eitt. Dessutan vert det ikkje teke omsyn til rørsla til risikoorgan. For å berekne dose til risikoorgana er det vanleg å anta at forma til organa i planleggingsskannet er representative for forma deira under behandling. Arbeidet i denne avhandlinga handlar om å bruka teknikkar frå Bayesiansk statistikk for å modellere korleis organ rører og deformerer seg mellom fraksjonane. Målet er å estimere nøyaktig den statistiske fordelinga av rørsle for eit eller fleire organ til ein pasient. Fordelinga gjev innsikt i korleis organa forandre seg medan behandlinga går for seg. Denne innsikta er nyttig for evaluering av stråleterapiplanar, statistisk prediksjon av biverknader, såkalla robust planlegging og å berekna størrelsen på marginar. Metodane som vert presentert er evaluerte for endetarmen (rektum) sine rørsler hjå prostatakreftpasientar. For desse pasientane er rektum eit viktig risikoorgan, som kan bli ramma både av akutte og seine biverknader, som lekkasje, bløding og smerter. Samanlikna med eksisterande metodar har den Bayesianske tilnærminga to fordelar: For det første gir kombinasjonen av populasjonsstatistikk og individuelle data meir nøyaktige anslag av den pasientspesifikke fordelinga. For det andre estimerer dei nye metodane den såkalla systematiske feilen i tillegg til variasjonar frå fraksjon til fraksjon. Den systematiske feilen er forskjellen mellom den estimerte forma på organet under planlegging, og gjennomsnittsforma til organet under bestråling. Denne typen feil var tema for artikkel I. Her fekk vi til å redusere den systematiske feilen til rektum hjå 33 av 37 prostatakreftpasientar ved å bruke ein metode som kombinerer forma på rektum under planlegginga og gjennomsnittsforma i populasjonen. Vi vurderte og om denne forbetringa hadde påverknad på estimering av summert dose til rektum. Metoden gav ikkje signifikant forbetring for to antatt relevante parametrar (ekvivalent uniform dose og D5%), men gav signifikant reduksjon av bias på det estimerte dose-volum-histogrammet i intervallet 52.5 Gy til 65 Gy. Hovudarbeidet i dette prosjektet er publisert i artikkel II. Der presenterer vi to modellar for organrørsle basert på Bayesianske metodar. Inndata til desse metodane er organformer som er henta frå 3D-skanningar. Metodane kan ta ulikt tal slike former, og produserer meir nøyaktige resultat jo fleire former dei får. Dei gjev anslag av gjennomsnittsforma og kor stor uvissa om denne forma er, i tillegg til anslag av fordelinga av variasjon av former frå fraksjon til fraksjon. Vi evaluerte metodane etter kor godt dei kunne berekne "dekningssannsyn", altså sannsynet for at organet skal dekke eit gitt punkt i pasientkoordinatsystemet til ei gitt tid. For denne berekninga måtte titusenvis av organformer gjerast om til såkalla binærmasker, som er 3D-matriser av punkter i pasient-koordinatsystemet der verdien til eit punkt er 1 dersom punktet er inne i organet, og 0 elles. Denne berekninga var mogleg på grunn av programvare som blei implementert for dette prosjektet, og som er presentert i artikkel III. Også her var det prostatakreftpasientar sitt rektum som vart brukt til evaluering. Berekningane til dei nye metodane var likare det sanne dekningssannsynet enn tilsvarande berekningar frå tidlegare metodar, i signifikant grad, i alle fall opp til tre input. Forskjellen mellom dei to nye algoritmane er i hovudsak kompleksiteten og nøyaktigheita, og valet mellom algoritmane i ein gitt bruk vil vere ei avveging mellom desse faktorane. Vi viste ein måte modellane kan verte brukte i artikkel IV, som handlar om pasientar som får re-bestråling for tilbakefall av prostatakreft. Her brukte vi modellane til å berekne forventa akkumulert dose til rektum frå dei to behandlingane, og også uvissa rundt den forventa dosen. Metoden er basert på representative former" av rektum, altså former som rektum kan ta som er sannsynlege, men lite fordelaktige. Desse formene kan brukast som visuell hjelp for onkologar og doseplanleggjarar, og metoden kan implementerast ved hjelp av eksisterande funksjonar i programvaren for behandlingsplanlegging. Overordna gir denne avhandlinga nye løysingar for den sentrale utfordringa med å redusere konsekvensar av organrørsle i stråleterapi. Dei presenterte modellane er dei første som utnyttar statistikk for populasjonen og data frå den enkelte pasienten samstundes, og som tar omsyn til både systematiske og tilfeldige feil.Modern radiotherapy tends to be highly conformal, meaning that a high and uniform dose is delivered to the target volume and as little dose as possible to the surrounding normal tissue. The total radiation dose is delivered across several smaller daily fractions, typically spanning several weeks. During and between these fractions, internal organs are constantly in motion due to factors such as breathing, changes to bladder filling state, intestinal movement and external influences. Nevertheless, the position of the target and relevant organs at risk (OARs) are determined based on a static 3D scan acquired before start of treatment. A common safeguard which is used to take such motion into account is the addition of margins around the target. These margins reduce the chance of missing parts of the target, yet increases dose to the healthy tissue surrounding the target. The margin size is based on statistics from previous patients. However, for the most part, the statistical methods used are very simple, and typically based on an assumption of rigid patient motion. Similarly, motion of the OARs is commonly neglected. For estimation of dose to the OARs, it is common to assume that the organ shape at the static scan is representative for its shape during treatment. The work in this thesis concerns the use of techniques from Bayesian statistics for modelling inter-fraction organ motion and deformation. The goal is to estimate accurately the statistical distribution of shapes for one or more organs for a given patient. The distribution provides knowledge of how the patient's organs might move and deform during the radiotherapy course. This information is useful for the evaluation of radiotherapy plans, prediction of adverse effects, so-called motion-robust radiotherapy planning, the generation of margins and more. The methods presented in this thesis have been evaluated for predicting deformations of the rectum of prostate cancer patients. For these patients, the rectum is a crucial OAR that is affected by both early and late side effects including leakage, bleeding and pain. Compared to existing methods, the Bayesian approach developed and implemented in this thesis offers two advantages: first, combining population statistics and individual data leads to more accurate estimates of the patient-specific distribution. Secondly, the new methods estimate the distribution of the so-called systematic error in addition to variations from fraction to fraction. The systematic error is the difference between the estimated shape/position of an organ at the planning stage and its average shape/position during therapy, and was the subject of paper I. Here, we were able to reduce the systematic error of the rectum in 33 out of 37 prostate cancer patients using a straightforward method to combine the shape of the rectum at the planning CT with the population mean shape. We also evaluated the impact of this improvement on the estimation of dose to the rectum. We found no significant improvement on the estimation of two presumably relevant dose parameters (equivalent uniform dose and D5%). However, we did find significant reduction in the bias of the estimated dose-volume histogram in the range from 52.5 Gy to 65 Gy. Paper II contains the central work of this project. It presents two organ deformation models based on Bayesian methods. The input data to these algorithms are organ shapes derived from 3D scans. The methods can take a varying number of such inputs from a given patient, and will produce more accurate results the more inputs they are given. They provide an estimate of the mean shape of the organ, as well as the uncertainty of this mean, in addition to the distribution of the variation of shapes from fraction to fraction. The methods were evaluated in the task of estimating coverage probabilities, i.e. the probability that the organ will cover a certain point in the patient coordinate system, for the rectum of prostate cancer patients. For this evaluation, tens of thousands of organ shapes needed to be converted to so-called binary masks, which are 3D arrays of points in the patient coordinate system where the value of each point is 1 if the point is inside the organ and 0 if it is outside. This was enabled by the highly efficient point-in-polyhedron software presented in paper III, which was developed for this project. The models were given varying number of scans, from 1 to 10, as input, and compared to two existing (non-Bayesian) models. The estimates of the coverage probability produced by the new models were significantly more similar to the ground truth than those produced by the existing models, at least up to three input scans. The main differences between the two new algorithms are their of conceptual complexity and accuracy, and the choice of method in a given application will therefore come down to a trade-off between these qualities. An application for the models derived in paper II, concerning patients receiving re-irradiation for recurrent prostate cancer, is presented in paper IV. We introduce a way of estimating the expectation and uncertainty of the accumulated dose to the rectum from the two treatment courses. The method is based on "representative shapes" of the rectum, that is, shapes that are probable and also particularly favourable or unfavourable in terms of dose. The advantage is that these shapes can be used as a visual aid for the oncologist or dose planner, and that the method can be implemented using existing features of treatment planning systems. Overall, this thesis provides novel solutions to the central challenge of organ motion mitigation in RT. The presented models are the first to simultaneously exploit population and patient specific organ motion and addressing both systematic and random errors.Doktorgradsavhandlin

Bayesian data assimilation in shape registration

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions\ud for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum p0 and the reparameterisation vector field v, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed MCMC methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem

Task adapted reconstruction for inverse problems

The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any task that is encodable as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation