The foot and ankle is a complex structure consisting of 28 bones and 30 joints that changes from being completely mobile when positioning the foot on the floor to a rigid closed pack position during propulsion such as when running or jumping. An understanding of this complex structure has largely been derived from cadaveric studies. In vivo studies have largely relied on skin surface markers and multi-camera systems that are unable to differentiate small motions between the bones of the foot. MRI and CT based studies have struggled to interpret functional weight bearing motion as imaging is largely static and non-load bearing. Arthritic diseases of the foot and ankle are treated either by fusion of the joints to remove motion, or joint replacement to retain motion. Until a better understanding of the biomechanics of these joints can be achieved

Betcke, M. (Marta)

Djurabekova, N. (Nargiza)

Goldberg, A. (Andrew)

Hauptmann, A. (Andreas)

Hawkes, D. (David)

Long, G. (Guy)

Lucka, F. (Felix)

English

CWI's Institutional Repository

PROCEEDINGS OF SPIESPIEDigitalLibrary.org/conference-proceedings-of-spieApplication of Proximal AlternatingLinearized Minimization (PALM) andinertial PALM to dynamic 3D CTNargiza Djurabekova, Andrew Goldberg, AndreasHauptmann, David Hawkes, Guy Long, et al.Nargiza Djurabekova, Andrew Goldberg, Andreas Hauptmann, David Hawkes,Guy Long, Felix Lucka, Marta Betcke, "Application of Proximal AlternatingLinearized Minimization (PALM) and inertial PALM to dynamic 3D CT," Proc.SPIE 11072, 15th International Meeting on Fully Three-Dimensional ImageReconstruction in Radiology and Nuclear Medicine, 1107208 (28 May 2019);doi: 10.1117/12.2534827Event: Fully Three-Dimensional Image Reconstruction in Radiology andNuclear Medicine, 2019, Philadelphia, United StatesDownloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 02 Sep 2019  Terms of Use: https://www.spiedigitallibrary.org/terms-of-useApplication of Proximal Alternating LinearizedMinimization (PALM) and inertial PALM todynamic 3D CTNargiza Djurabekova1, Andrew Goldberg1,3, Andreas Hauptmann1,5, DavidHawkes1, Guy Long2, Felix Lucka1,4, and Marta Betcke11University College London2Curvebeam3RNOH NHS Trust4Centrum Wiskunde & InformaticaKeywords: Cone beam CT, dynamic CT, non-convex optimisation, PALM, iPALM, opticalflow regularisation, variational methods1. INTRODUCTION1.1 Motivation and backgroundThe foot and ankle is a complex structure consisting of 28 bones and 30 joints that changesfrom being completely mobile when positioning the foot on the floor to a rigid closed packposition during propulsion such as when running or jumping. An understanding of this complexstructure has largely been derived from cadaveric studies.1 In vivo studies have largely relied onskin surface markers and multi-camera systems2 that are unable to differentiate small motionsbetween the bones of the foot . MRI and CT based studies have struggled to interpret functionalweight bearing motion as imaging is largely static and non-load bearing. Arthritic diseases of thefoot and ankle are treated either by fusion of the joints to remove motion, or joint replacement toretain motion. Until a better understanding of the biomechanics of these joints can be achieved,it is difficult to determine which treatment offers the best functional outcome (see Figure 1).Figure 1: Left: Ankle arthroplasty or ankle replacement. Right: Ankle arthrodesis or ankle fusion.Images courtesy of Mr Andrew Goldberg MD FRCS(Tr & Orth)The aim of the current project is to develop reconstruction techniques for a fully 4D (3D+ time) Cone beam CT (CBCT) of the foot and ankle using a weight-bearing CBCT machine(Curvebeam, Hatfield, PA, USA) such as PedCat or LineUP.15th International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, edited by Scott D. Metzler, Samuel Matej, Proc. of SPIE Vol. 11072, 1107208 · © 2019 SPIE · CCC code: 0277-786X/19/$18 · doi: 10.1117/12.2534827Proc. of SPIE Vol. 11072  1107208-1Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 02 Sep 2019Terms of Use: https://www.spiedigitallibrary.org/terms-of-use1.2 CT and motionMotion arises in many imaging applications whenever the imaging target exhibits dynamics attime scales of the order of image acquisition time. This can lead to blurring and other artifactswhich can affect diagnostic validity of the reconstruction from the acquired data. To this end, alot of research concentrates on motion correction3 although new application relying on dynamicimaging are constantly emerging, particularly in the field of cardiac imagining.In line with this development the goal of the present paper is to reconstruct the dynamicsof the more rigid foot and ankle structure. This is done by using optical flow (or its naturalextension into the 3D space - scene flow). Optical flow, or the transport term, is the assumptionof brightness constancy over all time-frames. That is, we assume that all elements of the footpresent in one frame are going to appear in the next frame, with only some fairly small changesin location and orientation. Such changes can be described using a vector field, also known asthe motion field, which is reconstructed along the linear attenuation coefficient.2. METHODOLOGY2.1 SetupAs it is unrealistic to expect to reconstruct the full 3D foot with just a single projection pertime-frame, we assume that the foot can move in a semi-consistent periodic motion. This wouldallow us to bin several projections to help reconstruct individual time-frames of the full motion.This situation is simulated as follows. A set of bones, extracted from a segmented full datareconstruction of the foot and ankle, are arranged and animated to create the phantom that isused to test the algorithms described in this paper (see Acknowledgements). ASTRA toolbox4emulates the forward conebeam projections at quasi-randomly selected angles. The angles weresampled using stratified random sampling, i.e. to get three projection angles, each would bechosen randomly from a designated third of all available angles.2.2 Algorithms and the problem formulationWe begin by formulating our problem, similarly to 5,6, as a minimization problem where regu-larization parameters are used to find balance between the data fidelity term, the optical flowterm (briefly mentioned in sect. 1) as well as TV regularization on the image u and the motionfield v. So, the discretized spatio-temporal CT problem can be stated as(u∗,v∗) = arg minu≥0,v{ T∑t12||Aut − ft||22 + αTV (ut) + βTVs(vt) +γ2||∇tut + (∇ut) · vt||22}, (1)where A is the forward operator, f is the measurement, the last term is the aforementionedoptical flow term and α, β and γ are the regularization parameters. Lastly TVs is just TV that’sapplied to each individual component of the motion field vThis problem is non-convex and can be re-arranged to satisfy the general functionminimizex,yF (x, y) := f1(x) + f2(y) +H(x, y), (2)Proc. of SPIE Vol. 11072  1107208-2Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 02 Sep 2019Terms of Use: https://www.spiedigitallibrary.org/terms-of-usethat can be solved by PALM7 or iPALM,8 with f1 and f2 being the TV and TVs respectivelyand H, the coupling function with Lipschitz continuous gradient, being the sum of the datafidelity and the transport terms. Then the PALM algorithm is described in Alg.1. iPALM,the inertial counterpart of PALM is similar but with some of the momentum saved from theprevious iterates. The main way the two algorithms differ is in their convergence, with PALMconverging in a strictly decreasing fashion and iPALM usually converging faster, as can be seenin figure 2.Algorithm 1 PALM1: Initialization: Choose any u0 ∈ R2(×R)× Nt and v0 ∈ R2(×R)× Nt × Ns, where Nt is the timestep dimension and Ns is the spatial component dimension with s = 1, 2(, 3).2: For every k = 0, 1, ... generate a sequence {(uk,vk)}3: Repeat4: For some γ1 > 1, let ck = γ1L1(vk)uk+1 ∈ proxf1ck(uk − 1ck∇uH(uk,vk))(3)5: For some γ2 > 1, let dk = γ2L2(uk+1)vk+1 ∈ proxf2dk(uk − 1dk∇vH(uk+1,vk))(4)3. RESULTS AND DISCUSSIONThe algorithms were implemented in MATLAB 2017a, using gpuArrays of resolution 1283 pixels,with a total of 19 time-frames. The appropriate regularization parameters (from (1)) were foundin a systematic manner. First setting γ = 1, which assumes that the transport term has thesame weight as the data fidelity term, ignoring β by setting it to 0 and just finding the right α.Once an α that denoises the image sufficiently well is found, β is adjusted to denoise the motionfield. Lastly, γ can be slightly adjusted if needed to improve the overall quality of both outputs.In order to find the appropriate step lengths, the respective Lipschitz constants are esti-mated in every iteration for the image u and the motion field v using power iteration. Thisis computationally quite costly, which is why in practice the Lipschitz constant for the image(which barely changes over the course of all iterations) is estimated once and remains fixed forthe subsequent iterations.We highlight a couple of conclusions that can be drawn from results in figures 3 and 2.Firstly, we note that using 3 projection angles, both PALM and iPALM seem to produce betterreconstruction than FISTA (an excellent reconstruction algorithm for static objects). Secondly,iPALM converges already on the 1100th iteration with a tolerance of 10−4 , while PALM doesn’treach full convergence even after 3000 iterations. And finally it is interesting to observe that theover-smoothing effect of PALM is likely to dissapear with full convergence, as is the case wuithProc. of SPIE Vol. 11072  1107208-3Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 02 Sep 2019Terms of Use: https://www.spiedigitallibrary.org/terms-of-useFigure 2: Objective function/energy values plotted over alliterations. (a) for PALM and (b) for iPALM. Note thatPALM was stopped due to reaching the maximum iterationof 3000, while iPALM converged at about the 1100th itera-tion with the tolerance of 10−4 between consecutive imageiterations.Figure 3: Isosurface plots of the phan-tom and its reconstructions using 3projection angles at time-frame 11.(a) Original phantom, (b) FISTA,9 (c)PALM, (d) iPALM.iPALM (see figure 3), returning a reconstruction that has similar unevenness as the originalphantom.While even iPALM reconstruction is not perfect and computationally quite heavy, it isencouraging to see that dynamic 3D CT is not completely out of reach. By getting betterreconstruction techniques, we are likely to better understand the complex motion in the ankleand foot and design better treatment to help patients.AcknowledgmentsThe authors would like to extend gratitude to NVIDIA Corporation for the GeForce TitanXp GPU used in this research. We are also grateful to CurveBeam for financial support andcollaboration on this project. For the phantom’s model our thanks go out to Dr Francois Lintzfrom Clinique de L’Union for the data and Prof Sorin Siegler from Drexel University for thesegmentation.REFERENCES[1] C. Nester, “Lessons from dynamic cadaver and invasive bone pin studies: do we know how the footreally moves during gait?,” Journal of Foot and Ankle Research 2(1), 18 (2009).[2] J. Leitch, J. Stebbins, and A. Zavatsky, “Subject-specific axes of the ankle joint complex,” Journalof Biomechanics 43(15), 2923 – 2928 (2010).[3] S. Jang, S. Kim, M. Kim, et al., “Head motion correction based on filtered backprojection for x-rayct imaging,” Medical Physics 45(2), 589–604 (2018).Proc. of SPIE Vol. 11072  1107208-4Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 02 Sep 2019Terms of Use: https://www.spiedigitallibrary.org/terms-of-use[4] W. van Aarle, W. Palenstijn, J. Cant, et al., “Fast and flexible x-ray tomography using the astratoolbox,” Opt. Express 24, 25129–25147 (2016).[5] M. Burger, H. Dirks, L. Frerking, et al., “A variational reconstruction method for undersampleddynamic x-ray tomography based on physical motion models,” Inverse Problems 33(12), 124008(2017).[6] M. Burger, H. Dirks, and C.-B. Schonlieb, “A variational model for joint motion estimation andimage reconstruction,” SIAM J. Imaging Sciences 11(1), 94 – 128 (2018).[7] J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvexand nonsmooth problems,” Mathematical Programming 146, 459–494 (2014).[8] T. Pock and S. Sabach, “Inertial proximal alternating linearized minimization (ipalm) for nonconvexand nonsmooth problems,” 9, 1756–1787 (2016).[9] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverseproblems,” SIAM Journal on Imaging Sciences 2(1), 183–202 (2009).Proc. of SPIE Vol. 11072  1107208-5Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 02 Sep 2019Terms of Use: https://www.spiedigitallibrary.org/terms-of-use

Application of Proximal Alternating Linearized Minimization (PALM) and inertial PALM to dynamic 3D CT

Application of Proximal Alternating Linearized Minimization (PALM) and inertial PALM to dynamic 3D CT

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