{"title": "Learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data", "book": "Advances in Neural Information Processing Systems", "page_first": 1152, "page_last": 1160, "abstract": "We introduce a hierarchical model which allows to estimate a group-average piecewise-geodesic trajectory in the Riemannian space of measurements and individual variability. This model falls into the well defined mixed-effect models. The subject-specific trajectories are defined through spatial and temporal transformations of the group-average piecewise-geodesic path, component by component. Thus we can apply our model to a wide variety of situations. Due to the non-linearity of the model, we use the Stochastic Approximation Expectation-Maximization algorithm to estimate the model parameters. Experiments on synthetic data validate this choice. The model is then applied to the metastatic renal cancer chemotherapy monitoring: we run estimations on RECIST scores of treated patients and estimate the time they escape from the treatment. Experiments highlight the role of the different parameters on the response to treatment.", "full_text": "Learning spatiotemporal piecewise-geodesic\n\ntrajectories from longitudinal manifold-valued data\n\nJuliette Chevallier\n\nCMAP, \u00c9cole polytechnique\n\njuliette.chevallier@polytechnique.edu\n\nPr St\u00e9phane Oudard\nOncology Department\nUSPC, AP-HP, HEGP\n\nSt\u00e9phanie Allassonni\u00e8re\n\nCRC, Universit\u00e9 Paris Descartes\n\nstephanie.allassonniere@parisdescartes.fr\n\nAbstract\n\nWe introduce a hierarchical model which allows to estimate a group-average\npiecewise-geodesic trajectory in the Riemannian space of measurements and in-\ndividual variability. This model falls into the well de\ufb01ned mixed-effect models.\nThe subject-speci\ufb01c trajectories are de\ufb01ned through spatial and temporal trans-\nformations of the group-average piecewise-geodesic path, component by compo-\nnent. Thus we can apply our model to a wide variety of situations. Due to the\nnon-linearity of the model, we use the Stochastic Approximation Expectation-\nMaximization algorithm to estimate the model parameters. Experiments on syn-\nthetic data validate this choice. The model is then applied to the metastatic renal\ncancer chemotherapy monitoring: we run estimations on RECIST scores of treated\npatients and estimate the time they escape from the treatment. Experiments high-\nlight the role of the different parameters on the response to treatment.\n\n1\n\nIntroduction\n\nDuring the past few years, the way we treat renal metastatic cancer was profoundly changed: a new\nclass of anti-angiogenic therapies targeting the tumor vessels instead of the tumor cells has emerged\nand drastically improved survival by a factor of three (Escudier et al., 2016). These new drugs,\nhowever, do not cure the cancer, and only succeed in delaying the tumor growth, requiring the use of\nsuccessive therapies which must be continued or interrupted at the appropriate moment according to\nthe patient\u2019s response. The new medicine process has also created a new scienti\ufb01c challenge: how\nto choose the more ef\ufb01cient drug therapy. This means that one has to properly understand how the\npatient reacts to the possible treatments. Actually, there are several strategies and taking the right\ndecision is a contested issue (Rothermundt et al., 2015, 2017).\nTo achieve that goal, physicians took an interest in mathematical modeling. Mathematics has already\ndemonstrated its ef\ufb01ciency and played a role in the change of stop-criteria for a given treatment\n(Burotto et al., 2014). However, to the best of our knowledge, there only exists one model which\nwas designed by medical practitioners. Although, very basic mathematically, it seems to show that\nthis point of view may produce interesting results. Introduced by Stein et al. in 2008, the model\nperforms a non-linear regression using the least squares method to \ufb01t an increasing or/and decreasing\nexponential curve. This model is still used but suffers limitations. First, as the pro\ufb01le are \ufb01tted\nindividual-by-individual independently, the model cannot explain a global dynamic. Then, the choice\nof exponential growth avoids the emergence of plateau effects which are often observed in practice.\nThis opens the way to new models which would explain both a population and each individual with\nother constraints on the shape of the response.\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\fLearning models of disease progression from such databases raises great methodological challenges.\nWe propose here a very generic model which can be adapted to a large number of situations. For a\ngiven population, our model amounts to estimating an average trajectory in the set of measurements\nand individual variability. Then we can de\ufb01ne continuous subject-speci\ufb01c trajectories in view of the\npopulation progression. Trajectories need to be registered in space and time, to allow anatomical\nvariability (as different tumor sizes), different paces of progression and sensitivity to treatments. The\nframework of mixed-effects models is well suited to deal with this hierarchical problem. Mixed-\neffects models for longitudinal measurements were introduced in the seminal paper of Laird and Ware\n(1982) and have been widely developed since then. The recent generic approach of Schiratti et al.\n(2015) to align patients is even more suitable. First, anatomical data are naturally modeled as points on\nRiemannian manifolds while the usual mixed-effects models are de\ufb01ned for Euclidean data. Secondly,\nthe model was built with the aim of granting individual temporal and spatial variability through\nindividual variations of a common time-line grant and parallel shifting of the average trajectory.\nHowever, Schiratti et al. (2015) have made a strong hypothesis to build their model as they consider\nthat the mean evolution is a geodesic. This would mean in our targeted situation that the cancer\nwould either go on evolving or is always sensitive to the treatment. Unfortunately, the anti-angiogenic\ntreatments may be inef\ufb01cient, ef\ufb01cient or temporarily ef\ufb01cient, leading to a re-progression of the\nmetastasis. Therefore, we want to relax this assumption on the model.\nIn this paper, we propose a generic statistical framework for the de\ufb01nition and estimation of spatiotem-\nporal piecewise-geodesic trajectories from longitudinal manifold-valued data. Riemannian geometry\nallows us to derive a method that makes few assumptions about the data and applications dealt with.\nWe \ufb01rst introduce our model in its most generic formulation and then make it explicit for RECIST\n(Therasse et al., 2000) score monitoring, i.e. for one-dimension manifolds. Experimental results\non those scores are given in section 4.2. The introduction of a more general model is a deliberate\nchoice as we are expecting to apply our model to the corresponding medical images. Because of\nthe non-linearity of the model, we have to use a stochastic version of the Expectation-Maximization\nalgorithm (Dempster et al., 1977), namely the MCMC-SAEM algorithm, for which theoretical results\nregarding the convergence have been proved in Delyon et al. (1999) and Allassonni\u00e8re et al. (2010)\nand numerical ef\ufb01ciency has been demonstrated for these types of models (Schiratti et al. (2015),\nMONOLIX \u2013 MOd\u00e8les NOn LIn\u00e9aires \u00e0 effets miXtes).\n\n2 Mixed-effects model for piecewise-geodesically distributed data\nWe consider a longitudinal dataset obtained by repeating measurements of n \u2208 N\u2217 individuals,\n\nwhere each individual i \u2208(cid:74)1, n(cid:75) is observed ki \u2208 N\u2217 times, at the time points ti = (ti,j)1(cid:54)j(cid:54)ki\ndenote k = (cid:80)n\nand where yi = (yi,j)1(cid:54)j(cid:54)ki denotes the sequence of observations for this individual. We also\ni=1 ki the total numbers of observations. We assume that each observation yi,j\nis a point on a d-dimensional geodesically complete Riemannian manifold (M, g), so that y =\n(yi,j)1(cid:54)i(cid:54)n, 1(cid:54)j(cid:54)ki \u2208 M k.\nWe generalize the idea of Schiratti et al. (2015) and build our model in a hierarchical way. We see\nour data points as samples along trajectories and suppose that each individual trajectory derives\nfrom a group-average scenario through spatiotemporal transformations. Key to our model is that the\ngroup-average trajectory in no longer assumed to be geodesic but piecewise-geodesic.\n\n2.1 Generic piecewise-geodesic curves model\nR < . . . < tm\u22121\n\nLet m \u2208 N\u2217 and tR =(cid:0)\u2212\u221e < t1\nup times sequence. Let M0 a d-dimensional geodesically complete manifold and(cid:0)\u00af\u03b3(cid:96)\n\nR < +\u221e(cid:1) a subdivision of R, called the breaking-\n\n(cid:1)\n\n1(cid:54)(cid:96)(cid:54)m a\nfamily of geodesics on M0. To completely de\ufb01ne our average trajectory, we introduce m isometries\n0 : M0 \u2192 M (cid:96)\n0 \u2013 by\n\u03c6(cid:96)\n0 ensures that the manifolds\nsetting down \u03b3(cid:96)\n0 remains\nM (cid:96)\nparametrizable (Gallot et al., 2004). We de\ufb01ne the average trajectory by\n\n0(M0). This de\ufb01nes m new geodesics \u2013 on the corresponding space M (cid:96)\n0 \u25e6 \u00af\u03b30\n\n0 remain Riemannian and that the curves \u03b3(cid:96)\n\n0 remain geodesic. In particular, each \u03b3(cid:96)\n\n(cid:96). The isometric nature of the mapping \u03c6(cid:96)\n\n0 := \u03c6(cid:96)\n0 = \u03c6(cid:96)\n\n0\n\n\u2200t \u2208 R,\n\n\u03b30(t) = \u03b31\n\n0 (t) 1\n\n]\u2212\u221e,t1\n\nR](t) +\n\n\u03b3(cid:96)\n0(t) 1\n\n]t(cid:96)\u22121\nR ,t(cid:96)\n\nR](t) + \u03b3m\n\n0 (t) 1\n\n]tm\u22121\n\n,+\u221e[(t) .\n\nR\n\nm\u22121(cid:88)\n\n(cid:96)=2\n\n2\n\n\ft1 \u2208 R. We impose1 that for all (cid:96) \u2208(cid:74)1, m \u2212 1(cid:75), \u00af\u03b31\n\nIn this framework, M0 may be understood as a manifold-template of the geodesic components of the\ncurve \u03b30.\nBecause of the piecewise nature of our average-trajectory \u03b30, constraints have to be formulated on\neach interval of the subdivision tR. Following the formulation of the local existence and uniqueness\ntheorem (Gallot et al., 2004), constraints on geodesics are generally formulated by forcing a value\nand a tangent vector at a given time-point. However, such an approach cannot ensure the curve \u03b30\nto be at least continuous. That is why we re-formulate these constraints in our model as boundary\nconditions. Let a sequence \u00afA = ( \u00afA0, . . . , \u00afAm) \u2208 (M0)m+1, an initial time t0 \u2208 R and a \ufb01nal time\nR) = \u00afA(cid:96) and\n0 (t1) = \u00afAm. Notably, the 2m constraints are de\ufb01ned step by step. In one dimension (cf section\n\u00af\u03b3m\n2.2), the geodesics could be written explicitly and such constraints do not complicate the model so\nmuch. In higher dimension, we have to use shooting or matching methods to enforce this constraint.\nIn practice, the choice of the isometries \u03c6(cid:96)\n0 have to be done with the aim to be\n\"as regular as possible\" (at least continuous as said above) at the rupture points t(cid:96)\nR. In one dimension\nfor instance, we build trajectories that are continuous, not differentiable but with a very similar slope\non each side of the breaking-points.\nWe want the individual trajectories to represent a wide variety of behaviors and to derive from the\ngroup average path by spatiotemporal transformations. To do that, we de\ufb01ne for each component (cid:96) of\nthe piecewise-geodesic curve \u03b30 a couple of transformations (\u03c6(cid:96)\ni ). These transformations, namely\nthe diffeomorphic component deformations and the time component reparametrizations, characterize\nrespectively the spatial and the temporal variability of propagation among the population. Thus,\nindividual trajectories may write in the form of\n\n0 and the geodesics \u00af\u03b3(cid:96)\n\nR) = \u00afA(cid:96), \u00af\u03b3(cid:96)+1\n\n0 (t0) = \u00afA0, \u00af\u03b3(cid:96)\n\ni , \u03c8(cid:96)\n\n0(t(cid:96)\n\n(t(cid:96)\n\n0\n\n\u2200t \u2208 R,\n\n\u03b3i(t) = \u03b31\n\ni (t) 1\n\n]\u2212\u221e,t1\n\nR,i](t) +\n\n\u03b3(cid:96)\ni (t) 1\n\n]t(cid:96)\u22121\nR,i ,t(cid:96)\n\nR,i](t) + \u03b3m\n\ni (t) 1\n\n]tm\u22121\n\nR,i\n\n,+\u221e[(t)\n\n((cid:63))\n\nm\u22121(cid:88)\n\nof rupture times tR,i =(cid:0)t(cid:96)\n\n(cid:96)=2\n0 through the applications of the two transformations\ni described below. Note that, in particular, each individual possesses his own sequence\n. Moreover, we require the fewest constraints possible in the\n\nwhere the functions \u03b3(cid:96)\ni and \u03c8(cid:96)\n\u03c6(cid:96)\n\ni are obtained from \u03b3(cid:96)\n\n(cid:1)\n\nR,i\n\n1(cid:54)(cid:96) 2p and m\u03c3 > 2. In practice, we yet use degenerate priors but with correct posteriors\n.To be consistent with the one-dimension inverse Wishart distribution, we de\ufb01ne the density function\nof distribution of higher dimension as\n\n(cid:1) (cid:32) (cid:112)|V |\n2(cid:112)|\u03a3| exp\n\n2\n\np\n\n(cid:18)\n\n(cid:0) m\u03a3\n\n1\n\n2\n\ntr(cid:0)V \u03a3\u22121(cid:1)(cid:19)(cid:33)m\u03a3\n\n\u2212 1\n2\n\nfW\u22121(V,m\u03a3)(\u03a3) =\n\n\u0393p\n\nwhere \u0393p is the multivariate gamma function. The maximization step is straightforward given the\nsuf\ufb01cient statistics of our exponential model: we update the parameters by taking a barycenter\nbetween the corresponding suf\ufb01cient statistic and the prior. See the supplementary material for\nexplicit equations.\n\n3.2 Existence of the Maximum a Posteriori\n\nThe next theorem ensures us that the model is well-posed and that the maximum we are looking for\nthrough the MCMC-SAEM algorithm exists. Let \u0398 the space of admissible parameters :\n\n(cid:17) \u2208 R5 \u00d7 SpR \u00d7 R+(cid:12)(cid:12)(cid:12) \u03a3 positive-de\ufb01nite\n\n(cid:111)\n\n.\n\n\u0398 =\n\n0 , \u03b3escap\n\u03b3init\n\n0\n\n, \u03b3\ufb01n\n\n0 , tR, t1, \u03a3, \u03c3\n\n(cid:110) (cid:16)\n\nTheorem 1 (Existence of the MAP). Given the piecewise-logistic model and the choice of\nprobability distributions for the parameters and latent variables of the model, for any dataset\n\n(ti,j, yi,j)i\u2208(cid:74)1,n(cid:75), j\u2208(cid:74)1,ki(cid:75), there exist(cid:98)\u03b8M AP \u2208 argmax\n\nq(\u03b8|y).\n\n\u03b8\u2208\u0398\n\nA detailed proof is postponed to the supplementary material.\n\n4 Experimental results\n\nThe piecewise-logistic model has been designed for chemotherapy monitoring. More speci\ufb01cally,\nwe have met radiologists of the H\u00f4pital Europ\u00e9en Georges-Pompidou (HEGP \u2013 Georges Pompidou\nEuropean Hospital) to design our model. In practice, patients suffer from the metastatic kidney cancer\nand take a drug each day. Regularly, they come to the HEGP to check the tumor evolution. The\nresponse to a given treatment has generally two distinct phases: \ufb01rst, tumor\u2019s size reduces; then, the\ntumor grows again. A practical question is to quantify the correlation between both phases and to\ndetermine as accurately as possible the individual rupture times ti\nR which are related to an escape of\nthe patient\u2019s response to treatment.\n\n4.1 Synthetic data\n\nIn order to validate our model and numerical scheme, we \ufb01rst run experiments on synthetic data.\nWe well understood that the covariance matrix \u03a3 gives a lot of information on the health status of a\npatient: pace and amplitude of tumor progression, individual rupture times. . . Therefore, we have to\npay special attention to the estimation of \u03a3 in this paragraph.\nAn important point was to allow a lot of different individual behaviors. In our synthetic example,\nFigure 1a illustrates this variability. From a single average trajectory (\u03b30 in bold plain line), we can\ngenerate individuals who are cured at the end (dot-dashed lines: \u03b33 and \u03b34), some whose response to\nthe treatment is bad (dashed lines: \u03b35 and \u03b36), some who only escape (no positive response to the\ntreatments \u2013 dotted lines: \u03b37). Likewise, we can generate \"patients\" with only positive responses or\nno response at all. The case of individual 4 is interesting in practice: the tumor still grows but so\nslowly that the growth is negligible, at least in the short-run.\nFigure 2 illustrates the qualitative performance of the estimation. We are notably able to understand\nvarious behaviors and \ufb01t subjects which are far from the average path, such as the orange and the\ngreen curves. We represent only \ufb01ve individuals but 200 subjects have been used to perform the\nestimation.\nTo measure the in\ufb02uence of the sample size on our model/algorithm, we generate synthetic datasets\nof various size and perform the estimation 50 times for each dataset. Means and standard deviations\n\n6\n\n\f200\n\n100\n\n)\ns\ns\ne\nl\nn\no\ni\nt\nn\ne\nm\ni\nd\n(\n\ne\nr\no\nc\ns\nT\nS\nI\nC\nE\nR\n\n0\n\n0\n\n200\n\n100\n\n)\ns\ns\ne\nl\nn\no\ni\nt\nn\ne\nm\ni\nd\n(\n\ne\nr\no\nc\ns\nT\nS\nI\nC\nE\nR\n\n0\n\n0\n\n500\n\n1,000\n\nTimes (in days)\n\n1,500\n\n(b) After 600 iterations.\n\n500\n\n1,000\n\nTimes (in days)\n\n1,500\n\n(a) Initialisation.\n\nFigure 2: Initialisation and \"results\". On both \ufb01gures, the estimated trajectories are in plain lines and\nthe target curves in dashed lines. The (noisy) observations are represented by crosses. The average\npath is in bold black line, the individuals in color. Figure 2a: Population parameters zpop and latent\nvariables zpop are initialized at the empirical mean of the observations; individual trajectories are\ninitialized on the average trajectory (P = 0, \u03a3 = 0.1Ip, \u03c3 = 1). Figure 2b: After 600 iterations,\nsometime less, the estimated curves \ufb01t very well the observations. As the algorithm is stochastic,\nestimated curves \u2013 and effectively the individuals \u2013 still wave around the target curves.\n\nTable 1: Mean (standard deviation) of relative error (expressed as a percentage) for the population\nparameters zpop and the residual standard deviation \u03c3 for 50 runs according to the sample size n.\nSample\nsize n\n50\n100\n150\n\n11.58 (1.64)\n13.62 (1.31)\n9.24 (1.63)\n\n9.45 (5.40)\n9.07 (5.19)\n11.40 (5.72)\n\n25.24 (12.84)\n10.35 (3.96)\n2.83 (2.31)\n\n1.63 (1.46)\n2.42 (1.50)\n2.14 (1.17)\n\n6.23 (2.25)\n7.82 (2.43)\n5.82 (2.55)\n\n4.41 (0.75)\n5.27 (0.60)\n3.42 (0.71)\n\n\u03b3escap\n0\n\n\u03b3init\n0\n\n\u03b3\ufb01n\n0\n\ntR\n\nt1\n\n\u03c3\n\n0 , \u03b3escap\n\n0\n\n, \u03b3\ufb01n\n\nof the relative errors for the real parameters, namely \u03b3init\n0 , tR, t1 and \u03c3, are compiled\nin Table 1. To compare things which are comparable, we have generated a dataset of size 200 and\nshortened them to the desired size. Moreover, to put the algorithm on a more realistic situation, the\nsynthetic individual times are non-periodically spaced, individual sizes vary between 12 and 18 and\nthe observed values are noisy (\u03c3 = 3).\nWe remark that our algorithm is stable and that the bigger the sample size, the better we learn the\nresidual standard deviation \u03c3. The parameters tR and \u03b3escap\nare quite dif\ufb01cult to learn as they occur\non the \ufb02at section of the trajectory. However, the error we made is not crippling as the most important\nfor clinicians is the dynamic along both phases. As the algorithm enables to estimate both the mean\ntrajectory and the individual dynamic, it succeeds in estimating the inter-individual variability. This\nends in a good estimate of the covariance matrix \u03a3 (see \ufb01gure 4).\n\n0\n\n4.2 Chemotherapy monitoring: RECIST score of treated patients\n\nWe now run our estimation algorithm on real data from HEGP.\nThe RECIST (Response Evaluation Criteria In Solid Tumors) score (Therasse et al., 2000) measures\nthe tumoral growth and is a key indicator of the patient survival. We have performed the estimation\nover a drove of 176 patients of the HEGP. There is an average of 7 visits per subjects (min: 3, max:\n22), with an average duration of 90 days between consecutive visits.\nWe have run the algorithm several times, with different proposal laws for the sampler (a Symmetric\nRandom Walk Hasting-Metropolis within Gibbs one) and different priors. We present here a run with\na low residual standard variation in respect to the amplitude of the trajectories and complexity of the\ndataset: \u03c3 = 14.50 versus max(\u03b3init\n0 = 452.4. Figure 3a illustrates the performance of\nthe model on the \ufb01rst eight patients. Although we cannot explain all the paths of progression, the\nalgorithm succeeds in \ufb01tting various types of curves: from the yellow curve \u03b33 which is rather \ufb02at\nand only escape to the red \u03b37 which is spiky. From Figure 3b, it seems that the rupture times occur\nearly in the progression in average. Nevertheless , this result is to be considered with some reserve:\nthe rupture time generally occurs on a stable phase of the disease and the estimation may be dif\ufb01cult.\n\n0 ) \u2212 \u03b3escap\n\n0 , \u03b3\ufb01n\n\n7\n\n\f400\n\n200\n\n)\ns\ns\ne\nl\nn\no\ni\nt\nn\ne\nm\ni\nd\n(\n\ne\nr\no\nc\ns\nT\nS\nI\nC\nE\nR\n\n0\n\n0\n\n\u03b30\n\u03b31\n\u03b32\n\u03b33\n\u03b34\n\u03b35\n\u03b36\n\u03b37\n\u03b38\n\n40\n\n20\n\n0\n\n0\n\n1,000\n\n2,000\n\n3,000\n\n4,000\n\n5,000\n\nIndividual rupture times (in days)\n\n(b) Individual rupture times ti\nR.\n\n100\n\n200\n300\nTimes (in days)\n\n400\n\n500\n\n(a) After 600 iterations.\n\nFigure 3: RECIST score. We keep conventions of the previous \ufb01gures. Figure 3a is the result of a 600\niterations run. We represent here only the \ufb01rst 8 patients among the 176. Figure 3b is the histogram of\nthe rupture times ti\nR for this run. In black bold line, the estimated average rupture time tR is a good\nestimate of the average of the individual rupture times although there exists a large range of escape.\n\n10\n\n0\n\n\u03c4\n\nt\nf\ni\nh\ns\n\ne\nm\nT\n\ni\n\n\u2192\n\nt\ne\ns\nn\no\nt\ns\na\nL\n\nt\ne\ns\nn\no\ny\nl\nr\na\nE\n\n\u2190\n\n\u221210\n\u22124\n\n\u22122\n1st acceleration factor \u03be1\n\n\u2190Slow response Fast response\u2192\n\n0\n\n2\n\n4\n\n2\n\n0\n\n\u22125\n\nSlowprogress\u2192\nnd accelerationfactor\u03be\n\u2190 Fastprogress\n\n5\n\n2\n\n(a) The time warp.\n\n\u2192\n\n500\n\ne\nr\no\nc\ns\nh\ng\ni\nH\n\n\u03b4\n\nt\nf\ni\nh\ns\n\ne\nc\na\np\nS\n\ne\nr\no\nc\ns\nw\no\nL\n\n\u2190\n\nIndividual\n\nrupture times ti\nR\n\n(in days)\n\n4,000\n\n2,000\n\n0\n\n0\n\n\u221210\n0\n1st amplitude factor \u03c11\n\u2190Low step High step\u2192\n\n(b) The space warp.\n\n2\n\n0\n\n\u22125\n\nnd amplitudefactor\u03c1\nLowstep\u2192\n\u2190 Highstep\n\n2\n\n5\n\nFigure 4: Individual random effects. Figure 4a: log-acceleration factors \u03be1\nshifts \u03c4i. Figure 4b: log-amplitude factors \u03c11\ncorresponds to the individual rupture time ti\n\ni against times\ni and \u03c12\ni against space shifts \u03b4i. In both \ufb01gure, the color\nR. These estimations hold for the same run as Figure 3.\n\ni and \u03be2\n\nIn Figure 4, we plot the individual estimates of the random effects (obtained from the last iteration) in\ncomparison to the individual rupture times. Even though the parameters which lead the space warp,\ni.e. \u03c11\ni and \u03b4i are correlated, the correlation with the rupture time is not clear. In other words, the\nvolume of the tumors seems to not be relevant to evaluate the escape of a patient. On the contrary,\nwhich is logical, the time warp strongly impacts the rupture time.\n\ni , \u03c12\n\n4.3 Discussion and perspective\n\nWe propose here a generic spatiotemporal model to analyze longitudinal manifold-valued measure-\nments. Contrary to Schiratti et al. (2015), the average trajectory is not assumed to be geodesic\nanymore. This allows us to apply our model to more complex situations: in chemotherapy monitoring\nfor example, where the patients are treated and tumors may respond, stabilize or progress during\nthe treatment, with different conducts for each phase. At the age of personalized medicine, to give\nphysicians decision support systems is really important. Therefore learning correlations between both\nphases is crucial. This has been taken into account here.\nFor purpose of working with more complicated data, medical images for instance, we have \ufb01rst\npresented our model in a very generic version. Then we made it explicit for RECIST scores monitoring\nand performed experiments on data from the HEGP. However, we have studied that dataset as if all\npatients behave similarly, which is not true in practice. We believe that a possible amelioration of our\nmodel is to put it into a mixture model.\nLastly, the SAEM algorithm is really sensitive to initial conditions. This phenomenon is emphasized\nby the non-independence between the individual variables: actually, the average trajectory \u03b30 is not\nexactly the trajectory of the average parameters. Fortunately, the more the sample size n increases,\nthe more \u03b30 draws closer to the trajectory of the average parameters.\n\n8\n\n\fAcknowledgments\n\nCe travail b\u00e9n\u00e9\ufb01cie d\u2019un \ufb01nancement public Investissement d\u2019avenir, reference ANR-11-LABX-\n0056-LMH. This work was supported by a public grant as part of the Investissement d\u2019avenir, project\nreference ANR-11-LABX-0056-LMH.\nTravail r\u00e9alis\u00e9 dans le cadre d\u2019un projet \ufb01nanc\u00e9 par la Fondation de la Recherche M\u00e9dicale,\n\"DBI20131228564\". Work performed as a part of a project funded by the Fondation of Medical\nResearch, grant number \"DBI20131228564\".\n\nReferences\nSt\u00e9phanie Allassonni\u00e8re, Estelle Kuhn, and Alain Trouv\u00e9. Construction of bayesian deformable models via a\n\nstochastic approximation algorithm: A convergence study. Bernoulli, 16(3):641\u2013678, 08 2010.\n\nMauricio Burotto, Julia Wilkerson, Wilfred Stein, Motzer Robert, Susan Bates, and Tito Fojo. Continuing a\ncancer treatment despite tumor growth may be valuable: Sunitinib in renal cell carcinoma as example. PLoS\nONE, 9(5):e96316, 2014.\n\nBernard Delyon, Marc Lavielle, and Eric Moulines. Convergence of a stochastic approximation version of the\n\nem algorithm. The Annals of Statistics, 27(1):94\u2013128, 1999.\n\nArthur Dempster, Nan M. Laird, and Donald B. Rubin. Maximum likelihood from incomplete data via the em\n\nalgorithm. Journal of the Royal Statistical Society. 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