Integrated Biophysical Modeling and Image Analysis: Application to Neuro-Oncology.

Central nervous system (CNS) tumors come with vastly heterogeneous histologic, molecular, and radiographic landscapes, rendering their precise characterization challenging. The rapidly growing fields of biophysical modeling and radiomics have shown promise in better characterizing the molecular, spatial, and temporal heterogeneity of tumors. Integrative analysis of CNS tumors, including clinically acquired multi-parametric magnetic resonance imaging (mpMRI) and the inverse problem of calibrating biophysical models to mpMRI data, assists in identifying macroscopic quantifiable tumor patterns of invasion and proliferation, potentially leading to improved (a) detection/segmentation of tumor subregions and (b) computer-aided diagnostic/prognostic/predictive modeling. This article presents a summary of (a) biophysical growth modeling and simulation,(b) inverse problems for model calibration, (c) these models' integration with imaging workflows, and (d) their application to clinically relevant studies. We anticipate that such quantitative integrative analysis may even be beneficial in a future revision of the World Health Organization (WHO) classification for CNS tumors, ultimately improving patient survival prospects.

Coupling brain-tumor biophysical models and diffeomorphic image registration.

We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for joint image registration and biophysical inversion and we apply it to analyze MR images of glioblastomas (primary brain tumors). We have two applications in mind. The first one is normal-to-abnormal image registration in the presence of tumor-induced topology differences. The second one is biophysical inversion based on single-time patient data. The underlying optimization problem is highly non-linear and non-convex and has not been solved before with a gradient-based approach. Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we determine tumor growth parameters and a registration map so that if we "grow a tumor" (using our tumor model) in the normal brain and then register it to the patient image, then the registration mismatch is as small as possible. This "coupled problem" two-way couples the biophysical inversion and the registration problem. In the image registration step we solve a large-deformation diffeomorphic registration problem parameterized by an Eulerian velocity field. In the biophysical inversion step we estimate parameters in a reaction-diffusion tumor growth model that is formulated as a partial differential equation (PDE). In SIBIA, we couple these two sub-components in an iterative manner. We first presented the components of SIBIA in "Gholami et al., Framework for Scalable Biophysics-based Image Analysis, IEEE/ACM Proceedings of the SC2017", in which we derived parallel distributed memory algorithms and software modules for the decoupled registration and biophysical inverse problems. In this paper, our contributions are the introduction of a PDE-constrained optimization formulation of the coupled problem, and the derivation of a Picard iterative solution scheme. We perform extensive tests to experimentally assess the performance of our method on synthetic and clinical datasets. We demonstrate the convergence of the SIBIA optimization solver in different usage scenarios. We demonstrate that using SIBIA, we can accurately solve the coupled problem in three dimensions (2563 resolution) in a few minutes using 11 dual-x86 nodes.

An inverse problem formulation for parameter estimation of a reaction-diffusion model of low grade gliomas.

We present a numerical scheme for solving a parameter estimation problem for a model of low-grade glioma growth. Our goal is to estimate the spatial distribution of tumor concentration, as well as the magnitude of anisotropic tumor diffusion. We use a constrained optimization formulation with a reaction-diffusion model that results in a system of nonlinear partial differential equations. In our formulation, we estimate the parameters using partially observed, noisy tumor concentration data at two different time instances, along with white matter fiber directions derived from diffusion tensor imaging. The optimization problem is solved with a Gauss-Newton reduced space algorithm. We present the formulation and outline the numerical algorithms for solving the resulting equations. We test the method using a synthetic dataset and compute the reconstruction error for different noise levels and detection thresholds for monofocal and multifocal test cases.

Biophysical modeling of brain tumor progression: from unconditionally stable explicit time integration to an inverse problem with parabolic PDE constraints for model calibration.

PURPOSE: A novel unconditionally stable, explicit numerical method is introduced to the field of modeling brain cancer progression on a tissue level together with an inverse problem (IP) based on optimal control theory that allows for automated model calibration with respect to observations in clinical imaging data. METHODS: Biophysical models of cancer progression on a tissue level are in general based on the assumption that the spatiotemporal spread of cancerous cells is determined by cell division and net migration. These processes are typically described in terms of a parabolic partial differential equation (PDE). In the present work a parallelized implementation of an unconditionally stable, explicit Euler (EE(⋆)) time integration method for the solution of this PDE is detailed. The key idea of the discussed EE(⋆) method is to relax the strong stability requirement on the spectral radius of the coefficient matrix by introducing a subdivision regime for a given outer time step. The performance is related to common implicit numerical methods. To quantify the numerical error, a simplified model that has a closed form solution is considered. To allow for a systematic, phenomenological validation a novel approach for automated model calibration on the basis of observations in medical imaging data is developed. The resulting IP is based on optimal control theory and manifests as a large scale, PDE constrained optimization problem. RESULTS: The numerical error of the EE(⋆) method is at the order of standard implicit numerical methods. The computing times are well below those obtained for implicit methods and by that demonstrate efficiency. Qualitative and quantitative analysis in 12 patients demonstrates that the obtained results are in strong agreement with observations in medical imaging data. Rating simulation success in terms of the mean overlap between model predictions and manual expert segmentations yields a success rate of 75% (9 out of 12 patients). CONCLUSIONS: The discussed EE(⋆) method provides desirable features for image-based model calibration or hybrid image registration algorithms in which the model serves as a biophysical prior. This is due to (i) ease of implementation, (ii) low memory requirements, (iii) efficiency, (iv) a straightforward interface for parameter updates, and (v) the fact that the method is inherently matrix-free. The explicit time integration method is confirmed via experiments for automated model calibration. Qualitative and quantitative analysis demonstrates that the proposed framework allows for recovering observations in medical imaging data and by that phenomenological model validity.

CLAIRE-Parallelized Diffeomorphic Image Registration for Large-Scale Biomedical Imaging Applications.

We study the performance of CLAIRE-a diffeomorphic multi-node, multi-GPU image-registration algorithm and software-in large-scale biomedical imaging applications with billions of voxels. At such resolutions, most existing software packages for diffeomorphic image registration are prohibitively expensive. As a result, practitioners first significantly downsample the original images and then register them using existing tools. Our main contribution is an extensive analysis of the impact of downsampling on registration performance. We study this impact by comparing full-resolution registrations obtained with CLAIRE to lower resolution registrations for synthetic and real-world imaging datasets. Our results suggest that registration at full resolution can yield a superior registration quality-but not always. For example, downsampling a synthetic image from 10243 to 2563 decreases the Dice coefficient from 92% to 79%. However, the differences are less pronounced for noisy or low contrast high resolution images. CLAIRE allows us not only to register images of clinically relevant size in a few seconds but also to register images at unprecedented resolution in reasonable time. The highest resolution considered are CLARITY images of size 2816×3016×1162. To the best of our knowledge, this is the first study on image registration quality at such resolutions.

CLAIRE: Constrained Large Deformation Diffeomorphic Image Registration on Parallel Computing Architectures.

CLAIRE (Mang & Biros, 2019) is a computational framework for Constrained LArge deformation diffeomorphic Image REgistration (Mang et al., 2019). It supports highly-optimized, parallel computational kernels for (multi-node) CPU (Gholami et al., 2017; Mang et al., 2019; Mang & Biros, 2016) and (multi-node multi-)GPU architectures (Brunn et al., 2020, 2021). CLAIRE uses MPI for distributed-memory parallelism and can be scaled up to thousands of cores (Mang et al., 2019; Mang & Biros, 2016) and GPU devices (Brunn et al., 2020). The multi-GPU implementation uses device direct communication. The computational kernels are interpolation for semi-Lagrangian time integration, and a mixture of high-order finite difference operators and Fast-Fourier-Transforms (FFTs) for differentiation. CLAIRE uses a Newton-Krylov solver for numerical optimization (Mang & Biros, 2015, 2017). It features various schemes for regularization of the control problem (Mang & Biros, 2016) and different similarity measures. CLAIRE implements different preconditioners for the reduced space Hessian (Brunn et al., 2020; Mang et al., 2019) to optimize computational throughput and enable fast convergence. It uses PETSc (Balay et al., n.d.) for scalable and efficient linear algebra operations and solvers and TAO (Balay et al., n.d.; Munson et al., 2015) for numerical optimization. CLAIRE can be downloaded at https://github.com/andreasmang/claire.

Fast GPU 3D diffeomorphic image registration.

3D image registration is one of the most fundamental and computationally expensive operations in medical image analysis. Here, we present a mixed-precision, Gauss-Newton-Krylov solver for diffeomorphic registration of two images. Our work extends the publicly available CLAIRE library to GPU architectures. Despite the importance of image registration, only a few implementations of large deformation diffeomorphic registration packages support GPUs. Our contributions are new algorithms to significantly reduce the run time of the two main computational kernels in CLAIRE: calculation of derivatives and scattered-data interpolation. We deploy (i) highly-optimized, mixed-precision GPU-kernels for the evaluation of scattered-data interpolation, (ii) replace Fast-Fourier-Transform (FFT)-based first-order derivatives with optimized 8th-order finite differences, and (iii) compare with state-of-the-art CPU and GPU implementations. As a highlight, we demonstrate that we can register 2563 clinical images in less than 6 seconds on a single NVIDIA Tesla V100. This amounts to over 20× speed-up over the current version of CLAIRE and over 30× speed-up over existing GPU implementations.

Estimating Glioblastoma Biophysical Growth Parameters Using Deep Learning Regression.

Glioblastoma ( GBM ) is arguably the most aggressive, infiltrative, and heterogeneous type of adult brain tumor. Biophysical modeling of GBM growth has contributed to more informed clinical decision-making. However, deploying a biophysical model to a clinical environment is challenging since underlying computations are quite expensive and can take several hours using existing technologies. Here we present a scheme to accelerate the computation. In particular, we present a deep learning ( DL )-based logistic regression model to estimate the GBM's biophysical growth in seconds. This growth is defined by three tumor-specific parameters: 1) a diffusion coefficient in white matter ( Dw ), which prescribes the rate of infiltration of tumor cells in white matter, 2) a mass-effect parameter ( Mp ), which defines the average tumor expansion, and 3) the estimated time ( T ) in number of days that the tumor has been growing. Preoperative structural multi-parametric MRI ( mpMRI ) scans from n = 135 subjects of the TCGA-GBM imaging collection are used to quantitatively evaluate our approach. We consider the mpMRI intensities within the region defined by the abnormal FLAIR signal envelope for training one DL model for each of the tumor-specific growth parameters. We train and validate the DL-based predictions against parameters derived from biophysical inversion models. The average Pearson correlation coefficients between our DL-based estimations and the biophysical parameters are 0.85 for Dw, 0.90 for Mp, and 0.94 for T, respectively. This study unlocks the power of tumor-specific parameters from biophysical tumor growth estimation. It paves the way towards their clinical translation and opens the door for leveraging advanced radiomic descriptors in future studies by means of a significantly faster parameter reconstruction compared to biophysical growth modeling approaches.

Multi-Node Multi-GPU Diffeomorphic Image Registration for Large-Scale Imaging Problems.

We present a Gauss-Newton-Krylov solver for large deformation diffeomorphic image registration. We extend the publicly available CLAIRE library to multi-node multi-graphics processing unit (GPUs) systems and introduce novel algorithmic modifications that significantly improve performance. Our contributions comprise (i) a new preconditioner for the reduced-space Gauss-Newton Hessian system, (ii) a highly-optimized multi-node multi-GPU implementation exploiting device direct communication for the main computational kernels (interpolation, high-order finite difference operators and Fast-Fourier-Transform), and (iii) a comparison with state-of-the-art CPU and GPU implementations. We solve a 2563-resolution image registration problem in five seconds on a single NVIDIA Tesla V100, with a performance speedup of 70% compared to the state-of-the-art. In our largest run, we register 20483 resolution images (25 B unknowns; approximately 152× larger than the largest problem solved in state-of-the-art GPU implementations) on 64 nodes with 256 GPUs on TACC's Longhorn system.

IMAGE-DRIVEN BIOPHYSICAL TUMOR GROWTH MODEL CALIBRATION.

We present a novel formulation for the calibration of a biophysical tumor growth model from a single-time snapshot, multiparametric magnetic resonance imaging (MRI) scan of a glioblastoma patient. Tumor growth models are typically nonlinear parabolic partial differential equations (PDEs). Thus, we have to generate a second snapshot to be able to extract significant information from a single patient snapshot. We create this two-snapshot scenario as follows. We use an atlas (an average of several scans of healthy individuals) as a substitute for an earlier, pretumor, MRI scan of the patient. Then, using the patient scan and the atlas, we combine image-registration algorithms and parameter estimation algorithms to achieve a better estimate of the healthy patient scan and the tumor growth parameters that are consistent with the data. Our scheme is based on our recent work (Scheufele et al., Comput. Methods Appl. Mech. Engrg., to appear), but we apply a different and novel scheme where the tumor growth simulation in contrast to the previous work is executed in the patient brain domain and not in the atlas domain yielding more meaningful patient-specific results. As a basis, we use a PDE-constrained optimization framework. We derive a modified Picard-iteration-type solution strategy in which we alternate between registration and tumor parameter estimation in a new way. In addition, we consider an ℓ 1 sparsity constraint on the initial condition for the tumor and integrate it with the new joint inversion scheme. We solve the sub-problems with a reduced space, inexact Gauss-Newton-Krylov/quasi-Newton method. We present results using real brain data with synthetic tumor data that show that the new scheme reconstructs the tumor parameters in a more accurate and reliable way compared to our earlier scheme.

CLAIRE: A DISTRIBUTED-MEMORY SOLVER FOR CONSTRAINED LARGE DEFORMATION DIFFEOMORPHIC IMAGE REGISTRATION.

With this work we release CLAIRE, a distributed-memory implementation of an effective solver for constrained large deformation diifeomorphic image registration problems in three dimensions. We consider an optimal control formulation. We invert for a stationary velocity field that parameterizes the deformation map. Our solver is based on a globalized, preconditioned, inexact reduced space Gauss‒Newton‒Krylov scheme. We exploit state-of-the-art techniques in scientific computing to develop an eifective solver that scales to thousands of distributed memory nodes on high-end clusters. We present the formulation, discuss algorithmic features, describe the software package, and introduce an improved preconditioner for the reduced space Hessian to speed up the convergence of our solver. We test registration performance on synthetic and real data. We Demonstrate registration accuracy on several neuroimaging datasets. We compare the performance of our scheme against diiferent flavors of the Demons algorithm for diifeomorphic image registration. We study convergence of our preconditioner and our overall algorithm. We report scalability results on state-of-the-art supercomputing platforms. We Demonstrate that we can solve registration problems for clinically relevant data sizes in two to four minutes on a standard compute node with 20 cores, attaining excellent data fidelity. With the present work we achieve a speedup of (on average) 5× with a peak performance of up to 17× compared to our former work.

A SEMI-LAGRANGIAN TWO-LEVEL PRECONDITIONED NEWTON-KRYLOV SOLVER FOR CONSTRAINED DIFFEOMORPHIC IMAGE REGISTRATION.

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20× speedup for a two dimensional, real world multi-subject medical image registration problem.

A LAGRANGIAN GAUSS-NEWTON-KRYLOV SOLVER FOR MASS- AND INTENSITY-PRESERVING DIFFEOMORPHIC IMAGE REGISTRATION.

We present an efficient solver for diffeomorphic image registration problems in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and non-stationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming mass-preservation). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of characteristic curves. We approximate these curves using a fourth-order Runge-Kutta method. We also present an efficient algorithm for computing the derivatives of the final state of the system with respect to the velocity field. This allows us to use fast Gauss-Newton based methods. We present quickly converging iterative linear solvers using spectral preconditioners that render the overall optimization efficient and scalable. Our method is embedded into the image registration framework FAIR and, thus, supports the most commonly used similarity measures and regularization functionals. We demonstrate the potential of our new approach using several synthetic and real world test problems with up to 14.7 million degrees of freedom.

Constrained H1-regularization schemes for diffeomorphic image registration.

We propose regularization schemes for deformable registration and efficient algorithms for their numerical approximation. We treat image registration as a variational optimal control problem. The deformation map is parametrized by its velocity. Tikhonov regularization ensures well-posedness. Our scheme augments standard smoothness regularization operators based on H1- and H2-seminorms with a constraint on the divergence of the velocity field, which resembles variational formulations for Stokes incompressible flows. In our formulation, we invert for a stationary velocity field and a mass source map. This allows us to explicitly control the compressibility of the deformation map and by that the determinant of the deformation gradient. We also introduce a new regularization scheme that allows us to control shear. We use a globalized, preconditioned, matrix-free, reduced space (Gauss-)Newton-Krylov scheme for numerical optimization. We exploit variable elimination techniques to reduce the number of unknowns of our system; we only iterate on the reduced space of the velocity field. Our current implementation is limited to the two-dimensional case. The numerical experiments demonstrate that we can control the determinant of the deformation gradient without compromising registration quality. This additional control allows us to avoid oversmoothing of the deformation map. We also demonstrate that we can promote or penalize shear whilst controlling the determinant of the deformation gradient.

An Inexact Newton-Krylov Algorithm for Constrained Diffeomorphic Image Registration.

We propose numerical algorithms for solving large deformation diffeomorphic image registration problems. We formulate the nonrigid image registration problem as a problem of optimal control. This leads to an infinite-dimensional partial differential equation (PDE) constrained optimization problem. The PDE constraint consists, in its simplest form, of a hyperbolic transport equation for the evolution of the image intensity. The control variable is the velocity field. Tikhonov regularization on the control ensures well-posedness. We consider standard smoothness regularization based on H1- or H2-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field (control variable) rendering the deformation incompressible (Stokes regularization scheme) and thus ensuring that the determinant of the deformation gradient is equal to one, up to the numerical error. We use a Fourier pseudospectral discretization in space and a Chebyshev pseudospectral discretization in time. The latter allows us to reduce the number of unknowns and enables the time-adaptive inversion for nonstationary velocity fields. We use a preconditioned, globalized, matrix-free, inexact Newton-Krylov method for numerical optimization. A parameter continuation is designed to estimate an optimal regularization parameter. Regularity is ensured by controlling the geometric properties of the deformation field. Overall, we arrive at a black-box solver that exploits computational tools that are precisely tailored for solving the optimality system. We study spectral properties of the Hessian, grid convergence, numerical accuracy, computational efficiency, and deformation regularity of our scheme. We compare the designed Newton-Krylov methods with a globalized Picard method (preconditioned gradient descent). We study the influence of a varying number of unknowns in time. The reported results demonstrate excellent numerical accuracy, guaranteed local deformation regularity, and computational efficiency with an optional control on local mass conservation. The Newton-Krylov methods clearly outperform the Picard method if high accuracy of the inversion is required. Our method provides equally good results for stationary and nonstationary velocity fields for two-image registration problems.

Identification of crucial parameters in a mathematical multiscale model of glioblastoma growth.

Glioblastomas are highly malignant brain tumours. Mathematical models and their analysis provide a tool to support the understanding of the development of these tumours as well as the design of more effective treatment strategies. We have previously developed a multiscale model of glioblastoma progression that covers processes on the cellular and molecular scale. Here, we present a novel nutrient-dependent multiscale sensitivity analysis of this model that helps to identify those reaction parameters of the molecular interaction network that influence the tumour progression on the cellular scale the most. In particular, those parameters are identified that essentially determine tumour expansion and could be therefore used as potential therapy targets. As indicators for the success of a potential therapy target, a deceleration of the tumour expansion and a reduction of the tumour volume are employed. From the results, it can be concluded that no single parameter variation results in a less aggressive tumour. However, it can be shown that a few combined perturbations of two systematically selected parameters cause a slow-down of the tumour expansion velocity accompanied with a decrease of the tumour volume. Those parameters are primarily linked to the reactions that involve the microRNA-451 and the thereof regulated protein MO25.

A computational multiscale model of glioblastoma growth: regulation of cell migration and proliferation via microRNA-451, LKB1 and AMPK.

A new computational multiscale model of glioblastoma growth is introduced. This model combines an agent-based model for representing processes on the cellular level with a molecular interaction network for each cell on the subcellular scale. The network is based on recently published work on the interaction of microRNA-451, LKB1 and AMPK in the regulation of glioblastoma cell migration and proliferation. We translated this network into a mathematical description by the use of 17 ordinary differential equations. In our model, we furthermore establish a link from the molecular interaction network of a single cell to cellular actions (e.g. chemotactic movement) on the microscopic level. First results demonstrate that the computational model reproduces a tumor cell development comparable to that observed in in vitro experiments.

A novel method for simulating the extracellular matrix in models of tumour growth.

A novel hybrid continuum-discrete model to simulate tumour growth on a cellular scale is proposed. The lattice-based spatiotemporal model consists of reaction-diffusion equations that describe interactions between cancer cells and their microenvironment. The fundamental ingredients that are typically considered are the nutrient concentration, the extracellular matrix (ECM), and matrix degrading enzymes (MDEs). The in vivo processes are very complex and occur on different levels. This in turn leads to huge computational costs. The main contribution of the present work is therefore to describe the processes on the basis of simplified mathematical approaches, which, at the same time, depict realistic results to understand the biological processes. In this work, we discuss if we have to simulate the MDE or if the degraded matrix can be estimated directly with respect to the cancer cell distribution. Additionally, we compare the results for modelling tumour growth using the common and our simplified approach, thereby demonstrating the advantages of the proposed method. Therefore, we introduce variations of the positioning of the nutrient delivering blood vessels and use different initializations of the ECM. We conclude that the novel method, which does not explicitly model the matrix degrading enzymes, provides means for a straightforward and fast implementation for modelling tumour growth.

In-silico oncology: an approximate model of brain tumor mass effect based on directly manipulated free form deformation.

PURPOSE: The present work introduces a novel method for approximating mass effect of primary brain tumors. METHODS: The spatio-temporal dynamics of cancerous cells are modeled by means of a deterministic reaction-diffusion equation. Diffusion tensor information obtained from a probabilistic diffusion tensor imaging atlas is incorporated into the model to simulate anisotropic diffusion of cancerous cells. To account for the expansive nature of the tumor, the computed net cell density of malignant cells is linked to a parametric deformation model. This mass effect model is based on the so-called directly manipulated free form deformation. Spatial correspondence between two successive simulation steps is established by tracking landmarks, which are attached to the boundary of the gross tumor volume. The movement of these landmarks is used to compute the new configuration of the control points and, hence, determines the resulting deformation. To prevent a deformation of rigid structures (i.e. the skull), fixed shielding landmarks are introduced. In a refinement step, an adaptive landmark scheme ensures a dense sampling of the tumor isosurface, which in turn allows for an appropriate representation of the tumor shape. RESULTS: The influence of different parameters on the model is demonstrated by a set of simulations. Additionally, simulation results are qualitatively compared to an exemplary set of clinical magnetic resonance images of patients diagnosed with high-grade glioma. CONCLUSIONS: Careful visual inspection of the results demonstrates the potential of the implemented model and provides first evidence that the computed approximation of tumor mass effect is sensible. The shape of diffusive brain tumors (glioblastoma multiforme) can be recovered and approximately matches the observations in real clinical data.