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Title: Where did the tumor start? An inverse solver with sparse localization for tumor growth models

Abstract

In this work, we present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction–diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of magnetic resonance imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one.more » We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.« less

Authors:
ORCiD logo [1];  [2];  [2];  [1]
+ Show Author Affiliations
  1. Univ. of Texas, Austin, TX (United States). Oden Inst. for Computational Engineering and Sciences
  2. Univ. Stuttgarg (Germany). Inst. for Parallel and Distributed Systems
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFOSR); National Institutes of Health (NIH); National Science Foundation (NSF)
OSTI Identifier:
1803775
Grant/Contract Number:  
SC0019393; R01NS042645-14; CCF-1817048; CCF-1725743; FA9550-17-1-0190
Resource Type:
Accepted Manuscript
Journal Name:
Inverse Problems
Additional Journal Information:
Journal Volume: 36; Journal Issue: 4; Journal ID: ISSN 0266-5611
Publisher:
IOPscience
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics; Physics

Citation Formats

Subramanian, Shashank, Scheufele, Klaudius, Mehl, Miriam, and Biros, George. Where did the tumor start? An inverse solver with sparse localization for tumor growth models. United States: N. p., 2020. Web. doi:10.1088/1361-6420/ab649c.
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Subramanian, Shashank, Scheufele, Klaudius, Mehl, Miriam, & Biros, George. Where did the tumor start? An inverse solver with sparse localization for tumor growth models. United States. https://doi.org/10.1088/1361-6420/ab649c
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Subramanian, Shashank, Scheufele, Klaudius, Mehl, Miriam, and Biros, George. Wed . "Where did the tumor start? An inverse solver with sparse localization for tumor growth models". United States. https://doi.org/10.1088/1361-6420/ab649c. https://www.osti.gov/servlets/purl/1803775.
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@article{osti_1803775,
title = {Where did the tumor start? An inverse solver with sparse localization for tumor growth models},
author = {Subramanian, Shashank and Scheufele, Klaudius and Mehl, Miriam and Biros, George},
abstractNote = {In this work, we present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction–diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of magnetic resonance imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.},
doi = {10.1088/1361-6420/ab649c},
journal = {Inverse Problems},
number = 4,
volume = 36,
place = {United States},
year = {Wed Feb 26 00:00:00 EST 2020},
month = {Wed Feb 26 00:00:00 EST 2020}
}
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    Works referencing / citing this record:

    Image-Driven Biophysical Tumor Growth Model Calibration
    journal, January 2020

    • Scheufele, Klaudius; Subramanian, Shashank; Mang, Andreas
    • SIAM Journal on Scientific Computing, Vol. 42, Issue 3
    • DOI: 10.1137/19m1275280

    Multiatlas Calibration of Biophysical Brain Tumor Growth Models with Mass Effect
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