EF3 – Model-based Imaging



Quantitative Tissue Pressure Imaging via PDE-Informed Assimilation of MR Data

Project Heads

Alfonso Caiazzo, Karsten Tabelow, Ingolf Sack

Project Members

Felipe Galarce Marín

Project Duration

01.01.2022 − 31.12.2023

Located at




Medical imaging of tissues based on MRI enables the non-invasive quantitative characterization of important biomarkers, and it is therefore a pillar of clinical diagnostic of tissue diseases. Biological tissues are by nature poroelastic and the understanding of biphasic aspects such as porosity and interstitial pressure might provide the identification of abnormal pressure increase in the cardiovascular system or cerebro-spinal fluid. To this purpose, the mechanical behavior of the tissue is modeled as a biphasic (poroelastic) material.

Magnetic resonance elastography (MRE) is an imaging technique sensitive to the stiffness properties of tissues. In MRE, the propagation of shear waves in the audible frequency range (10 – 1000 Hz) through the tissue – recorded as a three-dimensional image of the inner displacement field. MRE has been largely used in the context of inverse problems for elastic and viscoelastic tissues. However, to characterize pathological pressure gradients, the data assimilation problem shall take into account (i) a poroelastic underlying physical model and (ii) both the estimation of mechanical parameters and the reconstruction of the physical state (i.e., the IFP – interstitial fluid pressure). In particular, the estimation of pressure gradients is extremely demanding, due to the high dimensionality of the unknown space and to the limited availability and quality of experimental data. In fact, MRE acquisition is practically constrained by the length of examination time, and displacement data are typically available only on a subregion of the tissue of interest and with a limited resolution.


Within the MATH+ project EF3-9 we established a new statistical framework to infer on porosity maps from inversion recovery MR data and we developed a pipeline to setup a patient-specific poroelastic model from MPRAGE and IRMRI data. The project EF3-11 targets the quantitative estimation of interstitial pressure from medical imaging. To this purpose, we aim at developing a mathematical framework to solve the data assimilation problem in a physics-informed reduced space, i.e., a manifold of numerical solutions of the poroelastic equations modeling the tissue mechanics.



We tackle data assimilation of MRE data combining (i) a state reconstruction over the whole domain and (ii) a variational projection method for the estimation of pressure gradients. The first step allows to exploit the underlying physical model to extend the available MRE measurement to the full 3D space and avoid the issue of setting internal pressure boundaries.

The displacement reconstruction algorithm will be based on the so-called parametrized- background data-weak (PBDW) method [Maday et al. 2014], an approach designed to reconstruct a physical solution over the whole domain given only partial information. Originally introduced for wave equations, the PBDW has been recently extended in the context of ultrasound images of blood flow [Galarce et al., 2021]. In this approach, one generates a manifold of suitable solutions of an underlying PDE dependent on few physical parameters, by solving the problem on a finite approximation of the parameter space. Then, the reconstruction is sought on a low-dimensional subspace of this PDE- informed manifold, solving an optimization problem depending on the available observations and including additional terms to account for the model bias


Project Publications

F. Galarce, K. Tabelow, J. Polzehl, C. Panagiotis Papanikas, V. Vavourakis, L. Lilaj, I. Sack, A. Caiazzo.

Displacement and pressure reconstruction from magnetic resonance elastography images: Application to an in silico brain model [WIAS Preprint 2933 (2021) ]

Abstract: Magnetic resonance elastography is a motion sensitive image modality that allows to measure in vivo tissue displacement fields in response to mechanical excitations. This paper investigates a data assimilation approach for reconstructing tissue displacement and pressure fields in an in silico brain model from partial elastography data. The data assimilation is based on a parametrized-background data weak methodology, in which the state of the physical system — tissue displacements and pressure fields — is reconstructed from the available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges, to simulate the corresponding poroelastic problem, and compute a reduced basis. Displacements and pressure reconstruction is sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics on a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physics-informed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions, by appropriately training the reduced-order model.

News & Events

Computational & Mathematical Biomedical Engineering (CMBE 2022)

When & Where: June 27-29, in Milan (Politecnico)

Minisymposium: Inverse problems in biofluids and -solids

More info here


SIAM Imaging Science (IS) Conference

When: March 21-25, Online

Minisymposium: Biomechanical Model-Based MR Imaging, Inverseproblems, and Applications



MRE & Mechanics Session:

  • Jing Guo (MRE Group, Charité, Berlin)
  • Felipe Galarce (WIAS Berlin)
  • Federica Caforio (Medical U of Graz)
  • Sebastien Imperiale (INRIA Saclay)

Cardiovascular MRI Session:

  • David Nordsletten (U Michigan)
  • Katerina Skardova (U Prag)
  • Jorge Aguayo (U Chile)
  • Jeremía Garay (U Groningen)

Relevant Publications

F. Galarce, J.-F. Gerbeau, D. Lombardi, and O. Mula. Fast reconstruction of 3d blood flows from doppler ultrasound images and reduced models. Comp. Meth. Appl. Mech. Engnr., 375:113559, 2021.

L. Lilaj, H. Herthum, T. Meyer, M. Shahryari, G. Bertalan, A. Caiazzo, J. Braun, T. Fischer, S. Hirsch, and I. Sack. Inversion recovery MR elastography of the human brain for improved stiffness quantification near fluid-solid boundaries. Mag. Res. Med.  2021 86(5):2552-2561

Y. Maday, A.T. Patera, J.D. Penn, and M. Yano. A Parameterized-Background Data-Weak approach to variational data assimilation: formulation, analysis, and application to acoustics. Int. J. Num. Methods Engnr., 102(5):933–965, 2014

I. Sack and T. Schäffter, editors. Quantification of Biophysical Parameters in Medical Imaging. Springer, 2018

Related Pictures

View of MPRAGE Brain image. The red area indicates the subregion where MRE data are available

Patient-specific computational mesh created from the anatomical images


Example of forward simulation: displacement field

Example of forward simulation: pressure field