1st Supervisor: Jack Lee, King’s College London

2nd Supervisor: Alistair Young, King’s College London

Clinical Champion: Keshthra Satchithananda, King’s College London

Additional Supervisor: Ralph Sinkus, King’s College London

**Aim of the PhD Project:**

- Magnetic resonance elastography enables safe non-invasive analysis of lesion stiffness, which is very useful in the diagnosis and assessment of treatment of cancer.
- However, the estimation is difficult because it involves an inverse solution of partial differential equation systems.
- We will apply deep learning and Bayesian inference to robustly estimate stiffness from noisy images

**Project Description / Background:**

Magnetic Resonance Elastography (MRE) is an imaging technique that is capable of non-invasively estimating biomechanical parameters (elasticity and viscosity) in tissues. MRE has found several clinical applications, as alterations in tissue biomechanical properties occur in many forms of cancer and pathologies. MRE works by introducing periodic mechanical vibrations into the tissue of interest at a controlled frequency. The imaging sequence is phase-locked to the resulting standing waves at the same frequency to allow the displacement of the tissue to be measured at selected discrete phases throughout the wave period.

The estimation of the apparent tissue elasticity requires the inverse solution of the elastic wave equation at each voxel, which is challenging by virtue of it being a 3D partial differential equation, as detailed in our recent review (Fovargue et al., 2018). The tumours are often positioned deep within the tissue that are difficult for the waves to penetrate, thereby reducing the SNR. The presence of high order derivatives in the governing PDEs tend to greatly amplify the noise present in the raw data. To deal with this, global integral approaches have been proposed, but in general, FEM-based point estimates inherently lack the facility to deal with uncertainty. In addition, the accompanied application of filtering (e.g. Gaussian, Savitzky-Golay) lead to a modification of the true underlying elasticity. The target tumours also may be small or oddly-shaped, surrounded by a heterogeneous biomechanical environment giving rise to complex wave patterns. In practice these complications are often neglected because they would dramatically increase the cost of reconstruction.

Regarding uncertainties in inverse solutions, Bayesian inference methods have been highly successful in parameter estimation problems including PDE-based quantifications. We have previously applied these approaches in flow quantification from first-pass perfusion MR imaging (Scannell et al., 2019) as well as fitting cardiac electromechanical finite element models to personalised clinical data (Fovargue et al, 2016). Our application of a hierarchical Bayesian method to perfusion MR significantly reduced spurious numerical divergence in the estimated perfusion map; the use of spatial priors via generalised Gaussian Markov random field helped to enforce smoothness, while the use of hyperpriors for the kinetic parameters prevented the smaller discontinuous regions of perfusion defect from merging with the surroundings. These advantages are attractive for the present application to MRE where similar challenges exist, and accurate prior cannot be identified.

Markov Chain Monte Carlo Bayesian inference techniques are generally expensive, because posterior distribution has to be sampled many times until convergence. Although there are newer schemes in development for efficient sampling (e.g. NUTS (Hoffman & Gelman, 2014)), they are unlikely to render the current work suitable for clinical timeframe. Instead, a solution may be offered by deep learning; recently, a neural network trained to purely simulation-based data has been shown to produce high correlation with classical reconstruction method (R^{2}>0.9) on simulated data, although it was limited to a purely elastic one without viscosity (Murphy et al., 2017). We have recently demonstrated a similar accuracy by training a network to replicate the elasticity maps of in-vivo brain tissue estimated by a FEM-based reconstruction technique (Darwish et al., 2019). These advances demonstrate that a deep neural network may capably learn the physics represented by the PDEs, thus allowing a rapid model evaluation and/or reconstruction. Furthermore, this approach broadens the scope to anisotropic reconstruction, in which identification of tensorial unknown parameters is currently intractable.

The candidate for this project will have a background in mathematics, engineering, physics or computer science. Experience in programming (C++, Python, or similar) is necessary.

**References:**

- Darwish, Omar I., et al. “AI to improve elasticity reconstruction in MRE using in-vivo data”
*ISMRM*2020. - Fovargue, Daniel, et al. “Stiffness reconstruction methods for MR elastography.”
*NMR in Biomedicine.*(2018) 31:e3935. - Fovargue, Daniel et al., “Robust MR elastography stiffness quantification using a localized divergence free finite element reconstruction.”
*Medical Image Analysis*(2018b) 44:126-142. - Fovargue, Lauren, et al., “Non-invasive prediction of acute hemodynamics in cardiac resynchronisation therapy through patient specific modelling”,
*European Heart Journal.*(2016) 37:62. - Hoffman, Matthew D., Gelman, Andrew. “The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.”
*Journal of Machine Learning Research.*(2014) 15:1593-1623. - Lee, Jack, et al., “Multiphysics Computational Modelling in CHeart”
*SIAM**Journal of Scientific Computing*(2016) 38(3): C150-C178. - Murphy, Matthew C., et al. “Artificial neural networks for stiffness estimation in magnetic resonance elastography.”
*Magnetic resonance in medicine*80.1 (2018): 351-360. - Raissi, M., et al., “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.”
*Journal of Computational Physics*(2019) 378: 686-707. - Scannell, Cian, et al. “Hierarchical Bayesian myocardial perfusion quantification.”
*Medical Image Analysis.*(2019) accepted.