Data-driven discovery of mechanical models directly from MRI spectral data

TL;DR

A reconstruction framework for data-driven discovery of dynamical models from experimentally obtained undersampled MRI spectral data that makes use of the previously developed spectro-dynamic framework and is successfully validated using spectral data of a dynamic phantom gathered on a clinical MRI scanner.

Abstract

Finding interpretable biomechanical models can provide insight into the functionality of organs with regard to physiology and disease. However, identifying broadly applicable dynamical models for in vivo tissue remains challenging. In this proof of concept study we propose a reconstruction framework for data-driven discovery of dynamical models from experimentally obtained undersampled MRI spectral data. The method makes use of the previously developed spectro-dynamic framework which allows for reconstruction of displacement fields at high spatial and temporal resolution required for model identification. The proposed framework combines this method with data-driven discovery of interpretable models using Sparse Identification of Non-linear Dynamics (SINDy). The design of the reconstruction algorithm is such that a symbiotic relation between the reconstruction of the displacement fields and the model identification is created. Our method does not rely on periodicity of the motion. It is successfully validated using spectral data of a dynamic phantom gathered on a clinical MRI scanner. The dynamic phantom is programmed to perform motion adhering to 5 different (non-linear) ordinary differential equations. The proposed framework performed better than a 2-step approach where the displacement fields were first reconstructed from the undersampled data without any information on the model, followed by data-driven discovery of the model using the reconstructed displacement fields. This study serves as a first step in the direction of data-driven discovery of in vivo models.

Figures (8)

• Figure 1: Schematic representation of the $k$-space sampling scheme with 12 phase encodes. The numbers indicate the temporal order in which the read-outs are sampled. The dimensions of $k$-space during the actual dynamical experiments is $|\Omega_k| = N_1\times N_2=64\times64$. In this figure $k_1$ is the read-out direction and $k_2$ the phase encode direction. For the reconstruction, 4 consecutively acquired read-out lines are grouped to form one time instance $t_i$. This results in an effective temporal resolution of $\Delta t=4\text{TR}$.
• Figure 2: (a) The Quasar™ MRI$^{4D}$ Motion Phantom. A large water compartment surrounds a stationary cylinder (off-center) and a dynamic one (center), driven by a mechanical motor. The dynamic cylinder is loaded with two smaller signal generating, gel-filled tubes. (b) Phantom positioned in the clinical MRI scanner during the experiment.
• Figure 3: The 4 basis functions $\vec{\phi}(\vec{r})$ used for spatial parameterisation conform \ref{['eq: param']}. Two basis functions are defined for the dynamic compartment ($\Omega_D$, top row), and two for the static compartment ($\Omega_S$, bottom row). The generalized coordinates $q_p(t)$ corresponding to each spatial basis function, are estimated during the reconstruction. Only $q_1(t)$ corresponding to basis function $\vec{\phi}_1(\vec{r})$ should be nonzero.
• Figure 4: Model identification for dynamic 1-5. The bars refer to the absolute value of the reconstructed system parameters displayed together with the absolute value of the ground truth. The specific values for the reconstructed system parameters are reported in table S1 of the Supplementary materials. The color of the bars match the color of the corresponding $y$-axis. The error flags denote the standard deviation from 5 repetitions of the experiment.
• Figure 5: Estimate for the displacement ($q_1$). The 95% interval is calculated from 5 repetitions of the experiment.