Multimodal neural operators for real-time biomechanical modelling of traumatic brain injury.

Traumatic brain injury modeling requires integrating volumetric neuroimaging, demographic parameters, and acquisition metadata. Finite element solvers are too computationally expensive for clinical settings. Neural operators offer much faster inference. Their ability to integrate volumetric imaging with scalar metadata remains underexplored for biomechanical predictions. This study evaluates multimodal neural operator architectures for brain biomechanics. We test strategies fusing volumetric anatomical imaging, demographic features, and acquisition parameters to predict full-field brain displacement from MRE data. We framed TBI modeling as a multimodal operator learning problem. Two fusion strategies were tested. Field projection was applied for Fourier Neural Operator (FNO) architectures. Branch decomposition was used for Deep Operator Networks (DeepONet). Four models (FNO, Factorized FNO, Multi-Grid FNO, DeepONet) were evaluated on 249 in vivo MRE datasets across frequencies from 20 to 90 Hz. DeepONet achieved the highest accuracy on real displacement fields (MSE = 0.0039, 90.0% accuracy) with the fastest inference (3.83 it/s) and fewest parameters (2.09M). MG-FNO performed best on imaginary fields (MSE = 0.0058, 88.3% accuracy) requiring the lowest GPU memory among FNO variants (7.12 GB). No single architecture dominated all criteria. This reveals distinct trade-offs between accuracy, spatial fidelity, and computational cost. Neural operators augmented with multimodal fusion can accurately predict full-field brain displacement from heterogeneous inputs. They offer inference times orders of magnitude faster than finite element solvers. This comparison provides guidance for selecting operator learning approaches in biomedical settings.

Magnetic resonance elastography is a noninvasive imaging technique capable of quantifying and spatially resolving the shear stiffness of soft tissues by visualization of synchronized mechanical wave displacement fields. However, magnetic resonance elastography inversions generally assume that the measured tissue motion consists primarily of shear waves propagating in a uniform, infinite medium. This assumption is not valid in organs such as the heart, eye, bladder, skin, fascia, bone and spinal cord, in which the shear wavelength approaches the geometric dimensions of the object. The aim of this study was to develop and test mathematical inversion algorithms capable of resolving shear stiffness from displacement maps of flexural waves propagating in bounded media such as beams, plates, and spherical shells, using geometry-specific equations of motion. Magnetic resonance elastography and finite element modeling of beam, plate, and spherical shell phantoms of various geometries were performed. Mechanical testing of the phantoms agreed with the stiffness values obtained from finite element modeling and magnetic resonance elastography data, and a linear correlation of r(2) >or= 0.99 was observed between the stiffness values obtained using magnetic resonance elastography and finite element modeling data. In conclusion, we have demonstrated new inversion methods for calculating shear stiffness that may be more appropriate for waves propagating in bounded media.

Traumatic brain injuries (TBI) are common, and often lead to permanent cognitive impairment. Despite the prevalence and severity of TBI, the condition remains poorly understood. Computer simulations of injury mechanics offer enormous potential for the study of TBI; however, computer models require accurate descriptions of tissue constitutive behavior and brain-skull boundary conditions. Magnetic resonance elastography (MRE) is a non-invasive imaging modality that provides quantitative spatial maps of tissue stiffness in vivo. MRE is performed by inducing micron-amplitude propagating shear waves into tissue and imaging the resulting motion with a specialized “motion-sensitive” MRI pulse sequence. Invoking a restricted form of Navier’s equation these data can be inverted to estimate material stiffness. As such, clinical interest in MRE has largely been driven by the direct empirical relationship between tissue stiffness and health. However, the so-called “raw” MRE data themselves (3-D displacement measurements) and calculated strains can elucidate loading paths, anatomic boundaries and the dynamic response of the intact human head. In this study, we use the MRE imaging technique to measure in vivo displacement fields of brain motion as the cranium is exposed to acoustic frequency pressure excitation (45, 60, 80 Hz) and calculate the resulting shear-strain fields (2-D).

Traumatic brain injuries (TBI) due to blast are common in modern combat situations, and often lead to permanent cognitive impairment. Despite the prevalence and severity of blast-induced TBI, the condition remains poorly understood. Computer simulations of blast and blast injury mechanics offer enormous potential; however, computer models require accurate descriptions of tissue mechanics and boundary conditions in vivo. To gain insight into the mechanisms of blast injury, we applied direct (light) oscillatory pressure loading to the skulls of human volunteers, and measured displacement and strain fields using the methodology of magnetic resonance elastography (MRE). MRE is a non-invasive imaging modality that provides quantitative spatial maps of tissue stiffness. MRE is performed by inducing micron-amplitude propagating shear waves into tissue and imaging the resulting harmonic motion with standard clinical MRI hardware. Shear waves are initiated by an MR-compatible actuator and detected by a specialized “motion-sensitive” MRI pulse sequence (software). Motion sensitized MR images provide displacement field data which can be inverted to estimate material stiffness by invoking a restricted form of Navier’s equation. Clinical interest in MRE has largely been driven by the empirical relationship between tissue stiffness and health. However, the “raw” MRE data (3-D displacement measurements) themselves can elucidate loading paths, anatomic boundaries and the dynamic response of the intact human head. In this study, we use the MRE imaging technique to measure in vivo displacement fields of brain motion as the cranium is exposed to acoustic frequency pressure excitation (45, 60, 80 Hz) and we calculate the resulting shear-strain fields (2-D). We estimate the Poynting vector (energy flux) field to illuminate the directions of internal wave propagation, and to identify the energy absorbing and reflecting regions within the brain.

While MRI allows to encode the motion of tissue in the magnetization's phase, it remains yet a challenge to obtain high fidelity motion images due to wraps in the phase for high encoding efficiencies. Therefore, we propose an optimal multiple motion encoding method (OMME) and exemplify it in Magnetic Resonance Elastography (MRE) data. OMME is formulated as a non-convex least-squares problem for the motion using an arbitrary number of phase-contrast measurements with different motion encoding gradients (MEGs). The mathematical properties of OMME are proved in terms of standard deviation and dynamic range of the motion's estimate for arbitrary MEGs combination which are confirmed using synthetically generated data. OMME's performance is assessed on MRE data from in vivo human brain experiments and compared to dual encoding strategies. The unwrapped images are further used to reconstruct stiffness maps and compared to the ones obtained using conventional unwrapping methods. OMME allowed to successfully combine several MRE phase images with different MEGs, outperforming dual encoding strategies in either motion-to-noise ratio (MNR) or number of successfully reconstructed voxels with good noise stability. This lead to stiffness maps with greater resolution of details than obtained with conventional unwrapping methods. The proposed OMME method allows for a flexible and noise robust increase in the dynamic range and thus provides wrap-free phase images with high MNR. In MRE, the method may be especially suitable when high resolution images with high MNR are needed.

This paper deals with an inverse problem for recovering the viscoelasticity of a living body from MRE (Magnetic Resonance Elastography) data. Based on a viscoelastic partial differential equation whose solution can approximately simulate MRE data, the inverse problem is transformed to a least square variational problem. This is to search for viscoelastic coefficients of this equation such that the solution to a boundary value problem of this equation fits approximately to MRE data with respect to the least square cost function. By computing the Gateaux derivatives of the cost function, we minimize the cost function by the projected gradient method is proposed for recovering the unknown coefficients. The reconstruction results based on simulated data and real experimental data are presented and discussed.

Integrating material properties from magnetic resonance elastography into subject-specific computational models for the human brain

In traumatic brain injury (TBI), membranes such as the dura mater, arachnoid mater, and pia mater play a vital role in transmitting motion from the skull to brain tissue. Magnetic resonance elastography (MRE) is an imaging technique developed for noninvasive estimation of soft tissue material parameters. In MRE, dynamic deformation of brain tissue is induced by skull vibrations during magnetic resonance imaging (MRI); however, skull motion and its mode of transmission to the brain remain largely uncharacterized. In this study, displacements of points in the skull, reconstructed using data from an array of MRI-safe accelerometers, were compared to displacements of neighboring material points in brain tissue, estimated from MRE measurements. Comparison of the relative amplitudes, directions, and temporal phases of harmonic motion in the skulls and brains of six human subjects shows that the skull-brain interface significantly attenuates and delays transmission of motion from skull to brain. In contrast, in a cylindrical gelatin "phantom," displacements of the rigid case (reconstructed from accelerometer data) were transmitted to the gelatin inside (estimated from MRE data) with little attenuation or phase lag. This quantitative characterization of the skull-brain interface will be valuable in the parameterization and validation of computer models of TBI.

Magnetic resonance elastography (MRE) allows the visualization of displacement patterns from induced harmonic mechanical waves propagating in tissue. Strain and mechanical properties can be computed from these displacement patterns. Mechanical properties of tissue are affected by various disease processes. MRE has shown brain tumor to differ in stiffness in comparison to normal tissue. MRE is currently being offered as an upgrade on most conventional MRI scanners. However, the actuator supplied by vendors is a drum driver designed primarily for hepatic MRE scan. The goals of the project was to design and build an ergonomic flexible driver for use in MRE of the brain, to assess the Scan-Rescan reproducibility of shear modulus measurements, and to investigate the relationship between shear modulus measurements and driver frequency. An ergonomic flexible driver was constructed to induce mechanical waves in the brain. MRE of the brain was performed in 10 healthy volunteers. MRE data was collected at frequencies of 60 Hz, 50 Hz, and 40 Hz. After the scans were completed, the subjects were removed from the table, and then repositioned and rescanned using the same process. All subjects were scanned and rescanned within an hour. The within-subject coefficient of variance (CV) and inter-subject CV were calculated for shear modulus measurements of white matter, grey matter, and whole brain. A one-way analysis of variance (ANOVA) was applied to test for any difference between shear modulus measurements made at different frequencies. The within-subject CVs of white matter, grey matter, and whole brain shear modulus measurements for all frequencies ranged from 3.7-4.1%, 4.7-6.0%, and 1.8-3.5% respectively. A significant statistical difference was found between measurements made at different frequencies. This study demonstrated the ability to make in vivo shear modulus measurements of brain tissue. MRE was shown to be able to differentiate white matter from grey matter using the shear modulus. Measured white and grey matter shear modulus values were within the range of values reported in literature. A dependence of shear modulus measurements on frequency was observed; Standardization of MRE imaging parameters is recommended to facilitate the interpretation of brain MRE results.

This paper deals with an inverse problem for recovering the piecewise constant viscoelasticity of a living body from MRE (Magnetic Resonance Elastography) data. Based on a scalar partial differential equation whose solution can approximately simulate MRE data, our inverse coefficient problem is considered as a statistical inverse problem of reconstructing the posterior distribution of unknown viscoelastic modulus. For sampling this distribution, one usually can use the Metropolis-Hastings Markov chain Monte Carlo (MH-MCMC) algorithm. However, without an appropriate “proposal” distribution given artificially, the MH-MCMC algorithm is hard to draw samples efficiently. To avoid this, a so-called slice sampling algorithm is introduced in this paper and applied for solving our problem. The performance of these statistical inversion algorithms is numerically tested basing on simulated data.

Objective. Magnetic resonance elastography (MRE) is widely adopted as a biomarker of liver fibrosis. However, in vivo MRE accuracy is difficult to assess. Approach. Finite element model (FEM) simulation was employed to evaluate liver MRE accuracy and inform methodological optimisation. MRE data was simulated in a 3D FEM of the human torso including the liver, and compared with spin-echo echo-planar imaging MRE acquisitions. The simulated MRE results were compared with the ground truth magnitude of the complex shear modulus (∣G*∣) for varying: (1) ground truth liver ∣G*∣; (2) simulated imaging resolution; (3) added noise; (4) data smoothing. Motion and strain-based signal-to-noise (SNR) metrics were evaluated on the simulated data as a means to select higher-quality voxels for preparation of acquired MRE summary statistics of ∣G*∣. Main results. The simulated MRE accuracy for a given ground truth ∣G*∣ was found to be a function of imaging resolution, motion-SNR and smoothing. At typical imaging resolutions, it was found that due to under-sampling of the MRE wave-field, combined with motion-related noise, the reconstructed simulated ∣G*∣ could contain errors on the scale of the difference between liver fibrosis stages, e.g. 54% error for ground truth ∣G*∣ = 1 kPa. Optimum imaging resolutions were identified for given ground truth ∣G*∣ and motion-SNR levels. Significance. This study provides important knowledge on the accuracy and optimisation of liver MRE. For example, for motion-SNR ≤ 5, to distinguish between liver ∣G*∣ of 2 and 3 kPa (i.e. early-stage liver fibrosis) it was predicted that the optimum isotropic voxel size is 4–6 mm.

Quantification of tissue stiffness with magnetic resonance elastography and finite difference time domain (FDTD) simulation-based spatiotemporal neural network.

Characterisation of soft tissue mechanical properties is a topic of increasing interest in translational and clinical research. Magnetic resonance elastography (MRE) has been used in this context to assess the mechanical properties of tissues in vivo noninvasively. Typically, these analyses rely on linear viscoelastic wave equations to assess material properties from measured wave dynamics. However, deformations that occur in some tissues (e.g. liver during respiration, heart during the cardiac cycle, or external compression during a breast exam) can yield loading bias, complicating the interpretation of tissue stiffness from MRE measurements. In this paper, it is shown how combined knowledge of a material’s rheology and loading state can be used to eliminate loading bias and enable interpretation of intrinsic (unloaded) stiffness properties. Equations are derived utilising perturbation theory and Cauchy’s equations of motion to demonstrate the impact of loading state on periodic steady-state wave behaviour in nonlinear viscoelastic materials. These equations demonstrate how loading bias yields apparent material stiffening, softening and anisotropy. MRE sensitivity to deformation is demonstrated in an experimental phantom, showing a loading bias of up to twofold. From an unbiased stiffness of 4910.4 pm 635.8 Pa in unloaded state, the biased stiffness increases to 9767.5 pm ,1949.9 Pa under a load of approx 34% uniaxial compression. Integrating knowledge of phantom loading and rheology into a novel MRE reconstruction, it is shown that it is possible to characterise intrinsic material characteristics, eliminating the loading bias from MRE data. The framework introduced and demonstrated in phantoms illustrates a pathway that can be translated and applied to MRE in complex deforming tissues. This would contribute to a better assessment of material properties in soft tissues employing elastography.

Magnetic resonance elastography (MRE) maps the viscoelastic properties of soft tissues for diagnostic purposes. However, different MRE inversion methods yield different results, which hinder comparison of values, standardization, and establishment of quantitative MRE markers. Here, we introduce an expandable, open-access, webserver-based platform that offers multiple inversion techniques for multifrequency, 3D MRE data. The platform comprises a data repository and standard MRE inversion methods including local frequency estimation (LFE), direct-inversion based multifrequency dual elasto-visco (MDEV) inversion, and wavenumber-based (k-) MDEV. The use of the platform is demonstrated in phantom data and in vivo multifrequency MRE data of the kidneys and brains of healthy volunteers. Detailed maps of stiffness were generated by all inversion methods showing similar detail of anatomy. Specifically, the inner renal cortex had higher shear wave speed (SWS) than renal medulla and outer cortex without lateral differences. k-MDEV yielded higher SWS values than MDEV or LFE (full kidney/brain k-MDEV: 2.71 ± 0.19/1.45 ± 0.14 m/s, MDEV: 2.14 ± 0.16/0.99 ± 0.11 m/s, LFE: 2.12 ± 0.15/0.89 ± 0.06 m/s). The freely accessible platform supports the comparison of MRE results obtained with different inversion methods, filter thresholds, or excitation frequencies, promoting reproducibility in MRE across community-developed methods.

Imaging phantoms with known anisotropic mechanical properties are needed to evaluate magnetic resonance elastography (MRE) methods to estimate anisotropic parameters. The aims of this study were to fabricate mechanically anisotropic MRE phantoms, characterize their mechanical behavior by direct testing, then assess the accuracy of MRE estimates of anisotropic properties using a transversely isotropic nonlinear inversion (TI-NLI) algorithm. Directionally scaled and unscaled lattices were designed to exhibit anisotropic or isotropic mechanical properties. Lattices were three-dimensionally printed in poly(ethelyne glycol) diacrylate using a commercial digital light processing printer, then infilled with gelatin to form a composite material. Benchtop testing determined two shear stiffnesses, and , governing loading parallel and perpendicular to the symmetry axis, and two analogous Young's moduli and . From these measures, shear anisotropy = and tensile anisotropy = were calculated. Three phantoms were driven by a central actuator and imaged with MRE at frequencies from 300 to 500 Hz. From MRE data, the TI-NLI algorithm estimated maps of , , and . In benchtop tests, geometrically scaled lattice composites exhibited the following anisotropic properties: = 6.1 ± 0.7 kPa, = 0.83 ± 0.13, = 0.78 ± 0.09} (mean ± standard deviation). MRE of scaled lattice composites revealed elliptical wavefields; TI-NLI analysis identified the following median property ranges: = 11-19 kPa, = 0.6-1.0, = 0.8-1.6}. Anisotropic MRE phantoms are created by embedding anisotropic three-dimensionally printed lattices into a softer matrix. The TI-NLI algorithm accurately estimates spatial contrast in anisotropic properties.