A computational approach will be presented for the estimation of the in vivo magnitude and spatial distribution of mechanical material properties of organs and other soft tissues from standard clinical imaging data. To this end, a new shape-based objective function, quantifying the difference between the measured and predicted tissue mechanical response is introduced. By utilizing shape, rather than deformation or related quantity, standard clinical imaging data can be utilized (e.g., without tagging) without further manipulation to approximate such measures from the images first. This objective function can then be implemented into an optimization-based inverse solution approach to estimate mechanical properties of tissues from initial and final shapes derived from clinical imaging data. This approach is an extension of prior work by the authors that used a standard discretized version of the Hausdorff distance as an objective function in an iterative approach to material parameter estimation [1]. A key component of the new inverse approach is constructing the geometry of the region of interest using a signed distance function. As such, a novel level-set framework is introduced for the objective function that is easily differentiable, and thus, able to be implemented into an optimization framework to estimate the material parameters that minimize the objective function with respect to a target shape with relative computational efficiency. A set of simulated inverse problems was used to evaluate the inverse solution estimation procedure based on estimating the passive elasticity of human ventricular walls from standard cardiac imaging data and corresponding hemodynamic measurements. In evaluating the results, emphasis will be placed on not just the accuracy of the material parameter estimates, but also on the computational expense (e.g., number of forward finite element analyses) required to approximate the target response. Various levels of heterogeneity will be considered in terms of the effect on solution accuracy and/or need for regularization. Additionally, sensitivity to model error will be explored.

REFERENCES

[1] Xu, J., Wong, T.C., Simon, M.A., and Brigham, J.C. A Clinically Applicable Strategy to Estimate the In Vivo Distribution of Mechanical Material Properties of the Right Ventricular Wall. International Journal for Numerical Methods in Biomedical Engineering, (2021)

For physics-based simulations, forward (UQ) and inverse uncertainty propagation still generate challenges for computationally demanding, real-world applications, like the ones appearing in many scenarios in Bioengineering. Two main obstacles pertain to a high stochastic dimension and many necessary forward solver calls caused by the statistical nature of the underlying algorithms.

A high stochastic dimension usually precludes the reliable use of surrogate-based approaches to reduce computational costs. Instead, we propose a Bayesian multi-fidelity procedure that exploits conditional distributions between two or more model fidelities in a small data regime (100 to 300 high-fidelity evaluations are necessary to train the conditional). The required online sampling is entirely shifted to the low-fidelity models. The latter can, for example, use simplified physics and coarser numerical discretization and only need to share a (nonlinear) statistical relationship with the high-fidelity model, giving a high degree of flexibility in selecting and creating such low-fidelity models. In both cases, the forward UQ and the Bayesian inverse problem, our approach results in an accurate posterior distribution, despite the inaccurate and noisy information the low-fidelity models provide. In the case of forward UQ, we call our approach Bayesian Multi Fidelity Monte Carlo (BMFMC), and for the inverse problem Bayesian Multi-Fidelity Inverse Analysis (BMFIA).

Especially for the Bayesian inverse problem, BMFIA brings further advantages. It shifts the necessary derivative evaluations to the low-fidelity model, allowing the analyst to exploit adjoint implementations for simplified physical models. We then perform Bayesian inference with efficient stochastic variational methods, which require solely evaluations of the lower-fidelity model. If the low-fidelity models cannot provide model gradients, we propose new developments in sequential black box variational inference schemes. The latter poses the variational optimization problem without needing model gradients. The formulation is more efficient than gradient-free sampling schemes, such as sequential Monte Carlo or Markov Chain Monte Carlo methods. It gives reliable posterior approximations for moderate stochastic dimensions (up to 50) where surrogate approaches are already prohibitive.

Another possibility for some challenging problems in Bioengineering is to drastically reduce the number of forward model calls via approximating the log-likelihood by a surrogate model. The advantage of such a log-likelihood approximation over the outputs of the forward model is that instead of a potentially highly dimensional model output, only a scalar value has to be approximated. To allow the scalability of the approach to higher stochastic dimensions, the training samples of the surrogate are adaptively selected, and the uncertainties of the surrogate are taken into account.

In this presentation, we will show the proposed methods' essential aspects and their application to different challenging problems in Biomechanics.

Changes in our body that occur due to illness and disease also bring mechanical changes, which are often only observed by clinicians poking and prodding tissues. These present opportunities for new diagnosis and monitoring technologies. In our work we have taken a tissue biomechanics approach to study the wound healing process. This area is particularly important due to the large costs that wounds place on healthcare resources (>£4.7 billion in the UK annually) and pain experienced by patients.

To assess how wound healing relates to mechanical changes, we used a mouse model of acute wound healing. We made 4 mm diameter wounds in the skin using a biopsy punch and then the skin was allowed to recover with mechanical and histological assessment at days 1, 3, 7 and 14 post-wounding. These correlated with the stages of wound healing – haemostasis, inflammation, proliferation, and remodelling. We excised the tissue and undertook tensile testing for global properties, image-based local strain assessment using Digital Image Correlation (DIC), and Optical Coherence Topography Elastography (OCE) to characterise the material changes in the skin. We correlated these to histology with H&E and Picrosirius red staining for wound-staging and collagen fibre characterisation respectively. Concurrently we developed a finite element model to aid the mechanistic interpretation of the data.

Our tensile testing results showed no discernible change in the hyperelastic moduli/coefficients when a wound was present in skin, compared to intact skin. This contrasted with literature which had shown the opposite. We believe that this is due to the more physiologically relevant strains we used to test the tissue, rather than the “test to failure” approach of others. To understand this further, we measured the local in-plane strains in the skin during tension. We observed re-organisation of the tissue strains with a compliant ring which reduces stress on the wound. This mechanism during healing appears to ensure that wounds are protected whilst healing. When skin was simulated by finite element models it became clear that a bulk hyperelastic approach did not sufficiently account for these changes, and fibrous models would be required. Our histology data showed how the fibrous components in the skin vary during healing, which indicates models will require time-variant structural changes. Our analysis of the wound mechanics by OCE helps identify sub-surface mechanical changes and we will share our progress on this.

The changes that occur in tissues during physiological changes, presents a substantial opportunity for our bioengineering community but we need to have both experiments and models that reflect the reality of tissue changes. Our work has shown that the experimentally derived data is most useful for computational model development only if both global and local structural changes are considered. Furthermore, the need for a model that adapts to the time-variant changes during a disease’s progression become more central. These, set against a backdrop of a varied population, increase the challenge of accurate physiological modelling but present a huge opportunity for the computational and experimental communities to work together.

Considerable research efforts have been devoted to the development of microstructurally-based anisotropic continuum constitutive models of biological soft tissues. These formulations typically rely on the definition of one or more vector fields representing the local orientation of biological fibres (i.e. collagen fibres) within a tissue. Methods to incorporate such a structural information into anatomically realistic micromechanical finite element (FE) models of soft tissues such as skin are still lagging behind, particularly when it comes to models aiming to capture the complex local three-dimensional (3D) architecture of the collagen fibres network.

In order to improve the predictive power of the next generation of biophysical models of soft tissues it is essential to develop robust methods and methodologies to seamlessly integrate the microstructural characteristics of the collagen network. Here, we developed such an approach combining high-resolution imaging of human dermis, fibre orientation image analysis, voxel-based mesh generation and micromechanical FE analyses.

Fresh full-thickness abdominal skin extracted during a cosmetic surgery procedure from a 39 years-old female Caucasian patient was commercially obtained (TCS Cellworks, Buckingham, UK). The skin sample was cut into 1 mm slices and processed for serial block-face scanning electron microscopy using a high contrast fixation protocol (SBEM Protocol v7_01_10, https://ncmir.ucsd.edu/sbem-protocol) and embedded in Spurr resin (Agar Scientific, Stansted, UK). The tissue block was then trimmed, glued onto an aluminium pin, sputter coated in gold/palladium and imaged in the serial block-face imaging system 3View® (Gatan, Inc., Pleasanton, CA, USA) mounted inside a Zeiss Sigma VP field emission scanning electron microscope with variable pressure mode (Carl Zeiss Microscopy GmbH, Jena, Germany) and imaged at 2.5kV. The acquisition was done at a sampling XY resolutions of 2500 x 2500 pixels (i.e. 8 nm pixel size) every 50 nm, resulting in 376 images. For development purpose, and due to the large size of the data set, the original image stack was cropped to generate a sub-stack with a 600×600×50 voxel dimension. The stack of 50 images was processed using a 3D orientation analysis algorithm based on the calculation of local structure tensors. A series of Python and Fortran programmes were written to generate a voxel-based hexahedral FE mesh with the option to assign fibre vectors at node, element or integration point level, import it into the FE package Abaqus® (Simulia, Dassault Systèmes, Johnston, RI, USA), code constitutive equations for invariant-based anisotropic hyperelasticity via Abaqus® UMAT subroutines, assign material properties to matrix and fibre phases, and run numerical analyses to study the micromechanics of the dermis. The main objective of our analyses was to quantify and understand the effects of faithfully capturing collagen fibre orientation on the homogenised micromechanics of collagen assemblies, and compare our technique to approaches considering a uniform orientation of collagen fibres, with or without fibre dispersion. Uniaxial/biaxial extensions and shear tests were simulated. The results of our numerical analyses did not only demonstrate the criticality of accounting for local fibre orientation but also the importance of distinguishing between the matrix and fibre phases as spatially independent domains within a tissue block.