Simulation tools for biomechanical applications with PGD-based reduced order models

Cotutela Universitat Politècnica de Catalunya i Università degli Studi di PaviaNumerical simulation tools are generally used in all modern engineering fields, especially those having difficulties in performing large number of practical experiments, such as biomechanics. Among the computational methods, Finite Element (FE) is an essential tool. Nowadays, the fast-growing computational techniques, from the upgrading hardware to the emerging of novel algorithm, have already enabled extensive applications in biomechanics. For applications that require fast response and/or multiple queries, Reduced Order Modelling (ROM) methods have been developed based on existing methods such as FE, and have eventually enabled real-time numerical simulation for a large variety of engineering problems. In this thesis, several novel computational techniques are developed to explore the capability of Proper Generalised Decomposition (PGD), which is an important approach of ROM. To assess the usability of the PGD-based ROM for biomechanical applications, a real human femur bone is chosen to study its mechanical behaviour as an example. Standard image-based modelling procedure in biomechanics is performed to create an FE model which is then validated with in vitro experimental results. As a basis of this work, the medical image processing has to be performed, in order to generate an available FE model. This model is validated according to data collected from a previously performed \textit{in vitro} experimental test. The full procedure of image-based model generation and the validation of generated model is described in Chapter 2. As a major objective of this thesis, a non-intrusive scheme for the PGD framework is developed in Chapter 3. It is implemented using in-house developed Matlab (Mathworks, USA) code to conduct the PGD work flow, and calling Abaqus as an external solver for devised fictitious mechanical problems. The transformation of data from computed tomography (CT) image set to FE model including inhomogeneous material properties is subjected to some physical constraints, and when applying the load, there are also geometric constraints limiting the locations where load could be applied. These constraints will lead to a constrained parameter space, which possibly has difficulty to be separated in a Cartesian fashion. Therefore, a novel strategy to separate the parameters in a collective manner is proposed in Chapter 4. Chapter 5 details a comprehensive application in biomechanics, the methodologies proposed in Chapter 3 and 4 are applied on the practical model generated in Chapter 2. As a typical application of the PGD vademecum, a material property identification problem is discussed. Further PGD vademecum is generated using the identified material properties with variable loading locations, and with this vademecum, real-time mechanical response of the femur is available. In addition, for the purpose of extending the methodologies to orthotropic materials, which is commonly used in biomechanics, in Chapter 6 another linear elastic model is investigated with the non-intrusive PGD scheme. Nowadays, isogeometric analysis (IGA) is a very popular tool in computational mechanics. It is appealing to take advantage of non-uniform rational B-splines (NURBS) to discretise the model. For PGD, using B-splines for the discretisation of the parameter space could improve the quality of vademecum, especially for problems involving sensitivities with respect to the parameters during the online computations. It is important and necessary to extend the PGD framework to nonlinear solid mechanics, because most biological soft tissues have been observed nonlinear mechanical behaviours. Consequently, in Chapter 7 we have developed a PGD framework for the St.Venant-Kirchhoff constitutive model using the Picard linearisation which is consistent with the fixed-point iteration algorithm commonly used in PGD. In Chapter 8, conclusive remarks are addressed as well as forecasts of possible future works.Postprint (published version

Multiscale Geometric Methods for Isolating Exercise Induced Morphological Adaptations in the Proximal Femur

The importance of skeletal bone in the functioning of the human body is well-established and acknowledged. Less pervasive among the populace, is the understanding of bone as an adaptive tissue which modulates itself to achieve the most construction sufficient for the role it is habituated to. These mechanisms are more pronounced in the long load bearing bones such as the femur. The proximal femur especially, functions under significant loads and does so with high degree of articulation, making it critical to mobility. Thus, exercising to buttress health and reinforce tissue quality is just as applicable to bone as it is to muscles. However, the efficiency of the adaptive (modelling/remodelling) processes is subdued after maturity, which makes the understanding of its potential even more important. Classically, studies have translated the evaluation of strength in terms of its material and morphology. While the morphology of the femur is constrained within a particular phenotype, minor variations can have a significant bearing on its capability to withstand loads. Morphology has been studied at different scales and dimensions wherein parameters quantified as lengths, areas, volumes and curvatures in two and three dimensions contribute towards characterising strength. The challenge has been to isolate the regions that show response to habitual loads. This thesis seeks to build on the principles of computational anatomy and develop procedures to study the distribution of mechanically relevant parameters. Methods are presented that increase the spatial resolution of traditional cross-sectional studies and develop a conformal mapping procedure for proximal femur shape matching. In addition, prevalent methods in cross-sectional analyses and finite element simulations are employed to analyse the morphology of the unique dataset. The results present the spatial heterogeneity and a multi-scale understanding of the adaptive response in the proximal femur morphology to habitual exercise loading

Visualization of Tensor Fields in Mechanics

Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large-scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application-oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio-, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context

Studies on Spinal Fusion from Computational Modelling to ‘Smart’ Implants

Low back pain, the worldwide leading cause of disability, is commonly treated with lumbar interbody fusion surgery to address degeneration, instability, deformity, and trauma of the spine. Following fusion surgery, nearly 20% experience complications requiring reoperation while 1 in 3 do not experience a meaningful improvement in pain. Implant subsidence and pseudarthrosis in particular present a multifaceted challenge in the management of a patient’s painful symptoms. Given the diversity of fusion approaches, materials, and instrumentation, further inputs are required across the treatment spectrum to prevent and manage complications. This thesis comprises biomechanical studies on lumbar spinal fusion that provide new insights into spinal fusion surgery from preoperative planning to postoperative monitoring. A computational model, using the finite element method, is developed to quantify the biomechanical impact of temporal ossification on the spine, examining how the fusion mass stiffness affects loads on the implant and subsequent subsidence risk, while bony growth into the endplates affects load-distribution among the surrounding spinal structures. The computational modelling approach is extended to provide biomechanical inputs to surgical decisions regarding posterior fixation. Where a patient is not clinically pre-disposed to subsidence or pseudarthrosis, the results suggest unilateral fixation is a more economical choice than bilateral fixation to stabilise the joint. While finite element modelling can inform pre-surgical planning, effective postoperative monitoring currently remains a clinical challenge. Periodic radiological follow-up to assess bony fusion is subjective and unreliable. This thesis describes the development of a ‘smart’ interbody cage capable of taking direct measurements from the implant for monitoring fusion progression and complication risk. Biomechanical testing of the ‘smart’ implant demonstrated its ability to distinguish between graft and endplate stiffness states. The device is prepared for wireless actualisation by investigating sensor optimisation and telemetry. The results show that near-field communication is a feasible approach for wireless power and data transfer in this setting, notwithstanding further architectural optimisation required, while a combination of strain and pressure sensors will be more mechanically and clinically informative. Further work in computational modelling of the spine and ‘smart’ implants will enable personalised healthcare for low back pain, and the results presented in this thesis are a step in this direction

An Assessment of Hallux Valgus

The foot is an essential component for human gait and begins the propagation of forces in the lower extremity of the body. One of the most common conditions that produce forefoot pain is hallux valgus (HV). HV alters or restricts normal body kinematics, influences physical mobility and increases the risk of falling. The root cause of HV has not been fully determined. While the principal kinematics are known and understood, the etiology still remains unclear. Clinically standard planar radiographs are employed but cannot accurately capture first metatarsal pronation, which is known to occur in the onset of hallux valgus. Previous research has also shown changes occur in bone density near the midfoot of cadavers with hallux valgus. Plantar pressure models have shown patients with hallux valgus have increased loading at the big toe and metatarsal head. In this study, we analyzed the forefoot of normal and HV patients groups to measure in vivo density and bone orientation. We also developed patient specific three-dimensional finite element models of the first and second rays of the foot to develop predictions of stress on the metatarsal in the progression of the HV. We found changes in the density profile in patients with hallux valgus. We quantified pronation in the first metatarsal and found differences in the patients with hallux valgus. The pronation reported here is the first true three-dimensional measurement of metatarsal rotation due to the hallux valgus deformity. We found differences in contact forces at the metatarsal head and metatarsal base due to hallux valgus. This study is the first to report an estimate of pressure at the metatarsal sesamoid interface. We found increased pressure due to the altered kinematics as a result of HV, which can lead to pain and erosion at the metatarsal head

COUPLING CELLS COMPETITION, GROWTH AND REMODELLING IN MECHANICS OF BIOLOGICAL SYSTEMS

The biomechanical behavior and the mechanobiology of cells, tissues and organs have been intensively investigated in the last decades, with the aim of discovering the key feedback mechanisms governing the ways in which cascades of chemical signals are transmitted within the hierarchically organized living structures and interplay with physical events at different scale levels. Continuum Mechanics has deeply contributed to develop this research area and to meet related challenges, by creating the physically and mathematically consistent ground on which large deformation, stresses, evolving constitutive laws, growth, remodeling and morphogenesis do interact. The needed multiphysics vision in analysing the complex behavior of the living matter has in particular consolidated Tissue Mechanics theoretical approaches and related modeling strategies which are currently recognized as indispensable tools for explaining experimental evidences, for predicting dynamics of living systems as well as for supporting the design of prostheses for both soft and hard tissues. Further impulse to these studies is then given by the rapidly growing advances of the research in tissue engineering which continuously redraw new scenarios for applications in medicine and lead to envisage innovative drug delivery systems and biomaterials. Within this vivid multidisciplinary debate, an increasing interest has been recently registered in the Literature for the mechanical properties of living cells -and for the understanding of the dynamics to which they obey at different scale levels- also motivated by some recent discoveries which seem to allow to envisage new horizons for therapy and diagnosis of human diseases like cancer, by for example exploiting the different in-frequency response of single healthy and tumor cells stimulated by Utrasound. However, at the macroscopic scale -say at the tissue level- the feedback mechanisms and the cascade of bio-chemical and physical signals characterizing the complex interaction of dynamics occuring at different scales significantly complicates the biomechanical response of living matter and growing tumor masses, thus requiring enriched models which encorporate the mechanobiology at the micro- and meso-scale levels. Cancer diseases in fact occur when in a healthy tissue the cell-cell and cells-ECM (the Extra-Cellular Matrix) interactions are altered, and hyperplasia is generated as effect of sudden and often unforeseeable genetic modifications followed by a cascade of biochemical events leading to abnormal cell growth, lost of apoptosis, back-differentiation and metastasis. As a consequence, the determination of models capable to macroscopically describe how tumor masses behave and evolve in living tissues by embodying tumor invasion dynamics determined by cell-cell and cells-environment to date still remains an open issue. Growth of biological tissues has been recently treated within the framework of Continuum Mechanics, by adopting heterogeneous poroelastic models where the interaction between soft matrix and interstitial fluid flow is additionally coupled with inelastic effects ad hoc introduced to simulate the macroscopic volumetric growth determined by cells division, cells growth and extracellular matrix changes occurring at the micro-scale level. These continuum models seem to overcome some limitations intrinsically associated to other alternative approaches based on mass balances in multiphase systems, because the crucial role played by residual stresses accompanying growth and nutrients walkway is preserved. Nevertheless, when these strategies are applied to analyse solid tumors, mass growth is usually assigned in a prescribed form that essentially copies the in vitro measured intrinsic growth rates of the cell species. As a consequence, some important cell-cell dynamics governing mass evolution and invasion rates of cancer cells, as well as their coupling and feedback mechanisms associated to in situ stresses, are inevitably lost and hence the spatial distribution and the evolution with time of the growth inside the tumor --which would be results rather than input-- are forced to simply be data. In order to solve this sort of paradox, the present Thesis work, within a consistent thermodynamic framework, builds up an enhanced multi-scale poroelastic model undergoing large deformation and embodying inelastic growth, where the net growth terms directly result from the "interspecific" predator-prey (Volterra/Lotka-like) competition occurring at the micro-scale level between healthy and abnormal cell species. In this way, a system of fully-coupled non-linear PDEs is derived to describe how the fight among cell species to grab the available common resources, stress field, pressure gradients, interstitial fluid flows driving nutrients and inhomogeneous growth do all simultaneously interact to decide the tumor fate. The stability of the predator-prey dynamics and some original theoretical results for the non-linear mechanics of growing media are also developed and discussed in detail. The general approach -that is the coupling of growth, large deformation and competitive cell dynamics- is therefore applied to actual biomechanical problems (in particular analyzing growth and stress in tumor spheroids and arterial walls) and the theoretical outcomes are finally compared with in vivo experiments and animal models to validate the effectiveness and the robustness of the proposed strategy