Biophysical constraints on the evolution of tissue structure and function.

Phylogenetic analyses based on models of molecular sequence evolution have driven to industrial scale the generation, cataloguing and modelling of nucleic acid and polypeptide structure. The recent application of these techniques to study the evolution of protein interaction networks extends this analytical rigour to the study of nucleic acid and protein function. Can we further extend phylogenetic analysis of protein networks to the study of tissue structure and function? If the study of tissue phylogeny is to join up with mainstream efforts in the molecular evolution domain, the continuum field description of tissue biophysics must be linked to discrete descriptions of molecular biochemistry. In support of this goal we discuss tissue units, and biophysical constraints to molecular function associated with these units, to present a rationale with which to model tissue evolution. Our rationale combines a multiscale hierarchy of functional tissue units (FTUs) with the corresponding application of physical laws to describe molecular interaction networks and flow processes over continuum fields within these units. Non-dimensional numbers, derived from the equations governing biophysical processes in FTUs, are proposed as metrics for comparative studies across individuals, species or evolutionary time. We also outline the challenges inherent to the systematic cataloguing and phylogenetic analysis of tissue features relevant to the maintenance and regulation of molecular interaction networks. These features are key to understanding the core biophysical constraints on tissue evolution.

Using smooth particle hydrodynamics to investigate femoral cortical bone remodelling at the Haversian level.

In the neck of the femur, about 70% of the strength is contributed by the cortical bone, which is the most highly stressed part of the structure and is the site where failure is almost certainly initiated. A better understanding of cortical bone remodelling mechanisms can help discern changes at this anatomical site, which are essential if an understanding of the mechanisms by which hips weaken and become vulnerable to fracture is to be gained. The aims of this study were to (i) examine a hypothesis that low strain fields arise because of subject-specific Haversian canal distributions causing bone resorption and reduced bone integrity and (ii) introduce the use of a meshless particle-based computational modelling approach SPH to capture bone remodelling features at the level of the Haversian canals. We show that bone remodelling initiated by strain at the Haversian level is highly influenced by the subject-specific pore distribution, bone density, loading and osteocyte density. SPH is shown to be effective at capturing the intricate bone pore shapes that evolved over time.

Mechanics of the foot Part 1: a continuum framework for evaluating soft tissue stiffening in the pathologic foot.

Soft tissue stiffening is a common mechanical observation reported in foot pathologies including diabetes mellitus and gout. These material changes influence the spatial distribution of stress and affect blood flow, which is essential to nutrient entry and waste removal. An anatomically-based subject-specific foot model was developed to explore the influence of tissue stiffening on plantar pressure and internal von Mises stress at heel-strike, midstance and toe-off. This work draws on the model database developed for the Physiome project consisting of muscles, bones, soft tissue and other structures such as sensory nerves. The anisotropic structure of soft tissue was embedded in a single continuum as an efficient model for finite soft tissue deformation, and customisation methods were used to capture the unique foot profile. The model was informed by kinetics from an instrumented treadmill and kinematics from motion capture, synchronised together. Foot sole pressure predictions were evaluated against a commercial pressure platform. Key outcomes showed that internal stress can be up to 1.6 times the surface pressure with implications for internal soft tissue damage not observed at the surface. The main nerve branch stimulated during gait was the lateral plantar nerve. This subject-specific modelling framework can play an integral part in therapeutic treatments by informing assistive strategies such as mechanical noise stimulation and orthotics.

Mechanics of the foot Part 2: A coupled solid-fluid model to investigate blood transport in the pathologic foot.

A coupled computational model of the foot consisting of a three-dimensional soft tissue continuum and a one-dimensional (1D) transient blood flow network is presented in this article. The primary aim of the model is to investigate the blood flow in major arteries of the pathologic foot where the soft tissue stiffening occurs. It has been reported in the literature that there could be up to about five-fold increase in the mechanical stiffness of the plantar soft tissues in pathologic (e.g. diabetic) feet compared with healthy ones. The increased stiffness results in higher tissue hydrostatic pressure within the plantar area of the foot when loaded. The hydrostatic pressure acts on the external surface of blood vessels and tend to reduce the flow cross-section area and hence the blood supply. The soft tissue continuum model of the foot was modelled as a tricubic Hermite finite element mesh representing all the muscles, skin and fat of the foot and treated as incompressible with transversely isotropic properties. The details of the mechanical model of soft tissue are presented in the companion paper, Part 1. The deformed state of the soft tissue continuum because of the applied ground reaction force at three foot positions (heel-strike, midstance and toe-off) was obtained by solving the Cauchy equations based on the theory of finite elasticity using the Galerkin finite element method. The geometry of the main arterial network in the foot was represented using a 1D Hermite cubic finite element mesh. The flow model consists of 1D Navier-Stokes equations and a nonlinear constitutive equation to describe vessel radius-transmural pressure relation. The latter was defined as the difference between the fluid and soft tissue hydrostatic pressure. Transient flow governing equations were numerically solved using the two-step Lax-Wendroff finite difference method. The geometry of both the soft tissue continuum and arterial network is anatomically-based and was developed using the data derived from visible human images and magnetic resonance images of a healthy male volunteer. Simulation results reveal that a two-fold increase in tissue stiffness leads to about 28% reduction in blood flow to the affected region.

Minimum Information about a Cardiac Electrophysiology Experiment (MICEE): standardised reporting for model reproducibility, interoperability, and data sharing.

Cardiac experimental electrophysiology is in need of a well-defined Minimum Information Standard for recording, annotating, and reporting experimental data. As a step towards establishing this, we present a draft standard, called Minimum Information about a Cardiac Electrophysiology Experiment (MICEE). The ultimate goal is to develop a useful tool for cardiac electrophysiologists which facilitates and improves dissemination of the minimum information necessary for reproduction of cardiac electrophysiology research, allowing for easier comparison and utilisation of findings by others. It is hoped that this will enhance the integration of individual results into experimental, computational, and conceptual models. In its present form, this draft is intended for assessment and development by the research community. We invite the reader to join this effort, and, if deemed productive, implement the Minimum Information about a Cardiac Electrophysiology Experiment standard in their own work.

The effect of hoof angle variations on dorsal lamellar load in the equine hoof.

REASONS FOR PERFORMING STUDY: In the treatment of laminitis it is believed that reducing tension in the deep digital flexor tendon by raising the palmar angle of the hoof can reduce the load on the dorsal lamellae, allowing them to heal or prevent further damage. OBJECTIVE: To determine the effect of alterations in hoof angle on the load in the dorsal laminar junction. METHODS: Biomechanical finite element models of equine hooves were created with palmar angles of the distal phalanx varying from 0-15°. Tissue material relations accounting for anisotropy and the effect of moisture were used. Loading conditions simulating the stages in the stance where the vertical ground reaction force, midstance joint moment and breakover joint moment were maximal, were applied to the models. The loads were adjusted to account for the reduction in joint moment caused by increasing the palmar angle. Models were compared using the stored elastic energy, an indication of load, which was sampled in the dorsal laminar junction. RESULTS: For all loading cases, increasing the palmar angle increased the stored elastic energy in the dorsal laminar junction. The stored elastic energy near the proximal laminar junction border for a palmar angle of 15° was between 1.3 and 3.8 times that for a palmar angle of 0°. Stored elastic energy at the distal laminar junction border was small in all cases. For the breakover case, stored elastic energy at the proximal border also increased with increasing palmar angle. CONCLUSIONS AND POTENTIAL RELEVANCE: The models in this study predict that raising the palmar angle increases the load on the dorsal laminar junction. Therefore, hoof care interventions that raise the palmar angle in order to reduce the dorsal lamellae load may not achieve this outcome.

Coupling multi-physics models to cardiac mechanics.

We outline and review the mathematical framework for representing mechanical deformation and contraction of the cardiac ventricles, and how this behaviour integrates with other processes crucial for understanding and modelling heart function. Building on general conservation principles of space, mass and momentum, we introduce an arbitrary Eulerian-Lagrangian framework governing the behaviour of both fluid and solid components. Exploiting the natural alignment of cardiac mechanical properties with the tissue microstructure, finite deformation measures and myocardial constitutive relations are referred to embedded structural axes. Coupling approaches for solving this large deformation mechanics framework with three dimensional fluid flow, coronary hemodynamics and electrical activation are described. We also discuss the potential of cardiac mechanics modelling for clinical applications.

A finite element study of invariant-based orthotropic constitutive equations in the context of myocardial material parameter estimation.

A previous study investigated a number of invariant-based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. The study was based on the assumption of a homogeneous deformation. Here, we extend the previous study by performing an inverse finite element material parameter estimation. This ensures a more realistic deformation state and material parameter estimates. The constitutive relations were compared on the basis of (i) 'goodness of fit': how well they fit a set of six shear deformation tests and (ii) 'variability': how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive relations. It was found that a specific form of the polyconvex type as well as the exponential Fung-type equations were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.

Myocardial material parameter estimation: a comparison of invariant based orthotropic constitutive equations.

This study investigated a number of invariant based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. A number of orthotropic laws based on Green strain components and one microstructurally based law have previously been investigated to fit experimental measurements of stress-strain behaviour. Here we extend this investigation to include several recently proposed functional forms, i.e. invariant based orthotropic and transversely isotropic constitutive relations. These laws were compared on the basis of (i) 'goodness of fit': how well they fit a set of six shear deformation tests, (ii) 'variability': how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive laws. It was found that a specific form of the polyconvex type as well as the exponential Fung-type law from the previous study were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.

Myocardial material parameter estimation: a non-homogeneous finite element study from simple shear tests.

The passive material properties of myocardium play a major role in diastolic performance of the heart. In particular, the shear behaviour is thought to play an important mechanical role due to the laminar architecture of myocardium. We have previously compared a number of myocardial constitutive relations with the aim to extract their suitability for inverse material parameter estimation. The previous study assumed a homogeneous deformation. In the present study we relaxed the homogeneous assumption by implementing these laws into a finite element environment in order to obtain more realistic measures for the suitability of these laws in both their ability to fit a given set of experimental data, as well as their stability in the finite element environment. In particular, we examined five constitutive laws and compare them on the basis of (i) "goodness of fit": how well they fit a set of six shear deformation tests, (ii) "determinability": how well determined the objective function is at the optimal parameter fit, and (iii) "variability": how well determined the material parameters are over the range of experiments. Furthermore, we compared the FE results with those from the previous study.It was found that the same material law as in the previous study, the orthotropic Fung-type "Costa-Law", was the most suitable for inverse material parameter estimation for myocardium in simple shear.

A computationally efficient optimization kernel for material parameter estimation procedures.

Estimating material parameters is an important part in the study of soft tissue mechanics. Computational time can easily run to days, especially when all available experimental data are taken into account. The material parameter estimation procedure is examplified on a set of homogeneous simple shear experiments to estimate the orthotropic constitutive parameters of myocardium. The modification consists of changing the traditional least-squares approach to a weighted least-squares. This objective function resembles a L(2)-norm type integral which is approximated using Gaussian quadrature. This reduces the computational time of the material parameter estimation by two orders of magnitude.

Myocardial material parameter estimation-a comparative study for simple shear.

The study of ventricular mechanics-analyzing the distribution of strain and stress in myocardium throughout the cardiac cycle-is crucially dependent on the accuracy of the constitutive law chosen to represent the highly nonlinear and anisotropic properties of passive cardiac muscle. A number of such laws have been proposed and fitted to experimental measurements of stress-strain behavior. Here we examine five of these laws and compare them on the basis of (i) "goodness of fit:" How well they fit a set of six shear deformation tests, (ii) "determinability:" How well determined the objective function is at the optimal parameter fit, and (iii) "variability:" How well determined the material parameters are over the range of experiments. These criteria are utilized to discuss the advantages and disadvantages of the constitutive laws.

The Noble cardiac ventricular electrophysiology models in CellML.

We present a review of the cardiac ventricular cell electrophysiology models developed by Prof. Denis Noble and colleagues as an example of how models may be published using a web-based CellML publication framework. The models reviewed have been marked-up in CellML and then used to compute all results presented here. The models are freely available from a website as are the specific numerical experiments discussed in this review and the tools used to perform the simulations.

A quantitative analysis of cardiac myocyte relaxation: a simulation study.

The determinants of relaxation in cardiac muscle are poorly understood, yet compromised relaxation accompanies various pathologies and impaired pump function. In this study, we develop a model of active contraction to elucidate the relative importance of the [Ca2+]i transient magnitude, the unbinding of Ca2+ from troponin C (TnC), and the length-dependence of tension and Ca2+ sensitivity on relaxation. Using the framework proposed by one of our researchers, we extensively reviewed experimental literature, to quantitatively characterize the binding of Ca2+ to TnC, the kinetics of tropomyosin, the availability of binding sites, and the kinetics of crossbridge binding after perturbations in sarcomere length. Model parameters were determined from multiple experimental results and modalities (skinned and intact preparations) and model results were validated against data from length step, caged Ca2+, isometric twitches, and the half-time to relaxation with increasing sarcomere length experiments. A factorial analysis found that the [Ca2+]i transient and the unbinding of Ca2+ from TnC were the primary determinants of relaxation, with a fivefold greater effect than that of length-dependent maximum tension and twice the effect of tension-dependent binding of Ca2+ to TnC and length-dependent Ca2+ sensitivity. The affects of the [Ca2+]i transient and the unbinding rate of Ca2+ from TnC were tightly coupled with the effect of increasing either factor, depending on the reference [Ca2+]i transient and unbinding rate.

Anatomically based modelling of the human skull and jaw.

We present here an anatomically based model of the human masticatory system that provides a framework for simulating the complex chewing process. The initial motivation for creating this model was the desire to have a computational model of the human jaw that can be used to simulate the action of simple bites, and to calculate the stresses and forces on the teeth that are involved. The model created also provides a platform that can be used to investigate other features of the masticatory system. To construct this global model, individual models of the bones of the skull and jaw were created from generic data sets. Geometric models of the muscles of mastication were also created and attached to the appropriate bones. To complete this initial model, representations of the crowns of the teeth were created and a basic model of the temporomandibular joint (TMJ) was included. The finite element method was used to solve for the stresses and strains created by the loading conditions during a clenching simulation involving the mandible bone. The model presented here is also discussed in relation to a model of the entire musculo-skeletal system being developed as part of the Physiome Project.

An anatomically based patient-specific finite element model of patella articulation: towards a diagnostic tool.

A 3D anatomically based patient-specific finite element (FE) model of patello-femoral (PF) articulation is presented to analyse the main features of patella biomechanics, namely, patella tracking (kinematics), quadriceps extensor forces, surface contact and internal patella stresses. The generic geometries are a subset from the model database of the International Union of Physiological Sciences (IUPS) (http://www.physiome.org.nz) Physiome Project with soft tissue derived from the widely used visible human dataset, and the bones digitised from an anatomically accurate physical model with muscle attachment information. The models are customised to patient magnetic resonance images using a variant of free-form deformation, called 'host-mesh' fitting. The continuum was solved using the governing equation of finite elasticity, with the multibody problem coupled through contact mechanics. Additional constraints such as tissue incompressibility are also imposed. Passive material properties are taken from the literature and implemented for deformable tissue with a non-linear micro-structurally based constitutive law. Bone and cartilage are implemented using a 'St-Venant Kirchoff' model suitable for rigid body rotations. The surface fibre directions have been estimated from anatomy images of cadaver muscle dissections and active muscle contraction was based on a steady-state calcium-tension relation. The 3D continuum model of muscle, tendon and bone is compared with experimental results from the literature, and surgical simulations performed to illustrate its clinical assessment capabilities (a Maquet procedure for reducing patella stresses and a vastus lateralis release for a bipartite patella). Finally, the model limitations, issues and future improvements are discussed.

A cerebral palsy assessment tool using anatomically based geometries and free-form deformation.

A geometrical analysis tool for investigating muscle length change in cerebral palsy (CP) patients is presented. A subset of anatomically based geometries from the International Union of Physiological Sciences (IUPS) Physiome Project is used, which is derived from the visible human (VH) data set with muscle attachment information, and customised using volume-preserving free-form deformation (FFD), the 'host-mesh' technique. The model's intended use is to provide pre- and post-surgery assessment for muscle lengthening, a surgery performed to help slacken tight muscles and improve gait. The model is illustrated using healthy patient data from motion capture as a validation followed by three CP case studies to highlight its use. The methodology is presented in three stages, (1) a FFD of the complete lower limb, (2) a focused geometric study on the semimembranosus (SM) and gastrocnemius (GT) muscles, and (3) an improved hybrid mechanics-FFD approach as an improvement for future analysis, with differentiation between muscle and tendon lengthening, and contact detection between sliding muscles. Finally, the issues, limitations, in particular with the marker system, and model improvements are discussed.

Modelling the passive and nerve activated response of the rectus femoris muscle to a flexion loading: a finite element framework.

A muscle modelling framework is presented which relates the mechanical response of the rectus femoris muscle (at the organ level) to tissue level properties, with the capability of linking to the cellular level as part of the IUPS Physiome Project. This paper will outline our current approach to muscle modelling incorporating micro-structural passive and active properties including fibre orientations and nerve innervation. The technique is based on finite deformation (using FE analysis) coupled to electrical nerve initiated muscle activation, and we present the influence of active tension through an eccentric contraction at specific flexion angles. Finally we discuss the future goals of incorporating cell mechanics and validating at the organ level to provide a complete diagnostic tool with the ability to relate mechanisms of failure across spatial scales.