A mixed finite element formulation for a non-linear, transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

The finite polygenic mixed model: An alternative formulation for the mixed model of inheritance

This paper presents a mixed model of inheritance with a finite number of polygenic loci. This model leads to a likelihood that can be calculated using efficient algorithms developed for oligogenic models. For comparison, likelihood profiles were obtained for the finite polygenic mixed model, the usual mixed model, with exact and approximate calculations, and for a class D regressive model. The profiles for the finite polygenic mixed model were closest to the profiles for the usual mixed model with exact calculations.     

Updated Lagrangian finite element formulations of various biological soft tissue non-linear material models: a comprehensive procedure and review

Simplified material models are commonly used in computational simulation of biological soft tissue as an approximation of the complicated material response and to minimize computational resources. However, the simulation of complex loadings, such as long-duration tissue swelling, necessitates complex models that are not easy to formulate. This paper strives to offer the updated Lagrangian formulation comprehensive procedure of various non-linear material models for the application of finite element analysis of biological soft tissues including a definition of the Cauchy stress and the spatial tangential stiffness. The relationships between water content, osmotic pressure, ionic concentration and the pore pressure stress of the tissue are discussed with the merits of these models and their applications.     

A finite element model of myocardial infarction using a composite material approach

Computational models are effective tools to study cardiac mechanics under normal and pathological conditions. They can be used to gain insight into the physiology of the heart under these conditions while they are adaptable to computer assisted patient-specific clinical diagnosis and therapeutic procedures. Realistic cardiac mechanics models incorporate tissue active/passive response in conjunction with hyperelasticity and anisotropy. Conventional formulation of such models leads to mathematically-complex problems usually solved by custom-developed non-linear finite element (FE) codes. With a few exceptions, such codes are not available to the research community. This article describes a computational cardiac mechanics model developed such that it can be implemented using off-the-shelf FE solvers while tissue pathologies can be introduced in the model in a straight-forward manner. The model takes into account myocardial hyperelasticity, anisotropy, and active contraction forces. It follows a composite tissue modeling approach where the cardiac tissue is decomposed into two major parts: background and myofibers. The latter is modelled as rebars under initial stresses mimicking the contraction forces. The model was applied in silico to study the mechanics of infarcted left ventricle (LV) of a canine. End-systolic strain components, ejection fraction, and stress distribution attained using this LV model were compared quantitatively and qualitatively to corresponding data obtained from measurements as well as to other corresponding LV mechanics models. This comparison showed very good agreement.     

A computational study of invariant I5 in a nearly incompressible transversely isotropic model for white matter

The aligned axonal fiber bundles in white matter make it suitable to be modeled as a transversely isotropic material. Recent experimental studies have shown that a minimal form, nearly incompressible transversely isotropic (MITI) material model, is capable of describing mechanical anisotropy of white matter. Here, we used a finite element (FE) computational approach to demonstrate the significance of the fifth invariant (I₅) when modeling the anisotropic behavior of white matter in the large-strain regime. We first implemented and validated the MITI model in an FE simulation framework for large deformations. Next, we applied the model to a plate-hole structural problem to highlight the significance of the invariant I₅ by comparing with the standard fiber reinforcement (SFR) model. We also compared the two models by fitting the experiment data of asymmetric indentation, shear test, and uniaxial stretch of white matter. Our results demonstrated the significance of I₅ in describing shear deformation/anisotropy, and illustrated the potential of the MITI model to characterize transversely isotropic white matter tissues in the large-strain regime.     

Computational efficiency of numerical approximations of tangent moduli for finite element implementation of a fiber-reinforced hyperelastic material model

In this study, we evaluated computational efficiency of finite element (FE) simulations when a numerical approximation method was used to obtain the tangent moduli. A fiber-reinforced hyperelastic material model for nearly incompressible soft tissues was implemented for 3D solid elements using both the approximation method and the closed-form analytical method, and validated by comparing the components of the tangent modulus tensor (also referred to as the material Jacobian) between the two methods. The computational efficiency of the approximation method was evaluated with different perturbation parameters and approximation schemes, and quantified by the number of iteration steps and CPU time required to complete these simulations. From the simulation results, it can be seen that the overall accuracy of the approximation method is improved by adopting the central difference approximation scheme compared to the forward Euler approximation scheme. For small-scale simulations with about 10,000 DOFs, the approximation schemes could reduce the CPU time substantially compared to the closed-form solution, due to the fact that fewer calculation steps are needed at each integration point. However, for a large-scale simulation with about 300,000 DOFs, the advantages of the approximation schemes diminish because the factorization of the stiffness matrix will dominate the solution time. Overall, as it is material model independent, the approximation method simplifies the FE implementation of a complex constitutive model with comparable accuracy and computational efficiency to the closed-form solution, which makes it attractive in FE simulations with complex material models.     

Growth of mixed cultures on mixtures of substitutable substrates: the operating diagram for a structured model

The growth of mixed microbial cultures on mixtures of substrates is a problem of fundamental biological interest. In the last two decades, several unstructured models of mixed-substrate growth have been studied. It is well known, however, that the growth patterns in mixed-substrate environments are dictated by the enzymes that catalyse the transport of substrates into the cell. We have shown previously that a model taking due account of transport enzymes captures and explains all the observed patterns of growth of a single species on two substitutable substrates (J. Theor. Biol. 190 (1998) 241). Here, we extend the model to study the steady states of growth of two species on two substitutable substrates. The model is analysed to determine the conditions for existence and stability of the various steady states. Simulations are performed to determine the flow rates and feed concentrations at which both species coexist. We show that if the interaction between the two species is purely competitive, then at any given flow rate, coexistence is possible only if the ratio of the two feed concentrations lies within a certain interval; excessive supply of either one of the two substrates leads to annihilation of one of the species. This result simplifies the construction of the operating diagram for purely competing species. This is because the two-dimensional surface that bounds the flow rates and feed concentrations at which both species coexist has a particularly simple geometry: It is completely determined by only two coordinates, the flow rate and the ratio of the two feed concentrations. We also study commensalistic interactions between the two species by assuming that one of the species excretes a product that can support the growth of the other species. We show that such interactions enhance the coexistence region.     

A mixed model for the effects of single gene,polygenes and their interaction on quantitative traits

Summary A model for the effects of single gene (SG), polygenes (PG) and their interaction on quantitative traits was developed. It is a mixed model where the SG is a fixed effect and the PG is a random effect. A two-way factorial experiment, in which the SG and the PG are the main effects, is proposed. The experimental material is comprised of F₃ families derived from F₂ plants heterozygous for the SG. For this experiment an ANOVA table with expected mean square is proposed, which facilitates estimation of the components of the model and testing of their significance. A detailed method for the interpretation of results from such an experiment is proposed, with emphasis on the analysis of the SG × PG interaction. Theoretical and applied aspects of SG × PG interaction is discussed.This paper is part of a Ph.D. Thesis of the senior author to be submitted to the Hebrew University of Jerusalem     

A validated model of GAG deposition,cell distribution,and growth of tissue engineered cartilage cultured in a rotating bioreactor

In this work a new phenomenological model of growth of cartilage tissue cultured in a rotating bioreactor is developed. It represents an advancement of a previously derived model of deposition of glycosaminoglycan (GAG) in engineered cartilage by (i) introduction of physiological mechanisms of proteoglycan accumulation in the extracellular matrix (ECM) as well as by correlating (ii) local cell densities and (iii) tissue growth to the ECM composition. In particular, previously established predictions and correlations of local oxygen concentrations and GAG synthesis rates are extended to distinguish cell secreted proteoglycan monomers free to diffuse in cell surroundings and outside from the engineered construct, from large aggrecan molecules, which are constrained within the ECM and practically immovable. The model includes kinetics of aggregation, that is, transformation of mobile GAG species into immobile aggregates as well as maintenance of the normal ECM composition after the physiological GAG concentration is reached by incorporation of a product inhibition term. The model also includes mechanisms of the temporal evolution of cell density distributions and tissue growth under in vitro conditions. After a short initial proliferation phase the total cell number in the construct remains constant, but the local cell distribution is leveled out by GAG accumulation and repulsion due to negative molecular charges. Furthermore, strong repulsive forces result in expansion of the local tissue elements observed macroscopically as tissue growth (i.e., construct enlargement). The model is validated by comparison with experimental data of (i) GAG distribution and leakage, (ii) spatial‐temporal distributions of cells, and (iii) tissue growth reported in previous works. Validation of the model predictive capability—against a selection of measured data that were not used to construct the model—suggests that the model successfully describes the interplay of several simultaneous processes carried out during in vitro cartilage tissue regeneration and indicates that this approach could also be attractive for application in other tissue engineering systems. Biotechnol. Bioeng. 2010. 105: 842–853. © 2009 Wiley Periodicals, Inc.     

On the importance of 3D,geometrically accurate,and subject-specific finite element analysis for evaluation of in-vivo soft tissue loads

Pressure ulcers are a type of local soft tissue injury due to sustained mechanical loading and remain a common issue in patient care. People with spinal cord injury (SCI) are especially at risk of pressure ulcers due to impaired mobility and sensory perception. The development of load improving support structures relies on realistic tissue load evaluation e.g. using finite element analysis (FEA). FEA requires realistic subject-specific mechanical properties and geometries. This study focuses on the effect of geometry. MRI is used for the creation of geometrically accurate models of the human buttock for three able-bodied volunteers and three volunteers with SCI. The effect of geometry on observed internal tissue deformations for each subject is studied by comparing FEA findings for equivalent loading conditions. The large variations found between subjects confirms the importance of subject-specific FEA.     

A functionally graded material model for the transmural stress distribution of the aortic valve leaflet

Heterogeneities in structure and stress within heart valve leaflets are of significant concern to their functional physiology, as they affect how the tissue constituents remodel in response to pathological and non-pathological (e.g. exercise, pregnancy) alterations in cardiac function. Indeed, valve interstitial cells (VICs) are known to synthesize and degrade leaflet extracellular matrix (ECM) components in a manner specific to their local micromechanical environment. Quantifying local variations in ECM structure and stress is thus necessary to understand homeostatic valve maintenance as well as to develop predictive models of disease progression and post-surgical outcomes. In the aortic valve (AV), transmural variations in stress have previously been investigated by modeling the leaflet as a composite of contiguous but mechanically distinct layers. Based on previous findings about the bonded nature of these layers (Buchanan and Sacks, BMMB, 2014), we developed a more generalized structural constitutive model by treating the leaflet as a functionally graded material (FGM), whose properties vary continuously over the thickness. We informed the FGM model using high-resolution morphological measurements, which demonstrated that the composition and fiber structure change gradually over the thickness of the AV leaflet. For validation, we fit the model against an extensive database of whole-leaflet and individual-layer mechanical responses. The FGM model predicted large stress variations both between and within the leaflet layers at end-diastole, with low-collagen regions bearing significant radial stress. These novel results suggest that the continually varying structure of the AV leaflet has an important purpose with regard to valve function and tissue homeostasis.     

Validation of a full body finite element model (THUMS) for running-type impacts to the lower extremity

The purpose of this study was to determine whether modifying an existing, highly biofidelic full body finite element model [total human model for safety (THUMS)] would produce valid amplitude and temporal shock wave characteristics as it travels proximally through the lower extremity. Modifying an existing model may be more feasible than developing a new model, in terms of cost, labour and expertise. The THUMS shoe was modified to more closely simulate the material properties of a heel pad. Relative errors in force and acceleration data from experimental human pendulum impacts and simulated THUMS impacts were 22% and 54%, respectively, across the time history studied. The THUMS peak acceleration was attenuated by 57.5% and took 19.7 ms to travel proximally along the lower extremity. Although refinements may be necessary to improve force and acceleration timing, the modified THUMS represented, to a certain extent, shock wave propagation and attenuation demonstrated by living humans under controlled impact conditions.     

A computational study of the EN 1078 impact test for bicycle helmets using a realistic subject-specific finite element head model

Abstract

In the present study, the free fall impact test in accordance with the EN1078 standard for certification of bicycle helmets is replicated using numerical simulations. The impact scenario is simulated using an experimentally validated, patient-specific head model equipped with and without a bicycle helmet. Head accelerations and intracranial biomechanical injury metrics during the impacts are recorded. It is demonstrated that wearing the bicycle helmet during the impact reduces biomechanical injury metrics, with the biggest reduction seen in the metric for skull fracture.     

A temperature-dependent developmental model for a nucleopolyhedrosis virus of the velvetbean caterpillar,Anticarsia gemmatalis (Lepidoptera: Noctuidae)

Developmental times and rates of nucleopolyhedrosis virus infection in third-instar velvetbean caterpillar, Anticarsia gemmatalis, larvae were studied in the laboratory at a variety of constant and variable temperatures. Developmental time was considered to be time from inoculation until death. Viral infection exhibited a temperature optimum of ca. 30°C and was inhibited at 10 and 40°C. Mean developmental time of the virus ranged from 18.1 days (15°C) to 5.5 days (30°C). Means and standard deviations of viral developmental rates (= development time^?1) were used as inputs into a previously derived absolute reaction rate model designed to generate a set of kinetic constants usable in predicting developmental times. Actual distributions of viral cohort developmental times were compared to distributions generated by the model. Reasonable agreement between predicted and actual distributions was found at three of four temperatures tested.     

Phase locking,period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias

A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincaré or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincaré map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincaré map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 22 and 42 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.     

K317, R319, and E320 within the proximal C-terminus of the bradykinin B2 receptor form a motif important for phospholipase C and phospholipase A2 but not connective tissue growth factor related signaling

We showed previously that large domain exchanges between the bradykinin B2 (BKB2) and angiotensin II type 1a (AT1a) receptors can result in functional hybrids. However, when we proceeded to exchange the entire bradykinin B2 receptor (BKB2R) C-terminal tail with the AT1aR C-terminus, the hybrid, while continuing to bind BK and be endocytosed as wild type (WT) BKB2R, lost much of its ability to activate phosphatidylinositol (PI) turnover or the release of arachidonic acid (ARA). In this study, we constructed chimeric receptors within the proximal C-terminus between the BKB2R and AT1aR or bradykinin B1 receptor (BKB1R). The mutant and WT receptor cDNAs were stably transfected into Rat-1 cells. Also, point mutations were generated to evaluate the role of the individual residues within this region. These chimeric studies revealed that the proximal portion of the BKB2R C-tail is crucial for G protein-linked BKB2R functions. This region could not be swapped with the AT1aR to obtain a BK activated PI turnover or ARA release. Further studies demonstrated that the distal portion (325-330) of this region is exchangeable; however, the middle portion (317-324) is not. Small motif exchanges within this section identified the KSR and EVY motifs as crucial for G(alphaq), G(alphai) related signaling of the BKB2R. Point mutations then showed that the charged amino acids K317, R319, and E320 are the residues critical for linking to PI turnover and ARA release. However, these proximal chimeras showed normal receptor uptake. Interestingly, while apparently not activating G protein-linked signaling, the proximal tail AT1aR exchange mutant and the entire C-terminus exchange hybrid continued to cause a substantial bradykinin effected increase in connective tissue growth factor (CTGF) mRNA level, as WT BKB2R.