A mechanical analysis of the closed Hancock heart valve prosthesis

In order to obtain mechanical specifications for the design of an artificial leaflet valve prosthesis, a geometrically non-linear numerical model is developed of a closed Hancock leaflet valve prosthesis. In this model, the fibre reinforcement of the leaflet and the viscoelastic properties of frame and leaflets are incorporated. The calculations are primarily restricted to 1/6 part of the valve and a time varying pressure load is applied. The calculations are verified experimentally by measuring the commissure displacements and leaflet centre displacement of a Hancock valve. The numerically obtained commissure displacements are found to be linearly dependent on the pressure load, and the slope of the curves is hardly dependent on loading type and loading velocity. Experimentally a difference is found between the three commissure displacements, which is also predicted numerically using a simplified asymmetric total valve model. Besides, experimentally a clear dependency of commissure displacements on frame size is found. For the leaflet centre displacement, a qualitative agreement exists between numerical prediction and experimental result, although the numerical predicted values are systematically higher. The numerically obtained stress distributions revealed that the maximum von Mises intensity in the membranes occurs in the vicinity of the commissure in the free leaflet area (0.2 N mm-2). Wrinkling of the membranes may occur in the coaptation area near the leaflet suspension. The maximum fibre stress is found near the aortic ring in the fibres which form the boundaries of the coaptation area (0.64 N mm-2). These locations seem to correlate with some common regions of tissue valve failure.     

Aortic valve leaflet mechanical properties facilitate diastolic valve function

This work was concerned with the numerical simulation of the behaviour of aortic valves whose material can be modelled as non-linear elastic anisotropic. Linear elastic models for the valve leaflets with parameters used in previous studies were compared with hyperelastic models, incorporating leaflet anisotropy with pronounced stiffness in the circumferential direction through a transverse isotropic model. The parameters for the hyperelastic models were obtained from fits to results of orthogonal uniaxial tensile tests on porcine aortic valve leaflets. The computational results indicated the significant impact of transverse isotropy and hyperelastic effects on leaflet mechanics; in particular, increased coaptation with peak values of stress and strain in the elastic limit. The alignment of maximum principal stresses in all models follows approximately the coarse collagen fibre distribution found in aortic valve leaflets. The non-linear elastic leaflets also demonstrated more evenly distributed stress and strain which appears relevant to long-term scaffold stability and mechanotransduction.     

Stability and parameter dependency analysis of a facilitation tectal column (FTC) model

Mathematical models and computer simulations have been widely used to study the spatio-temporal characteristics of the processing of information carried out by the central nervous system. When trying to show whether or not a neural model accounts for the phenomena under study, if the number of parameters whose values need to be calculated becomes large, then computer simulations alone become very inefficient to define such values. Here, we developed stability and parameter dependency analyses of the mathematical representation of a single facilitation tectal column (FTC) model, to show how by using techniques from non-linear systems theory we can define the ranges of parameter values under which the model would explain the required performance of the neural net model. The benefits of these analyses can be grouped in two parts: first, the advantage of using non-linear systems techniques to analyze, analytically, the dynamics of neural net models; and second, we get a deeper understanding of why the hypotheses embedded in the models yield the appropriate behaviors and what are the critical situations (parametric combinations) under which these behaviors are displayed.     

Computed modeling of humeral mid-shaft fracture treated by bundle nailing

Elastic bundle nailing is a method for simple humeral mid-shaft fracture osteosynthesis. The aim of our subsequent numerical simulations was to find out torsional and bending stiffness of an elastic bundle nailed humerus. Parametrical 3D numerical model was developed. The diameter of nails was the varying parameter of 1.8, 2.5, 3 and 4 mm. From our results can be seen that the bending stiffness in bundle nailing technique does not depend on nail diameter. On the contrary the torsional stiffness does highly depend on nail diameter. The dependency of the maximal stress on a nail diameter during bending and torsion of the humerus is non-linear. It can be seen that the higher diameter is used the higher stress occurs. Achieved results allow us for the recommendation of optimal nail diameter for this method, which lies between 2 and 3 mm.     

Non-linear rotation-free shell finite-element models for aortic heart valves

Hyperelastic material models have been incorporated in the rotation-free, large deformation, shell finite element (FE) formulation of (Stolarski et al., 2013) and applied to dynamic simulations of aortic heart valve. Two models used in the past in analysis of such problem i.e. the Saint-Venant and May-Newmann–Yin (MNY) material models have been considered and compared. Uniaxial tests for those constitutive equations were performed to verify the formulation and implementation of the models. The issue of leaflets interactions during the closing of the heart valve at the end of systole is considered. The critical role of using non-linear anisotropic model for proper dynamic response of the heart valve especially during the closing phase is demonstrated quantitatively. This work contributes an efficient FE framework for simulating biological tissues and paves the way for high-fidelity flow structure interaction simulations of native and bioprosthetic aortic heart valves.     

Dose-response time modelling for highly pathogenic avian influenza A (H5N1) virus infection

Aims: To develop time‐dependent dose–response models for highly pathogenic avian influenza A (HPAI) of the H5N1 subtype virus. Methods and Results: A total of four candidate time‐dependent dose–response models were fitted to four survival data sets for animals (mice or ferrets) exposed to graded doses of HPAI H5N1 virus using the maximum‐likelihood estimation. A beta‐Poisson dose–response model with the N₅₀ parameter modified by an exponential‐inverse‐power time dependency or an exponential dose–response model with the k parameter modified by an exponential‐inverse time dependency provided a statistically adequate fit to the observed survival data. Conclusions: We have successfully developed the time‐dependent dose–response models to describe the mortality of animals exposed to an HPAI H5N1 virus. The developed model describes the mortality over time and represents observed experimental responses accurately. Significance and Impact of the Study: This is the first study describing time‐dependent dose–response models for HPAI H5N1 virus. The developed models will be a useful tool for estimating the mortality of HPAI H5N1 virus, which may depend on time postexposure, for the preparation of a future influenza pandemic caused by this lethal virus.     

A Mathematical Model for Leaf Photosynthesis of Mulberry

A mathematical model of leaf photosynthesis has been established. In this model, the processes of photosynthesis are divided into two parts, ie., the carboxylation process driven by light which is dependent on temperature and CO2 concentration, and the diffusion of CO2 from atmosphere to the carboxylation site. Finatly, CO2 uptake by the leaf is understood as dependent on 1), the CO2 response curve of the leaf mesophyll and 2). the CO2 partial pressure in the intercellular space in leaf. The COs response curve of the leaf photosynthesis is described mathematically in terms of carboxylation efficiency (Ca) or its initial slope and the photosynthetic capacity (Pm) or the CO2-saturated uptake rate of CO2 uptake, and dark respiration (Rd). The dependency of photosynthesis on leaf temperature and incident light intensity is incorporated into variations of those parameters which establish an appropriate response to internal CO2 pressure for particular light and temperature conditions prevailing at any time. Secondly the interactiion of stomata with photosynthesis is represented as an empirical relation between stomatal conductance and a combined environmental physiological index, APn·Hx/CaThe parameters used in the modelwere estimated with Marquardt-Newton method for non-linear function. Field measurements of mulberry leaf photosynthesis provided a data set for model testing. The resuks show that the simulated values of the model agree well with observed data. The model was used to analyse the response surface of leaf conductance and photosynthesis to environmental factors—Applications and limitations of the model are discussed     

Modelling amperometric enzyme electrode with substrate cyclic conversion

A mathematical model of amperometric enzyme electrodes in which chemical amplification by cyclic substrate conversion takes place in a single enzyme membrane has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis-Menten kinetic of the enzymatic reaction. The digital simulation was carried out using the finite difference technique. The influence of the substrate concentration, the maximal enzymatic rate as well as the membrane thickness on the biosensor response was investigated. The numerical experiments demonstrate significant (up to dozens of times) gain in biosensor sensitivity at low concentrations of substrate when the biosensor response is under diffusion control.     

Theory of antimicrobial combinations: biocide mixtures - synergy or addition?

AIMS: To demonstrate the effect that non-linear dose responses have on the appearance of synergy in mixtures of antimicrobials. METHODS AND RESULTS: A mathematical model, which allows the prediction of the efficacy of mixtures of antimicrobials with non-linear dose responses, was produced. The efficacy of antimicrobial mixtures that would be classified as synergistic by time-kill methodology was shown to be a natural consequence of combining antimicrobials with non-linear dose responses. CONCLUSIONS: The effectiveness of admixtures of biocides and other antimicrobials with non-linear dose responses can be predicted. If the dose response (or dilution coefficient) of any biocidal component, in a mixture, is other than one, then the time-kill methodology used to ascertain the existence of synergy in antimicrobial combinations is flawed. SIGNIFICANCE AND IMPACT OF THE STUDY: The kinetic model developed allows the prediction of the efficacy of antimicrobial combinations. Combinations of known antimicrobials, which reduce the time taken to achieve a specified level of microbial inactivation, can be easily assessed once the kinetic profile of each component has been obtained. Most patented cases of antimicrobial synergy have not taken into account the possible effect of non-linear dose responses of the component materials. That much of the earlier literature can now be predicted, suggests that future cases will require more thorough proof of the alleged synergy.     

Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.     

Dynamic finite element implementation of nonlinear,anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.     

Dynamic response of the brain with vasculature: a three-dimensional computational study

To date, the influence of the vasculature on the dynamic response of the brain has not been studied with a complete three-dimensional (3D) finite element head model. In this study, short duration rotational (10,000 rad/s(2) with a duration of 5 ms) and translational (100G with a duration of 5 ms) acceleration impulses were applied to the 3D finite element models to study the dynamic response of the brain. The hypothesis of this study was that due to the convoluted organization and non-linear material properties of cerebral vasculature, the difference in maximum principle strain between models with and without vasculature should be minimal. The effects of non-linear material properties and the convoluted structure of the vasculature were examined by comparing the results from the 3D finite element models. The peak average strain reduction in a model with non-linear elastic vasculature and a model with linear elastic vasculature compared to a model without vasculature was 2% and 5%, respectively, indicating that the influence of the vasculature on the dynamic response of the brain is minimal.     

Membrane damage to bacteria caused by single and combined biocides

AIMS: To examine the effect on the leakage of low molecular weight cytoplasmic constituents from Staphylococcus aureus using phenolics singly and in combination, and to see if the observations could be modelled using a non-linear dose response. METHODS AND RESULTS: The rate of potassium, phosphate and adenosine triphosphate leakage was examined in the presence of chlorocresol and m-cresol. Individually, leakage was observed only at long contact times or high concentrations. Combined at these ineffective concentrations, the cytoplasmic pool of all constituents studied was released within minutes. Both chlorocresol and m-cresol were shown to have non-linear dose responses. A rate model for the combinations, which takes account of these non-linear responses, accurately predicted the observations. CONCLUSIONS: Antimicrobials, which when used alone exhibit a non-linear dose response, will also give a non-linear dose response in combination. The simple linear-additive model ignores the concept of the dilution coefficient and will always describe the phenomenon of synergy for combinations where one or more of the components has a dilution coefficient greater than one. This has been borne out by examination of the purported prime lesion of chlorocresol and m-cresol, alone and in combination. SIGNIFICANCE AND IMPACT OF THE STUDY: Studies aimed at producing synergistic mixtures of antimicrobials, which ignore the non-linear additive effect, may waste valuable research effort looking for a physiological explanation for an apparent synergy, where none, in-fact, exists. Patents granted on the basis of analyses using the linear-additive model for combinations of compounds with non-linear dose responses may no longer be supportable.     

Incorporating time postinoculation into a dose–response model of Yersinia pestis in mice

Aims:  To develop a time-dependent dose–response model for describing the survival of animals exposed to Yersinia pestis.
Methods and Results:  Candidate time-dependent dose–response models were fitted to a survival data set for mice intraperitoneally exposed to graded doses of Y. pestis using the maximum likelihood estimation method. An exponential dose–response model with the model parameter modified by an inverse-power dependency of time postinoculation provided a statistically adequate fit to the experimental survival data. This modified model was verified by comparison with prior studies.
Conclusions:  The incorporated time dependency quantifies the expected temporal effect of in vivo bacteria growth in the dose–response relationship. The modified model describes the development of animal infectious response over time and represents observed responses accurately.
Significance and Impact of the Study:  This is the first study to incorporate time in a dose–response model for Y. pestis infection. The outcome may be used for the improved understanding of in vivo bacterial dynamics, improved postexposure decision making or as a component to better assist epidemiological investigations.     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

Systolic fluid–structure interaction model of the congenitally bicuspid aortic valve: assessment of modelling requirements

A transient fluid–structure interaction (FSI) model of a congenitally bicuspid aortic valve has been developed which allows simultaneous calculation of fluid flow and structural deformation. The valve is modelled during the systolic phase (the stage when blood pressure is elevated within the heart to pump blood to the body). The geometry was simplified to represent the bicuspid aortic valve in two dimensions. A congenital bicuspid valve is compared within the aortic root only and within the aortic arch. Symmetric and asymmetric cusps were simulated, along with differences in mechanical properties. A moving arbitrary Lagrange–Euler mesh was used to allow FSI. The FSI model requires blood flow to induce valve opening and induced strains in the region of 10%. It was determined that bicuspid aortic valve simulations required the inclusion of the ascending aorta and aortic arch. The flow patterns developed were sensitive to cusp asymmetry and differences in mechanical properties. Stiffening of the valve amplified peak velocities, and recirculation which developed in the ascending aorta. Model predictions demonstrate the need to take into account the category, including any existing cusp asymmetry, of a congenital bicuspid aortic valve when simulating its fluid flow and mechanics.     

Stoichiometric modeling of Clostridium acetobutylicum fermentations with non-linear constraints.

A stoichiometric model of Clostridium acetobutylicum and related strains has been previously derived. The stoichiometric matrix of the model contains a singularity which has prevented the calculation of a unique set of fluxes which describe the primary metabolic activity. To resolve the singularity, we have developed a non-linear constraint relating the acetate and butyrate uptake fluxes. Subsequently, we developed a software package utilizing a model independent heuristic global optimization approach to solve the resultant non-linear problem. We have validated the use of the non-linear constraint by correlating calculated butyrate production pathway flux profiles with measured intracellular pH profiles. Finally, we examined a controlled batch fermentation to determine that the acid formation pathways play critical roles throughout solventogenesis. The broader usefulness of reformulating the stoichiometric model as a constrained minimization problem is discussed.