Phm Peter Bovendeerd

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.

Steady entry flow in a curved pipe

Laser-Doppler velocity measurements were performed on the entry flow in a 90° bend of circular cross-section with a curvature ratio a / R = 1/6. The steady entry velocity profile was parabolic, having a Reynolds number Re = 700, with a corresponding Dean number κ = 286. Both axial and secondary velocities were measured, enabling a detailed description of the complete flow field. The secondary flow at the entrance of the bend was measured to be directed completely towards the inner bend. Significant disturbance of the axial velocity field was not measured until a downstream distance ( aR ) ½ . Maximum secondary velocities were measured at 1.7 ( aR ) ½ downstream from the inlet. The development of the axial flow field can be quite well explained from the secondary velocity field.

A finite element approach for skeletal muscle using a distributed moment model of contraction

Abstract The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.

A model for arterial adaptation combining microstructural collagen remodeling and 3D tissue growth

Long-term adaptation of soft tissues is realized through growth and remodeling (G&R). Mathematical models are powerful tools in testing hypotheses on G&R and supporting the design and interpretation of experiments. Most theoretical G&R studies concentrate on description of either growth or remodeling. Our model combines concepts of remodeling of collagen recruitment stretch and orientation suggested by other authors with a novel model of general 3D growth. We translate a growth-induced volume change into a change in shape due to the interaction of the growing tissue with its environment. Our G&R model is implemented in a finite element package in 3D, but applied to two rotationally symmetric cases, i.e., the adaptation towards the homeostatic state of the human aorta and the development of a fusiform aneurysm. Starting from a guessed non-homeostatic state, the model is able to reproduce a homeostatic state of an artery with realistic parameters. We investigate the sensitivity of this state to settings of initial parameters. In addition, we simulate G&R of a fusiform aneurysm, initiated by a localized degradation of the matrix of the healthy artery. The aneurysm stabilizes in size soon after the degradation stops.

Modeling cardiac electromechanics and mechanoelectrical coupling in dyssynchronous and failing hearts : insight from adaptive computer models

Computer models have become more and more a research tool to obtain mechanistic insight in the effects of dyssynchrony and heart failure. Increasing computational power in combination with increasing amounts of experimental and clinical data enables the development of mathematical models that describe electrical and mechanical behavior of the heart. By combining models based on data at the molecular and cellular level with models that describe organ function, so-called multi-scale models are created that describe heart function at different length and time scales. In this review, we describe basic modules that can be identified in multi-scale models of cardiac electromechanics. These modules simulate ionic membrane currents, calcium handling, excitation–contraction coupling, action potential propagation, and cardiac mechanics and hemodynamics. In addition, we discuss adaptive modeling approaches that aim to address long-term effects of diseases and therapy on growth, changes in fiber orientation, ionic membrane currents, and calcium handling. Finally, we discuss the first developments in patient-specific modeling. While current models still have shortcomings, well-chosen applications show promising results on some ultimate goals: understanding mechanisms of dyssynchronous heart failure and tuning pacing strategy to a particular patient, even before starting the therapy.

Insight into variable fetal heart rate decelerations from a mathematical model

During labor and delivery, variable decelerations in the fetal heart rate (FHR) are commonly seen on the cardiotocogram (CTG) that is used to monitor fetal welfare. These decelerations are often induced by umbilical cord compression from uterine contractions. Via changes in oxygenation and blood pressure, umbilical cord compression activates the chemo- and baroreceptor reflex, and thus affects FHR. Since the relation between the CTG and fetal oxygenation is complex, assessment of fetal welfare from the CTG is difficult. We investigated umbilical cord compression-induced variable decelerations with a mathematical model. For this purpose, we extended our model for decelerations originating from caput compression and reduced uterine blood flow with the possibility to induce umbilical venous, arterial and total cord occlusion. Model response during total occlusion is evaluated for varying contractions (duration and amplitude) and sensitivity of the umbilical resistance to the uterine pressure. A clinical scenario is used to simulate a labor CTG with variable decelerations. Simulation results show that fetal mean arterial pressure increases during umbilical cord occlusion, while fetal oxygenation drops. There is a clear relation between these signals and the resulting FHR. The extent of umbilical compression and thus FHR deceleration is positively related to increased contraction duration and amplitude, and increased sensitivity of the umbilical resistance to uterine pressure. No relation is found between contraction interval and FHR response, which can probably be ascribed to the lack of catecholamines in the model. The simulation model provides insight into the complex relation between uterine pressure, umbilical cord compression, fetal oxygenation, blood pressure and heart rate. The model can be used for individual learning, and incorporated in a simulation mannequin, be used to enhance obstetric team training.

New insights from a computational model on the relation between pacing site and CRT response

AIMS Cardiac resynchronization therapy (CRT) produces clinical benefits in chronic heart failure patients with left bundle-branch block (LBBB). The position of the pacing site on the left ventricle (LV) is considered an important determinant of CRT response, but the mechanism how the LV pacing site determines CRT response is not completely understood. The objective of this study is to investigate the relation between LV pacing site during biventricular (BiV) pacing and cardiac function. METHODS AND RESULTS We used a finite element model of BiV electromechanics. Cardiac function, assessed as LV dp/dtmax and stroke work, was evaluated during normal electrical activation, typical LBBB, fascicular blocks and BiV pacing with different LV pacing sites. The model replicated clinical observations such as increase of LV dp/dtmax and stroke work, and the disappearance of a septal flash during BiV pacing. The largest hemodynamic response was achieved when BiV pacing led to best resynchronization of LV electrical activation but this did not coincide with reduction in total BiV activation time (∼ QRS duration). Maximum response was achieved when pacing the mid-basal lateral wall and this was close to the latest activated region during intrinsic activation in the typical LBBB, but not in the fascicular block simulations. CONCLUSIONS In these model simulations, the best cardiac function was obtained when pacing the mid-basal LV lateral wall, because of fastest recruitment of LV activation. This study illustrates how computer modeling can shed new light on optimizing pacing therapies for CRT. The results from this study may help to design new clinical studies to further investigate the importance of the pacing site for CRT response.

Effects of activation pattern and active stress development on myocardial shear in a model with adaptive myofiber reorientation

It has been hypothesized that myofiber orientation adapts to achieve a preferred mechanical loading state in the myocardial tissue. Earlier studies tested this hypothesis in a combined model of left ventricular (LV) mechanics and remodeling of myofiber orientation in response to fiber cross-fiber shear, assuming synchronous timing of activation and uniaxial active stress development. Differences between computed and measured patterns of circumferential-radial shear strain E(cr) were assumed to be caused by limitations in either the LV mechanics model or the myofiber reorientation model. Therefore, we extended the LV mechanics model with a physiological transmural and longitudinal gradient in activation pattern and with triaxial active stress development. We investigated the effects on myofiber reorientation, LV function, and deformation. The effect on the developed pattern of the transverse fiber angle α(t,0) and the effect on global pump function were minor. Triaxial active stress development decreased amplitudes of E(cr) towards values within the experimental range and resulted in a similar base-to-apex gradient during ejection in model computed and measured E(cr). The physiological pattern of mechanical activation resulted in better agreement between computed and measured strain in myofiber direction, especially during isovolumic contraction phase and first half of ejection. In addition, remodeling was favorable for LV pump and myofiber function. In conclusion, the outcome of the combined model of LV mechanics and remodeling of myofiber orientation is found to become more physiologic by extending the mechanics model with triaxial active stress development and physiological activation pattern.

A mathematical model to simulate the cardiotocogram during labor. Part A: Model setup and simulation of late decelerations

The cardiotocogram (CTG) is commonly used to monitor fetal well-being during labor and delivery. It shows the input (uterine contractions) and output (fetal heart rate, FHR) of a complex chain of events including hemodynamics, oxygenation and regulation. Previously we developed a mathematical model to obtain better understanding of the relation between CTG signals and vital, but clinically unavailable signals such as fetal blood pressure and oxygenation. The aim of this study is to improve this model by reducing complexity of submodels where parameter estimation is complicated (e.g. regulation) or where less detailed model output is sufficient (e.g. cardiac function), and by using a more realistic physical basis for the description of other submodels (e.g. vessel compression). Evaluation of the new model is performed by simulating the effect of uterine contractions on FHR as initiated by reduction of uterine blood flow, mediated by changes in oxygen and blood pressure, and effected by the chemoreflex and baroreflex. Furthermore the ability of the model to simulate uterine artery occlusion experiments in sheep is investigated. With the new model a more realistic FHR decrease is obtained during contraction-induced reduction of uterine blood flow, while the reduced complexity and improved physical basis facilitate interpretation of model results and thereby make the model more suitable for use as a research and educational tool.

A mathematical model to simulate the cardiotocogram during labor. Part B:parameter estimation and simulation of variable decelerations

During labor and delivery the cardiotocogram (CTG), the combined registration of fetal heart rate (FHR) and uterine contractions, is used to monitor fetal well-being. In part A of our study we introduced a new mathematical computer model for CTG simulation in order to gain insight into the complex relation between these signals. By reducing model complexity and by using physically more realistic descriptions, this model was improved with respect to our previous model. Aim of part B of this study is to gain insight into the cascade of events from uterine contractions causing combined uterine flow reduction and umbilical cord compression, resulting in blood and oxygen pressure variations, which lead to changes in FHR via the baro- and chemoreflex. In addition, we extensively describe and discuss the estimation of model parameter values. Simulation results are in good agreement with sheep data and show the ability of the model to describe variable decelerations. Despite reduced model complexity, parameter estimation still remains difficult due to limited clinical data.