Transient finite element modeling of functional electrical stimulation

Transcutaneous functional electrical stimulation is commonly used for strengthening muscle. However, transient effects during stimulation are not yet well explored. The effect of an amplitude change of the stimulation can be described by static model, but there is no differency for different pulse duration. The aim of this study is to present the finite element (FE) model of a transient electrical stimulation on the forearm. Discrete FE equations were derived by using a standard Galerkin procedure. Different tissue conductive and dielectric properties are fitted using least square method and trial and error analysis from experimental measurement. This study showed that FE modeling of electrical stimulation can give the spatial-temporal distribution of applied current in the forearm. Three different cases were modeled with the same geometry but with different input of the current pulse, in order to fit the tissue properties by using transient FE analysis. All three cases were compared with experimental measurements of intramuscular voltage on one volunteer.     

A nonlinear finite element model of cartilage growth

The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MATLAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.     

The feasibility of computational modelling technique to detect the bladder cancer

A numerical technique, finite element analysis (FEA) was used to model the electrical properties, the bio impedance of the bladder tissue in order to predict the bladder cancer. This model results showed that the normal bladder tissue have significantly higher impedance than the malignant tissue that was in opposite with the impedance measurements or the experimental results. Therefore, this difference can be explained using the effects of inflammation, oedema on the urothelium and the property of the bladder as a distensible organ. Furthermore, the different current distributions inside the bladder tissue (in histological layers) in normal and malignant cases and finally different applied pressures over the bladder tissue can cause different impedances for the bladder tissue. Finally, it is believed that further studies have to be carried out to characterise the human bladder tissue using the electrical impedance measurement and modelling techniques.     

Analysis of oxygen delivery and uptake relationships in the Krogh tissue model

Normally, tissue O2 uptake (VO2) is set by metabolic activity rather than O2 delivery (QO2 = blood flow X arterial O2 content). However, when QO2 is reduced below a critical level, VO2 becomes limited by O2 supply. Experiments have shown that a similar critical QO2 exists, regardless of whether O2 supply is reduced by progressive anemia, hypoxemia, or reduction in blood flow. This appears inconsistent with the hypothesis that O2 supply limitation must occur by diffusion limitation, since very different mixed venous PO2 values have been seen at the critical point with hypoxic vs. anemic hypoxia. The present study sought to begin clarifying this paradox by studying the theoretical relationship between tissue O2 supply and uptake in the Krogh tissue cylinder model. Steady-state O2 uptake was computed as O2 delivery to tissue representative of whole body was gradually lowered by anemic, hypoxic, or stagnant hypoxia. As diffusion began to limit uptake, the fall in VO2 was computed numerically, yielding a relationship between QO2 and VO2 in both supply-independent and O2 supply-dependent regions. This analysis predicted a similar biphasic relationship between QO2 and VO2 and a linear fall in VO2 at O2 deliveries below a critical point for all three forms of hypoxia, as long as intercapillary distances were less than or equal to 80 microns. However, the analysis also predicted that O2 extraction at the critical point should exceed 90%, whereas real tissues typically extract only 65-75% at that point. When intercapillary distances were larger than approximately 80 microns, critical O2 extraction ratios in the range of 65-75% could be predicted, but the critical point became highly sensitive to the type of hypoxia imposed, contrary to experimental findings. Predicted gas exchange in accord with real data could only be simulated when a postulated 30% functional peripheral O2 shunt (arterial admixture) was combined with a tissue composed of Krogh cylinders with intercapillary distances of less than or equal to 80 microns. The unrealistic efficacy of tissue O2 extraction predicted by the Krogh model (in the absence of postulated shunt) may be a consequence of the assumed homogeneity of tissues, because real tissues exhibit many forms of heterogeneity among capillary units. Alternatively, the failure of the original Krogh model to fully predict tissue O2 supply dependency may arise from basic limitations in the assumptions of that model.     

Spatial distribution of cardiac transmembrane potentials around an extracellular electrode: dependence on fiber orientation.

Recent theoretical models of cardiac electrical stimulation or defibrillation predict a complex spatial pattern of transmembrane potential (Vm) around a stimulating electrode, resulting from the formation of virtual electrodes of reversed polarity. The pattern of membrane polarization has been attributed to the anisotropic structure of the tissue. To verify such model predictions experimentally, an optical technique using a fluorescent voltage-sensitive dye was used to map the spatial distribution of Vm around a 150-microns-radius extracellular unipolar electrode. An S1-S2 stimulation protocol was used, and vm was measured during an S2 pulse having an intensity equal to 10x the cathodal diastolic threshold of excitation. The recordings were obtained on the endocardial surface of bullfrog atrium in directions parallel and perpendicular to the cardiac fibers. In the longitudinal fiber direction, the membrane depolarized for cathodal pulses (and hyperpolarized for anodal pulses) but only in a region within 445 +/- 112 microns (and 616 +/- 78 microns for anodal pulses) from the center of the electrode (n = 9). Outside this region, vm reversed polarity and reached a local maximum at 922 +/- 136 microns (and 988 +/- 117 microns for anodal pulses) (n = 9). Beyond this point vm decayed to zero over a distance of 1.5-2 mm. In the transverse fiber direction, the membrane depolarized for cathodal pulses (and hyperpolarized for anodal pulses) at all distances from the electrode. The amplitude of the response decreased with distance from the electrode with an exponential decay constant of 343 +/- 110 microns for cathodal pulses and 253 +/- 91 microns for anodal pulses (n = 7). The results were qualitatively similar in both fiber directions when the atrium was bathed in a solution containing ionic channel blockers. A two-dimensional computer model was formulated for the case of highly anisotropic cardiac tissue and qualitatively accounts for nearly all the observed spatial and temporal behavior of vm in the two fiber directions. The relationships between vm and both the "activating function" and extracellular potential gradient are discussed.     

A mechano-regulatory bone-healing model incorporating cell-phenotype specific activity

Phenomenological computational models of tissue regeneration and bone healing have been only partially successful in predicting experimental observations. This may be a result of simplistic modeling of cellular activity. Furthermore, phenomenological models are limited when considering the effects of combined physical and biological interventions. In this study, a new model of cell and tissue differentiation, using a more mechanistic approach, is presented and applied to fracture repair. The model directly couples cellular mechanisms to mechanical stimulation during bone healing and is based on the belief that the cells act as transducers during tissue regeneration. In the model, the cells within the matrix proliferate, differentiate, migrate, and produce extracellular matrix, all at cell-phenotype specific rates, based on the mechanical stimulation they experience. The model is assembled from coupled partial differentiation equations, which are solved using a newly developed finite element formulation. The evolution of four cell types, i.e. mesenchymal stem cells, fibroblasts, chondrocytes and osteoblasts, and the production of extracellular matrices of fibrous tissue, cartilage and bone are calculated. The material properties of the tissues are iteratively updated based on actual amounts of extracellular matrix in material elements at progressive time points. A two-dimensional finite element model of a long bone osteotomy was used to evaluate the model's potential. The additional value of the presented model and the importance of including cell-phenotype specific activities when modeling tissue differentiation and bone healing, were demonstrated by comparing the predictions with phenomenological models. The model's capacity was established by showing that it can correctly predict several aspects of bone healing, including cell and tissue distributions during normal fracture healing. Furthermore, it was able to predict experimentally established alterations due to excessive mechanical stimulation, periosteal stripping and impaired effects of cartilage remodeling.     

Do Cost Functions for Tracking Error Generalize across Tasks with Different Noise Levels?

Control of human-machine interfaces are well modeled by computational control models, which take into account the behavioral decisions people make in estimating task dynamics and state for a given control law. This control law is optimized according to a cost function, which for the sake of mathematical tractability is typically represented as a series of quadratic terms. Recent studies have found that people actually use cost functions for reaching tasks that are slightly different than a quadratic function, but it is unclear which of several cost functions best explain human behavior and if these cost functions generalize across tasks of similar nature but different scale. In this study, we used an inverse-decision-theory technique to reconstruct the cost function from empirical data collected on 24 able-bodied subjects controlling a myoelectric interface. Compared with previous studies, this experimental paradigm involved a different control source (myoelectric control, which has inherently large multiplicative noise), a different control interface (control signal was mapped to cursor velocity), and a different task (the tracking position dynamically moved on the screen throughout each trial). Several cost functions, including a linear-quadratic; an inverted Gaussian, and a power function, accurately described the behavior of subjects throughout this experiment better than a quadratic cost function or other explored candidate cost functions (p<0.05). Importantly, despite the differences in the experimental paradigm and a substantially larger scale of error, we found only one candidate cost function whose parameter was consistent with the previous studies: a power function (cost ∝ error^α) with a parameter value of α = 1.69 (1.53–1.78 interquartile range). This result suggests that a power-function is a representative function of user’s error cost over a range of noise amplitudes for pointing and tracking tasks.     

Synthesis of voltage-sensitive optical signals: application to panoramic optical mapping

Fluorescent photon scattering is known to distort optical recordings of cardiac transmembrane potentials; however, this process is not well quantified, hampering interpretation of experimental data. This study presents a novel model, which accurately synthesizes fluorescent recordings over the irregular geometry of the rabbit ventricles. Using the model, the study aims to provide quantification of fluorescent signal distortion for different optical characteristics of the preparation and of the surrounding medium. A bi-domain representation of electrical activity is combined with finite element solutions to the photon diffusion equation simulating both the excitation and emission processes, along with physically realistic boundary conditions at the epicardium, which allow simulation of different experimental setups. We demonstrate that distortion in the optical signal as a result of fluorescent photon scattering is truly a three-dimensional phenomenon and depends critically upon the geometry of the preparation, the scattering properties of the tissue, the direction of wavefront propagation, and the specifics of the experimental setup. Importantly, we show that in an anatomically accurate model of ventricular geometry and fiber orientation, the morphology of the optical signal does not provide reliable information regarding the intramural direction of wavefront propagation. These findings underscore the potential of the new model in interpreting experimental data.     

Modelling and simulation of porcine liver tissue indentation using finite element method and uniaxial stress–strain data

We hypothesize that both compression and elongation stress–strain data should be considered for modeling and simulation of soft tissue indentation. Uniaxial stress–strain data were obtained from in vitro loading experiments of porcine liver tissue. An axisymmetric finite element model was used to simulate liver tissue indentation with tissue material represented by hyperelastic models. The material parameters were derived from uniaxial stress–strain data of compressions, elongations, and combined compression and elongation of porcine liver samples. in vitro indentation tests were used to validate the finite element simulation. Stress–strain data from the simulation with material parameters derived from the combined compression and elongation data match the experimental data best. This is due to its better ability in modeling 3D deformation since the behavior of biological soft tissue under indentation is affected by both its compressive and tensile characteristics. The combined logarithmic and polynomial model is somewhat better than the 5-constant Mooney–Rivlin model as the constitutive model for this indentation simulation.     

Cytokine mediation of experimental heart failure-induced anhedonia

Immune system dysfunction is hypothesized to influence several disease states, including cardiovascular disease and psychological depression. The comorbidity of depression and coronary artery disease may be influenced by immune system-brain interactions involving proinflammatory cytokines. The present studies evaluated an index of depression in a rodent model of heart failure by measuring responses to rewarding electrical brain stimulation, which provides an experimental procedure to operationally define anhedonia in rats. Heart failure led to a rightward shift in the current-response relationship in the brain stimulation paradigm, indicative of reduced rewarding properties of the brain stimulation (i.e., anhedonia). Acute treatment with a tumor necrosis factor antagonist, etanercept, reduced circulating tumor necrosis factor- levels in rats with heart failure and restored responding for electrical brain stimulation. The current findings have implications for the study of pathophysiological mechanisms underlying the association of cardiovascular disease and depression.     

Current injection into a two-dimensional anisotropic bidomain.

A two-dimensional sheet of anisotropic cardiac tissue is represented with the bidomain model, and the finite element method is used to solve the bidomain equations. When the anisotropy ratios of the intracellular and extracellular spaces are not equal, the injection of current into the tissue induces a transmembrane potential that has a complicated spatial dependence, including adjacent regions of depolarized and hyperpolarized tissue. This behavior may have important implications for the electrical stimulation of cardiac tissue and for defibrillation.     

Finite Element Analysis of Hepatic Radiofrequency Ablation Probes using Temperature-Dependent Electrical Conductivity

Background

Few finite element models (FEM) have been developed to describe the electric field, specific absorption rate (SAR), and the temperature distribution surrounding hepatic radiofrequency ablation probes. To date, a coupled finite element model that accounts for the temperature-dependent electrical conductivity changes has not been developed for ablation type devices. While it is widely acknowledged that accounting for temperature dependent phenomena may affect the outcome of these models, the effect has not been assessed.

Methods

The results of four finite element models are compared: constant electrical conductivity without tissue perfusion, temperature-dependent conductivity without tissue perfusion, constant electrical conductivity with tissue perfusion, and temperature-dependent conductivity with tissue perfusion.

Results

The data demonstrate that significant errors are generated when constant electrical conductivity is assumed in coupled electrical-heat transfer problems that operate at high temperatures. These errors appear to be closely related to the temperature at which the ablation device operates and not to the amount of power applied by the device or the state of tissue perfusion.

Conclusion

Accounting for temperature-dependent phenomena may be critically important in the safe operation of radiofrequency ablation device that operate near 100°C.

     

A model of motor and sensory axon activation in the median nerve using surface electrical stimulation

Surface electrical stimulation has the potential to be a powerful and non-invasive treatment for a variety of medical conditions but currently it is difficult to obtain consistent evoked responses. A viable clinical system must be able to adapt to variations in individuals to produce repeatable results. To more fully study the effect of these variations without performing exhaustive testing on human subjects, a system of computer models was created to predict motor and sensory axon activation in the median nerve due to surface electrical stimulation at the elbow. An anatomically-based finite element model of the arm was built to accurately predict voltages resulting from surface electrical stimulation. In addition, two axon models were developed based on previously published models to incorporate physiological differences between sensory and motor axons. This resulted in axon models that could reproduce experimental results for conduction velocity, strength-duration curves and activation threshold. Differences in experimentally obtained action potential shape between the motor and sensory axons were reflected in the models. The models predicted a lower threshold for sensory axons than motor axons of the same diameter, allowing a range of sensory axons to be activated before any motor axons. This system of models will be a useful tool for development of surface electrical stimulation as a method to target specific neural functions.     

The micromechanical environment of intervertebral disc cells determined by a finite deformation,anisotropic, and biphasic finite element model

Cellular response to mechanical loading varies between the anatomic zones of the intervertebral disc. This difference may be related to differences in the structure and mechanics of both cells and extracellular matrix, which are expected to cause differences in the physical stimuli (such as pressure, stress, and strain) in the cellular micromechanical environment. In this study, a finite element model was developed that was capable of describing the cell micromechanical environment in the intervertebral disc. The model was capable of describing a number of important mechanical phenomena: flow-dependent viscoelasticity using the biphasic theory for soft tissues; finite deformation effects using a hyperelastic constitutive law for the solid phase; and material anisotropy by including a fiber-reinforced continuum law in the hyperelastic strain energy function. To construct accurate finite element meshes, the in situ geometry of IVD cells were measured experimentally using laser scanning confocal microscopy and three-dimensional reconstruction techniques. The model predicted that the cellular micromechanical environment varies dramatically between the anatomic zones, with larger cellular strains predicted in the anisotropic anulus fibrosus and transition zone compared to the isotropic nucleus pulposus. These results suggest that deformation related stimuli may dominate for anulus fibrosus and transition zone cells, while hydrostatic pressurization may dominate in the nucleus pulposus. Furthermore, the model predicted that micromechanical environment is strongly influenced by cell geometry, suggesting that the geometry of IVD cells in situ may be an adaptation to reduce cellular strains during tissue loading.     

Sensitivity of damage predictions to tissue level yield properties and apparent loading conditions

High-resolution finite element models of trabecular bone failure could be used to augment current techniques for measuring damage in trabecular bone. However, the sensitivity of such models to the assumed tissue yield properties and apparent loading conditions is unknown. The goal of this study was to assess the sensitivity of the amount and mode (tension vs. compression) of tissue level yielding in trabecular bone to these factors. Linear elastic, high-resolution finite element models of nine bovine tibial trabecular bone specimens were used to calculate the fraction of the total tissue volume that exceeded each criterion for apparent level loading to the reported elastic limit in both on-axis and transverse compression and tension, and in shear. Four candidate yield criteria were studied, based on values suggested in the literature. Both the amount and the failure mode of yielded tissue were sensitive to the magnitudes of the tissue yield strains, the degree of tension-compression asymmetry of the yield criterion, and the applied apparent loads. The amount of yielded tissue was most sensitive to the orientation of the applied apparent loading, with the most tissue yielding for loading along the principal trabecular orientation and the least for loading perpendicular to it, regardless of the assumed tissue level yield criterion. Small changes in the magnitudes and the degree of asymmetry of the tissue yield criterion resulted in much larger changes in the amount of yielded tissue in the model. The results indicate that damage predictions based on high-resolution finite element models are highly sensitive to the assumed tissue yield properties. As such, good estimates of these values are needed before high-resolution finite element models can be applied to the study of trabecular bone damage. Regardless of the assumed tissue yield properties, the amount and type of damage that occurs in trabecular bone depends on the relative orientations of the applied apparent loads to the trabecular architecture, and this parameter should be controlled for both experimental and computational damage studies.     

Electric and magnetic fields from two-dimensional anisotropic bisyncytia.

Cardiac tissue can be considered macroscopically as a bidomain, anisotropic conductor in which simple depolarization wavefronts produce complex current distributions. Since such distributions may be difficult to measure using electrical techniques, we have developed a mathematical model to determine the feasibility of magnetic localization of these currents. By applying the finite element method to an idealized two-dimensional bisyncytium with anisotropic conductivities, we have calculated the intracellular and extracellular potentials, the current distributions, and the magnetic fields for a circular depolarization wavefront. The calculated magnetic field 1 mm from the tissue is well within the sensitivity of a SQUID magnetometer. Our results show that complex bisyncytial current patterns can be studied magnetically, and these studies should provide valuable insight regarding the electrical anisotropy of cardiac tissue.     

A case for poroelasticity in skeletal muscle finite element analysis: experiment and modeling

Finite element models of skeletal muscle typically ignore the biphasic nature of the tissue, associating any time dependence with a viscoelastic formulation. In this study, direct experimental measurement of permeability was conducted as a function of specimen orientation and strain. A finite element model was developed to identify how various permeability formulations affect compressive response of the tissue. Experimental and modeling results suggest the assumption of a constant, isotropic permeability is appropriate. A viscoelastic only model differed considerably from a visco-poroelastic model, suggesting the latter is more appropriate for compressive studies.     

It's all in the timing: Modeling isovolumic contraction through development and disease with a dynamic dual electromechanical bioreactor system

This commentary discusses the rationale behind our recently reported work entitled “Mimicking isovolumic contraction with combined electromechanical stimulation improves the development of engineered cardiac constructs,” introduces new data supporting our hypothesis, and discusses future applications of our bioreactor system. The ability to stimulate engineered cardiac tissue in a bioreactor system that combines both electrical and mechanical stimulation offers a unique opportunity to simulate the appropriate dynamics between stretch and contraction and model isovolumic contraction in vitro. Our previous study demonstrated that combined electromechanical stimulation that simulated the timing of isovolumic contraction in healthy tissue improved force generation via increased contractile and calcium handling protein expression and improved hypertrophic pathway activation. In new data presented here, we further demonstrate that modification of the timing between electrical and mechanical stimulation to mimic a non-physiological process negatively impacts the functionality of the engineered constructs. We close by exploring the various disease states that have altered timing between the electrical and mechanical stimulation signals as potential future directions for the use of this system.     

A Vital Microscopic Description of the Effects of Electrical Stimulation of Bone Tissue

An implantable, titanium, optical chamber that allows vital microscopic observations of bone tissue during continuous electrical stimulation is described. Tissue reactions to direct current stimulation of a defined bone tissue compartment may be repeatedly observed and recorded on film for indefinite follow-up periods. In this study 5, 20 or 50 μA direct current were applied to chambers inserted into the rabbit tibia. The bone tissue was found to increase in volume after stimulation with 5 and 20 μA, while 50 μA caused bone resorption. Fat-cell concentration was independent of the current levels as used in the present study. Vital microscopy revealed no signs of acute microvascular alterations in the hours following onset of the current. Over an eleven-week follow-up, vessels close to bone borders were found to increase in size from 15–25 microns to 30–50 microns, independent of the level of stimulation. The amount of vessels increased after stimulation with 5 and 20 μA, while in the 50 μA group the vessels decreased to 50% or less of the vascular density observed before start of the stimulation.     

Steady-state analysis of self-heated thermistors using finite elements

The purpose of this work is to validate, using numerical, finite element methods, the thermal assumptions made in the analytical analysis of a coupled thermistor probe-tissue model upon which a thermal conductivity measurement scheme has been based. Analytic, closed form temperature profiles generated by the self-heated thermistors can be found if three simplifying assumptions are made: the thermistor is spherical; heat is generated in all regions of the bead; and heat is generated uniformly in the bead. This analytic solution is used to derive a linear relationship between tissue thermal conductivity and the ratio of thermistor temperature rise over electrical power required to maintain that temperature rise. This derived, linear relationship is used to determine thermal conductivity from the observed experimental data. However, in reality, the thermistor bead is a prolate spheroid surrounded by a passive shell, and the heating pattern in the bead is highly nonuniform. In the physical system, the exact relationship between the tissue thermal conductivity and parameters measured by the thermistor is not known. The finite element method was used to calculate the steady-state temperature profiles generated by thermistor beads with realistic geometry and heating patterns. The results of the finite element analysis show that the empirical, linear relationship remains valid when all three simplified assumptions are significantly relaxed.