Effects of rotational myocardial anisotropy in forward potential computations with equivalent heart dipoles

Rotating fibers in the heart lead to a myocardium of inhomogeneous anisotropic conductivity. Besides affecting the activation isochrones, this anisotropy modifies the equivalent dipoles used in calculating extracardiac potentials, rendering them oblique rather than normal to the activation wavefront due to an added axial dipole component oriented along the fibers. Herein, however, consequences of the assumption usually made in forward potential calculations that the equivalent dipoles act in a myocardium that is homogeneous and isotropic are examined. A layered inner block representing the heart was placed inside an outer block representing an isotropic volume conductor. Fiber direction in the inner block rotated uniformly from layer to layer. Current dipoles of different orientations were placed in the inner block and the potentials calculated everywhere. Effects of the anisotropy of the inner block were gauged by computing an equivalent dipole that best fit the outer block surface potentials. For volume conductor conductivities close to that of the torso, the anisotropy diminished dipoles oriented along the fibers. Since the intraventricular blood masses in the heart also diminish such dipoles, these reductions of the axial component may explain the success of heart model simulations that ignore this component.     

A finite difference method with reciprocity used to incorporate anisotropy in electroencephalogram dipole source localization

Many implementations of electroencephalogram (EEG) dipole source localization neglect the anisotropical conductivities inherent to brain tissues, such as the skull and white matter anisotropy. An examination of dipole localization errors is made in EEG source analysis, due to not incorporating the anisotropic properties of the conductivity of the skull and white matter. First, simulations were performed in a 5 shell spherical head model using the analytical formula. Test dipoles were placed in three orthogonal planes in the spherical head model. Neglecting the skull anisotropy results in a dipole localization error of, on average, 13.73 mm with a maximum of 24.51 mm. For white matter anisotropy these values are 11.21 mm and 26.3 mm, respectively. Next, a finite difference method (FDM), presented by Saleheen and Kwong (1997 IEEE Trans. Biomed. Eng. 44 800-9), is used to incorporate the anisotropy of the skull and white matter. The FDM method has been validated for EEG dipole source localization in head models with all compartments isotropic as well as in a head model with white matter anisotropy. In a head model with skull anisotropy the numerical method could only be validated if the 3D lattice was chosen very fine (grid size < or = 2 mm).     

A Controllably Anisotropic Conductivity or Diffusion Phantom Constructed from Isotropic Layers

Phantoms with controllable and well-defined anisotropy are needed to test methods for imaging electrical anisotropy. We developed and tested a phantom that had properties similar to a homogeneous anisotropic conductive medium. The phantom was constructed with alternate slices of isotropic gel having different conductivities. The degree of anisotropy in the phantom could be varied easily by changing the relative conductivity of the two gels. We tested the stability of several phantoms and found their properties were maintained for approximately 8 h following construction. The phantom has application to electrical impedance tomography, magnetic resonance electrical impedance tomography, EEG and ECG source imaging and diffusion tensor imaging.     

Cardiac anisotropy in boundary-element models for the electrocardiogram

The boundary-element method (BEM) is widely used for electrocardiogram (ECG) simulation. Its major disadvantage is its perceived inability to deal with the anisotropic electric conductivity of the myocardial interstitium, which led researchers to represent only intracellular anisotropy or neglect anisotropy altogether. We computed ECGs with a BEM model based on dipole sources that accounted for a “compound” anisotropy ratio. The ECGs were compared with those computed by a finite-difference model, in which intracellular and interstitial anisotropy could be represented without compromise. For a given set of conductivities, we always found a compound anisotropy value that led to acceptable differences between BEM and finite-difference results. In contrast, a fully isotropic model produced unacceptably large differences. A model that accounted only for intracellular anisotropy showed intermediate performance. We conclude that using a compound anisotropy ratio allows BEM-based ECG models to more accurately represent both anisotropies. Computational resources for this work were provided by the Réseau québécois de calcul de haute performance (RQCHP). M. Potse was supported by the Research Center of Sacré-Coeur Hospital, Montreal, QC, Canada.     

Anisotropic conductivity imaging with MREIT using equipotential projection algorithm

Magnetic resonance electrical impedance tomography (MREIT) combines magnetic flux or current density measurements obtained by magnetic resonance imaging (MRI) and surface potential measurements to reconstruct images of true conductivity with high spatial resolution. Most of the biological tissues have anisotropic conductivity; therefore, anisotropy should be taken into account in conductivity image reconstruction. Almost all of the MREIT reconstruction algorithms proposed to date assume isotropic conductivity distribution. In this study, a novel MREIT image reconstruction algorithm is proposed to image anisotropic conductivity. Relative anisotropic conductivity values are reconstructed iteratively, using only current density measurements without any potential measurement. In order to obtain true conductivity values, only either one potential or conductivity measurement is sufficient to determine a scaling factor. The proposed technique is evaluated on simulated data for isotropic and anisotropic conductivity distributions, with and without measurement noise. Simulation results show that the images of both anisotropic and isotropic conductivity distributions can be reconstructed successfully.     

Finite element modeling of mitral leaflet tissue using a layered shell approximation

The current study presents a finite element model of mitral leaflet tissue, which incorporates the anisotropic material response and approximates the layered structure. First, continuum mechanics and the theory of layered composites are used to develop an analytical representation of membrane stress in the leaflet material. This is done with an existing anisotropic constitutive law from literature. Then, the concept is implemented in a finite element (FE) model by overlapping and merging two layers of transversely isotropic membrane elements in LS-DYNA, which homogenizes the response. The FE model is then used to simulate various biaxial extension tests and out-of-plane pressure loading. Both the analytical and FE model show good agreement with experimental biaxial extension data, and show good mutual agreement. This confirms that the layered composite approximation presented in the current study is able to capture the exponential stiffening seen in both the circumferential and radial directions of mitral leaflets.     

组织电导率性质对磁感应磁声耦合成像声源强度的影响

针对部分生物组织电导率分布具有各向异性的特点,对磁感应磁声耦合成像的声源强度特征进行理论分析,推导不同电导率性质的声源强度公式,并运用COMSOL Multiphysics5.5建立生物组织电导率模型,进行电磁场仿真分析求解,用Matlab 2016a计算其振动声源。结果证明,在同样磁场激励条件下,电导率各向同性和各向异性的声源分布都能反映生物组织的层析结构,但其强度不同。本研究为磁感应磁声耦合成像逆问题声源的精确重建提供了理论基础。     

Development of a model for point source electrical fibre bundle stimulation

A model is presented for determining the excitation (transmembrane) potentials on nerve and muscle fibres in a cylindrical bundle from an external point source electrode at the surface and within the preparation. The fibre bundle is considered to be immersed in an infinite isotropic conductive medium and is idealised as an infinitely extending cylinder. This cylinder is initially represented as an isotropic monodomain. A subsequent degree of complexity introduces anisotropy in the monodomain, and finally the bundle is represented as an anisotropic bidomain comprised of the interstitial radial and longitudinal conductivities, the intracellular longitudinal conductivity and the fibre membrane between the two domains. In this latter model, electrical coupling from extracellular to intracellular space is included by means of the bidomain formulation. Computational aspects are discussed, and preliminary results for prescribed conditions are presented.     

Dipole estimation errors due to differences in modeling anisotropic conductivities in realistic head models for EEG source analysis

To improve the EEG source localization in the brain, the conductivities used in the head model play a very important role. In this study, we focus on the modeling of the anisotropic conductivity of the white matter. The anisotropic conductivity profile can be derived from diffusion weighted magnetic resonance images (DW-MRI). However, deriving these anisotropic conductivities from diffusion weighted MR images of the white matter is not straightforward. In the literature, two methods can be found for calculating the conductivity from the diffusion weighted images. One method uses a fixed value for the ratio of the conductivity in different directions, while the other method uses a conductivity profile obtained from a linear scaling of the diffusion ellipsoid. We propose a model which can be used to derive the conductivity profile from the diffusion tensor images. This model is based on the variable anisotropic ratio throughout the white matter and is a combination of the linear relationship as stated in the literature, with a constraint on the magnitude of the conductivity tensor (also known as the volume constraint). This approach is stated in the paper as approach A. In our study we want to investigate dipole estimation differences due to using a more simplified model for white matter anisotropy (approach B), while the electrode potentials are derived using a head model with a more realistic approach for the white matter anisotropy (approach A). We used a realistic head model, in which the forward problem was solved using a finite difference method that can incorporate anisotropic conductivities. As error measures we considered the dipole location error and the dipole orientation error. The results show that the dipole location errors are all below 10 mm and have an average of 4 mm in gray matter regions. The dipole orientation errors ranged up to 66.4 degrees, and had a mean of, on average, 11.6 degrees in gray matter regions. In a qualitative manner, the results show that the orientation and location error is dependent on the orientation of the test dipole. The location error is larger when the orientation of the test dipole is similar to the orientation of the anisotropy, while the orientation error is larger when the orientation of the test dipole deviates from the orientation of the anisotropy. From these results, we can conclude that the modeling of white matter anisotropy plays an important role in EEG source localization. More specifically, accurate source localization will require an accurate modeling of the white matter conductivity profile in each voxel.     

Influence of Skull Modeling Approaches on EEG Source Localization

Electroencephalographic source localization (ESL) relies on an accurate model representing the human head for the computation of the forward solution. In this head model, the skull is of utmost importance due to its complex geometry and low conductivity compared to the other tissues inside the head. We investigated the influence of using different skull modeling approaches on ESL. These approaches, consisting in skull conductivity and geometry modeling simplifications, make use of X-ray computed tomography (CT) and magnetic resonance (MR) images to generate seven different head models. A head model with an accurately segmented skull from CT images, including spongy and compact bone compartments as well as some air-filled cavities, was used as the reference model. EEG simulations were performed for a configuration of 32 and 128 electrodes, and for both noiseless and noisy data. The results show that skull geometry simplifications have a larger effect on ESL than those of the conductivity modeling. This suggests that accurate skull modeling is important in order to achieve reliable results for ESL that are useful in a clinical environment. We recommend the following guidelines to be taken into account for skull modeling in the generation of subject-specific head models: (i) If CT images are available, i.e., if the geometry of the skull and its different tissue types can be accurately segmented, the conductivity should be modeled as isotropic heterogeneous. The spongy bone might be segmented as an erosion of the compact bone; (ii) when only MR images are available, the skull base should be represented as accurately as possible and the conductivity can be modeled as isotropic heterogeneous, segmenting the spongy bone directly from the MR image; (iii) a large number of EEG electrodes should be used to obtain high spatial sampling, which reduces the localization errors at realistic noise levels.     

Conductivities of three-layer live human skull

Electrical conductivities of compact, spongiosum, and bulk layers of the live human skull were determined at varying frequencies and electric fields at room temperature using the four-electrode method. Current, at higher densities that occur in human cranium, was applied and withdrawn over the top and bottom surfaces of each sample and potential drop across different layers was measured. We used a model that considers variations in skull thicknesses to determine the conductivity of the tri-layer skull and its individual anatomical structures. The results indicate that the conductivities of the spongiform (16.2-41.1 milliS/m), the top compact (5.4-7.2 milliS/m) and lower compact (2.8-10.2 milliS/m) layers of the skull have significantly different and inhomogeneous conductivities. The conductivities of the skull layers are frequency dependent in the 10-90 Hz region and are non-ohmic in the 0.45-2.07 A/m² region. These current densities are much higher than those occurring in human brain.     

Estimation of electrical conductivity distribution within the human head from magnetic flux density measurement

We have developed a new algorithm for magnetic resonance electrical impedance tomography (MREIT), which uses only one component of the magnetic flux density to reconstruct the electrical conductivity distribution within the body. The radial basis function (RBF) network and simplex method are used in the present approach to estimate the conductivity distribution by minimizing the errors between the 'measured' and model-predicted magnetic flux densities. Computer simulations were conducted in a realistic-geometry head model to test the feasibility of the proposed approach. Single-variable and three-variable simulations were performed to estimate the brain-skull conductivity ratio and the conductivity values of the brain, skull and scalp layers. When SNR = 15 for magnetic flux density measurements with the target skull-to-brain conductivity ratio being 1/15, the relative error (RE) between the target and estimated conductivity was 0.0737 +/- 0.0746 in the single-variable simulations. In the three-variable simulations, the RE was 0.1676 +/- 0.0317. Effects of electrode position uncertainty were also assessed by computer simulations. The present promising results suggest the feasibility of estimating important conductivity values within the head from noninvasive magnetic flux density measurements.     

A Smectic A Liquid Single Crystal Elastomer (LSCE): Phase Behavior and Mechanical Anisotropy

The analysis of the phase behavior of a smectic A (S_A) elastomer reveals a nematic phase existing within a small temperature range below the isotropic state. Stress‐optical measurements in the pretransformational regime of the isotropic state indicate smectic as well as nematic fluctuations yielding a critical exponent of γ = 0.65. The formation of the liquid single crystal elastomer (LSCE) at the isotropic to liquid crystalline phase transformation equals a nematic LSCE. At the nematic to S_A phase transformation, the orientation of the director remains constant while the tendency of the network strands towards an oblate equilibrium conformation is suppressed by the high modulus parallel to the smectic layer normal. The mechanical anisotropy of the S_A‐LSCE as a function of the temperature is characterized by entropy elasticity perpendicular to the smectic layer normal. Parallel to the layer normal the mechanical response is determined by the enthalpy elastic response of the smectic layers having a modulus larger by about two orders of magnitude. In this direction the modulus decreases linearly with increasing temperature and reflects the falling stability of the layers. Accordingly, above a deformation of about 2% the homogeneous layered structure breaks down at a threshold stress that also falls linearly with increasing temperature while the threshold strain remains constant at about 2% elongation.     

Anisotropic Mechanical Responses of Poly(Ethylene Oxide)‐Based Lithium Ions Containing Solid Polymer Electrolytes

Development of mechanically deformable solid state devices is receiving tremendous attention, and high ionic conductivity solid polymer electrolytes (SPEs) are highly sought after for their development. The process history‐induced polymer alignment anisotropy can lead to anisotropic conductivity to the SPEs. Here, a Li ion SPE membrane developed using poly(ethylene oxide) (PEO) and LiClO₄ is demonstrated for its microstructure variations while applying external stress and the corresponding variations in the ionic conductivity are also calculated. The microstructural evolution shows that larger strain values induce large dislocations in the crystallites of PEO leading to the formation of larger amorphous regions which soften the matrix. The anisotropic mechanical responses are observed while applying cyclic strain to thicker SPEs, where the compressive measurements show softening of the matrix while tensile measurements harden the matrix. The ionic conductivities of the softened matrix are found to be enhanced while those of toughened matrix are found to be decreased. This detailed mechanical analysis along with the in situ ionic conductivity studies of PEO‐based Li ion SPE show that along with the thermal history of the polymers, process history and the anisotropic mechanical responses of the polymers also need to be considered while developing SPEs for flexible devices.     

Bone regeneration and infiltration of an anisotropic composite scaffold: an experimental study of rabbit cranial defect repair

Tissue formation on scaffold outer edges after implantation may restrict cell infiltration and mass transfer to/from the scaffold center due to insufficient interconnectivity, leading to incidence of a necrotic core. Herein, a nano-hydroxyapatite/polyamide66 (n-HA/PA66) anisotropic scaffold with axially aligned channels was prepared with the aim to enhance pore interconnectivity. Bone tissue regeneration and infiltration inside of scaffold were assessed by rabbit cranial defect repair experiments. The amount of newly formed bone inside of anisotropic scaffold was much higher than isotropic scaffold, e.g., after 12 weeks, the new bone volume in the inner pores was greater in the anisotropic scaffolds (>50%) than the isotropic scaffolds (<30%). The results suggested that anisotropic scaffolds could accelerate the inducement of bone ingrowth into the inner pores in the non-load-bearing bone defects compared to isotropic scaffolds. Thus, anisotropic scaffolds hold promise for the application in bone tissue engineering.     

A modelling error approach for the estimation of optical absorption in the presence of anisotropies

Optical tomography is an emerging method for non-invasive imaging of human tissues using near-infrared light. Generally, the tissue is assumed isotropic, but this may not always be true. In this paper, we present a method for the estimation of optical absorption coefficient allowing the background to be anisotropic. To solve the forward problem, we model the light propagation in tissue using an anisotropic diffusion equation. The inverse problem consists of the estimation of the absorption coefficient based on boundary measurements. Generally, the background anisotropy cannot be assumed to be known. We treat the uncertainties in the background anisotropy parameter values as modelling error, and include this in our model and reconstruction. We present numerical examples based on simulated data. For reference, examples using an isotropic inversion scheme are also included. The estimates are qualitatively different for the two methods.     

Influence of heterogeneous and anisotropic tissue conductivity on electric field distribution in deep brain stimulation

The aim was to quantify the influence of heterogeneous isotropic and heterogeneous anisotropic tissue on the spatial distribution of the electric field during deep brain stimulation (DBS). Three finite element tissue models were created of one patient treated with DBS. Tissue conductivity was modelled as (I) homogeneous isotropic, (II) heterogeneous isotropic based on MRI, and (III) heterogeneous anisotropic based on diffusion tensor MRI. Modelled DBS electrodes were positioned in the subthalamic area, the pallidum, and the internal capsule in each tissue model. Electric fields generated during DBS were simulated for each model and target-combination and visualized with isolevels at 0.20 (inner), and 0.05 V mm⁻¹ (outer). Statistical and vector analysis was used for evaluation of the distribution of the electric field. Heterogeneous isotropic tissue altered the spatial distribution of the electric field by up to 4% at inner, and up to 10% at outer isolevel. Heterogeneous anisotropic tissue influenced the distribution of the electric field by up to 18 and 15% at each isolevel, respectively. The influence of heterogeneous and anisotropic tissue on the electric field may be clinically relevant in anatomic regions that are functionally subdivided and surrounded by multiple fibres of passage.     

The influence of skull-conductivity misspecification on inverse source localization in realistically shaped finite element head models

Summary The electric conductivities of different tissues are important parameters of the head model and their precise knowledge appears to be a prerequisite for the localization of electric sources within the brain. To estimate the error in source localization due to errors in assumed conductivity values, parameter variations on skull conductivities are examined. The skull conductivity was varied in a wide range and, in a second part of this paper, the effect of a nonhomogeneous skull conductivity was examined. An error in conductivity of lower than 20% appears to be acceptable for fine finite element head models with average discretization errors down to 3mm. Nonhomogeneous skull conductivities, e.g., sutures, yield important mislocalizations especially in the vincinty of electrodes and should be modeled.The authors wish to thank the VW — Foundation for financial support.     

Methods for high-resolution anisotropic finite element modeling of the human head: automatic MR white matter anisotropy-adaptive mesh generation

This study proposes an advanced finite element (FE) head modeling technique through which high-resolution FE meshes adaptive to the degree of tissue anisotropy can be generated. Our adaptive meshing scheme (called wMesh) uses MRI structural information and fractional anisotropy maps derived from diffusion tensors in the FE mesh generation process, optimally reflecting electrical properties of the human brain. We examined the characteristics of the wMeshes through various qualitative and quantitative comparisons to the conventional FE regular-sized meshes that are non-adaptive to the degree of white matter anisotropy. We investigated numerical differences in the FE forward solutions that include the electrical potential and current density generated by current sources in the brain. The quantitative difference was calculated by two statistical measures of relative difference measure (RDM) and magnification factor (MAG). The results show that the wMeshes are adaptive to the anisotropic density of the WM anisotropy, and they better reflect the density and directionality of tissue conductivity anisotropy. Our comparison results between various anisotropic regular mesh and wMesh models show that there are substantial differences in the EEG forward solutions in the brain (up to RDM=0.48 and MAG=0.63 in the electrical potential, and RDM=0.65 and MAG=0.52 in the current density). Our analysis results indicate that the wMeshes produce different forward solutions that are different from the conventional regular meshes. We present some results that the wMesh head modeling approach enhances the sensitivity and accuracy of the FE solutions at the interfaces or in the regions where the anisotropic conductivities change sharply or their directional changes are complex. The fully automatic wMesh generation technique should be useful for modeling an individual-specific and high-resolution anisotropic FE head model incorporating realistic anisotropic conductivity distributions towards more accurate analysis of bioelectromagnetic problems.     

基于头部组织不同电导率特性的阻抗成像正问题仿真研究

目的 通过建立5层有限元真实头模型,研究了各层组织非均质和颅骨、脑白质各向异性电特性对电阻抗成像问题中电磁场分布的影响.方法 对头部各组织建立4种电导率分布模型:均质分布、非均质分布以及颅骨和脑白质各向异性电导率模型;通过正问题数值求解得到不同模型下的磁场分布和电场分布,并通过定量的统计分析研究非均质和各向异性电导率特...