Analytic modeling of conductively anisotropic neural tissue.

The abdominal ganglion of the Aplysia californica is an established in vitro model for studying neuroelectric behavior in the presence of an applied electrical current and recently used in studies of magnetic resonance electrical impedance tomography (MREIT) which allows for quantitative visualization of spatially distributed current and magnetic flux densities. Understanding the impact the Aplysia geometry and anisotropic conductivity have on applied electromagnetic fields is central to intepreting and refining MREIT data and protocols, respectively. Here we present a simplified bidomain model of an in vitro experimental preparation of the Aplysia abdominal ganglion, describing the tissue as a radially anisotropic sphere with equal anisotropy ratios, i.e., where radial conductivities in both intra- and extra-cellular regions are ten times that of their polar and azimuthal conductivities. The fully three dimensional problem is validated through comparisons with limiting examples of 2D isotropic analyses. Results may be useful in validating finite element models of MREIT experiments and have broader relevance to analysis of MREIT data obtained from complex neural architecture in the human brain.

Analytic Modeling of Neural Tissue: I. A Spherical Bidomain.

Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates [Formula: see text]. The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains. The membrane and monodomain are defined by the scalar Helmholtz and Laplace equations, respectively, which are both separable in spherical coordinates. Product solutions are thus assumed and given through certain transcendental functions. From these electrical potentials, analytic expressions for current density are derived and from those fields the magnetic flux density is calculated. Numerical examples are considered wherein the interstitial conductivity is varied, as well as the limiting case of the problem simplifying to two dimensions due to azimuthal independence. Finally, future modeling work is discussed.

Mechanical analysis of an axially symmetric cylindrical phantom with a spherical heterogeneity for MR elastography.

Cylindrical homogenous phantoms for magnetic resonance (MR) elastography in biomedical research provide one way to validate an imaging systems performance, but the simplified geometry and boundary conditions can cloak complexity arising at tissue interfaces. In an effort to develop a more realistic gel tissue phantom for MRE, we have constructed a heterogenous gel phantom (a sphere centrally embedded in a cylinder). The actuation comes from the phantom container, with the mechanical waves propagating toward the center, focusing the energy and thus allowing for the visualization of high-frequency waves that would otherwise be damped. The phantom was imaged and its stiffness determined using a 9.4 T horizontal MRI with a custom build piezo-elastic MRE actuator. The phantom was vibrated at three frequencies, 250, 500, and 750 Hz. The resulting shear wave images were first used to reconstruct material stiffness maps for thin (1 mm) axial slices at each frequency, from which the complex shear moduli µ were estimated, and then compared with forward modeling using a recently developed theoretical model which took µ as inputs. The overall accuracy of the measurement process was assessed by comparing theory with experiment for selected values of the shear modulus (real and imaginary parts). Close agreement is shown between the experimentally obtained and theoretically predicted wave fields.

Scattering and Diffraction of Elastodynamic Waves in a Concentric Cylindrical Phantom for MR Elastography.

AIM: The focus of this paper is to report on the design and construction of a multiply connected phantom for use in magnetic resonance elastography (MRE)-an imaging technique that allows for the noninvasive visualization of the displacement field throughout an object from externally driven harmonic motion-as well as its inverse modeling with a closed-form analytic solution which is derived herein from first principles. METHODS: Mathematically, the phantom is described as two infinite concentric circular cylinders with unequal complex shear moduli, harmonically vibrated at the exterior surface in a direction along their common axis. Each concentric cylinder is made of a hydrocolloid with its own specific solute concentration. They are assembled in a multistep process for which custom scaffolding was designed and built. A customized spin-echo-based MR elastography sequence with a sinusoidal motion-sensitizing gradient was used for data acquisition on a 9.4 T Agilent small-animal MR scanner. Complex moduli obtained from the inverse model are used to solve the forward problem with a finite-element method. RESULTS: Both complex shear moduli show a significant frequency dependence (p 0.001) in keeping with previous work. CONCLUSION: The novel multiply connected phantom and mathematical model are validated as a viable tool for MRE studies. SIGNIFICANCE: On a small enough scale much of physiology can be mathematically modeled with basic geometric shapes, e.g., a cylinder representing a blood vessel. This study demonstrates the possibility of elegant mathematical analysis of phantoms specifically designed and carefully constructed for biomedical MRE studies.

Analysis of bipolar external excitation of spherical tissue by spatially opposed current source and sink points.

The recently increasing role in medical imaging that electrophysiology plays has spurned the need for its quantitative analysis at all scales-ions, cells, tissues, organs, etc.; so, here is presented a model of nerve tissue in a spherical volume excited by a point current source at one pole and a point current sink at the opposite pole. The sphere of tissue is described as an isotropic bidomain, consisting of the intra- and extra-cellular regions and the membrane that separates them, and is immersed in an infinite isotropic conductive bath. The system of coupled differential equations is solved by redefining the domains to be in terms of a monodomain and a membrane. The solution takes the form of an infinite sum of the product of certain transcendental functions. The study concludes with a numeric example in which the boundary conditions are shown to be satisfied, validating this analysis, paving the way for more sophisticated models of excitable tissue.

Effects of central airway shunting on the mechanical impedance of the mouse lung.

The mechanical properties of the lung are embodied in its mechanical input impedance, which it is interpreted in physiological terms by being fit with a mathematical model. The normal lung is extremely well described by a model consisting of a single uniformly ventilated compartment comprised of tissue having a constant-phase impedance, but to describe the abnormal lung it frequently becomes necessary to invoke additional compartments. To date, all evidence of regional mechanical heterogeneity in the mouse lung has been assumed to be of the parallel variety. We therefore investigated the use of a serial heterogeneity model, relative to parallel heterogeneity and homogeneous models, for describing impedance spectra in mice subjected to a variety of interventions designed to make their lungs heterogeneous. We found that functional evidence of the finite stiffness of the airway wall in mice with airways obstruction can sometimes be apparent in lung impedance below 20 Hz. The model estimates of airway stiffness were smaller than direct estimates obtained from micro-CT images of the lung in vivo, suggesting that the conducting airways alone are likely not the precise anatomical correlate of proximal functional stiffness in the lung. Nevertheless, we conclude that central airway shunting in mice can sometimes be an important physiological phenomenon.