Ph.D. University of North Carolina at Chapel Hill
B.A. University of Virginia
- Department of Mathematics, Cal Poly San Luis Obispo
- Mathematical Biology, University of Utah Department of Mathematics, University of Utah
- Carolina Center for Interdisciplinary and Applied Mathematics Department of Mathematics, UNC Chapel Hill
Electrical Activity in Myocardial Tissue
Joint work with Jim Keener (University of Utah Department of Mathematics) and Steven Poelzing (Virginia Tech Carilion Research Institute).
Electrical stimulation of cardiac cells causes an action potential wave to propagate through myocardial tissue, resulting in muscular contraction and pumping blood through the body. Con- duction failure in the heart has been linked to ventricular arrhythmia and cardiac death. While gap junctional proteins connecting cardiac cells are traditionally considered to be the main mode of cellular coupling, recent experimental studies have found that down-regulation of gap junctional proteins did not necessarily decrease conduction velocity, implying another mode of coupling. Additionally, our collaborators Steven Poelzing and Rengasayee Veeraraghavan at the Virginia Tech Carilion Research Institute have experimental evidence that modification of tissue structure alone is sufficient to substantially affect conduction.
A small change in transmembrane potential (the difference between the intracellular and extracellular potentials) causes sodium ion flux through voltage-gated channels, constituting the main upstroke of an action potential. The initial change may occur due to gap junctions, however, the narrow extracellular spaces have high resistances and could result in strong field coupling (or ephaptic effects) between neighboring cells . To explore ephaptic effects, microdomains of the complex geometric cardiac tissue structure must be modeled. Existing mathematical models of action potential propagation assume homogenized extracellular spaces and cannot capture these microdomain effects. More detailed numerical models show the importance of the cellular geometry, but are too computationally expensive to be used on a larger scale.
Our preliminary model uses the simplifying assumptions of isopotential intracellular spaces and collapses the extracellular space to a two-dimensional structure. Conservation of current gives rise to equations that can be numerically integrated through a careful modification of known algorithms.
In contrast to classical cable theory for cardiac conduction, we found that the distribution of transmembrane sodium ion channels and the resistance of the extracellular space has a significant impact on propagation velocity in a single strand of cylindrical cells. Cable theory (green surface in Figure 1(a)) monotonically increases with increasing extracellular lateral conductivity (in the space between the sides of cells) and does not change with extracellular junctional conductivity (in the space between the ends of cells). With the model we developed, we found that when the sodium ion channels are located primarily on the ends of cells, (brown surface in Figure 1(a)) the propagation velocity is greatly enhanced. Even when the sodium ion channels are mostly on the sides of the cells, we see a small enhancement in the propagation velocity with small extracellular lateral conductivities. Figure 1(b), in which propagation velocities are plotted over a larger range of extracellular conductivities with sodium ion channels primarily in the ends of the cells, shows that these ephaptic effects occur in all areas of high extracellular resistances. The field coupling for small extracellular junctional conductivities is much stronger than that for small extracellular lateral conductivities, but field effects are evident in both. Thus, microdomains in cardiac tissue modeling cannot be ignored.
Polar Sea Ice
Joint work with Ken Golden (University of Utah Department of Mathematics) and Cynthia Furse ((University of Utah Department of Electrical and Computer Engineering)
Fluid Permeability of Sea Ice
As a boundary layer between the ocean and atmosphere, sea ice plays an important role in the global ecosystem. Sea ice is composed of a porous ice matrix with vertically oriented brine inclusions or channels. Depending on the temperature, salinity, and microstructure, sea ice can allow for the transport of fluid such as sea water or surface meltwater. This fluid transport plays a role in snow-ice formation, heat exchange, nutrient transport, and melt pond evolution by mechanisms that are not well understood.
Along with three other researchers from the University of Utah (Ken Golden, Cynthia Furse, and David Lubbers), I traveled to Antarctica as part of a research expedition in 2010. Stationed in a field camp near Ross Island, we extracted ice cores in McMurdo Sound and measured the electrical properties of the ice as a function of depth. The resulting boreholes were used for permeability measurements, in which the rate at which the water rose was recorded. The crystallographic structure of the ice was also sampled and recorded as a function of ice depth. The vertical permeability and resistivities were computed as a function of brine volume fraction. Our measurements were further corroborated by numerical simulations and comparisons with existing theoretical models. This correlation between electrical resistance and fluid permeability is the basis for further extension into remote sensing techniques for monitoring fluid transport through sea ice, which will be used to improve existing predictive climate models.
Polycrystalline structure of sea ice:
Using tomographic data from ice samples, we numerically generated ice crystals and computed the effective permittivity of these crystals. The data from hundreds of ice crystals was compared against bounds on the effective permittivity of a polycrystalline structure built from these individual crystals. These theoretical polycrystalline bounds and the numerical comparative data, in a joint work with Adam Gully, Elena Cherkaev, and Ken Golden, are presented in a paper, currently in preparation, for the Journal of Geophysical Research.
Recovery of microstructural parameters:
To explore algorithms that recover the microstructural parameters of composite materials, we created random configurations of disks in a bounded domain. Given the area fraction of the disks and a specific parameter of minimum separation between disks, we numerically computed the effective permittivity of hundreds of configuration over a range of frequencies. This frequency-dependent data was inverted using two different, theoretical and numerical algorithms to recover the area fraction and separation between disks. These results, in a joint work with Chris Orum, Elena Cherkaev, and Ken Golden, are presented in a paper, currently in preparation, for the Journal of Computational Physics.
Electrorheological (ER) Fluids:
We have studied the behavior of ER fluids under the influence of an electric field. Using a finite element commercial package COMSOL, we have studied the energy landscape of different, static configurations for a small number of spheres in a small grid and a large number of spheres in a small grid. Ranking configurations in order of increasing energy, we explored the process of how spheres in ER fluids create chains and the possible presence of an energy barrier. Using previously studied equations of motion that take into account the fluid medium and the effect of surrounding spheres, we numerically modeled the movement of dielectric spheres in an insulating fluid. Taking this a step further, we vary the repulsion and attraction forces. We found that vertical connections were easily simulated, and different degrees of branching were attained depending on the repulsion and attraction forces used. These time varying simulations were also used to study the energy landscape during the process of chaining. Our collaborators, Shunbo Li and Ping Sheng, at the Hong Kong University of Science and Technology have provided experiments of ER fluids in the dilute concentrations to study simple interactions. We’ve used these experiments to compare with computer simulations and found similar behavior.
Sedimentation in Stratified Fluids
Joint work with Roberto Camassa (UNC Chapel Hill Mathematics Department) and Richard M. McLaughlin (UNC Chapel Hill Mathematics Department)
Consider a solid body moving at low Reynolds number through a sharp, stable stratification of miscible fluids. We study the hydrodynamics of the forces in the system, including the fluid flow and the resultant behavior of the sphere.
As the sphere passes through the interface, the importance of the entrained fluid is due to its buoyancy in the lower, denser fluid. We find that the sphere will slow down dramatically as it passes through the density transition. We develop a model from first principles of the highly coupled system to capture this effect.
Our model, with no adjustable parameters, can predict the velocity field of the fluid, and thus the shape of the interface between the stratified fluids. The velocity profile, which shows this non-monotonic transition between terminal velocities of the upper layer and lower layer, can also be compared with the experimental data.
We are currently studying the extensions of our model to linear or continuous stratifications, multibody sedimentation, and infinite fluid medium.
R. Veeraraghavan, J. Lin, J. P. Keener, R. G. Gourdie, and S. Poelzing, Potassium Channels in the Cx43 Gap Junction Perinexus Modulate Ephaptic Coupling: An Experimental and Modeling Study, Pfluger's Archiv - European Journal of Physiology, Aug 11 (2016), 1651-1661.
S. A. George, K. J. Sciuto, J. Lin, M. E. Salama, J. P. Keener, R. G. Gourdie, and S. Poelzing, Extracellular sodium and potassium levels modulate cardiac conduction in mice heterozygous null for the Connexin43 gene, Pfluger's Archiv - European Journal of Physiology, Mar 14 (2015), 1 – 11.
A. Gully, J. Lin, E. Cherkaev, and K. M. Golden, Bounds on the complex permittivity of polycrystalline materials by analytic continuation, Proc. R. Soc. A, 471(2015)
R. Veeraraghavan, J. Lin, G. S. Hoeker, J. P. Keener, R. G. Gourdie, and S. Poelzing, Sodium channels in the Cx43 gap junction perinexus may constitute a cardiac ephapse: an experimental and modeling study, Pfluger's Archiv - European Journal of Physiology, Jan 13 (2015), 1 – 13.
J. Lin and J. P. Keener, Microdomain effects on transverse cardiac propagation, Biophys. J. 106(2014), 925 – 931. (New and Notable)
J. Lin and J. P. Keener, Ephaptic coupling in cardiac myocytes, IEEE Trans. Biomed. Eng. 60(2012), 576 – 582.
J. Lin and J. P. Keener, A model for electrical activity of myocardial cells
incorporating the effects of ephaptic coupling, PNAS 107(2010), 20935—40.
R. Camassa, C. Falcon, J. Lin, R. M. McLaughlin, and N. Mykins, A first principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number, J. Fluid Mech. 664(2010), 436—465.
R. Camassa, C. Falcon, J. Lin, R. M. McLaughlin, and R. Parker, Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime, Phys. Fluids 21(2009), 031702-1–4.
J. Lin, An experimental and mathematical study on the prolonged residence time of a sphere falling through stratified fluids at low Reynolds number, PhD thesis, University of North Carolina at Chapel Hill (2009).