Peeyush Shrivastava

Ohio State University

Publication Date: January 8, 2015

1. Abstract

The study of physiology and biophysics had previously thought to be largely experimental, with a relatively cynical possibility of effective modeling of disease. However, in the late 1940s and early 1950s, the revolutionary Hodgkin-Huxley experiments and equations gave birth to a field known as quantitative physiology and effectively paved a marriage between mathematics and physiology. The inherent value associated with this series of papers was the newly recognized value of mathematical models in simplifying and becoming more predictive of function in an altered state (e.g. disease). In the contemporary age of science, seemingly unrelated functionalities and relationships can be dissolved into significant complexes and networks through computational and mathematical models of analysis. This review seeks to elucidate some of the computational and mathematical techniques used to quantify and analyze cardiac physiology in control and disease states.

1.1. Mathematical Models for Electrophysiological Discovery

Human physiology represents a mosaic of a seemingly simply, yet profoundly complex series of interrelated activities, networks and functions. For example, the individual neuron’s action potential is simple to delineate and measure, whereas the net electrical activity of the brain is rather difficult to summarize in any simple series of electrical equations. Furthermore, voltage gated ion channels in the heart and brain have been shown to have significant relationships, as seen in conditions such as Timothy’s Syndrome in which there are neurological states such as Autism Spectrum Disorders, but also frequent cardiac arrhythmias and QT prolongation.

In a series of breakthrough papers written by Alan Hodgkin and Andrew Huxley, the giant axon of the squid was separated into three major ionic currents contributing to the excitability of the membrane: the potassium current (I_{K}), the sodium current (I_{Na}) and the leak current (I_{L}). In order to quantify the relationship between voltage, time and ionic currents in one major expression of conductance and excitability, let us consider the individual cardiomyocyte or neuron to be a simple circuit with capacitors and resistors. As such, Hodgkin and Huxley define electrical activity in the following manner:

Here, Cm represents membrane capacitance, dVm/dt represents intracellular field potential, measured in mV, as a function of time, *I*ion as the current of interest, and *I*ext as the externally contributed current.

In order to quantify *I*ion in experimental terminology, we write:

Here, the ionic current is equated to the product of ion channel conductance and driving force. From an electrochemical standpoint, this can also be considered as an electrical work simulation, in which case it would be equivalent to write:

Here we equate electrical work to the product of intracellular potential and change in the magnitude of charge.

To simplify understanding of the Hodgkin-Huxley implications in electrophysiology, one must utilize a contemporary understanding of ion channel physiology. In order for a single channel to be considered open, all corresponding m, h and n gates must be open. The probability of a single channel being open is related to the membrane potential (measured in mV), and its implementation is written as:

Here, the activity of a voltage-gated channel is described as being the product of the open probability of the channel (which is, by association, dependent on membrane potential) and the number of active channels. For each of the voltage gates (m, n and h), the generalized Hodgkin-Huxley differential equation is as follows:

Here, alpha and beta and represent derived rate constants, p is the probability that a channel is in the open state, and V, as usually written, is membrane potential. The above mathematical proofs, equations and principles define the initiation and propagation of the action potential, as it relates to conductance, current and membrane potential.

1.2. Electrocardiographic Imaging (ECGI)

The differential equations which describe the complex gating activity and membrane excitability as defined in the above equations have indeed revolutionized computational physiology. However, these models and analyses have had much less prominence in point of care solutions like the electrocardiogram (ECG), which has remained almost unchanged for the last century.

Other than the ECG, traditional forms of cardiac interaction for patients include echocardiography and the less widely available forms of catheter based ablation, such as radiofrequency ablation. From a diagnostic standpoint, these invasive methods are moderate in their efficaciousness, with anti-arrhythmic drug agents remaining even more limited in success. For this reason, ventricular tachycardia and ventricular fibrillation remain as common and lethal diseases, even and especially in, the developed world. As such, there has recently been a push towards innovation in non-invasive cardiac mapping, giving rise to the electrocardiographic imaging (ECGI) methodology. In this system, functional imaging of the heart is perfomed by a marriage of CT scan and multieletrode based ECG, which provides a prominent geometric vector based map of the heart’s electrical activity. This area of analysis has proven quite effective as a diagnostic, and is a fast-moving area for computational analysis of ions as vector expressions.

1.3. Dipole and Current Density Analyses

As the ECG has been the primary diagnostic tool to assess cardiac activity, there have been several innovations like the ECGI and other forms of real-time cardiac mapping using imaging and electrophysiological analyses. However, one common feature of all of these analyses is the ubiquitous use of voltage as an assessment of cardiac electrical activity. The fundamental setback with this methodology is that such analysis includes dipole density, which is the area of interest, but also far field voltage, which promote background noise, and inherently disrupts the accuracy of detection.

As such, technologies have been develop to assess current density using magnetocardiography (MCG) to computationally analyze magnetic fields which can pick up information and data that may be untraceable by the ECG. This represents the movement of computational techniques into the clinical point of care space, and is an important transition for the future of healthcare. In addition, several contemporary technologies hinge upon the fact that current density maps of bioelectrical activity should be functional images of real-time change. Such a change would enable physicians and medical professionals to be more proactive in identifying at-risk patients for sudden cardiac death.

1.4. Conclusion and Future Work

The rise of mathematical models and computational biology have made it possible to interpret seemingly unrelated functions, map complex gene networks, and become more predictive when it comes to diagnosis of cardiac electrical activity. In the next few decades, cardiac pharmacology, which remains a difficult pharmacogenomics based problem, should become a prime area of focus using techniques similar to the ECGI and current density maps created by the MCG. In doing so, scientists will be able to provide physicians with the tools necessary to become more predictive and proactive of cardiac rhythm disturbance, thereby enabling them to begin to win the battle over sudden cardiac death.

1.5. References

1) Hodgkin, A.L., A.F. Huxley, and B. Katz, Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J Physiol, 1952. 116(4): p. 424-48.

2) Hodgkin, A.L. and A.F. Huxley, Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J Physiol, 1952. 116(4): p. 449-72.

3) Hodgkin, A.L. and A.F. Huxley, The components of membrane conductance in the giant axon of Loligo. J Physiol, 1952. 116(4): p. 473-96.

4) Rudy Y. Noninvasive Electrocardiographic Imaging (ECGI) of Arrhythmogenic Substrates in Humans. Circulation research 2013;112(5):863-874. doi:10.1161/CIRCRESAHA.112.279315.

5) Wang Y, Cuculich PS, Zhang J, et al. Noninvasive Electroanatomic Mapping of Human Ventricular Arrhythmias Using ECG Imaging (ECGI). Science translational medicine 2011;3(98):98ra84. doi:10.1126/scitranslmed.3002152.

6) Wang Y, Cuculich PS, Zhang J, Desouza KA, *et al*. Noninvasive electroanatomic mapping of human ventricular arrhythmias with electrocardiographic imaging. Sci Transl Med. 2011;3:98ra84.