Talk:Vulnerability of cardiac dynamics

This article deals with the vulnerability of cardiac dynamics.

It seems clear from the text that the intention is to make useful observations about both mathematical models of cardiac dynamics as well as dynamics in actual cardiac tissue. Yet the current draft deals almost exclusively with mathematical models. The confusion arises in many places. For example, the second paragraph of the article contains the sentence "Reentrant arrhythmias represent a class of cardiac arrhythmias where excitation can be altered by premature excitation within a region of vulnerability". The thought is not clear. Do the authors refer to premature excitation during a reentrant arrhythmia or premature excitation to induce an arrhythmia? If the latter, which seems to be the case, then it would be important to make clear exactly what this has to do with real cardiac arrhythmias. For example, in the clinical context, premature stimuli delivered to the ventricles is often used to assess the susceptibility to ventricular tachycardia. However, from the work of Josephson, Stevenson and other, the proposed mechanism for induction of reentrant monomorphic ventricular tachycardia in this setting involves unidirectional block in an section of the reentrant pathway, and does not appear to relate to the types of vulnerability described in this article. An easy way for the authors to fix the paper would be to make it clear at the outset that the paper deals with properties of mathematical equations. Then in the last paragraph, they could offer some hypotheses of the connections of this mathematical work with actual clinical issues.

Response to 1st paragraph The confusion, I believe, arises from terminology. The two mechanisms described above are identical as documented by many investigators (specifically see Spach's work on reentry and decremental conduction). Figure 11 illustrates the decremental antegrade propagation followed by collapse = unidirectional conduction. Whether premature excitation occurs following a sinus initiated front or a reentrant front or a premature excitation process such as EADs and DADs - the results are alterations in rhythms that in both cases frequently result in reentrant arrhythmias. Both cases reflect excitation within a vulnerable region. In Case 1 (above), excitation occurs within the vulnerable region trailing an existing reentrant front. In case 2, the vulnerable region trails the s1 front. Case 3 is when premature excitation is due to spontaneous oscillation of a group of cells, for example, EADs, that fall within a vulnerable region.

These results are quite compatible with Josephson, Stephenson and others - and the issue you raise seems more a matter of terminology of the concept of unidirectional block - which is typically the result of decremental propagation and eventual collapsing front - which has been documented in many mapping studies.

In Figure 2 there is a graph of excitability and associated text, "... transiently passes through an absolute refractory state and then through a relative refractory state where excitability is gradually restored until returning to the rest state." It is unclear what an excitability of 0.5 represents. The use of the term "relative refractory period", though often used, is not precisely defined here, and using this term will not help a reader understand what is happening without a more careful definition. I believe that the key concept is that a stimulus of a fixed amplitude delivered over some spatial domain, would be able to incite a new stimulus at some places in space, but not others. Reworking the figure indicating that the authors will plot the amplitude of stimulation needed to induce a new wave, would make this clear.

Response to 2nd paragraph The issues raised about the definition of refractory period are very important - as the classical relative refractory period is stimulus amplitude dependent. Thanks very much for pointing out these important issues. I have included several new graphs that illustrate the stimulus intensity relationship with respect to single cell properties, initiating propagation and the most premature boundary of the vulnerable period. Except in a resting preparation, these sensitivities to stimulus amplitude reflect the excitability (generic and fuzzy term). I have included text related to the critical nucleus which is the physical concept essential for understanding initiating propagation in uniform and non-uniform media. As an aside - the notion of unidirectional block, as used in descriptions of initiating reentry, terminating reentry around a ring or resetting the phase of reentry in a ring is really not block but a reflection that the stimulus amplitude was insufficient to excite a region greater than the critical nucleus required for the degree of excitability (refractory) of tissue adjacent to the stimulus location. The issue with 0.5 in the figure, finding the boundary between excitable and inexcitable tissue is a function of stimulus amplitude. I arbitrarily indicated a point that was about half way up the recovery curve - but it was arbitrarily placed and not meant to imply anything magic about 0.5.

On p. 4, the sentence "Fractionation of the reentrant wave will lead to multiple spirals that will further desynchronize ventricular contraction, a condition knows as ventricular fibrillation" uses the term "fibrillation". Fibrillation is a term in cardiology. Many believe that certain properties of mathematical equations such as breaking up of waves provide a good mathematical model for fibrillation, but the sentence is mixing up the math and the cardiology. The writing could be more precise.

Response Good point - I'll alter the text to unmix the cardiology and physics of the process.

Glass and Josephson presented a different approach to analyzing the vulnerable period in excitable media based on a consideration of a resetting a pulse circulating on a one-dimensional ring (L. Glass, M.E. Josephson. Resetting and annihilation of reentrant abnormally rapid heartbeat. Physical Review Letters 75, 2059-2063 (1995)). It might be suitable to indicate that this more topological approach, might also be useful for thinking about the vulnerable period for one-dimensional excitable media.

Response This is a good point as I had overlooked the resetting aspect of excitation within the vulnerable region. The concept of phase resetting, though, has a down side. As used by Glass and Josephson, a new cycle was initiated and the old cycle terminated. For me, the resetting process is more of a phase switching process in that at the moment of S2 excitation after the vulnerable period, there are 3 circulating waves - the S1 initiated wave (or wave initiated by some other process, and both an antegrade and retrograde wave initiated by th S2. Thus does phase mean anything during this transition from the primary wave to a circulating S2 initiated wave? Following collision between the primary circulating wave and the retrograde S2 wave, both these waves are annihilated leaving only the circulating S2 antegrade wave. Glass has typically referred to this as phase resetting, but this seems more like a phase switch - where the phase reflects the phase of the first circulating wave until annihilation and then describes S2 circulation. Because the period of vulnerability depends on the s2 amplitude, the resetting profiles shown in Glass and Josephson illustrate a single case that depends on the amplitude of the s2 stimulus.

In a clinical context in cardiology, there is a large literature analyzing the effects of single and multiple stimuli in inducing ventricular tachycardia and fibrillation, e.g. see B Avitall, J McKinnie, M Jazayeri, M Akhtar, AJ Anderson and P Tchou, "Induction of ventricular fibrillation versus monomorphic ventricular tachycardia during programmed stimulation. Role of premature beat conduction delay" Circulation, Vol 85, 1271-1278 (1992). In parallel, there is a large experimental literature, e.g. PS Chen, PD Wolf, EG Dixon, ND Danieley, DW Frazier, WM Smith and RE Ideker Mechanism of ventricular vulnerability to single premature stimuli in open-chest dogs. Circulation Research, Vol 62, 1191-1209 (1988). Although it might be beyond the scope of this article to review the literature in detail, the authors should at least point to some of this other work. They should also offer some guidance on the relevance of this work to the initiation of arrhythmias such as torsade de pointes, monomorphic ventricular tachycardia, and ventricular fibrillation in the context of the present ideas on the vulnerable period.

Response Thanks for the suggestion - these references have been included. I view these works as well as the much earlier work by Josephson showing the role of Na blockade in inducing monomorphic reentry in patients, where, under drug free conditions, exhibited TdP. I have not included either Mark's paper nor ours as they deal with the role of the wave front charge as a determinant either monomorphic reentry (low front charge associated with Na channel blockade) or polymorphic reentry (high front charge reflecting the normal Na charge and amplified by the reduction in late K currents - leading to a higher apparent charge.

Minor comments:

The acronyms VP and VR are not standard and it would be better to suppress acronyms. I have replaced these although, the repetitive use of the words instead of the abbreviated version, for me, is a distraction.

On the top of the second page the sentence starting: "Altering the excitation sequence of the heart.." does not make sense - something has been omitted. Fixed

The terms "antegrade" and "retrograde" should probably be defined. I have defined them .

On p. 5, there is a mispelling "comutations" instead of "computations". Fixed

On p. 5 the sentence starting, "When a second stimulus..." has omitted some words.

Top of p. 6, typo? "a the stimulus field"

Figs. 10-17. Don't these figures show exactly the same material as in Fig. 1? If so, perhaps it should made clear why they are being presented in this format, and some more descriptive text should be added in the text. It is not clear why the graphics here should be different from those in Fig. 1.

Fig. 16 - check spelling of "boundary"

Comments by Leon Glass

Thanks for considering my suggestions.

The article seems fine now. I will comment further on the relevance of the 1995 paper with Josephson. One of the main points of the article is that "vulnerability is a generic property of excitable media." Well, how do you know? It is possible to do experiments and look at simple models and find vulnerability, but a rigorous mathematical demonstration is certainly lacking. The 1995 article, as well as a subsequent article, T. Gedeon, L. Glass. Continuity of resetting curves for FitzHugh-Nagumo equations on the circle. In: Fields Institute Communications Volume 21: Differential Equations with Applications to Biology, 225-236 (1998) sketched a strategy, based on continuity arguments, to prove vulnerability from a formal mathematical perspective. However, it was not possible to fill in all parts of the proof and this may be a challenge for some who are mathematically inclined. The relevance to vulnerability is that there is an emphasis on topological arguments, and that it might be possible to establish a more rigorous proof for the generic properties of vulnerability, at least for one dimensional propagation. An analysis of phase resetting on a pulse circulating in a one dimensional ring of excitable media is a crucial step. There is no problem about describing phase resetting in this context provided care is taken to define the phase of the reset wave based on its asymptotic phase after all transients have died out. The basic argument is: if a pulse is circulating on a ring of excitable media it is impossible to have a continuous phase resetting curve (appropriately defined) when resetting is induced by a brief, spatially localized suprathreshold stimulus. Therefore, there must be some stimulus or stimuli that would lead to a transition to another basin of attraction other than the single circulating wave. One such basin of attraction could be the annihilation of the original circulating wave by a single retrogradely circulating wave induced as a consequence of vulnerability as described by Starmer and colleagues.

These matters are sufficiently abstruse that they may not suitable for detailed discussion in the Vulnerability article, and I leave it to the author to decide what to do. But perhaps by keeping it on line in Scholarpedia, someone who is interested in this topic will be stimulated to look into this problem in the future.

Leon Glass

Response Leon: it is clear from your response that I've failed to produce a persuasive characterization of the generic nature of vulnerability. Even you you've agreed to accept the article as is, its clear that there must be a better way to either discredit the generic nature of vulnerability or accept it. My argument for the generic nature is based on the generic properties of an excitable medium: threshold of excitability, a refractory state followed by a transition back to the rest state. Within this framework, there are a number of papers addressing the initiation of a propagating wave - which requires that a region exceeding that of the critical nucleus be excited (see references below). Vulnerability, as I see it, requires two components: 1) a gradient of excitability and 2) a critical nucleus for initiating propagation. Because the critical nucleus depends on excitability, then there are conditions where, in the 1D case, undirectional conduction results from a properly timed and proper amplitude s2. Trailing every wave is a traveling gradient of excitability - ranging from inexcitable immediately behind the wave front to the rest state excitability far behind the wave front. Then, depending on the stimulus amplitude, there exists a vulnerable region where an s2 excited region excites an area that exceeds the critical nucleus requirement in the retrograde direction and excites an region less than the critical nucleus requirement in the antegrade direction. The result is retrograde propagation and decremental antegrade propagation. How to formalize this in a clean analytical framework is beyond me - but I believe this qualitative argument is correct and establishes the generic nature of vulnerability.

References:

Ostrovskii, L.A. and Yakhno, V.G. Formation of pulses in an excitable medium. Biofizika 20:489-493, 1975

Flores, G. The Stable manifold of the standing wave of the Nagumo equation, J. Differential Equations, 80: 306-314, 1989.

Flores, G. Stability analysis for the slow traveling pulse of the FitzHugh-Nagumo system, SIAM J. Math. Anal. 22: 392-399, 1991.

Starobin, J., Zilberter, Y.I. and Starmer, C.F. Vulnerability in one-dimensional excitable media. Physica D. 70:321-341, 1994

Neu, J. C., Preissig, R. S. and Krassowska, W. Initiation of propagation in a one-dimensional excitable medium, Physica D. 102:285-299, 1997.

Bountis, T., Bezerianos, T. and Starmer, C.F. Wavefront formation in an excitable medium by perturbation of solitary pulse solutions. Progress in Theoretical Physics, 139:12-33 (supplement), 2000