Source-Sink Mismatch Causing Functional Conduction Block in Re-Entrant Ventricular Tachycardia

Ventricular tachycardia (VT) caused by a re-entrant circuit is a life-threatening arrhythmia that at present cannot always be treated adequately. A realistic model of re-entry would be helpful to accurately guide catheter ablation for interruption of the circuit. In this review, models of electrical activation wavefront propagation during onset and maintenance of re-entrant VT are discussed. In particular, the relationship between activation mapping and maps of transition in infarct border zone thickness, which results in source-sink mismatch, is considered in detail and supplemented with additional data. Based on source-sink mismatch, the re-entry isthmus can be modeled from its boundary properties. Isthmus boundary segments with large transitions in infarct border zone thickness have large source-sink mismatch, and functional block forms there during VT. These alternate with segments having lesser thickness change and therefore lesser source-sink mismatch, which act as gaps, or entrance and exit points, to the isthmus during VT. Besides post-infarction substrates, the source-sink model is likely applicable to other types of volumetric changes in the myocardial conducting medium, such as when there is presence of fibrosis or dissociation of muscle fibers.

has not yet been solved. Often, a re-entrant circuit is the source of clinical VT in patients with healing or healed myocardial infarction (2). If the arrhythmia is refractory to antiarrhythmic drugs, the patient will often be treated with catheter ablation. The ablation catheter delivers radiofrequency energy to the presumed location or locations of arrhythmogenicity to prevent arrhythmia recurrence (3). Usually, in clinical cases, these areas are located on the endocardial surface or intramurally, although they can also be transmural or entirely epicardial in origin (4). In the electrophysiology laboratory, the goal is to determine the arrhythmogenic region and to ablate it to prevent electrical conduction there and thereby to stop reentrant VT from recurring. To do this efficaciously, the mechanism of re-entrant VT should be known. Although several mechanisms have been proposed, there is no general acceptance of a model that describes the electrophysiological events leading to the onset and maintenance of re-entrant VT. In this review, proposed mechanisms of re-entrant circuit formation and maintenance are compared and contrasted. Emphasis is given to the source-sink (current-load) model, in which localized volumetric changes in the conducting medium cause the activation wavefront to accelerate or decelerate, depending upon the availability of electrical current for distal activation. The work is complementary to that recently published elsewhere (5). In the prior work, a derivation of wavefront curvature equations describing source-sink mismatch was given and origins of source-sink mismatch were defined.
Herein, the shape of the superficial infarct region (i.e., scar tissue) with respect to the boundary characteristics of the re-entrant VT isthmus, and to optimal catheter ablation locations for preventing VT reinduction, is reviewed. Additional data are included to supplement the reviewed material.
A re-entrant circuit driving VT typically consists of the components shown in the diagram of Figure 1, and has a double-loop configuration. The electrical activation wavefront propagates through viable myocardial tissue comprising the infarct border zone (IBZ), which is a constrained area of the myocardial conducting medium that is bounded by infarct in proximity and often by the heart surface. The wavefront travels as a single impulse through a region known as the isthmus of the re-entrant circuit, also known as the diastolic pathway, inner pathway, and central common pathway (6). Functional or fixed areas of conduction block prevent electrical conduction laterally outward from the isthmus, as noted by the thick lines or surfaces. At the isthmus exit, the impulse bifurcates, and travels around as 2 distinct wavefronts in the opposite direction. This region outside the isthmus is known as the outer pathway (7). At the isthmus entrance, the wavefronts coalesce, and then travel again through the isthmus region as a single electrical impulse.
A radiofrequency ablation lesion across the isthmus, where the impulse is constrained, would be most effective to prevent recurrence of re-entrant VT; however, finding this region with a roving ablation catheter can be problematic (8). First, the particular re-entrant circuit morphology responsible for clinical VT may not be inducible by programmed electrical stimulation, or it may not be well-tolerated by the patient, meaning that it must be terminated before being completely mapped. Second, current mapping techniques are not always entirely accurate. Voltage mapping seeks areas of very low voltage to be used as candidate arrhythmogenic regions (9,10). The method of concealed entrainment seeks to pace within the circuit by electrical stimulation during VT, with a cycle length equal to the VT cycle length (2). The location where the activation wavefront of the re-entrant circuit is advanced in time, but the signal shapes of the 12-lead electrocardiogram are otherwise unaffected, is sought for catheter ablation. This spot tends to reside at the exit of the re-entry isthmus. Pace-mapping can also be done during sinus rhythm, using a cycle length similar to the cycle length of the clinical re-entrant tachycardia (11). However, all of these methods suffer from limited accuracy for localizing the best site and the best orientation of the lesion to create by catheter ablation. Thus, the circuit may be incompletely interrupted, raising the possibility of VT recurrence and the need for follow-up procedures, and morbidity when ablated areas interfere with normal heart function. Mapping also requires significant procedure time (12), increasing the amount of radiation the patient receives from fluoroscopy, and increasing the cost of the procedure.

RE-ENTRANT VT
For improved detection of optimal ablation sites to interrupt the re-entrant circuit, it is important to develop a model that accurately reflects the underlying electrophysiologic phenomena by which reentrant VT is initiated and maintained. Such a model could then be used to improve the explanation of the observed phenomena, and to plan the best strategy for rapid and accurate localization of arrhythmogenic regions prior to catheter ablation.
Various models have been proposed over the years to describe how re-entrant circuits form and are maintained. In Figure 2, 1 such model is shown, based on refractoriness (13). Suppose the substrate consists of islands with long refractory periods and surrounding regions with shorter refractory period as depicted ( Figure 2A). Further suppose that the region is being electrically paced with S1-S1 stimuli from the point shown by the pulse symbol. Let an S1-S2 stimulus pulse with short coupling interval then be delivered.
This premature stimulus wavefront travels to the refractory region, whereupon it blocks functionally because of refractoriness. The activation wavefront then bifurcates and travels around the refractory region. If recovery of excitability occurs prior to the distinct wavefronts propagate beyond the distal end of the refractory region, they can then enter it as shown (green arrow). Electrical activity thereupon proceeds through the formerly refractory region in the opposite direction. Supposing that recovery of excitability has occurred in proximity to the stimulus site, the impulse can then re-enter the previously excited region so that re-entry is initiated. The impulse bifurcates and forms a double-loop re-entrant circuit (lower drawing, Figure 2A). If the stimulus site is changed, the same basic circuit morphology will occur, as shown in Figures 2B and 2C, although some alterations in the timing and shape of the isthmus may result from anisotropic conduction (i.e., wavefront propagation that is slowed transverse to muscle fibers because of differences in gap junctional connections, as compared with longitudinal wavefront propagation). Based on the refractory mechanism of Figure 2, the number of possible re-entry morphologies is multitudinous, depending upon the programmed stimulus location. Induction of any particular morphology depends highly on stimulus site location. There may also be instability and polymorphic tachycardia resulting from spatial inhomogeneities in refractoriness (14). Undoubtedly, this model is valid and a likely cause for some cases of ischemic and nonischemic tachycardia. Often however, 1 or at most 2 or 3 re-entrant circuit morphologies are observed from any location driving ventricular tachycardia in both canine models and clinical studies (15,16); thus, it would not seem likely that this model would be valid in all cases.

Formation of Functional Conduction Block
Another possibility to explain the mechanism of re-entrant ventricular tachycardia is depicted in In Figure 3A, a straight channel structure of conducting medium exists between the unexcitable areas, as has been observed in a porcine model (17). In Figure 3B, the channel has a zigzag structure (18,19). there is activation throughout the entire arrhythmogenic region during sinus rhythm (20). Third, during re-entrant VT, conduction block often occurs along thin lines bounding the isthmus laterally, not over large surface areas as shown in Figure 3.
The models for VT circuit mechanism depicted in

SOURCE-SINK MISMATCH
Source-sink mismatch alters the conduction velocity at boundaries between inhomogeneous tissue volumes (21)(22)(23)(24)(25). Shown in Figure 4, the activation wavefront propagates though a volume of tissue, the source (light patterned gray color). If in the travel direction, the subsequent localized volume of conducting tissue, the sink, is smaller as compared with the source (Figure 4A), propagation of the activation wavefront will proceed; there is sufficient electrical current available to conduct in the distal direction, and the wavefront actually accelerates. When the source and sink are the same size ( Figure 4B), the wavefront propagates with no change in speed.
However, suppose that the activation wavefront propagates from a source region with relatively small volume. If the tissue going forward, the sink, is substantially larger in volume, the wavefront will slow ( Figure 4C) or even block functionally (double black lines, Figure 4D) because of the insufficiency of  shown to occur in atrial tissue (27), and by discontinuities in muscle fiber bundles (28).

SOURCE-SINK MODEL EQUATIONS
An equation that can be used to describe the conduction velocity of the activation wavefront at areas of source-sink mismatch is (5,26): where q is the conduction velocity of the propagating wavefront; q o is the conduction velocity that the propagating wavefront would have if there were no source-sink mismatch, and it can be approximated as a constant with value of 0.4 mm/ms (5); D is the diffusion coefficient, which can be approximated as a constant with value 0.2 mm 2 /ms (29); T is the thickness of the conducting medium; and DT is the spatial change in thickness of the conducting medium per unit distance (i.e., space step c) (26). Thus, to estimate q, only thickness T requires measurement, whereas DT is calculated and D and q o are approximated as constants. An example is shown in Figure 5.
For the configuration of panel A, there is no change in IBZ thickness T (DT ¼ 0) and therefore no change in wavefront curvature; the leading edge is rectilinear.
The wavefront conducts from right to left along the light gray IBZ region, superficial to the nonconducting darker gray infarct area. Thus, for the configuration shown in Figure 5A: In Figure 5B, however, there is a change from thin to thicker IBZ in the propagation direction (right to left). Over an interval of c ¼ 1 mm (1,000 mm), the

FIGURE 4 Source-Sink Mismatch and its Effect on Electrical Conduction at the Isthmus Boundary
When the source volume is larger (A) or the same (B) compared with the sink volume, activation proceeds because there is sufficient electrical current to activate myocardial cells in the distal direction. When the source volume is smaller than the sink volume, however, the activation wavefront slows (C) or blocks (D) (the latter phenomenon is denoted by the double black line) because there is less electric current available to activate myocardial cells in the distal direction. thickness changes from T ¼ 400 mm to T ¼ 1,200 mm. Therefore: Hence, for the configuration depicted in Figure 5B, the wavefront curvature becomes critically convex and the conduction velocity goes to zero. Thus, DT/T for conduction block to occur is w 2 when the space step c ¼ 1 mm. By calculating q based on the geometry of the conducting medium, it is therefore possible to predict where functional block will occur (i.e., at locations where q approaches or reaches a value of zero [q / 0]). This will transpire when the right-hand term in the equation is approximately as large or even larger than q o and thus where DT is large and T is small in magnitude. In canine post-infarction, the reentry isthmus location is the thinnest region of surviving IBZ, it tends to be located subepicardially (26,30,31), and has an average thickness T ¼ 231 mm (0.23 mm), with a much greater average thickness T ¼ 1440 mm (1.44 mm) in the outer pathway (26).
Thus, at the boundaries of isthmus to outer pathway, the sharp changes from very thin to thicker tissue would be in agreement with Figure 5B and meet the criteria for functional block to occur.
Of note, the ability of an activation wavefront to propagate through an isthmus of a given thinness is also dependent upon the frequency of stimulation  Figure 6C, there are 2 entrance points, whereas in Figure 6D   that could arise from this configuration are depicted in Figures 6E and 6F. In Figure 6E, the entrance points are adjacent to each other and the exit points are adjacent to each other, giving rise to a double-loop reentrant circuit. In Figure 6F, however, the entrance and exit points alternate, giving rise to a 4-loop (quatrefoil) re-entrant circuit, which has been observed in activation mapping (35,36).

ISTHMUS SHAPE
The polygonal isthmus shapes depicted in Figure 6 can originate from alterations in IBZ thickness as described in Figures 4 and 5. In Figure 7A, the isthmus shapes of Figure 6 are repeated (rectangular, hexagonal, and octagonal). Suppose that the superficial infarct beneath the IBZ has the corresponding shape of the isthmus boundary in 3 dimensions, as depicted in Figure 7B. Let the IBZ be directly above each of the resulting prisms shown in Figure 7B.

SOURCE-SINK MODEL
A complete rendering of an infarct, and the endocardial and epicardial surfaces of the left ventricle, with mathematical registration (38), are shown in Figure 8 for a post-infarction canine heart experiment. In the first panel, a view oriented from above the IBZ is provided. There is an evident sharp drop-off to thicker tissue at the lateral edges of the infarct. In Figure 8B, a view of the infarct is shown from the side. The IBZ is superficial to the infarct, toward the epicardial surface.
Notice the thinness of the viable substrate there.
Furthermore, there is a more gradual change to thicker IBZ away from the thinnest point, noted by an arrow.
The infarct shape in 3 dimensions geometrically resembles the trapezoidal prism of Figure 7C   Formation of Functional Conduction Block the isthmus, exits, bifurcates, and then travels as 2 distinct wavefronts along the outer pathway, which then coalesce and re-enter the isthmus region, as noted by the black arrows. The thickness map T is shown in Figure 9C, with the thinnest IBZ colored dark blue and the thickest in red. This map was constructed from histologic analysis of the area outlined as a green square in the VT activation map of  Although the thickness map in Figure 9C   there are sharp changes to thicker tissue (i.e., DT is maximized). Thus, it is at these locations that r ¼ DT/ c$T is maximized and functional block can occur.
Correspondingly, in Figure 9C, at thickness map T, there is a sharper change to other colors and thus sharper change in T, away from the lateral borders of the isthmus. Whereas, along the long axis of the prism in Figure 9F, there is a more gradual change to thicker tissue (areas with ramps in Figure 9F). This represents the more gradual change in thickness at the entrance and exit of the isthmus region, and a correspondingly more gradual change in color in Figure 9C, in correspondence to the isthmus long axis.

FOR ABLATION
In this review, several mechanisms leading to the onset and maintenance of re-entrant VT and how these affect isthmus shape have been discussed. An ablation lesion should optimally be made within the isthmus confines to interrupt the circuit because it is a constrained region of approximately 2 cm across (34). Although magnetic resonance or other imaging, such as computed tomography (40), would be needed to detect wall thinning and to determine the infarct shape for source-sink modeling, based on the model itself, the electrophysiologic properties determined when a programmed electrical stimulus is applied within the confines of the isthmus boundary might also be useful to deduce the re-entry isthmus shape and thereby the best ablation lesion. This is shown as an example in Figure 10 for a post-infarction canine experiment. During sinus rhythm, electrical activation occurs throughout the IBZ ( Figure 10A). When stimulating from within the isthmus with a coupling interval on the order of the VT cycle length (i.e., S1-S1) ( Figure 10B), functional block forms at segments of the isthmus boundary (41). Applying a premature stimulus S1-S2, as in Figure 10C, it is possible to deduce the re-entrant circuit orientation, with the protoisthmus entrance being that gap segment at the Activation maps show (A) sinus rhythm and (B) S1-S1 stimulation, (C) S1-S2 (premature) stimulation, and (D) ventricular tachycardia. At center, functional block lines forming during S1-S1, S1-S2, and ventricular tachycardia are shown overlapped for correspondence. (B to D, insets) A geometric model of the superficial infarct shape, structured as a hexagonal prism with extending ramps. Conduction block occurs at short right arrows in these insets, whereas propagation of the wavefront proceeds at longer white arrows in the directions shown. Partially redrawn from Ciaccio et al. (41). In the C activation map and its inset, a single asterisk denotes the area where unidirectional block forms during premature stimulation. In the activation map of C, a double asterisk denotes the area where bidirectional block occurs during premature stimulation. LAD ¼ left anterior descending artery; LAT ¼ lateral; SR ¼ sinus rhythm.
isthmus border across which the activation wavefront traverses during the longer coupling interval of S1-S1, but not during the short coupling interval of S1-S2, because of source-sink mismatch leading to functional block. This entrance region is noted by the single asterisk in Figure 10C. Ablation across this so-determined boundary would be used to prevent re-entrant VT reinduction. The estimate is in good agreement with the actual configuration of the reentry isthmus, shown in Figure 10D. The unidirectional block line noted by an asterisk in Figure 10C would overlie the isthmus entrance in Figure 10D, and could be ablated across to prevent reinduction of re-entrant VT. To show correspondence, the overlap of functional block lines from Figures 10B to 10D is delineated in the center panel of Figure 10.
Although no thickness data were obtained for the experiment depicted in Figure 10, the infarct configuration that would be expected from source-sink mismatch is shown in the lower right insets of Figures 10B to 10D. The isthmus shape is anticipated to be hexagonal, as in Figure 6C to 6D, and the 3-dimensional superficial infarct configuration would be expected to have the approximate shape of a hexagonal prism with ramps as shown in Figure 7 (lower center column). Gaps where propagation proceeds during S1-S1 are noted by long white arrows in the model configuration at lower right of Figure 10B.
Functional block occurs only where there are segments with sharp increase in IBZ thickness in the outward direction (short white arrows). At the shorter coupling interval of S1-S2 ( Figure 10C), functional block also occurs at the gap across the isthmus boundary closest to the stimulus site, which may be either due to its proximity and/or to a steeper DT.
This would be the expected protoisthmus entrance location (white asterisk, Figure 10C inset). After the stimulus wavefront exits the 2 other gap segments, it then bifurcates around. There is insufficient time for recovery of excitability and re-entry to occur at 1 segment (**, 85 ms, Figure 10C), but there is sufficient time for recovery at the other segment (*, 173 ms, Figure 10C), leading to re-entry ( Figure 10D). Propagation succeeds across all gap segments where lesser DT would be expected during re-entrant VT ( Figure 10D inset).
For completeness, examples of electrograms from the experiment of Figure 10A (sinus rhythm) at the level of the dashed arrow that is drawn, are shown in Figure 11. Electrogram channels from 209 to 225 correspond to the direction from tail to head of the arrow in Figure 10A. Fractionation occurs at areas where functional block lines form during S1-S1 and S1-S2 stimulation and VT (channels 218-221), in accord with a bipolar electrogram fractionation model described in detail previously (41).
Finally, an example of 4-loop (quatrefoil) re-entry from a post-infarction canine experiment is shown in Figure 12. The sinus rhythm and VT activation maps are presented in Figures 12A and 12B, respectively.
There is a degree of slowing, but not block, in prox-  Figure 12C.

EXPECTED CIRCUIT MORPHOLOGIES
In canine post-infarction, it is rare for both re-entrant circuit morphologies to be inducible in the 4-sided isthmus boundary case ( Figures 6A and B). Moreover, regardless of the polygonality of the isthmus boundary, 2 or at most 3 circuit morphologies are typically observed to be inducible at any 1 location (15,16). Thus, not all potential circuit morphologies are actually realized. For a unidirectional block line to form during premature stimulation at the FIGURE 11 Electrograms Acquired During Sinus Rhythm for the Experiment Depicted in Figure 10 The channel number is given just below each trace; the interval shown is 250 ms. Arrow in Figure 10A shows sequence from tail to head (channels 209-225). Fractionation occurs where block lines form during stimulation and re-entrant VT, in accord with a previously described model of source-sink mismatch causing electrogram fractionation (41).
Abbreviation as in Figure 9.
Ciaccio et al. protoisthmus entrance, the value of r must be sufficiently large to cause critically convex curvature at the S1-S2 coupling interval (32). If DT is insufficiently steep, r will not be large, and unidirectional block will be unlikely to occur. The figure provides an overview of source-sink mismatch as it applies to re-entrant ventricular tachycardia. Whether or not the activation wavefront will propagate within the infarct border zone region depends upon the availability of electrical current for downstream activation of the viable myocardial substrate. When the downstream volume (the sink) is of lesser or equal size as compared with the previously activating tissue (the source), there will be sufficient electrical current for activation (right column in the figure), which is applicable to most of the infarct border zone. However when the sink is substantially larger in size as compared with the source, the current available for activation downstream is likely to be insufficient. Slow conduction or block will result (left 3 columns in the figure) which are crucial components of the re-entrant ventricular tachycardia circuit and the double-loop configuration. VT ¼ ventricular tachycardia.
Ciaccio et al.