# Non-Invasive Imaging of Cardiac Activation and Recovery

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## Abstract

The sequences of activation and recovery of the heart have physiological and clinical relevance. We report on progress made over the last years in the method that images these timings based on an equivalent double layer on the myocardial surface serving as the equivalent source of cardiac activity, with local transmembrane potentials (TMP) acting as their strength. The TMP wave forms were described analytically by timing parameters, found by minimizing the difference between observed body surface potentials and those based on the source description. The parameter estimation procedure involved is non-linear, and consequently requires the specification of initial estimates of its solution. Those of the timing of depolarization were based on the fastest route algorithm, taking into account properties of anisotropic propagation inside the myocardium. Those of recovery were based on electrotonic effects. Body surface potentials and individual geometry were recorded on: a healthy subject, a WPW patient and a Brugada patient during an Ajmaline provocation test. In all three cases, the inversely estimated timing agreed entirely with available physiological knowledge. The improvements to the inverse procedure made are attributed to our use of initial estimates based on the general electrophysiology of propagation. The quality of the results and the required computation time permit the application of this inverse procedure in a clinical setting.

### Keywords

Fastest route algorithm Non-invasive imaging Activation Recovery Electrotonic interaction## Introduction

The sequence of activation and recovery of the heart, atria and ventricles, has physiological significance and clinical relevance. The standard 12-lead electrocardiogram (ECG), commonly used in diagnostic procedures, provides insufficient information for obtaining an accurate estimate of these sequences in all types of abnormality or disease.

Since the inception of electrocardiography12,49 several methods have been developed aimed at providing more information about cardiac electric activity on the basis of potentials observed on the body surface. The differences between these methods relate to the implied physical description of the equivalent generator representing the observed potential field. The earliest of these are the electric current dipole, a key element in vectorcardiography,3,14 and the multipole expansion.16 Neither of these source models offer a direct view on the timing of myocardial activation and recovery, or other electro-physiologically tinted features.

From the 1970s onwards, the potential of two other types of source descriptions have been explored.21,22 This development stemmed from increased insight in cardiac electrophysiology and advances in numerical methods and their implementation in ever more powerful computer systems. The results of both methods are scalar functions on a surface. The solving of the implied inverse problem may, accordingly, be viewed as a type of functional imaging, which has led to their characterization as, e.g., “Noninvasive Electrocardiographic Imaging (ECGI)”42 or “Myocardial Activation Imaging.”29 A brief characterization of both methods is as follows.

The first surface source model is that of the potential distribution on a closed surface closely surrounding the heart, somewhat like the pericardium, referred to here as the pericardial potential source (PPS) model. The model is based on the fact that, barring all modeling and instrumentation errors, a unique relation exists between the potentials on either of two nested surfaces, one being the body surface, the other the pericardium, provided that there are no active electric sources in the region in between. It was first proposed at Duke by Martin and Pilkington37; its potential has subsequently been developed by several other groups.4,5,20,23,46

The second type of surface source model evolved from the classic model of the double layer as an equivalent source of the currents generated at the cellular membrane during depolarization, described by Wilson *et al.*69 Initially, this current dipole layer model was used to describe the activity at the front of a depolarization wave propagating through the myocardium.11,50 Later, Salu47 expressed the equivalence between the double layer at the wave front and a uniform double layer at the depolarized part of the surface bounding the myocardium, based on solid angle theory.62 This source description has been explored by others.8,26,29,38,45

This paper reports on recent progress made in inverse procedures based on the second type of surface source model: the equivalent double layer on the heart’s surface as a model of the electric sources throughout the myocardium; we refer to it as the EDL model. In contrast with the PPS model, the EDL source model relates to the entire surface bounding the atrial or ventricular myocardium: epicardium, endocardium and their connection at the base. As mentioned in the previous paragraph, the EDL was initially used for modeling the currents at the depolarizing wave front only. Based on the theory proposed by Geselowitz,17,18 it was found to be also highly effective for describing the cardiac generator during recovery (the repolarization phase of the myocytes). The transmembrane potentials (TMPs) of myocytes close to the heart’s surface act as the local source strength of the double layer. Several examples of its effectiveness in forward simulations have appeared in the literature.28,31,51,63 A striking example of the model’s potential in inverse procedures is seen in the paper by Modre *et al.*,39 an application dedicated to the atrial activation sequence.

In our application the TMPs’ wave forms were specified by an analytical function involving just two parameters, markers for the timing of local depolarization and repolarization. These parameters were found by using a standard parameter estimation method, minimizing the difference between observed body surface potentials (BSPs) and those based on the source description. Since the body surface potentials depend non-linearly on these parameters, a non-linear parameter estimation technique is required, which demands the specification of initial estimates. It is here that some novel elements are reported on, and a major part of the paper is devoted to their handling. The initial parameters for the timing of local depolarization were based on the fastest route algorithm, taking into account global properties of anisotropic propagation inside the myocardium. Those pertaining to recovery were based on electrotonic interaction as being the driving force for the spatial differences in the local activation-recovery interval. With the other elements of the method being similar to those used in previous work, the emphasis of the paper is on the description of the novel elements.

In the Methods section, the entire inverse procedure is described and a justification given of those model parameters that are treated as constants during the optimization procedure. In the Results section, examples of inverse estimated ventricular activation and recovery sequences are presented: those of a healthy subject, a WPW patient and a Brugada patient during an Ajmaline provocation test. The value, application and limitations of this approach are considered in the Discussion.

## Materials

The results shown in this paper are based on data recorded in three subjects. The nature of this data is summarized below. More details can be found in previous publications,1,13,35,56 in which essentially the same material was used.

In each patient, a 64-lead ECG was recorded. For each subject, MRI-based geometry data was available from which individualized numerical volume conductor models were constructed, incorporating the major inhomogeneities in the conductive properties of the thorax, i.e., the lungs, the blood-filled cavities and the myocardium.

The first subject (NH) is a healthy subject.56 This subject was included to illustrate intermediate and final results of the described inverse procedure. The other two subjects are added to illustrate clinical applications.

The second subject (WPW) is a WPW patient for whom previously estimated activation times have been published.1,13 The recorded ECGs included episodes in which the QRS displayed the typical WPW pattern, i.e., a fusion beat in which the activation is initiated at both the AV node and the Kent bundle. The location of the latter was determined invasively. The ECGs were also recorded after an AV-nodal block had been induced by a bolus administration of adenosine, resulting in an activation sequence solely originating from the Kent bundle.

The third case was a Brugada patient in whom ECG data were recorded during infusion of a sodium channel blocker (Ajmaline),35 10 bolus infusions of 10 mg, administered at 1-min intervals,71 which changes the activation and/or recovery sequence. The beats selected for analysis were: the baseline beat 5 min prior to infusion and the beat after the last bolus had been administered.

Subjects for whom the ventricular depolarization and repolarization sequences were estimated

Subject | Type | # Nodes on the heart’s surface | QRS duration ms |
---|---|---|---|

WPW | WPW patient | 697 | 119 (fusion beat) 159 (AV blocked) |

BG | Brugada patient | 287 | 104 (baseline) 124 (peak infusion) |

NH | Healthy subject | 1500 | 101 |

## Methods

### Local EDL Strength

*β*/4 at

*t*=

*τ*.

*δ*the timing of local depolarization and the factor

*β*setting the steepness of the upstroke.

*ρ*sets the position of the inflection point of the TMPs down slope and

*β*

_{1}and

*β*

_{2}are constants setting its leading and trailing slope. Note that the action potential duration,

*α*, defined as

*ρ*and the timing of local depolarization

*δ*; it may be interpreted as the activation recovery interval.24

*β*

_{1}and

*β*

_{2}were found by fitting

*S*(

*t*;

*δ*,

*ρ*) to the STT segment of the

*rms*(

*t*) curve of the 64 ECGs of the individual subjects, as is described in van Oosterom63 and motivated in van Oosterom.61 Examples of the TMP wave forms obtained from (5) are shown in Fig. 1. These may be shifted in time as appropriate in individual cases. Moreover they may be scaled in amplitude to an arbitrary, uniform level in the application to non-ischemic tissue for which, based on solid angle theory applied to a closed double layer, a uniform strength produces no external field.

### Computing Body Surface Potentials

*S*

_{v}) taken to be the local transmembrane potential \( V_{\text{m}} (t,\vec{x}) \), the potential \( \phi (t,\vec{y}) \) generated at any location \( \vec{y} \) on the body surface is

*S*

_{v}and \( B(\vec{y},\vec{x}) \) is the transfer function expressing the full complexity of the volume conductor (geometry and tissue conductivity). Previous studies26,59 indicated that an appropriate volume conductor model requires the incorporation of the heart, blood cavities, lungs and thorax. In this study, the conductivity values

*σ*assigned to the individual compartments were: thorax and ventricular muscle: 0.2 S/m, lungs: 0.04 S/m and blood cavities: 0.6 S/m.

*ℓ*of the discretized (triangulated) body surface was computed as

*n*the number of nodes of the triangulated version of

*S*

_{v}. For each moment in time this amounts to the pre-multiplication of the instantaneous column vector (source)

*S*by a (transfer) matrix

**B**, which incorporates the solid angles subtended by source elements as viewed from the nodes of the triangulated surface, scaled by the relative jump in \( \left( {\sigma_{i}^{ + } - \sigma_{i}^{ - } } \right)/\left( {\sigma_{i}^{ + } + \sigma_{i}^{ - } } \right) \) of the local conductivity values

*σ*

_{i}

^{+}and

*σ*

_{i}

^{−}at either sides of the interfaces

*i*considered.21,58

### Inverse Computation of Activation and Recovery

**δ**and

**ρ**, the difference between the potentials computed on the basis of (7) and the corresponding body surface potentials

*V*(

*t*,

*ℓ*) observed in the subjects studied. Since the source strength depends non-linearly on the parameters, the minimization procedure needs to be carried out iteratively, for which we used a dedicated version of the Levenberg–Marquardt algorithm.36 The subsequent steps of this procedure alternated between updating the

**δ**and

**ρ**estimates. Updating

**δ**was carried out on the basis of solving

Matrix **L** represents a numerical form of the surface Laplacian operator,27 which serves to regularize the solution by guarding its (spatial) smoothness, *μ*^{2} its weight26 in the optimization process and \( ||\quad ||_{\text{F}}^{2} \) the square of the Frobenius norm. Matrix **L** is the Laplacian operator normalized by the surface.26 Consequently **Lδ** is proportional to the reciprocal of the propagation velocity. As the propagation velocity of activation and recovery are in the same order of magnitude the same value for *μ* can be used for activation and recovery.

Updating **ρ** was based on the same expression after interchanging **δ** and **ρ** in the regularization operator (latter part of (8)). Since the problem in hand is non-linear, initial estimates are required for both. In previous studies, related to the activation sequence (**δ**) only, the initial estimates involved were derived exclusively from the observed BSPs.8,26,38,45 The method employed here is based on the general properties of a propagating wave front. From this initial estimate for depolarization, an estimate of the initial values of the repolarization marker, **ρ**, is worked out by including the effect of electrotonic interaction on the repolarization process.

### The Initial Estimate of the Timing of Activation

During activation of the myocardium, current flows from the intracellular space of the depolarized myocytes to the intracellular space of any of its neighbors that are still at rest (polarized at their resting potential). The activation of the latter takes about 1 ms and is confined to about 2 mm. The boundary of this region (the activation wave front) propagates toward the tissue at rest until all of the myocardium has been activated. The propagation can be likened to the Huygens process. The local wave front propagates in directions dominated by the orientation of local fibers, at velocities ranging from 0.3 m/s across fibers to 1 m/s along fibers. Under normal circumstances, ventricular depolarization originates from the bundle of His, progresses through the Purkinje system, from which the myocardium is activated.57 In humans this Purkinje system is mainly located on the lower 2/3 of the endocardial wall.9,40 In other cases, ventricular activation originates from an ectopic focus or from a combination of the activity of the His-Purkinje system and an ectopic focus. The initial estimate of the inverse procedure was based on the identification of one or more sites of initial activation, from which activation propagates. This includes normal activation of the myocardium, which can be interpreted as originating from several foci representing the endpoints of the Purkinje system. The activation sequence resulting from a single focus was derived by using the fastest route algorithm.

#### The Fastest Route Algorithm

The fastest route algorithm (FRA) determines the fastest route between any pair of nodes of a fully connected graph.70 The term ‘fully connected’ signifies that all nodes of the graph may be reached from any of the other ones by traveling along line segments, called edges, that directly connect pairs of nodes. In the current application the term edges not only refers to the edges of the triangles constituting the numerical representation of *S*_{v}, but also to the paths connecting epicardial and endocardial nodes,66 provided that the straight line connecting them lies entirely within the interior of *S*_{v}.

**A**, which has elements

*a*

_{i,j}= 1 if nodes

*i*and

*j*are connected by an edge, otherwise

*a*

_{i,j}= 0. By specifying velocity

*v*

_{i,j}for every edge (

*i*,

*j*), i.e., for every non-zero element of

**A**, the travel time

*t*

_{i,j}along the edge is

*d*

_{i,j}is the distance along the edge.

#### Edge Velocities

In the MRI based geometry of the ventricles the node distances *d*_{i,j} are known. In the application of (9), the velocities along the edges need to be specified. For want of proper estimates on the myocardial penetration sites of the Purkinje system of the individual subjects, as will be case in the ultimate, clinical application of the proposed method, relatively crude edge velocity estimates were used. Values reported for the anisotropy ratio *v*_{ℓ}/*v*_{t} of longitudinal and transverse-fiber velocities show a wide range: from 2 to 6.34,43,48,52

*v*

_{ℓ}and

*v*

_{t}, respectively. Their ratio was taken to be 2, the lower end of the range, selected in order to account for transmural rotation of the fibers.54 For transmural edges that were not normal to the local surface, the travel time

*t*

_{i,j}was taken to be

*d*and

*h*the lengths of the projections of the edge along

*S*

_{v}and normal to it, respectively. This procedure approximates locally elliptical wave fronts. The factor 4 appearing on the right in (10) results from the assumed anisotropy ratio 2. Infinite travel times are assigned to pairs (

*i*,

*j*) that are not connected by an edge. The entire set of

*t*

_{i,j}constitutes a square, symmetric matrix

**A**

_{T}, on the basis of which the FRA computes the shortest travel time between

*arbitrary*node pairs. The results form a square, symmetric travel time matrix

**T**.

#### The Initial Activation Estimate: Multi-Foci Search

*j*of any row

*i*of matrix

**T**was interpreted as the activation time at node

*j*resulting from focal activity at node

*i*only. A search algorithm was designed, aimed at identifying one focus, or a number of foci, which identified the activation sequence yielding simulated body surface potentials that most closely resemble the recorded ones. If a focus at node

*i*is taken to be activated at time

*t*

_{i}

*,*the other nodes will be depolarized at

*δ*

_{j}=

*t*

_{i}+

*t*

_{i,j}. If multiple foci are considered, the activation sequence is computed by the “first come, first served” principle: if

*K*foci are involved, the depolarization time

*δ*

_{j}is taken to be

As is shown on the righthand side of (10), *v*_{ℓ} scales the elements of **T**, and consequently of **δ**. At each step the intermediate activation sequence, **δ**, *v*_{ℓ} was approximated by max(**δ**)/*T*_{QRS}, with *T*_{QRS} the QRS duration (see Table 1). For nodes known to represent myocardial tissue without Purkinje fibers the maximum velocity was set at 0.8 m/s. No maximum velocity was defined for nodes in a region potentially containing Purkinje fibers, the nodes in the lower 2/3 of the endocardial surfaces of the left and right ventricles.

For any activation sequence **δ** tested, ECGs were computed from (7) at each of the 65 electrode positions (64 lead signals + reference). Note that, with pre-computed matrices **B** and **T**, this requires merely the multiplication of the source vector by matrix **B**. The linear correlation coefficient *R* between all elements of the simulated data matrix **Φ** and those of the matrix of the measured ECGs, **V**, was taken as a measure for the suitability of **δ** for serving as an initial estimate. The lead signals were restricted to those pertaining to the QRS interval (about 100 samples spaced at 1 ms).

When taking a single node *i* of the *N* nodes on *S*_{v} as a focus, this results in *N* basic activation sequences **δ**_{i}, *i* = 1,…,*N*, and corresponding values *R*_{i}*.* The node exhibiting the maximum *R* value was selected as a focus.

The entire procedure was carried out iteratively. During the first iteration, the value *t*_{i} *=* 0 was used, corresponding to the timing of onset QRS. In any subsequent iteration *k*, the accepted values for *t*_{i} were set at 90% of their activation times found from the previous iteration. For the nodes corresponding to the above described ‘Purkinje system’ the values for *t*_{i} were set at 40% of the previous activation times. The Purkinje systems is largely insulated from the myocardial tissue. The propagation velocity in this system ranges between 2 and 4 m/s. The 40% value of *t*_{i} represents a 2.5 times higher velocity than the one used in the previous activation sequence, which is usually around 1 m/s. Within the myocardium the differences in velocity are much smaller, limited to approximately 0.7–1 m/s.

Note that a focus can be selected more than once, in which case its activation time decreases. The iteration process, identifying a focus at each step, was continued until the observed maximum value of *R* decreased.

### The Initial Estimate of the Timing of Recovery

#### Background

In *contrast* to the situation during the activation of the myocardium, local recovery may take up to some hundred ms, while the repolarization process starts almost directly after the local depolarization. *Similar* to the situation during activation, throughout the recovery period current flows from the myocytes to their neighbors. The spatial distribution of these currents is not confined to some “repolarization” boundary, but instead is present throughout all regions that are “recovering”. Even so, some measure of the timing of local recovery at node *n* can be introduced, such as the marker *ρ*_{n} introduced previously, and its distribution over *S*_{v} can be used to quantify the timing of the overall recovery process.

The intracellular current flowing toward a myocyte is positive when originating from neighbors that are at a less advance state of recovery, thus retarding the local repolarization stemming from ion kinetics. Conversely, the current flows away when originating from neighbors at a more advanced stage of recovery, thus advancing local repolarization. The size of the two domains determines the balance of these currents, and thus the magnitude of the electrotonic interaction. At a site where depolarization is initiated, the balance is positive, resulting locally in longer action potential durations than those at locations where depolarization ends. The extent of the two domains is determined by the location of the initial sites of depolarization and overall tissue geometry.65 As a consequence, local action potential duration, *α*, is a function of the timing of local depolarization and, expressed in the notation of (4), so is *ρ*(*δ*) = *δ* + *α*(*δ*). Literature reports on the nature of the function *α*(*δ*) as observed through invasive measurements are scarce. In some reports7,15 a linear function was suggested, for which a slope of −1.32 was reported. The function used in our study involved the subtraction of two exponential functions. For small distances between a local depolarization (its source) and local ending of activation (its sink) this function becomes linear in approximation.65

#### The Initial Recovery Estimate

**ρ**was found from

**ρ**(

**δ**) =

**δ**+

**α**(

**δ**), with

**δ**the initial estimate of the timing of depolarization. The value of

*α*

_{n}at any node

*n*was computed as

*ξ*a dimensionless shape constant.65 Sources and sinks of activation were identified as nodes of

*S*

_{v}for which all neighbors in a surrounding region of 2 cm were activated earlier or later, respectively.

If more than one source and sink was found within a distance of 4 cm, the average value of the parameters *x*_{n} and *κ*_{n} was assigned to the node *n* involved. The value of \( \bar{\alpha } \) was found through lining up \( \bar{\rho } = \bar{\delta } + \bar{\alpha } \) with the apex of the *rms*(*t*) signal of the observed ECG signals.64

#### Quality Measures

The differences between simulated and recorded potential data are quantified by using the *rd* measure: the root mean square value of all matrix elements involved relative to those of the recorded data. In addition, the linear correlation coefficient *R* between all elements of simulated and reference data is used.

## Results

For all three cases, the activation and recovery sequences as found by means of the inverse procedure are presented. Some intermediate results pertaining to the healthy subject are included.

### Healthy Subject

For all three cases considered, the weight parameter of the regularization operator, *μ* (see (8)), was empirically determined and set to 1.5 × 10^{−4}, both while optimizing activation and recovery.

The upstroke slope, *β* (see (1))*,* was set to 2, resulting in an upstroke slope of 50 mV/ms. The parameter *ξ* (12) was tuned such that, for the healthy subject (NH), the linear slope of **α**(**δ**) was −1.32 (see Franz *et al.*15). This resulted in a value of 7.9 × 10^{−3} for *ξ.*

*rd*value (0.12, Table 2).

*R* and *rd* values of the 3 subjects, computed over the segments indicated, pertaining to the initial estimate (focal search) and the final solution

Subject | Initial activation (QRS segment only) | Initial recovery (QRST segment) | Activation & recovery (QRST segment) | |||
---|---|---|---|---|---|---|

| Correlation |
| Correlation |
| Correlation | |

NH | 0.69 | 0.85 | 0.73 | 0.80 | 0.12 | 0.99 |

WPW | ||||||

Fusion | 2.46 | 0.87 | 2.39 | 0.84 | 0.19 | 0.98 |

AV blocked | 1.72 | 0.86 | 2.39 | 0.84 | 0.17 | 0.99 |

BG | ||||||

Baseline | 1.41 | 0.89 | 1.31 | 0.82 | 0.15 | 0.99 |

Peak infusion | 1.84 | 0.84 | 2.20 | 0.36 | 0.17 | 0.99 |

### Fusion Beat with Kent Bundle

*et al.*13) subsequently 3 focal areas were determined: one on the lower left septal area and two on the right ventricular wall. The estimated repolarization times have a small dispersion (32 ms, see Fig. 7b), which is in agreement with the fact that the heart is activated from both the Kent Bundle and the His-Purkinje system.

### Brugada Patient

*rms*signal of the recorded BSPs (insets Fig. 9): the QRS broadened and the ST segment became slightly elevated following the administration of Ajmaline.

The first focal area was found on the left side of the septum for the baseline beat and at peak Ajmaline. For the baseline beat two additional foci were found, one on the left and one on right lateral wall (see Fig. 9a). The estimated activation patterns of both analyzed beats are similar, although the activation times of the Ajmaline beat were later near the left and right base of the heart (Figs. 9a and 9b).

### Overall Performance of the Procedure

For each step in the inverse procedure the correlation and *rd* values are calculated between the measured ECG and the simulated ECG (see Table 2). For all subjects the resulting inverse procedure *rd* values were small. The initial estimates, however, showed high *rd* (>0.7) values despite the fact that the corresponding correlation was well above 80%.

**α**(

**δ**) in the initial solution was close to −1.32 for most subjects. After optimization the slope values decreased for all cases, except in the Brugada patient at peak Ajmaline (Table 3).

The range of ARI (*α*), repolarization times (*ρ*) and the slope of the linear regression between ARI and depolarization times (*δ*)

Subject | (Initial) Slope (mV/ms) | Range | Range |
---|---|---|---|

NH | (−1.32) −1.17 | 176–300 (124) | 275–324 (49) |

WPW | |||

Fusion | (−1.21) −0.83 | 168–263 (95) | 271–306 (35) |

AV blocked | (−1.08) −0.93 | 119–268 (149) | 248–336 (88) |

BG | |||

Baseline | (−1.27) −1.12 | 182–297 (116) | 244–317 (73) |

Peak infusion | (−1.26) −0.92 | 108–302 (193) | 183–358 (174) |

Computation times of the focal search algorithm and the optimization procedure

Subject | Focal search | Optimization | ||
---|---|---|---|---|

# Scans | Computation time (s) | # Iterations | Computation time (s) | |

NH | 9 | 233 | 10 | 1143 |

WPW | ||||

Fusion | 4 | 61 | 17 | 208 |

AV blocked | 2 | 23 | 85 | 993 |

BG | ||||

Baseline | 6 | 6 | 23 | 31 |

Peak infusion | 1 | 1 | 36 | 46 |

## Discussion

For all of the three cases presented, the inversely estimated timing of activation and recovery agreed well with available physiological knowledge. The resulting ECGs closely matched the measured ECGs (*rd* ≤ 0.19, correlation ≥ 0.98, see Table 2). The quality of the results and the required computation time hold promise for the application of this inverse procedure in a clinical setting.

In previous studies the required initial estimates were derived exclusively from the observed BSPs.8,26,38,45 The robustness of these initial estimates was limited in the sense that small variations in the parameters of the first estimate lead to quite different outcomes of the inverse procedure. In the study presented here, the initial estimates are based on knowledge about the electrophysiology of the heart. The results show that this improves the quality of the inverse procedure significantly. The major elements of the inverse procedure are discussed below.

### Activation

A first improvement in the initial estimation procedure was the incorporation of global anisotropic propagation in the simulation of ventricular activation. When using an uniform velocity the estimated activation wave revealed earlier epicardial activation for subject NH in the anterior part of the left ventricle.55 Although no data on individual fiber orientation was available, the global handling of trans-mural anisotropy, estimated using common accepted insights,53,54,67 improved the overall performance.

A second improvement concerns the selection of foci in the multi-foci search algorithm. Within this algorithm, the correlation *R* between measured and simulated ECGs was used, instead of the *rd* values used previously. The idea to investigate the appropriateness of the correlation arose from the observation that the overall morphology of the simulated wave forms closely corresponded to the measured data, in spite of relatively high *rd* values. This was first observed in an application to the relatively simple atrial activation sequence, frequently involving just the “focus” in the sinus node region.56 Subsequently, it also proved to be effective in applications to the ventricles.

Each simulated activation sequence of the initial multi-foci search was scaled by an estimation of the global propagation velocity (*v*_{ℓ}) derived from the QRS duration, taking into account the differences in propagation velocity in myocardial and Purkinje tissue. The QRS duration was derived from the *rms*(*t*) curve, computed from all leads referred to a zero-mean signal reference.30 This produces the optimal estimate of the global onset and completion of the activation process.

This initial estimation procedure proved to be very insensitive to slight variations in parameters settings. This can be observed from the first subject presented, NH, yielding an initial estimate that agrees well with literature data.10,32,72 The final activation times, resulting from the subsequently applied inverse procedure, globally resemble those of the initial activation sequence (Figs. 3a and 3b). Further visual inspection of the final activation sequence also showed no unphysiological phenomena.

The inverse solutions for the WPW patient have been published by Fisher *et al.*13 In their report, the solution presented solution was based an initial estimate derived from using the critical point theorem.29 The solution for the fusion beat clearly identified the actual, invasively determined location of the accessory pathway location, but the accompanying initiation of activation at the septum and the right ventricle were not found. In our current inverse procedure, the initial estimated activation sequence of the same beat (see Fig. 7a) the accessory pathway location was identified in the first run of the multi-focal search, followed by locations from where the ventricle normally is activated.10 The estimated activation sequence thus not only shows the correct position of the Kent bundle, but also a true fusion type of activation resulting from early activation in the right ventricle and left septum, as is to be expected in this situation. For the situation in which the AV node was blocked a single focus was found at the approximate location of the Kent bundle (see Fig. 7b).

A reduction in propagation velocity of 20–40% can be found44 after the administration of Ajmaline, which is reflected in the estimated activation sequences before and after Ajmaline administration (compare Figs. 9a and 9b). Note that the earliest site of activation in both sequences are approximately the same. These activation patterns are similar, suggesting that the sodium channel blocker has a global influence on the propagation velocity within the heart (Figs. 9a and 9b) and a more pronounced effect in the right basal area, in accordance with Linnenbank *et al.*35

### Recovery

In previous studies only the cardiac activation times were estimated from body surface potentials.8,35 In the current study the recovery sequence, as quantified by the timing of the steepest down slope of the local transmembrane at the heart’s surface, is included as well. For the initial estimate of the ventricular recovery sequence, **ρ**, a quantification of the effect of electrotonic interaction on the recovery process is used, expressed by its effect on the local the activation recovery interval (ARI). Few invasive data are available on the ventricular activation recovery intervals.60 Generally these are derived from potentials measured on the endocardial and epicardial aspects of the myocardium.19,32,42

The linear regression slope of the **α**(**δ**) curves for all 5 cases was within the range as found by Franz *et al.*15 (−1.3 ± 0.45). The estimated slopes revealed slightly smaller values in the right ventricle than those in the left ventricle and septum (Fig. 4). These results suggest that electrotonic interaction is a major determinant of the action potential duration. Consequently the ARI depends on the activation time, resulting in similar patterns for ARI and activation times. These findings are in contrast with the ARI values based on the PPS source model found by Ramathan *et al.*,42 in which local ARI is almost completely uncoupled from the local activation time. The differences in the ARI values estimated by both methods can be attributed to the fact that the local TMP waveforms cannot be extracted uniquely from the, more global, electrograms used in the PPS based inverse procedure to extract local recovery times.

The dispersion in the recovery times found was smaller than those of the activation times, in agreement with the negative slopes observed for the **α**(**δ**) function. The ranges of the activation times found were about twice as large as those of the repolarization times for the normal cases (NH and BG (1 min), Table 3). The apex-to-base differences in the recovery times were small (20–30 ms), which is in accordance to literature data.42 At several sites the local transmural recovery differences were more substantial (Figs. 6–9). Such large transmural recovery differences (frequently referred to by the misnomer recovery *gradients*), were found throughout the ventricles in all ‘normal’ subjects (BG baseline and NH), but not in the right ventricle of the Brugada patient (BG) after the administration of Ajmaline. It is unknown whether the recovery times of the Brugada patient match reality, but the locations having the largest deviations in recovery time do match common knowledge.71 An explanation for the short ARI value (and the advanced activation) might lie in the fact that this area is not activated at all due to structural changes,68 an option not permitted by the presented inverse procedure.

### Action Potential Wave Forms of the Source Model

The description of the transmembrane potential waveform used for driving the EDL source model (5), Fig. 1, proved to be adequate. When testing more refined variants only minor differences in the resulting isochrone patterns were observed, which is an indication of the robustness of the inverse procedure (see Appendix).

## Limitations

The EDL source model involved assumes a uniform equal anisotropy ratio throughout the myocardium.18 A quantitative judgment on the quality of the initial estimate and its influence on the estimate of activation and recovery times could not be given for want of a complete set of invasive data pertaining to the entire myocardium.

## Conclusions

In all three cases, the inversely estimated timing agreed with available physiological knowledge. The progress made in the inverse procedure using an equivalent double layer source model is attributed to our use of initial estimates based on the general electrophysiology of propagation. The quality of the results and the required computation time hold promise for a future application of this inverse procedure in a clinical setting.

## Acknowledgments

We would thank Gerald Fisher for providing us with the data of WPW patient. We would also like to thank Pascal van Dessel and his co-workers for permitting us to use the Brugada patient’s data, recorded during an Ajmaline provocation test.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.