I graduated on the Master “Operations Research” with the thesis titled “Non-invasive reconstruction of electrical heart activity”. This thesis handles the mathematical problem of reconstructing heart-surface potentials from body-surface potentials. We will see that this so called “inverse problem” is ill-posed, meaning that we need additional techniques to dampen the influence of noise in our reconstructions. Some regularization methods were implemented with this purpose, and a first try with human data was performed. The thesis can be found here. For a summary of this work, read on.

Summary

The procedure of reconstructing electrical heart-surface potentials from body-surface potentials. We start with a patient, take 256 potential recordings (ECGs) on his body surface, find the anatomical relationschip between his heart and body surface, and with mathematical techniques we reconstruct his heart-surface (epicardial) potentials.

In this thesis, we see that the heart is a pump that heavily depends on electric impulses to function. These electrical waves spread over the heart surface to induce contraction. The results of this electric heart activity reflect to the body surface, where we can record the electric potentials that for example are used in the standard electrocardiogram (ECG).

However, these ECGs lack sensitivity and localized detail. A novel imaging technique called non-invasive electrocardiographic imaging (ECGI) tries to reconstruct from these ECGs the corresponding heart-surface potentials, more specifically, the potentials at the outer heart wall (the epicardium). For this, we need geometrical information, for example obtained by computed tomography (CT). We also need many locations at the body surface where we measure the body-surface potentials. The reconstruction of heart-surface potentials from body-surface potentials is part of a framework that tries to enable the analysis of transmural (that is, “through the heart wall”) potentials in the heart.

Based on physical laws, we derived a method, based on previous work, that enables us to relate heart-surface potentials to body-surface potentials, to solve the so called forward problem of electrocardiography. For obtaining body-surface potentials, we need two separate sets of data: geometrical data, relating the anatomy of the heart to the anatomy of the torso; and the heart-surface potentials itself.

The Forward Problem: how to compute body-surface potentials from heart-surface potentials?

When considering a heart surface and a body surface with potentials distributed on it, as on the figure above, then the mathematical problem can be reformulated in the following two equations, with the corresponding model:

When we solve this problem, we get a formula for obtaining the potential at any location between your heart and your body surface:

The potential at any location within the volume V is determined by the potentials at all other locations.

Now, when we place the observer at all heart-surface locations and at all body-surface equations, we get a large set of equations, in which the potential at any surface location is a function of the potentials at all other surface locations. We can combine all these equations into one formula:

The transfer matrix is an essential part of the forward problem, as it captures the geometrical data to relate the heart-surface potentials to the body-surface potentials. Now we have derived a simple linear relation between the body-surface potentials and the heart-surface potentials, with the transfer matrix as key part. For various models and data sets, we validated our implementation of this linear relation and its corresponding framework. We saw a nice correspondence between our forwardly computed body-surface potentials, and the known body-surface potentials, except for some scaling deviation that did not influence the distribution itself.

However, this solution to the forward problem is of little use in the real world, as in medical practice, we will want to compute heart-surface potentials from body-surface potentials. Unfortunately, we cannot simply invert this linear relation to reconstruct heart-surface potentials from body-surface potentials, to solve this inverse problem of electrocardiography. The reason for this is the ill-posed nature of the inverse problem, meaning that a small disturbance in the data yields completely different reconstructions. As noise is always present in real-world applications, these small disturbances will surely influence our reconstructions extremely, as we have seen in our experiments.

The inverse problem of electrocardiography: how to compute heart-surface potentials from body-surface potentials?

Therefore, regularization methods are needed to gently direct the solutions into a feasible direction. Regularization methods use extra information about the solution to prefer physically more realistic solutions. We implemented Tikhonov regularization, that poses a bound on the norm of the solution, preferring low-valued solutions over extremely high-valued solutions. We also used the generalized minimal residual (GMRes) method, iterating over an expanding set of possible solutions until a reasonable solution is found.

In Tikhonov regularization, the potentials at the heart-surface are formulated as follows:

Here, the heart-surface potentials are the result of a process in which we want to minimize both the mismatch (between our regularized solution and the naive solution) and a penalty.

GMRes regularization only minimizes over the mismatch, but will only allow a certain set of solutions in this minimization process:

GMRes regularization, in which only solutions coming from a Krylov subspace are allowed.

Both methods require a regularization parameter: Tikhonov for the weight of the imposed bound, and GMRes as selection of the optimal iteration. For this, we used the L-curve method. This method plots a measure for the ill-posedness versus a measure for the mismatch. Both quantities should be as low as possible, but getting to decrease one typically increases the other. To find an optimal trade-off, we select the corner at which a decrease in one starts to have a large impact on the increase of the other.

The L-curve method (in this case for Tikhonov, but similar for GMRes) for selecting the optimal value for the regularization parameter. This parameter determines the balance between the mismatch and the influence of ill-posedness, both of which should be as low as possible.

We validated this inverse implementation in several data sets and models. Both Tikhonov and GMRes methods produced nice reconstructions under controlled circumstances. We were able to reconstruct two close maxima if not too much noise was present, but also discovered that the amount of noise present in the body-surface potentials is a key factor in determining the quality of reconstruction. Also, when the position of the heart within the torso has an uncertainty factor, this influences the reconstruction considerably. This also holds, although to a lesser extent, for uncertainty in the size of the heart.

In these difficult situations, we discovered that GMRes, by carefully selecting the correct iteration, might be able to provide reconstructions that are somewhat more detailed than Tikhonov reconstructions. In general, however, they often yield the same results, which validates both methods as good regularization methods.

First human reconstructions of heart-surface potentials. Regardings the previous experiments, these results are probably not valid, but still do represent a huge landmark in our research.

This thesis was ended with the results of our very first reconstructions based on own human data. Body-surface potential maps were measured on a healthy test person. An existing, digitized geometry (not belonging to our test person) was used as reference. Withstanding the influence of noise and geometrical uncertainty, the reconstructed heart-surface potentials during the depolarization phase of a heart beat showed resemblance with findings described in literature.