For solving the inverse problem of electrocardiography Tikhonov regularization  is still the standard approach. The main disadvantage is that no prior information about the excitation of the TMVs or the temporal behavior of an action potential is taken into account. Wang et al.  already showed with a complex diffusion TMV model that an UKF can improve the resolution of the solution. In contrast, we want to show the potential of an idealized parametric UKF with a simple spatio-temporal model that approximates the depolarization of ventricular cells and the TMV excitation propagation. Also, the ill-posedness of the inverse problem is decoupled from the filter equations by using the Tikhonov regularization solution as ”measurement vector”. Due to the complete knowledge of the TMV ground truth of the simulated dataset, we are able to calculate the exact measurement and process noise covariances.
2.1 State space framework
To improve the solution of reconstructed TMVs in the ventricles with a Kalman filter, we considered the following state space representation of the TMVs(1)(2)
Equation 1 as the state space difference equation contains the nonlinear dynamic model of the and the process noise of . The process noise is Gaussian distributed with zero mean and covariance .
for we used a recursive cellular automaton: the surface of the ventricles was discretised with an L = 2018 vertices mesh. Every vertex si (for i = 1, …,L) was able to activate its reachable neighbours sj (for i ≠ = j) at k + 1, if si had been above a threshold of −60 mV at time step k. If a vertice was not depolarized, its TMV was set to −80 mV. The temporal behaviour of the activation between k and k+1 (shape of the depolarization phase in an action potential) was modelled with a Heaviside step function. The cellular automaton had a constant electrical excitation conduction velocity over all time steps k.
The measurement model in Equation 2 with identity matrix describes a linear mapping between constrained Tikhonov regularisations of 2nd order  as data vector and the TMVs . The constrained Tikhonov approach had a regularisation parameter of λ2 = 10−1 for all time steps k and the TMVs were scaled to the range of [−80 mV, 20 mV]. The data vector had a Gaussian distributed additive measurement noise with zero mean and covariance .
2.2 Unscented Kalman filter
2.2.1 Unscented transform
A challenge in Kalman filtering is the highly nonlinear dynamic model . To model the Gaussian distribution of(3)
during the propagation of through , the distributions of the internal states had to be approximated. With sigma-points the unscented transform (UT) captures the first moment and the second moment of the probability distribution of , , , (4)
The term is the ith column of The scaling parameter ρ ∈ ℝ(5)
determines the spread of the sigma-points around and is influenced by(6)
The determination of the probability distribution moments of depends on the dimension L of the mesh, a primary scaling parameter α ∈ ℝ and a tertiary scaling parameter κ ∈ ℝ. The parameter was set to(7)(8)
2.2.2 UKF algorithm
– Initialization (with k = 1):
As a prior information we assumed completly repolarized ventricles (cells in rest)(9)
But as the TMV may range between −80 mV and 20 mV the initial first moment had an uncertainty ff of 100 mV. Therefore the covariance is(10)
For filtering the data at time step k = 1, the probability distribution of had to be approximated with the UT(11)
– Prediction step:
The sigma-points were propagated through(12)
The first and second moment were predicted with(13)(14)
and with constant weighting coefficients(15)(16)(17)(18)(19)
– Update step:
As a linear measurement model was used, see Equation 2, the update step was linear as well and could be implemented without an additional UT:(20)(21)(22)
The considered measurement model in Equation2 did not need a forward solution. This part is already integrated in the calculation of the data vector dk using Tikhonov reconstruction.
18.104.22.168 Process noise covariance
For calculating the process noise covariance, the ground truth at time step k of simulated TMVs was propagated through . The simulation was calculated with a cellular automaton as in . The difference between the prediction output and the ground truth was built(23)
Therefore, the process noise covariance had the form(24)
with : arithmetic mean of . Because was close to singular for all k and has a contribution to the inversion in Eq.2 0, only the diagonal of was used. The process noise in the UKF had the form(25)
22.214.171.124 Measurement noise covariance
With the output model in Equation 2, was built from the difference between the ground truth and the data(26)(27)
indicates the arithmetic mean of
2.3 Quality criterion
The results of the UKF were evaluated with a quality criterion Gk ∈ ℝ(28)
where is the mean error between and and the mean error between and . An improvement with respect to the constrained-Tikhonov regularisation of 2nd order is given for(29)
A premature ventricular contraction was simulated with a cellular automaton on the model of a healthy volunteer and corresponding BSPMs were computed , . With the optimal setting of the noise covariances and scaling parameters the UKF was able to estimate a TMV excitation spread even though the initialization assumes completely repolarized ventricles. Fig. 1 shows Gk of the UKF, which is above 0 mV for all time steps k. With an ideally parameterized UKF the standard Tikhonov approach was outperformed. With time step k = 12 as an example, Fig. 3 (b) reveals that the error in the Tikhonov solution (see Fig. 3 (c)) was almost completely suppressed. The estimation was very close to the ground truth, see Fig. 3 (a). In Fig. 2 different time behaviors of mesh point 283 were plotted. It shows the ground truth, marked as simulation, and the estimation of the UKF. Even after 1 time step the UKF converges torwards the simulation. The prediction is revealed to have an offset during the resting potential stage and a time shift during the depolarization stage. The offset and especially the time shift were corrected by the innovation of the UKF. The time shift correction can be seen between k = 10 and k = 12. The innovation is defined as(30)
and indicates that the prediction shift is forced by the data . Through Eq.20 the shift is forced by the noise process covariances and as well.
An UKF algorithm was proposed to improve previously obtained ECG imaging solutions (Tikhonov). It could be demonstrated to predict and correct estimates of TMVs in line with the expected behaviour of a Kalman filter. Results were strongly improved and very close to the ground truth using an ideal parameterization (i.e., space- and time-dependent covariance matrices that were trained on the ground truth). While this demonstrates the feasibility of the approach, reasonable assumptions have to be made for the parameterization in practical applications. Good estimates of the noise covariances will not be available in both time and space, and simplifying assumptions will need to be made, presumably also requiring an adjustment of the weighting parameters: In Eq.20 there is a strong connection between the measurement and process noise and the scaling parameters through s−k. With new parameters the reduced knowledge of the noise may partially be compensated and the filter may still do an appropriate approximation of the probability distribution with sigma-points. Still, a proper setting of the noise and scaling parameters is required to facilitate an excitation propagation with the simple spatio-temporal excitation model in use.
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About the article
Published Online: 2015-09-12
Published in Print: 2015-09-01
Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.