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An ideally parameterized unscented Kalman filter for the inverse problem of electrocardiography

  • Christian Ritter EMAIL logo , Walther H. W. Schulze , Danila Potyagaylo and Olaf Dössel
Published/Copyright: September 12, 2015
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Current Directions in Biomedical Engineering
From the journal Current Directions in Biomedical Engineering Volume 1 Issue 1

Abstract

ECG imaging as noninvasive method is aiming to reconstruct the distribution of the transmembrane voltage amplitudes (TMVs) from the body surface potential map (BSPM). Due to the ill-posedness, standard approaches like the Tikhonov regularization method cause blurring and artefacts in the solution. To suppress blurring and artefacts, this work investigated a model based approach, the unscented Kalman filter (UKF). The intention of this paper is to show the potential of an UKF approach by using an idealized parametrization.

Keywords: Inverse problem of electrocardiography; ECG Imaging; unscented Kalman filter

1 Introduction

For solving the inverse problem of electrocardiography Tikhonov regularization [1] 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. [2] 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 Methods

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)s¯k+1=f¯(s¯k)+q¯
(2)d¯k=H¯s¯k+n¯k=I¯s¯k+n¯k.

Equation 1 as the state space difference equation contains the nonlinear dynamic model f¯(⋅):ℝ2018→ℝ2018 of the TMVs s¯k∈ℝ2018×1 and the process noise q¯∈ℝ2018×1 of f¯(⋅). The process noise is Gaussian distributed with zero mean and covariance Σq¯k∈ℝ2018×2018.

for f¯(⋅) 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 I¯∈ℝL×L describes a linear mapping between constrained Tikhonov regularisations of 2nd order [4] as data vector d¯k∈ℝL×1 and the TMVs s¯k. 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 d¯k had a Gaussian distributed additive measurement noise n¯k∈ℝL×1 with zero mean and covariance Σn¯k∈ℝL×L.

2.2 Unscented Kalman filter

2.2.1 Unscented transform

A challenge in Kalman filtering is the highly nonlinear dynamic model f¯(⋅). To model the Gaussian distribution of

(3)s_∼N(μ_,Σs_)

during the propagation of s¯ through f¯(⋅), the distributions of the internal states had to be approximated. With sigma-points X¯∈ℝL×2L+1 the unscented transform (UT) captures the first moment μ¯∈ℝL×1 and the second moment Σs¯∈ℝL×L of the probability distribution of s¯ [5], [6], [7], [8]

(4)X¯=[X¯0X¯iX¯i+L]=[[μ¯][μ¯+ρ(Σ)i][μ¯−ρ(Σ)i]]

The term (Σs_)i is the ith column of (Σs_) The scaling parameter ρ ∈ ℝ

(5)ρ=L+δ

determines the spread of the sigma-points around μ¯ and is influenced by

(6)δ=α2(L+κ)−L.

The determination of the probability distribution moments of s¯ 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)α=1

The parameter can be used to support the accuracy of the approximation by catching the kurtosis of s [9], [10] and was set to

(8)κ=0.

2.2.2 UKF algorithm

The complete UKF algorithm [5], [6], [7], [8] consists of an initialization, the sigma-points calculation, a prediction step with constant weighting coefficients and an update step.

– Initialization (with k = 1):

As a prior information we assumed completly repolarized ventricles (cells in rest)

(9)μ^¯0=[−80⋮−80].

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 Σs^¯0 is

(10)Σs^¯0=σ2⋅I¯=10000⋅I¯.

– Sigma-points:

For filtering the data at time step k = 1, the probability distribution of s¯k−1 had to be approximated with the UT

(11)X¯k−1[μ^k−1μ^k−1+ρ(Σs^¯k−1)iμ^k−1−ρ(Σs^¯k−1)i].

– Prediction step:

The sigma-points X¯k−1 were propagated through f¯(⋅)

(12)X¯k|k−1=f¯(X¯k−1).

The first and second moment were predicted with

(13)μ¯k−=∑i=02LWimX¯i,k|k−1
(14)Σs¯−k=∑i=02Lwic(χ¯i,k|k−1−μk−)(χ¯i−μ¯k−)T+Σq¯

and with constant weighting coefficients

(15)W0m=δL+δ
(16)Wim=δ2(L+δ)
(17)w0c=δL+δ+(1−α2+β)
(18)wic=12(L+δ).

The secondary scaling parameter β ∈ ℝ can be used to minimize approximation errors of higher probability distribution moments by using prior knowledge about s¯ [7], [8]. We set to

(19)β=2.

– 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)K¯k=Σs¯k−H¯T(H¯Σs¯k−H¯T+Σn¯k)−1
(21)μ^¯k=μ¯k−+K¯k(d¯k−H¯μ¯k−)
(22)Σs^¯k=(I¯−K¯kH¯)Σs¯k−.

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.

2.2.2.1 Process noise covariance Σq¯

For calculating the process noise covariance, the ground truth s¯ at time step k of simulated TMVs was propagated through f¯(⋅). The simulation s¯k was calculated with a cellular automaton as in [11]. The difference ∈¯k+1∈ℝL×1 between the prediction output μ¯k+1− and the ground truth s¯k+1− was built

(23)∈¯k+1μ¯k+1−−s¯k+1.

Therefore, the process noise covariance had the form

(24)ρ¯k+1=(∈¯k+1−∈¯¯k+1)(∈¯k+1−∈¯¯k+1)T

with ∈¯¯k+1: arithmetic mean of ∈¯¯k+1. Because ρ¯k+1 was close to singular for all k and has a contribution to the inversion in Eq.2 0, only the diagonal of ρ¯k+1 was used. The process noise in the UKF had the form

(25)Σq¯k=[ρk,110¯⋱0¯ρk,LL].
2.2.2.2 Measurement noise covariance Σn¯k

With the output model in Equation 2, Σn¯k was built from the difference between the ground truth s¯k and the data d¯k

(26)v¯=d¯k−s¯k
(27)Σn¯k=(v¯k−v¯¯k)(v¯k−v¯¯k)T.

v¯¯k indicates the arithmetic mean of v¯

2.3 Quality criterion

The results of the UKF were evaluated with a quality criterion Gk ∈ ℝ

(28)Gk=gk,d¯k−gk,μ^¯k=∑i=1L‖di,k−si,k‖L−∑i=1L‖μ^¯i,k−si,k‖L,

where gk,μ^¯k∈ℝ is the mean error between μ^¯k and s¯k,gk,d¯k∈ℝ and the mean error between d¯k and s¯k. An improvement with respect to the constrained-Tikhonov regularisation of 2nd order is given for

(29)Gk>0.

3 Results

A premature ventricular contraction was simulated with a cellular automaton on the model of a healthy volunteer and corresponding BSPMs were computed [3], [11]. 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)innovation=K¯k(d¯k−H¯μ¯k−)

and indicates that the prediction shift is forced by the data d¯k. Through Eq.20 the shift is forced by the noise process covariances n¯k and q¯k as well.

Figure 1 The figure reveals the quality criterion of the UKF for all time steps k. As G > 0 for all k, the UKF was able to improve the resolution of the Tikhonov regularization method in all time steps k. The UKF was parameterized with α = 1, β = 2, κ = 0 and used Σs^¯0=10000⋅I¯$\Sigma _{{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\underline {\hat s} _0 } = 10000 \cdot {\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\underline {I}$ as initialization. Furthermore, the ventricles were assumed to be completly repolarized with −80 mV.
Figure 1

The figure reveals the quality criterion of the UKF for all time steps k. As G > 0 for all k, the UKF was able to improve the resolution of the Tikhonov regularization method in all time steps k. The UKF was parameterized with α = 1, β = 2, κ = 0 and used Σs^¯0=10000⋅I¯ as initialization. Furthermore, the ventricles were assumed to be completly repolarized with −80 mV.

4 Discussion

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.

Figure 2 The figure shows the time behavior of the simulation, prediction, estimation and innovation of mesh point 283. After one time step the estimation converges torwards the simulation. The offset and the time shift of the prediction as output of the spatio-temporal model in the UKF is reduced through the innovation in the update step. The innovation line shows that the Tikhonov regularization solutions d¯k${\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\underline {d} _k$ have a contribution to the UKF estimation.
Figure 2

The figure shows the time behavior of the simulation, prediction, estimation and innovation of mesh point 283. After one time step the estimation converges torwards the simulation. The offset and the time shift of the prediction as output of the spatio-temporal model in the UKF is reduced through the innovation in the update step. The innovation line shows that the Tikhonov regularization solutions d¯k have a contribution to the UKF estimation.

Figure 3 Figure (a), (b) and (c) show the TMV distribution in the heart at time point k = 12. It can be seen that the UKF suppresses the noise in the Tikhonov solution (c). Due to the prior knowledge about the measurement and process noise covariance, the UKF estimation (b) is very close to the simulation (a). The UKF was able to suppress the noise in (c).
Figure 3

Figure (a), (b) and (c) show the TMV distribution in the heart at time point k = 12. It can be seen that the UKF suppresses the noise in the Tikhonov solution (c). Due to the prior knowledge about the measurement and process noise covariance, the UKF estimation (b) is very close to the simulation (a). The UKF was able to suppress the noise in (c).

Author’s Statement

  1. 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.

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Published Online: 2015-9-12
Published in Print: 2015-9-1

© 2015 by Walter de Gruyter GmbH, Berlin/Boston

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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https://doi.org/10.1515/cdbme-2015-0096
Keywords for this article
Inverse problem of electrocardiography; ECG Imaging; unscented Kalman filter
Creative Commons
BY-NC-ND 3.0

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