SINGULAR VALUE DECOMPOSITION OF THE T WAVE:
ITS LINK WITH
 A BIOPHYSICAL MODEL OF REPOLARIZATION

A. van Oosterom
Department of Medical Physics, University of Nijmegen,
6525EZ Nijmegen  The Netherlands

Abstract: One of the more promising ways of analyzing the T wave uses the spectrum of the singular values of the matrix of body surface signals observed throughout the ST-T interval. The ratio of the sum of the minor singular values over the sum of all singular values is taken as an index of the dispersion of repolarization. At first sight this method seems to be an ad hoc procedure, derived from the domain of linear algebra in its application to abstract signal space. In this contribution it is shown that this index can in fact be directly linked to a biophysical model of the genesis of the T wave [1].

INTRODUCTION

The current interest in T wave abnormalities has given rise to various procedures, aimed at capturing the essence of these abnormalities. Initially, a promise seemed to lie in using the dispersion of the QT intervals as estimated from the QRS onset and the end of the T wave as observed in the 12-lead electrocardiogram. However, severe doubts have developed with respect to the usefulness of this method [2].

Another approach has been to compute the singular value decomposition (SVD) of the matrix of discretized body surface potentials as observed during the ST-T interval, a method that is equivalent to the Principle Component Analysis, the Karhunen-Loeve Expansion of signals and Factor Analysis.

The SVD method has a long history, stemming from the domain of linear algebra. In its application to the ECG, with the signals stored in a matrix Φ of (L × T) elements, with L the number of leads and T the number of discrete time samples, we may write Φ = U Σ V^T.

In this decomposition, the columns of matrix U represent the spatial information of the data, and the columns of matrix V represent their temporal aspect. The superscript ^T denotes the transpose of the matrix. The matrix Σ is diagonal, containing non-negative elements only: the singular values, σ, of the data matrix. In the usual representation, the singular values appear in decreasing order of magnitude along the diagonal. For the discussion to follow, we will consider separate columns of U and of V, which we denote as u_k and v_k, respectively. In this manner we write the decomposition as a summation of matrices of rank one:

Φ = σ1u1v1T + σ2u2v2T + σ3u3v3T+ …

(1)

If this summation is carried through to the maximum possible number of terms (equal to the rank of the data matrix) the representation of the data is perfect. If the spectrum of successive singular values exhibits a steep downward slope, the data can be approximated well by adding up just a few of the leading terms of (1). The associated representation error of the data can be expressed by the ratio of the sum of the discarded singular values over the sum of all singular values. Correspondingly, the ratio of the first singular value and the sum of all singular values indicates how well the data can be represented by taking just the first term. This ratio has been studied for its potential to quantify abnormal repolarization [3]. Below it is shown that this index has a direct link to the equivalent surface source (ESS) model of the cardiac electric generator [1].

THEORY

When using the equivalent surface source model the genesis of the ECG wave on the body surface can be expressed by

Φ = A S

(2)

In this discretized form of the formulation, row n (n=1,N) of the source matrix S represents the time course at T discrete time instants of the transmembrane potential at node n of a set of N nodes specifying the closed surface S_h bounding ventricular mass [1]. Each of these node potentials acts as the local strength of an elementary double layer element. Column a_n of matrix A represents the transfer of the source strength of node n to the potential at all L lead positions on the thorax, the transfer as resulting from the volume conduction properties of the thorax.

Equation (2) is a completely general formulation of the forward problem, which holds true as long as the transfer between sources and lead positions is independent of time. Here we will assume this to be the case. If the source matrix represents the ESS model, and if the maximum source strength is uniform over all nodes, the sum of all elements of any row of the transfer matrix is zero. This corresponds to the fact that in their fully polarized state myocardial cells do not produce an external field. Expressed in matrix notation:

A e = 0

(3)

with e the unit vector of dimension N.

            In [1] it is demonstrated that accurate simulated T waves are generated by the ESS model by assuming a uniform general shape, S(t), for the transmembrane potential during depolarization at all nodes on S_h, while specifying the timing of repolarization at these nodes by ρ_n , defined here as the moment at which the potential is half-way on its return to the fully polarized value. In this way we rewrite (2) as

φl(t) = Σn al,n sn,t =Σn al,n S(t-ρn)

The complete set of repolarization times at the nodes is denoted by a N-dimensional vector ρ, their mean value as  Next, we write ρ = +Δρ, in which Δρ represents the dispersion of the timing of repolarization. By using this notation we may write S(t-ρ_n) = S(t- -Δρ_n). The link between this biophysical model of the genesis of the T wave and the SVD then follows from the application of the Taylor expansion with respect to ρ_n to S(t-ρ_n) around . This reads, with the prime denoting differentiation,

S(t-ρn) = S(t- )–S′(t- )Δρn+ S″(t- )(Δρn)2-….    .

(4)

Applied to (2) and expressed in matrix form, (4) reads

Φ = A S{ } - A Δρ S′{ } + A (Δρ)2 S″{ }-….

(5)

Here the expressions involving Δρ are diagonal matrixes. All rows of S{ }are identical, likewise those of its derivatives. As a consequence we may write S{ }=es^T, with s^T any row of S, and similar expressions for its derivatives. By virtue of (3) the first term in the expansion is zero:

A S{ } = AesT = 0 .

The next term is -AΔρS′{ }=-AΔρes′^T The leading three factors on the right constitute a column vector of dimension L, the last factor a row vector of dimension T, hence the first term of the Taylor expansion is a matrix of rank one. After normalization of the row and the column vector we find an expression that is analogous to the leading term in the singular value decomposition (1). Similarly, all subsequent terms of the Taylor expansion are rank one matrixes. For small values of (the norm of) the dispersion Δρ one may expect the leading term of the singular value decomposition (1) to be identical to the first term of the Taylor expansion. For larger magnitudes of the dispersion, an increasing number of terms become significant in the Taylor expansion and, correspondingly, a larger number of terms are required in the expansion based on the SVD of the data matrix (1).

METHODS

ECG data (64 leads) in 160 normal subjects and various patient categories were collected in the course of previous studies. The data were baseline corrected and referred to a zero mean reference. SVD of the ST-T intervals was carried out. In addition, in each subject the shape of the function -S′(t- ) was estimated from the observed data matrix as e^TΦ^TΦ. This is a weighted mean of the T waves observed in all leads, the weights being the time integrals over the ST-T interval of the respective lead signals. The spectrum of singular values was analyzed. The shapes of the temporal aspects of the singular value decomposition (the vectors v_k ) were compared to the shapes of -S′(t- ) and its derivatives.

RESULTS

In normal subjects, the spectrum of singular values decreased very rapidly, and the shape of -S′(t- ) was highly correlated with that of the dominant singular vector v₁. In those cases where the initial ST segment showed small voltages, the functions representing the first and second derivatives of the estimated -S′(t- ) were highly correlated with the singular vectors v₂ and v₃. For all patient categories in which repolarization abnormalities were present, the spectrum of singular values decreased less rapidly.

DISCUSSION

The present analysis explains the link between the SVD of the ST-T signals and the underlying electrophysiology. Whenever repolarization dispersion becomes greater than normal, the ratio of the first singular value and the sum of all singular values decreases. Though not discussed here, it can be shown that other types of non-uniformity in repolarization also decrease this index.

Acknowledgments: The data analyzed were collected in various previous studies carried out in collaboration with the Department of Cardiology at our University.

REFERENCES

[1] A. van Oosterom, Genesis of the T wave as based on an Equivalent Surface Source Model. J. Electrocardiol, vol 34 S, 2001,217--227

[2] M. Malik, V. N. Batchvarov. ``Measurement, Interpretation and Clinical Potential of QT Dispersion’’, J. Am. Coll. Cardiology, vol 36/6, pp1749—1766, 2000.

[3] L. De Ambroggi, E. Aime, C. Ceriotti, M. Rovida and S. Negroni. ``Mapping of Ventricular Repolarization Potenrtials in Patients With Arrhythmogenic Right Ventricular Dysplasia;’’ Circulation; 96/12; 4314—4318; 1997.