IJBEM, Vol. 4, No. 2, 2002.

TISSUE CONDUCTIVITY AND ST DEPRESSION IN A CYLINDRICAL LEFT VENTRICLE

Peter R. Johnston
School of Science, Griffith University, Nathan, Queensland, AUSTRALIA, 4111

Abstract: Using a cylindrical model of the left ventricle, various sets of cardiac tissue conductivity values are used to study the relationship between the degree of subendocardial ischaemia and the observed potential distribution on the epicardium during the ST segment. The disparity between the observed potential distributions is quite marked and suggests that the measurement of cardiac tissue conductivity warrants further investigation.

INTRODUCTION

Models of the electric potential in the myocardium during the ST segment of the electrocardiogram can provide insight into the relationship between the degree of subendocardial ischaemia and the observed potential distribution on the epicardium [1]. In such models, there can be a large number of parameters which affect the observed potential distribution. Some parameters (eg. geometrical) are varied simply to show the effects of different physical situations, but others (eg. conductivity) are measured from experiments and should be fixed. Here, attention will be focused on the effect of the tissue conductivity by utilising several different parameter sets available in the literature [2, 3, 4]. The distributions obtained are also compared with those found experimentally [5].

METHODS

Consider a model of the left ventricle as an infinite cylinder (shown in cross-section at z =0 in Figure 1 with distances in cm). The inner region contains blood and the annular ring contains cardiac tissue. The z-axis extends into the page. Rotation of the fibres within the tissue is included in this model. The fibres on the endocardium are parallel to the z-axis, then they rotate linearly with r through an angle of 120º as they traverse the tissue. The model also includes an ischaemic region, shown at the cross-hatch region in Figure 1, with a narrow boundary [1], subtending an angle of 36º at the centre, as well as covering 75% of the tissue in the radial direction and extending from z = - 2 cm to z =2 cm. In this ischaemic region the resting transmembrane potential is lower than that of the normal tissue, hence a source of electrical potential is created around the boundary of this region [1].

Using the bidomain model [6] for electric potential in cardiac tissue, it can be shown that the equation governing the extracellular potential, Φ_e , is given by

where Φ_m is the transmembrane potential and M_i and M_e are conductivity tensors in the intracellular and extracellular spaces, respectively, and are based on the intra- and extracellular longitudinal and transverse conductivities σ_i^l , σ_i^t , σ_e^l and σ_e^t.

Note that the gradient operator,

, is given in cylindrical coordinates. The potential in the blood is governed by Laplace's equation.

The boundary conditions required to solve these differential equations are a bounded potential on the z-axis, continuity of potential and current on the endocardium and an insulated epicardium.

Briefly, the differential equations are solved using methods based on Fourier Series (in θ), Fourier Transforms (in z) and the finite difference method (in r).

Three sets of conductivity values are shown in Table I corresponding to the most popular values cited in the literature. The conductivity of blood is set at 0.0067 S/cm.

RESULTS

Fig. 2 shows the contour maps of epicardial potentials for the three different sets of conductivity data shown in Table I and the geometric model described above. For display purposes, the cylinder has been cut along the z direction opposite the region of subendocardial ischaemia and lain flat.

TABLE I
Measured bidomain conductivities (S/cm)

	Clerc [2]	Roberts et al. [3]	Roberts & Scher [4]
σ_i^l	0.0017	0.0028	0.0034
σ_i^t	0.00019	0.00026	0.0006
σ_e^l	0.0062	0.0022	0.0012
σ_e^t	0.0024	0.0013	0.0008

Epicardial Potential Distribution (mV)
Contour Interval = 1 mV, Minimum = -4.2 mV, Maximum = 1.7 mV

Epicardial Potential Distribution (mV)
Contour Interval= 1 mV, Minimum= -11.5 mV, Maximum = 4.2 mV

Epicardial Potential Distribution (mV)
Contour Interval = 1 mV, Minimum = -18.7 mV, Maximum = 0.005 mV

theta (radians)
(a) Conductivities from Clerc [2]

theta (radians)
(b) Conductivities from Roberts et al. [3]

theta (radians)
(c) Conductivities from Roberts & Scher [4]

Fig. 2: Contour maps of epicardial potentials for the differing conductivity parameters. Positive potentials are indicated by solid lines, negative potentials by dashed lines and the zero line is the thick solid line.

The data of Clerc [2] (Fig. 2(a)) results in regions of ST depression on two sides, in the angular direction, outside the region of ischaemia. There is also ST elevation above the region of ischaemia, as well as large potential gradients above the ischaemic boundary in the angular direction. The data of Roberts et al. [3] (Fig. 2(b)) results in a similar distribution of potential except that the ST depression is lower and the ST elevation is higher. Also larger potential gradients are observed. In contrast, the data of Roberts and Scher [4] (Fig. 2(c)) yields predominantly ST depression on the epicardial surface, which is much deeper than with the other two data sets, and only minimal ST elevation. The ST depression occurs as a valley over the region of ischaemia, as well as outside this region, as with the other two data sets. It does however, also show large potential gradients above the ischaemic boundary.

DISCUSSION

The three contour maps shown in Fig. 2 vary quite considerably, yet the only variation in the model is due to the tissue conductivity parameters, all of which have been measured experimentally at various times. The data sets of Clerc [2] and Roberts et al. [3] do at least give rise to similar distributions, although the magnitudes differ by a factor of about 2.75. Unfortunately, this number does not arise as a simple combination of the values in Table I, making it difficult to explain the difference.

When the data from Roberts and Scher [4] is used, a completely different pattern arises. This is somewhat surprising as the trend in the conductivity values, shown in columns 2 and 3 in Table I, continues into column 4. That is, the intracellular conductivities continue to increase and the extracellular conductivities continue to decrease. It might be expected that increases in ST depression for this data would be matched with an increase in ST elevation and a similar distribution to the first two data sets would be observed.

Contrasting this cylindrical model with an earlier slab model [1], it can be seen that the epicardial surface potentials have larger magnitudes with the cylindrical geometry. The reason for this is that the blood provides a smaller sink for the potentials, with this type of geometry.

It is also interesting to compare these simulations with experimental data obtained from the sheep model [5]. The data sets of Clerc [2] and Roberts et al. [3] both show ST elevation and ST depression on the epicardium and large potential gradients near the ischaemic border which qualitatively agree with the experimental data. In particular, the data set of Roberts et al. gives rise to potentials which are closer in magnitude to the experimental results.

These simulations suggest that further experimentation to determine conductivity values is perhaps warranted.

REFERENCES

[1] P. R. Johnston, D. Kilpatrick, and C. Y. Li, "The importance of anisotropy in modelling ST segment shift in subendocardial ischaemia," IEEE Transactions on Biomedical Engineering, vol. 48, pp. 1366-1376, December 2001.

[2] L. Clerc, "Directional differences of impulse spread in trabecular muscle from mammalian heart," Journal of Physiology, vol. 255, pp. 335-346, 1976.

[3] D. E. Roberts, L. T. Hersh, and A. M. Scher, "Influence of cardiac fiber orientation on wavefront voltage, conduction velocity and tissue resistivity in the dog," Circ. Res., vol. 44, pp. 701-712, 1979.

[4] D. E. Roberts and A. M. Scher, "Effects of tissue anisotropy on extracellular potential fields in canine myocardium in situ," Circ. Res., vol. 50, pp. 342-351, 1982.

[5] D. Li, C. Y. Li, A. C. Yong, and D. Kilpatrick, "Source of electrocardiographic ST changes in subendocardial ischemia," Circ. Res., vol. 82, pp. 957-970, 1998.

[6] O. H. Schmitt, "Biological information processing using the concept of interpenetrating domains," in Information Processing in the Nervous System (K. N. Leibovic, ed.), ch. 18, pp. 325-331, New York: Springer-Verlag, 1969.