Solid-angle theory and heart rate adjustment of ST-segment depression for the identification and quantification of...

Determinants of the ST-segment response to exercise can be mathematically modeled by solid-angle theory, and heart rate adjustment of the magnitude of exercise-induced ST-segment depression can remodel the solid-angle relationship to provide a theoretic and practical basis for application of heart rate-adjusted indexes of ST depression in exercise electrocardiography. Solid-angle theory indicates that the magnitude of ST depression recorded at a surface electrode (epsilon) can be described as the product of spatial and nonspatial determinants: epsilon = (omega/4 pi).(delta Vm).K (equation 1), where omega is the solid angle subtending the boundary of the ischemic territory, delta Vm is the difference in transmembrane voltage between the ischemic and adjacent nonischemic regions, and K is a term correcting for differences in intracellular and extracellular conductivity and changes in end-plate conductance. As a consequence, the magnitude of ST depression recorded by a surface electrode will be proportional both to the area of ischemic territory subtended by the recording electrode, which reflects the solid angle, and to the local transmembrane potential difference, which in turn reflects the electric consequences of the metabolic severity of ischemia at the level of the myocardial cell. It follows from equation 1 that the amplitude of ST depression can accurately reflect the area of ischemic boundary only when the severity of ischemia is constant or otherwise controlled, and differences in ST depression will only reflect varying areas of underlying ischemia when similar severity of ischemia is present. During exercise the severity of ischemia is directly proportional to changes in myocardial oxygen demand and coronary blood flow, which in turn are directly related to increasing heart rate (delta HR). Because the change in transmembrane voltage across the ischemic boundary is linearly proportional to delta HR, delta Vm/delta HR remains constant as ischemia develops. Dividing the solid-angle relationship in equation 1 by delta HR and making the appropriate substitution for a constant delta Vm/delta HR then indicates that epsilon/delta HR = (omega/4 pi).(c . K) [equation 2], where c is the new constant. Under conditions where changes in conductance are proportional or small, this simplified relationship reduces to delta ST/delta HR = c'.omega [equation 3], where delta ST reflects the magnitude of ST depression recorded by the surface electrode, delta HR the change in heart rate during developing ischemia, and c' the resulting empiric constant.