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Electrocardiography Theory

Field: Cardiology

Sequence of Expressions

Let ϕ(r,t)\phi(\mathbf{r}, t) be the transmembrane electrical potential field in the atrial myocardium, where r∈Ωatria⊂R3\mathbf{r} \in \Omega_{atria} \subset \mathbb{R}^3 and tt is time. The P-wave generation is governed by the modified cable equation, incorporating the local current density Jion\mathbf{J}_{ion} and the extracellular current Jext\mathbf{J}_{ext}: \n∇⋅Jext=∂ϕ∂t+∑i∂Iion,i∂t\nabla \cdot \mathbf{J}_{ext} = \frac{\partial \phi}{\partial t} + \sum_{i} \frac{\partial I_{ion, i}}{\partial t} \nThe morphology of the P-wave, ϕP(r,t)\phi_{P}(\mathbf{r}, t), reflects the spatio-temporal gradient of the potential, ∇ϕ\nabla \phi, originating from the Sinoatrial (SA) node source SSA\mathbf{S}_{SA}. The rate of depolarization is characterized by the local conduction velocity vcond=−∇ϕ/Jext\mathbf{v}_{cond} = -\nabla \phi / \mathbf{J}_{ext}.
Define the ventricular depolarization process by the change in transmembrane potential ΔV(r,t)=Vmax−Vrest\Delta V(\mathbf{r}, t) = V_{max} - V_{rest}. The QRS complex is modeled by the rapid propagation of the electrical wavefront E(r,t)\mathbf{E}(\mathbf{r}, t) through the ventricular myocardium Ωvent\Omega_{vent}. The governing equation for the potential change is: \n∂V∂t=12κ∇2V−vcond⋅∇V+Istim/Cm\frac{\partial V}{\partial t} = \frac{1}{2\kappa} \nabla^2 V - \mathbf{v}_{cond} \cdot \nabla V + I_{stim} / C_m \nWhere κ\kappa is the tissue conductivity and CmC_m is the membrane capacitance. The complex's duration ΔtQRS\Delta t_{QRS} is inversely proportional to the maximum conduction velocity vmax\mathbf{v}_{max} in the His-Purkinje system: ΔtQRS∝1/vmax\Delta t_{QRS} \propto 1 / \mathbf{v}_{max}.
The T-wave represents the repolarization phase, governed by the recovery of ion channel conductances. Let m,h,nm, h, n be the gating variables for the sodium, potassium, and calcium channels, respectively. The repolarization potential V(t)V(t) is determined by the steady-state solution of the biophysical model: \nV(t)=Erest−1Cm∫tdepoltIion(t′)dt′V(t) = E_{rest} - \frac{1}{C_m} \int_{t_{depol}}^{t} I_{ion}(t') dt' \nWhere Iion(t)=gˉNam(t)(V−ENa)+gˉKn(t)(V−EK)+gˉCah(t)(V−ECa)I_{ion}(t) = \bar{g}_{Na} m(t) (V-E_{Na}) + \bar{g}_{K} n(t) (V-E_{K}) + \bar{g}_{Ca} h(t) (V-E_{Ca}). The T-wave amplitude and symmetry are sensitive indicators of ion channel imbalances, particularly those affecting the potassium current IKI_K.
The PR interval measures the conduction delay ΔtPR\Delta t_{PR} across the Atrioventricular (AV) node. This delay is mathematically defined as the time integral of the potential gradient ∇ϕ\nabla \phi as the electrical impulse traverses the nodal tissue ΩAV\Omega_{AV}: \nΔtPR=∫rSArHisdt=∫rSArHisdsvcond(r)\Delta t_{PR} = \int_{\mathbf{r}_{SA}}^{\mathbf{r}_{His}} dt = \int_{\mathbf{r}_{SA}}^{\mathbf{r}_{His}} \frac{ds}{\mathbf{v}_{cond}(\mathbf{r})} \nWhere dsds is the differential path length and vcond(r)\mathbf{v}_{cond}(\mathbf{r}) is the local conduction velocity, which is significantly reduced in the AV node compared to atrial or ventricular tissue. Prolongation implies a decrease in vcond\mathbf{v}_{cond} along the path.
The total ventricular electrical activity, QTQT, is the time duration from the start of depolarization to the end of repolarization. It is defined by the integral of the transmembrane potential V(t)V(t) over the cardiac cycle: \nQT=trepol−tdepolQT = t_{repol} - t_{depol} \nTo normalize for heart rate, the corrected QT interval, QTcQTc, is calculated using the heart rate RR (in beats per minute): \nQTc=QTR=QT60/TcycleQTc = \frac{QT}{\sqrt{R}} = \frac{QT}{\sqrt{60/T_{cycle}}} \nThis correction assumes a square root relationship between the time constant and the heart rate, ensuring consistency across varying cardiac frequencies.