Calculations

Calculation methods used in BV Workbench.

Action potential characteristics

Where $F_0$ is the baseline, $F_max$ the peak intensity, $\Delta F$ the amplitude, $t_a$ the activation time, $t_{pk}$ the peak time, $t_r$ the decay time (repolarization time) and $\tau$ the decay constant.

note

No model is fitted to the data.

Alternans

$$\delta F(i) = \bigg|\frac{F_{max(i)}- F_{max(i-1)}}{F_{max(i-1)}} \bigg|$$

APD Alternans

$$\Delta t_{d(i)} = \big|t_{d(i)} - t_{d(i-1)}\big|$$

where $t_d = t_r - t_a$.

Diastolic Interval

$$DI(i) = t_{a(i)} - t_{r(i-1)}$$

Peak Interval

$$\Delta t_{pk(i)} = t_{pk(i)} - t_{pk(i-1)}$$

Conduction Velocity

The CV vectors come from the activation time gradient, which is calculated by a two-sided derivative in each direction.

$$\nabla t_a = \begin{bmatrix} \frac{\delta t_a}{\delta x} \\ \frac{\delta t_a}{\delta y} \\ \end{bmatrix} = \begin{bmatrix} t_{x2} - t_{x1} \\ t_{y2} - t_{y1} \\ \end{bmatrix}$$

tip

The activation time map may be smoothed by a gaussian filter of size $\omega$ and $\sigma = \omega / 2$ prior to derivation.

Frequency analysis

The Dominant Frequency Map shows the varation of the Fourier transform peak frequency.

Time-Frequency analysis

Spectrograms are generated by the the Short-time Fourier transform.

Phase singularties

See Bray et al. "Experimental and theoretical analysis of phase singularity dynamics in cardiac tissue." Journal of cardiovascular electrophysiology 12.6 (2001): 716-722.