Cardiovascular Engineering and Technology

, Volume 4, Issue 4, pp 520–534

A Simplified Approach for Simultaneous Measurements of Wavefront Velocity and Curvature in the Heart Using Activation Times

  • Nachaat Mazeh
  • David E. Haines
  • Matthew W. Kay
  • Bradley J. Roth

DOI: 10.1007/s13239-013-0158-2

Cite this article as:
Mazeh, N., Haines, D.E., Kay, M.W. et al. Cardiovasc Eng Tech (2013) 4: 520. doi:10.1007/s13239-013-0158-2


The velocity and curvature of a wave front are important factors governing the propagation of electrical activity through cardiac tissue, particularly during heart arrhythmias of clinical importance such as fibrillation. Presently, no simple computational model exists to determine these values simultaneously. The proposed model uses the arrival times at four or five sites to determine the wave front speed (v), direction (θ), and radius of curvature (ROC) (r0). If the arrival times are measured, then v, θ, and r0 can be found from differences in arrival times and the distance between these sites. During isotropic conduction, we found good correlation between measured values of the ROC r0 and the distance from the unipolar stimulus (r = 0.9043 and p < 0.0001). The conduction velocity (m/s) was correlated (r = 0.998, p < 0.0001) using our method (mean = 0.2403, SD = 0.0533) and an empirical method (mean = 0.2352, SD = 0.0560). The model was applied to a condition of anisotropy and a complex case of reentry with a high voltage extra stimulus. Again, results show good correlation between our simplified approach and established methods for multiple wavefront morphologies. In conclusion, insignificant measurement errors were observed between this simplified approach and an approach that was more computationally demanding. Accuracy was maintained when the requirement that ε (ε = b/r0, ratio of recording site spacing over wave fronts ROC) was between 0.001 and 0.5. The present simplified model can be applied to a variety of clinical conditions to predict behavior of planar, elliptical, and reentrant wave fronts. It may be used to study the genesis and propagation of rotors in human arrhythmias and could lead to rotor mapping using low density endocardial recording electrodes.


Cardiac musclePropagation velocityWave front curvatureElectrodeAnisotropy



Wave front speed


Angle specifying wave front velocity direction


Radius of curvature


Recording sites spacing (shortest distance)


Ratio of electrode spacing over radius of curvature


Intracellular conductivity in the x-direction


Intracellular conductivity in the y-direction


Extracellular conductivity in the x-direction


Extracellular conductivity in the y-direction


Activation time at electrode n, where n = 1, 2, 3, or 4


Difference of activation times between the ith and jth electrodes


Time for wave front to travel segment i


Stimulation protocol using stimulus of strength S1 and at a later time stimulus S2


Side of square inside the tissue where fibers curve


Point of interest


Distance formula


Line segments method


Our computational method


Ventricular fibrillation


Membrane potential

Supplementary material

13239_2013_158_MOESM1_ESM.doc (1.3 mb)
Supplementary material 1 (DOC 1283 kb)

Copyright information

© Biomedical Engineering Society 2013

Authors and Affiliations

  • Nachaat Mazeh
    • 1
  • David E. Haines
    • 2
  • Matthew W. Kay
    • 3
  • Bradley J. Roth
    • 4
  1. 1.Department of Cardiovascular MedicineBeaumont Health SystemRoyal OakUSA
  2. 2.Department of Cardiovascular MedicineOakland University William Beaumont School of MedicineRoyal OakUSA
  3. 3.Department of Electrical and Computer EngineeringGeorge Washington UniversityWashingtonUSA
  4. 4.Department of PhysicsOakland UniversityRochesterUSA