Fractional diffusion models of cardiac electrical propagation: Role of structural heterogeneity in dispersion of repolarization

Journal of the Royal Society Interface

Volumn 11, Issue 97, 2014, Pages

  Bueno Orovio, Alfonso ^a   Kay, David ^a   Grau, Vicente ^a   Rodriguez, Blanca ^a   Burrage, Kevin ^a,b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Cardiac tissue; Dispersion of repolarization; Electrotonic effects; Fractional diffusion; Riesz potential; Structural heterogeneity

Indexed keywords

DIFFUSION; ELECTRIC FIELDS; HEART; MODULATION; TISSUE; TISSUE ENGINEERING;

CARDIAC TISSUES; ELECTROTONIC EFFECTS; FRACTIONAL DIFFUSION; REPOLARIZATION; RIESZ POTENTIALS; STRUCTURAL HETEROGENEITY;

DISPERSION (WAVES);

ARTICLE; BIOPHYSICS; DIFFUSION; DISPERSION; EXTRASYSTOLE; FRACTIONAL DIFFUSION MODEL; HEART CONDUCTION; HEART MUSCLE; HEART MUSCLE POTENTIAL; HEART REPOLARIZATION; HUMAN; IN VIVO STUDY; NONHUMAN; SIMULATION; TISSUE HOMOGENATE; TISSUE LEVEL; ACTION POTENTIAL; ANIMAL; BIOLOGICAL MODEL; BIOLOGICAL RHYTHM; CARDIAC MUSCLE CELL; COMPUTER SIMULATION; CYTOLOGY; HEART MUSCLE CONDUCTION SYSTEM; MEMBRANE POTENTIAL; NERVE CONDUCTION; PHYSIOLOGY;

ACTION POTENTIALS; ANIMALS; BIOLOGICAL CLOCKS; COMPUTER SIMULATION; DIFFUSION; HEART CONDUCTION SYSTEM; HUMANS; MEMBRANE POTENTIALS; MODELS, CARDIOVASCULAR; MYOCYTES, CARDIAC; NEURAL CONDUCTION;

EID: 84903555093     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2014.0352     Document Type: Article

References (61)

1. Metzler R, Klafter J. 2000 The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1-77. (doi:10.1016/ S0370-1573(00)00070-3)
2. Metzler R, Klafter J. 2004 The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A, Math. Gen. 37, R161-R208. (doi:10.1088/0305-4470/37/31/ R01)
3. Clayton RH et al. 2011 Models of cardiac tissue electrophysiology: progress, challenges and open questions. Prog. Biophys. Mol. Biol. 104, 22-48. (doi:10.1016/j.pbiomolbio.2010.05.008)
4. Plonsey R, Barr R. 2007 Bioelectricity: a quantitative approach, 3rd edn. New York, NY: Springer.
5. Frank J, Langer G. 1974 The myocardial interstitium: its structure and its role in ionic exchange. J. Cell. Biol. 60, 586-601. (doi:10.1083/jcb.60.3. 586)
6. Spach MS, Heidlage JF, Dolber PC, Barr RC. 1998 Extracellular discontinuities in cardiac muscle: evidence for capillary effects on the action potential foot. Circ. Res. 83, 1144-1164. (doi:10.1161/01. RES.83.11.1144)
7. Kuhn T, Ihalainen TO, Hyvaluoma J, Dross N, Willman SF, Langowski J, Vihinen-Ranta M, Timonen J. 2011 Protein diffusion in mammalian cell cytoplasm. PLoS ONE 6, e22962. (doi:10.1371/ journal.pone.0022962)
8. Diaz PJ, Rudy Y, Plonsey R. 1983 Intercalated discs as a cause for discontinuous propagation in cardiac muscle: a theoretical simulation. Ann. Biomed. Eng. 11, 177-189. (doi:10.1007/BF02363285)
9. Luke RA, Saffitz JE. 1991 Remodeling of ventricular conduction pathways in healed canine infarct border zones. J. Clin. Invest. 87, 1594-1602. (doi:10.1172/ JCI115173)
10. Jeon J-H, Tejedor V, Burov S, Barkai E, Selhuber-Unkel C, Berg-Sørensen K, Oddershede L, Metzler R. 2011 In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106, 048103. (doi:10.1103/PhysRevLett.106.048103)
11. Barkai E, Garini Y, Metzler R. 2012 Strange kinetics of single molecules in living cells. Phys. Today 65, 29-35. (doi:10.1063/PT.3.1677)
12. Ali Tabei SM, Burov S, Kim HY, Kuznetsov A, Huynh T, Jureller J, Philipson LH, Dinner AR, Scherer NF. 2013 Intracellular transport of insulin granules is a subordinated random walk. Proc. Natl Acad. Sci. USA 110, 4911-4916. (doi:10.1073/pnas. 1221962110)
13. Camelliti P, Green C, LeGrice I, Kohl P. 2004 Fibroblast network in rabbit sinoatrial node: structural and functional identification of homogeneous and heterogeneous cell coupling. Circ. Res. 94, 828-835. (doi:10.1161/01.RES. 0000122382.19400.14)
14. Sperelakis N, McConnell K. 2002 Electric field interactions between closely abutting excitable cells. IEEE Eng. Med. Biol. Mag. 21, 77-89. (doi:10.1109/ 51.993199)
15. Spach MS, Miller III WT, Geselowitz DB, Barr RC, Kootsey JM, Johnson EA. 1981 The discontinuous nature of propagation in normal canine cardiac muscle. Evidence for recurrent discontinuities of intracellular resistance that affect the membrane currents. Circ. Res. 48, 39-54. (doi:10.1161/01.RES. 48.1.39)
16. Benson DA, Wheatcraft SW, Meerschaert MM. 2000 Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403-1412. (doi:10.1029/2000WR900031)
17. Capuani S, Palombo M, Gabrielli A, Orlandi A, Maraviglia B, Pastore FS. 2013 Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas. Magn. Reson. Imaging 31, 359-365. (doi:10.1016/j. mri.2012.08.012)
18. Kruse K, Iomin A. 2008 Superdiffusion of morphogens by receptor-mediated transport. New J. Phys. 10, 023019. (doi:10.1088/1367-2630/10/2/ 023019)
19. Hernández-Labrado GR, Contreras-Donayre RE, Collazos-Castro JE, Polo JL. 2011 Subdiffusion behavior in poly(3,4-ethylenedioxythiophene): polystyrene sulfonate (PEDOT:PSS) evidenced by electrochemical impedance spectroscopy. J. Electroanal. Chem. 659, 201-204. (doi:10.1016/j. jelechem.2011.05.008)
20. Papanicolaou GC. 1995 Diffusion in random media. In Surveys in applied mathematics (eds JB Keller, DW McLaughlin, GC Papanicolaou), pp. 205-253. New York, NY: Plenum Press.
21. Owhadi H. 2001 Anomalous diffusion and homogenization on an infinite number of scales. PhD dissertation, Thesis number 2340. É cole Polytechnique Fe'de'rale de Lausanne, Switzerland.
22. Ben Arous G, Owhadi H. 2002 Super-diffusivity in a shear flow model from perpetual homogenization. Commun. Math. Phys. 227, 281-302. (doi:10.1007/ s002200200640)
23. Riesz M. 1949 L'intégrale de Riemann-Liouville et le probleme de Cauchy. Acta Math. 81, 1-222. (doi:10.1007/BF02395016)
24. Karlsen KH, Petitta F, Ulusoy S. 2011 A duality approach to the fractional Laplacian with measure data. Publ. Mat. 55, 151-161. (doi:10.5565/ PUBLMAT-55111-07)
25. Engheta N. 1996 On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44, 554-566. (doi:10. 1109/8.489308)
26. Engheta N. 1997 On the role of fractional calculus in electromagnetic theory. IEEE Antennas Propag. Mag. 39, 35-46. (doi:10.1109/74.632994)
27. Mikhailov ID, Betancur FJ, Escorcia RA, Sierra-Ortega J. 2003 Shallow donors in semiconductor heterostructures: fractal dimension approach and the variational principle. Phys. Rev. B 67, 115317. (doi:10.1103/PhysRevB.67.115317)
28. Ketzmerick R. 1996 Fractal conductance fluctuations in generic chaotic cavities. Phys. Rev. B 54, 10 841-10 844. (doi:10.1103/PhysRevB.54.10841)
29. Marlow CA et al. 2006 Unified model of fractal conductance fluctuations for diffusive and ballistic semiconductor devices. Phys. Rev. B 73, 195318. (doi:10.1103/PhysRevB.73.195318)
30. Bueno-Orovio A, Cherry EM, Fenton FH. 2008 Minimal model for human ventricular action potentials in tissue. J. Theor. Biol. 253, 544-560. (doi:10.1016/j.jtbi.2008.03.029)
31. Decker KF, Heijman J, Silva JR, Hund TJ, Rudy Y. 2009 Properties and ionic mechanisms of action potential adaptation, restitution, and accommodation in canine epicardium. Am. J. Physiol. Heart Circ. Physiol. 296, H1017-H1026. (doi:10.1152/ajpheart.01216.2008)
32. Ten Tusscher KHWJ, Panfilov AV. 2006 Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291, H1088-H1100. (doi:10.1152/ajpheart.00109.2006)
33. Mahajan A et al. 2008 A rabbit ventricular action potential model replicating cardiac dynamics at rapid heart rates. Biophys. J. 94, 392-410. (doi:10. 1529/biophysj.106.98160)
34. Hanson B, Sutton P, Elameri N, Gray M, Critchley H, Gill JS, Taggart P. 2009 Interaction of activationrepolarization coupling and restitution properties in humans. Circ. Arrhythmia Electrophysiol. 2, 162-170. (doi:10.1161/circep. 108.785352)
35. Opthof T, Coronel R, Janse MJ. 2009 Is there a significant transmural gradient in repolarization time in the intact heart? Repolarization gradients in the intact heart. Circ. Arrhythmia Electrophysiol. 2, 89-96. (doi:10.1161/CIRCEP.108.825356)
36. Costard-Jackle A, Goetsch B, Antz M, Franz MR. 1989 Slow and long-lasting modulation of myocardial repolarization produced by ectopic activation in isolated rabbit hearts. Evidence for cardiac 'memory'. Circulation 80, 1412-1420. (doi:10.1161/01.CIR.80.5.1412)
37. Cherry EM, Fenton FH. 2011 Effects of boundaries and geometry on the spatial distribution of action potential duration in cardiac tissue. J. Theor. Biol. 285, 164-176. (doi:10.1016/j.jtbi.2011.06.039)
38. Laurita KR, Girouard SD, Rosenbaum DS. 1996 Modulation of ventricular repolarization by a premature stimulus. Role of epicardial dispersion of repolarization kinetics demonstrated by optical mapping of the intact guinea pig heart. Circ. Res. 79, 493-503. (doi:10.1161/01.RES.79.3.493)
39. Laurita KR, Girouard SD, Akar FG, Rosenbaum DS. 1998 Modulated dispersion explains changes in arrhythmia vulnerability during premature stimulation of the heart. Circulation 98, 2774-2780. (doi:10.1161/01.CIR.98.24.2774)
40. Fenton FH, Cherry EM, Hastings HM, Evans SJ. 2002 Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12, 852-892. (doi:10.1063/1.1504242)
41. Cherry EM, Fenton FH. 2004 Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory, and conduction velocity restitution effects. Am. J. Physiol. Heart Circ. Physiol. 286, H2332-H2341. (doi:10.1152/ajpheart. 00747.2003)
42. Engelman ZJ, Trew ML, Smaill BH. 2010 Structural heterogeneity alone is a sufficient substrate for dynamic instability and altered restitution. Circ. Arrhythmia Electrophysiol. 3, 195-203. (doi:10. 1161/CIRCEP.109.890459)
43. Coronel R, Wilms-Schopman FJG, Opthof T, Janse MJ. 2009 Dispersion of repolarization and arrhythmogenesis. Heart Rhythm 6, 537-543. (doi:10.1016/j.hrthm.2009.01.013)
44. Myles RC, Bernus O, Burton FL, Cobbe SM, Smith GL. 2010 Effect of activation sequence on transmural patterns of repolarization and action potential duration in rabbit ventricular myocardium. Am. J. Physiol. Heart Circ. Physiol. 299, H1812-H1822. (doi:10.1152/ajpheart.00518.2010)
45. Rutherford SL, Trew ML, Sands GB, LeGrice IJ, Smaill BH. 2012 High-resolution 3-dimensional reconstruction of the infarct border zone: impact of structural remodeling on electrical activation. Circ. Res. 111, 301-311. (doi:10.1161/circresaha.111.260943)
46. Plank G et al. 2009 Generation of histo-anatomically representative models of the individual heart: tools and application. Phil. Trans. R. Soc. A 367, 2257-2292. (doi:10.1098/rsta.2009.0056)
47. Costa CM, Campos FO, Prassl AJ, dos Santos RW, Sánchez-Quintana D, Ahammer H, Hofer E, Plank G. 2014 An efficient finite element approach for modeling fibrotic clefts in the heart. IEEE Trans. Biomed. Eng. 61, 900-910. (doi:10.1109/TBME. 2013.2292320)
48. Burrage K, Hale N, Kay D. 2012 An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34, A2145-A2172. (doi:10.1137/110847007)
49. Bueno-Orovio A, Kay D, Burrage K. In press. Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. (doi:10.1007/ s10543-014-0484-2)
50. Magin RL, Ingo C, Colon-Perez L, Triplett W, Mareci TH. 2013 Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy. Microporous Mesoporous Mater. 178, 39-43. (doi:10.1016/j.micromeso.2013.02.054)
51. Britton OJ, Bueno-Orovio A, Van Ammel K, Lu HR, Towart R, Gallacher DJ, Rodriguez B. 2013 Experimentally calibrated population of models predicts and explains intersubject variability in cardiac cellular electrophysiology. Proc. Natl Acad. Sci. USA 110, E2098-E2105. (doi:10.1073/pnas. 1304382110)
52. Walmsley J, Rodriguez JF, Mirams GR, Burrage K, Efimov IR, Rodriguez B. 2013 mRNA expression levels in failing human hearts predict cellular electrophysiological remodeling: a population-based simulation study. PLoS ONE 8, e56359. (doi:10. 1371/journal.pone.0056359)
53. Gemmell P, Burrage K, Rodriguez B, Quinn TA. 2014 Population of computational rabbit-specific ventricular action potential models for investigating sources of variability in cellular repolarisation. PLoS ONE 9, e90112. (doi:10.1371/journal.pone. 0090112)
54. Bueno-Orovio A, Pérez-García VM, Fenton FH. 2006 Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method. SIAM J. Sci. Comput. 28, 886-900. (doi:10. 1137/040607575)
55. Pavliotis GA, Stuart AM. 2008 Multiscale methods. Averaging and homogenization. New York, NY: Springer.
56. Berlyand L, Mityushev V. 2001 Generalized Clausius-Mossotti formula for random composite with circular fibers. J. Stat. Phys. 102, 115-145. (doi:10.1023/A:1026512725967)
57. Richardson LF. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110, 709-737. (doi:10.1098/rspa.1926.0043)
58. Kolmogorov AN. 1941 The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299-303. (doi:10.1098/rspa.1991.0075)
59. Cherstvy AG, Chechkin AV, Metzler R. 2013 Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes. New J. Phys. 15, 083039. (doi:10.1088/1367-2630/15/8/083039)
60. Monin AS. 1955 Equations of turbulent diffusion. Dokl. Akad. Nauk SSSR 105, 256-259.
61. Uchaikin VV. 2003 Anomalous diffusion and fractional stable distributions. J. Exp. Theor. Phys. 97, 810-825. (doi:10.1134/1.1625072)