ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Vakhlaeva A. ., Ishbulatov Y. M., Karavaev A. S., Ponomarenko V. I., Prokhorov M. D. Mathematical model of the photoplethysmogram for testing methods of biological signals analysis. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 5, pp. 586-596. DOI: 10.18500/0869-6632-003059, EDN: WGKVOM

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
Full text PDF(En):
Article type: 

Mathematical model of the photoplethysmogram for testing methods of biological signals analysis

Vakhlaeva Anna Mikhailovna, Saratov State University
Ishbulatov Yurii Michailovich, Saratov State University
Karavaev Anatolij Sergeevich, Saratov State University
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Prokhorov Mihail Dmitrievich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

The purpose of this study was to develop a mathematical model of the photoplethysmogram, which can be used to test methods that introduce the instantaneous phases of the modulating signals. The model must reproduce statistical and spectral characteristics of the real photoplethysmogram, and explicitly incorporate the instantaneous phases of the modulating signals, so they can be used as a reference during testing.

Methods. Anacrotic and catacrotic phases of the photoplethysmogram pulse wave were modeled as a sum of two density distributions for the skew normal distribution. The modulating signals were introduced as harmonic functions taken from the experimental instantaneous phases of the VLF (0.015...0.04 Hz), LF (0.04...0.15 Hz) and HF (0.15...0.4 Hz) oscillations in the real photoplethysmogram. The spectral power in the VLF, LF, and HF frequency ranges was calculated to compare the model and experimental data.

Results. The model qualitatively reproduces the shape of the experimental photoplethysmogram pulse wave and shows less than 1% error when simulating the spectral properties of the signal.

Conclusion. The proposed mathematical model can be used to test the methods for introduction of the instantaneous phases of the modulating signals in photoplethysmogram time-series.

The work was carried out within the framework of the state task of the Saratov Branch of the Institute of Radioengineering and Electronics of Russian Academy of Sciences
  1. Gorshkov O, Ombao H. Multi-chaotic analysis of inter-beat (R-R) intervals in cardiac signals for discrimination between normal and pathological classes. Entropy (Basel). 2021;23(1):112. DOI: 10.3390/e23010112.
  2. Fagard RH, Stolarz K, Kuznetsova T, Seidlerova J, Tikhonoff V, Grodzicki T, Nikitin Y, Filipovsky J, Peleska J, Casiglia E, Thijs L, Staessen JA, Kawecka-Jaszcz K. Sympathetic activity, assessed by power spectral analysis of heart rate variability, in white-coat, masked and sustained hypertension versus true normotension. J. Hypertens. 2007;25(11):2280–2285. DOI: 10.1097/HJH.0b013e3282efc1fe.
  3. Borovkova EI, Prokhorov MD, Kiselev AR, Hramkov AN, Mironov SA, Agaltsov MV, Ponomarenko VI, Karavaev AS, Drapkina OM, Penzel T. Directional couplings between the respiration and parasympathetic control of the heart rate during sleep and wakefulness in healthy subjects at different ages. Front. Netw. Physiol. 2022;2:942700. DOI: 10.3389/fnetp.2022.942700.
  4. Ponomarenko VI, Prokhorov MD, Karavaev AS, Kiselev AR, Gridnev VI, Bezruchko BP. Synchronization of low-frequency oscillations in the cardiovascular system: Application to medical diagnostics and treatment. The European Physical Journal Special Topics. 2013;222(10): 2687–2696. DOI: 10.1140/epjst/e2013-02048-1.
  5. Lefrandt JD, Smit AJ, Zeebregts CJ, Gans ROB, Hoogenberg KH. Autonomic dysfunction in diabetes: a consequence of cardiovascular damage. Current Diabetes Reviews. 2010;6(6):348–358. DOI: 10.2174/157339910793499128.
  6. Dimitriev DA, Saperova EV, Dimitriev AD. State anxiety and nonlinear dynamics of heart rate variability in students. PLoS ONE. 2016;11(1):e0146131. DOI: 10.1371/journal.pone.0146131.
  7. Deka B, Deka D. Nonlinear analysis of heart rate variability signals in meditative state: a review and perspective. BioMedical Engineering OnLine. 2023;22(1):35. DOI: 10.1186/s12938-023- 01100-3.
  8. de Abreu RM, Porta A, Rehder-Santos P, Cairo B, Sakaguchi CA, da Silva CD, Signini EF, Milan-Mattos JC, Catai AM. Cardiorespiratory coupling strength in athletes and non-athletes. Respiratory Physiology & Neurobiology. 2022;305:103943. DOI: 10.1016/j.resp.2022.103943.
  9. Delliaux S, Ichinose M, Watanabe K, Fujii N, Nishiyasu T. Muscle metaboreflex activation during hypercapnia modifies nonlinear heart rhythm dynamics, increasing the complexity of the sinus node autonomic regulation in humans. Pflugers Archiv - European Journal of Physiology. 2023;475(4):527–539. DOI: 10.1007/s00424-022-02780-x.
  10. Karavaev AS, Skazkina VV, Borovkova EI, Prokhorov MD, Hramkov AN, Ponomarenko VI, Runnova AE, Gridnev VI, Kiselev AR, Kuznetsov NV, Chechurin LS, Penzel T. Synchronization of the processes of autonomic control of blood circulation in humans is different in the awake state and in sleep stages. Front. Neurosci. 2022;15:791510. DOI: 10.3389/fnins.2021.791510.
  11. Goldstein DS, Bentho O, Park MY, Sharabi Y. Low-frequency power of heart rate variability is not a measure of cardiac sympathetic tone but may be a measure of modulation of cardiac autonomic outflows by baroreflexes. Exp. Physiol. 2011;96(12):1255–1261. DOI: 10.1113/expphysiol. 2010.056259.
  12. Natarajan A, Pantelopoulos A, Emir-Farinas H, Natarajan P. Heart rate variability with photoplethys-mography in 8 million individuals: a cross-sectional study. The Lancet Digital Health. 2020;2(12): E650–E657. DOI: 10.1016/S2589-7500(20)30246-6.
  13. Ringwood JV, Malpas SC. Slow oscillations in blood pressure via a nonlinear feedback model. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology. 2001;280(4): R1105–R1115. DOI: 10.1152/ajpregu.2001.280.4.R1105.
  14. Tang Q, Chen Z, Ward R, Elgendi M. Synthetic photoplethysmogram generation using two Gaussian functions. Sci. Rep. 2020;10(1):13883. DOI: 10.1038/s41598-020-69076-x.
  15. McSharry PE, Clifford GD, Tarassenko L, Smith LA. A dynamical model for generating synthetic electrocardiogram signals. IEEE Transactions on Biomedical Engineering. 2003;50(3):289–294. DOI: 10.1109/TBME.2003.808805.
  16. Cheng L, Khoo MCK. Modeling the autonomic and metabolic effects of obstructive sleep apnea: a simulation study. Front. Physiol. 2012;2:111. DOI: 10.3389/fphys.2011.00111.
  17. Mejıa-Mejıa E, May JM, Torres R, Kyriacou PA. Pulse rate variability in cardiovascular health: a review on its applications and relationship with heart rate variability. Physiol. Meas. 2020;41(7):07TR01. DOI: 10.1088/1361-6579/ab998c.
  18. Kotani K, Struzik ZR, Takamasu K, Stanley HE, Yamamoto Y. Model for complex heart rate dynamics in health and diseases. Phys. Rev. E. 2005;72(4):041904. DOI: 10.1103/PhysRevE. 72.041904.
Available online: