Abstract
This study develops non-pulsatile and pulsatile models for the prediction of blood flow and pressure during head-up tilt. This test is used to diagnose potential pathologies within the autonomic control system, which acts to keep the cardiovascular system at homeostasis. We show that mathematical modeling can be used to predict changes in cardiac contractility, vascular resistance, and arterial compliance, quantities that cannot be measured but are useful to assess the system’s state. These quantities are predicted as time-varying parameters modeled using piecewise linear splines. Having models with various levels of complexity formulated with a common set of parameters, allows us to combine long-term non-pulsatile simulations with pulsatile simulations on a shorter time-scale. We illustrate results for a representative subject tilted head-up from a supine position to a \(60^{\circ }\) angle. The tilt is maintained for 5 min before the subject is tilted back down. Results show that if volume data is available for all vascular compartments three parameters can be identified, cardiovascular resistance, vascular compliance, and ventricular contractility, whereas if model predictions are made against arterial pressure and cardiac output data alone, only two parameters can be estimated either resistance and contractility or resistance and compliance.
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References
Batzel J, Kappel F, Schneditz D, Tran HT (2007) Cardiovascular and respiratory systems: modeling, analysis, and control. SIAM, Philadelphia
Beneken J, Dewit B (1967) A physical approach to hemodynamic aspects of the human cardiovascular system. In: Reeve E, Guyton A (eds) Physical bases of circulatory transport: regulation and exchange. W.B. Saunders, Philadelphia, pp 1–45
Boron W, Boulpaep E (2017) Medical physiology, 3rd edn. Elsevier, Philadelphia
Ellwein L (2008) Cardiovascular and respiratory modeling, Ph.D. thesis, Department of Mathematics, NC State University, Raleigh
Fink M, Batzel J, Kappel F (2004) An optimal control approach to modeling the cardiovascular-respiratory system: an application to orthostatic stress. Cardiovsc Eng 4:27–38
Guyton A, Hall J (2013) Textbook of medical physiology, 13th edn. Elsevier, Philadelphia
Haario H, Laine M, Mira A, Saksman E (2006) DRAM: efficient adaptive MCMC. Stat Comput 16:339–354
Heldt T, Karmm R, Mark R (2002) Computational modeling of cardiovascular response to orthostatic stress. J Appl Phyisiol 92:1239–1254
Hoppensteadt F, Peskin C (2002) Modeling and simulation in medicine and the life sciences. Springer, New York
Kelley C (1999) Iterative methods for optimization. Applied mathematics, vol 6. SIAM, Philadelphia
Kenny R, O’Shea D, Parry S (2000) The Newcastle protocols for head-up tilt table testing in the diagnosis of vasovagal syncope, carotid sinus hypersensitivity, and related disorders. Heart 83:564–569
Lanier J, Mote M, Clay E (2011) Evaluation and management of orthostatic hypotension. Am Fam Physician 84:527–536
Lim E, Chan G, Dokos S, Ng S, Latif L, Vandenberghe S, Karunanithi M, Lovell N (2013) A cardiovascular mathematical model of graded head-up tilt. PLoS ONE 8:e77357
Lim E, Salamonsen R, Mansouri M, Gaddum N, Mason D, Timms D, Stevens M, Fraser J, Akmeliawati R, Lovell N (2014) Hemodynamic response to exercise and head-up tilt of patients implanted with a rotary blood pump: a computational modeling study. J Artif Organs 39:E24–E35
Marquis A, Arnold A, Dean-Bernhoft C, Carlson B, Olufsen M (2018) Practical identifiability and uncertainty quantification of a pulsatile cardiovascular model. Math Biosci 304:9–24
Matzuka B, Mehlsen J, Tran H, Olufsen M (2015) Using Kalman filtering to predict time-varying parameters in a model predicting baroreflex regulation during head-up tilt. IEEE Trans Biomed Eng 62:1992–2000
Melchior F, Srinivasen R, Charles J (1992) Mathematical modeling of human cardiovascular system for simulation of orthostatic response. Am J Physiol 262:H1920–H1933
Olufsen M, Ottesen J (2013) A practical approach to parameter estimation applied to model predicting heart rate regulation. J Math Biol 67:39–68
Olufsen M, Ottesen J, Tran H, Ellwein L, Lipsitz L, Novak V (2005) Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation. J Appl Physiol 99:1523–1537
Parry S, Reeve P, Lawson J, Shaw F, Norton M, Frearson R, Kerr S, Newton J (2009) The newcastle protocols 2008: an update on head-up tilt table testing and the management of vasovagal syncope and related disorders. Heart 95:416–420
Pope S, Ellwein L, Zapata C, Novak V, Kelley C, Olufsen M (2011) Estimation and identification of parameters in a lumped cerebrovascular model. Math Biosci Eng 6:93–115
Porta A, Bassani T, Bari V, Tobaldini E, Takahashi A, Catai A, Montano N (2011) Model-based assessment of baroreflex and cardiopulmonary couplings during graded head-up tilt. Comput Biol Med 42:298–305
Ramsey J, Silverman B (2002) Functional data analysis, 2nd edn. Springer, New York
Shoemaker W (1989) Fluids and electrolytes in the acutely ill adult. In: Shoemaker W, Ayres S, Greenvik A, Holbrook P (eds) Textbook of critical care. W.B. Saunders, Philadelphia
Silvani A, Magosso E, Bastianini S, Lenzi P, Ursino M (2011) Mathematical modeling of cardiovascular coupling: central autonomic commands and baroreflex control. Auton Neurosci 162:66–71
Starling E, Visscher M (1926) The regulation of the energy output of the heart. J Physiol 62:243–261
TenVoorde B, Kingma R (2000) A baroreflex model of short term blood pressure and heart rate variability. Stud Health Technol Inform 71:179–200
Ursino M (1998) Interaction between carotid baroregulation and the pulsating heart: a mathematical model. Am J Physiol 275:H1733–H1747
Ursino M (1999) A mathematical model of the carotid baroregulation in pulsating conditions. IEEE Trans Biomed Eng 46:382–392
van de Vooren H, Gademan M, Swenne C, TenVoorde B, Schalij M, der Wall EV (2007) Baroreflex sensitivity, blood pressure buffering, and resonance: what are the links? Computer simulation of healthy subjects and heart failure patients. J Appl Physiol 102:1348–1356
van Heusden K, Gisolf J, Stok W, Dijkstra S, Karemaker J (2006) Mathematical modeling of gravitational effects on the circulation: importance of the time course of venous pooling and blood volume changes in the lungs. Am J Phsyiol 291:H2152–H2165
Williams N, Wind-Willassen O, Program R, Mehlsen J, Ottesen J, Olufsen M (2014) Patient specific modeling of head-up tilt. Math Med Biol 31:365–392
Zaidi A, Benitez D, Gaydecki P, Vohra A, Fitzpatrick A (2000) Haemodynamic effects of increasing angle of head up tilt. Heart 83:181–184
Zhang J, Critchley L, Lee D, Khaw K (2015) The effect of head up tilting on bioreactance cardiac output and stroke volume readings using suprasternal transcutaneous doppler as a control in healthy young adults. J Clin Monit Comput 30:519–526
Acknowledgements
Results from this study was obtained with support from the Virtual Physiological Rat (VPR) project NIH-NIGMS #1P50GM094503, by the National Science Foundation under awards NSF-DMS #1022688, #1557761, and the NCSU Research Training Grant NSF-DMS #1246991.
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Williams, N.D., Brady, R., Gilmore, S. et al. Cardiovascular dynamics during head-up tilt assessed via pulsatile and non-pulsatile models. J. Math. Biol. 79, 987–1014 (2019). https://doi.org/10.1007/s00285-019-01386-9
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DOI: https://doi.org/10.1007/s00285-019-01386-9
Keywords
- Cardiovascular dynamics modeling
- Head-up tilt
- Pulsatile versus non-pulsatile modeling
- Parameter estimation
- Orthostatic intolerance