Annals of Biomedical Engineering

, Volume 46, Issue 11, pp 1722–1735 | Cite as

Estimating Left Ventricular Elastance from Aortic Flow Waveform, Ventricular Ejection Fraction, and Brachial Pressure: An In Silico Study

  • Stamatia Z. PagoulatouEmail author
  • Nikolaos Stergiopulos


Although left ventricular end-systolic elastance (Ees) serves as a major index of cardiac contractility, a widely-accepted noninvasive estimation of Ees does not exist. To overcome this limitation, we developed a two-step inverse method that allows for its noninvasive estimation from measurements of aortic flow and brachial pressure using a previously validated one-dimensional model of the cardiovascular system. In a first step, aortic flow is set as the model input and the output brachial pressure is compared with the “real” values. Subsequently, the basic properties of the arterial tree are tuned according to an optimization algorithm. In a second step, the same optimization method is used to estimate the elastance parameters that produce an aortic flow waveform that matches the “real” one. Additional knowledge of the ejection fraction can allow for the accurate estimation of the entire PV loop, including end-diastolic elastance. The method was tested on a database of 50 different in silico hemodynamic cases generated after varying cardiac and arterial model parameters. Implementation of the method yielded good agreement (r = 0.99) and accuracy (n-RMSE = 4%) between “real” and estimated values of Ees. Furthermore, a sensitivity analysis revealed that errors due to poor arterial adjustment and measurements are small (≤ 8% for Ees).


Cardiac contractility Noninvasive 1-D model Hemodynamics 


  1. 1.
    Adler, D., E. S. Monrad, E. H. Sonnenblick, O. M. Hess, and H. P. Krayenbuehl. Time to dp/dtmax, a useful index for evaluation of contractility in the catheterization laboratory. Clin. Cardiol. 19:397–403, 1996.CrossRefGoogle Scholar
  2. 2.
    Baan, J., and E. T. Van der Velde. Sensitivity of left ventricular end-systolic pressure–volume relation to type of loading intervention in dogs. Circ. Res. 62:1247–1258, 1988.CrossRefGoogle Scholar
  3. 3.
    Bland, J. M., and D. G. Altman. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet Lond. Engl. 1:307–310, 1986.CrossRefGoogle Scholar
  4. 4.
    Bonnet, B., F. Jourdan, G. du Cailar, and P. Fesler. Noninvasive evaluation of left ventricular elastance according to pressure–volume curves modeling in arterial hypertension. Am. J. Physiol. Heart Circ. Physiol. 313:H237–H243, 2017.CrossRefGoogle Scholar
  5. 5.
    Boutouyrie, P., S. Laurent, A. Benetos, X. J. Girerd, A. P. Hoeks, and M. E. Safar. Opposing effects of ageing on distal and proximal large arteries in hypertensives. J. Hypertens. Suppl. 10:S87–91, 1992.CrossRefGoogle Scholar
  6. 6.
    Burkhoff, D., P. P. De Tombe, and W. C. Hunter. Impact of ejection on magnitude and time course of ventricular pressure-generating capacity. Am. J. Physiol. 265:H899–H909, 1993.Google Scholar
  7. 7.
    Burkhoff, D., S. Sugiura, D. T. Yue, and K. Sagawa. Contractility-dependent curvilinearity of end-systolic pressure–volume relations. Am. J. Physiol. 252:H1218–H1227, 1987.Google Scholar
  8. 8.
    Chen, C.-H., B. Fetics, E. Nevo, C. E. Rochitte, K.-R. Chiou, P.-A. Ding, M. Kawaguchi, and D. A. Kass. Noninvasive single-beat determination of left ventricular end-systolic elastance in humans. J. Am. Coll. Cardiol. 38:2028–2034, 2001.CrossRefGoogle Scholar
  9. 9.
    Chen, C. H., M. Nakayama, E. Nevo, B. J. Fetics, W. L. Maughan, and D. A. Kass. Coupled systolic-ventricular and vascular stiffening with age: implications for pressure regulation and cardiac reserve in the elderly. J. Am. Coll. Cardiol. 32:1221–1227, 1998.CrossRefGoogle Scholar
  10. 10.
    Chirinos, J. A. Ventricular–arterial coupling: invasive and non-invasive assessment. Artery Res. 2013. Scholar
  11. 11.
    Davidson, S., C. Pretty, A. Pironet, S. Kamoi, J. Balmer, T. Desaive, and J. G. Chase. Minimally invasive, patient specific, beat-by-beat estimation of left ventricular time varying elastance. Biomed. Eng. Online 16(1):42, 2017.CrossRefGoogle Scholar
  12. 12.
    Feldman, M. D., P. H. Pak, C. C. Wu, H. L. Haber, C. M. Heesch, J. D. Bergin, E. R. Powers, T. D. Cowart, W. Johnson, A. M. Feldman, and D. A. Kass. Acute cardiovascular effects of OPC-18790 in patients with congestive heart failure. Time- and dose-dependence analysis based on pressure–volume relations. Circulation 93:474–483, 1996.CrossRefGoogle Scholar
  13. 13.
    Holenstein, R., P. Niederer, and M. Anliker. A viscoelastic model for use in predicting arterial pulse waves. J. Biomech. Eng. 102:318–325, 1980.CrossRefGoogle Scholar
  14. 14.
    Kass, D. A., R. Beyar, E. Lankford, M. Heard, W. L. Maughan, and K. Sagawa. Influence of contractile state on curvilinearity of in situ end-systolic pressure–volume relations. Circulation 79:167–178, 1989.CrossRefGoogle Scholar
  15. 15.
    Kass, D. A., and W. L. Maughan. From “Emax” to pressure–volume relations: a broader view. Circulation 77:1203–1212, 1988.CrossRefGoogle Scholar
  16. 16.
    Kelly, R. P., C. T. Ting, T. M. Yang, C. P. Liu, W. L. Maughan, M. S. Chang, and D. A. Kass. Effective arterial elastance as index of arterial vascular load in humans. Circulation 86:513–521, 1992.CrossRefGoogle Scholar
  17. 17.
    Kimoto, E., T. Shoji, K. Shinohara, M. Inaba, Y. Okuno, T. Miki, H. Koyama, M. Emoto, and Y. Nishizawa. Preferential stiffening of central over peripheral arteries in type 2 diabetes. Diabetes 52:448–452, 2003.CrossRefGoogle Scholar
  18. 18.
    Klotz, S., I. Hay, M. L. Dickstein, G.-H. Yi, J. Wang, M. S. Maurer, D. A. Kass, and D. Burkhoff. Single-beat estimation of end-diastolic pressure–volume relationship: a novel method with potential for noninvasive application. Am. J. Physiol. Heart Circ. Physiol. 291:H403–H412, 2006.CrossRefGoogle Scholar
  19. 19.
    Kono, A., W. L. Maughan, K. Sunagawa, K. Hamilton, K. Sagawa, and M. L. Weisfeldt. The use of left ventricular end-ejection pressure and peak pressure in the estimation of the end-systolic pressure–volume relationship. Circulation 70:1057–1065, 1984.CrossRefGoogle Scholar
  20. 20.
    Langewouters G. J. Visco-elasticity of the human aorta in vitro in relation to pressure and age. PhD Thesis, Free University of Amsterdam, 1982.Google Scholar
  21. 21.
    Mirsky, I., and W. W. Parmley. Assessment of passive elastic stiffness for isolated heart muscle and the intact heart. Circ. Res. 33:233–243, 1973.CrossRefGoogle Scholar
  22. 22.
    Mynard, J. P., M. R. Davidson, D. J. Penny, and J. J. Smolich. A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models. Int. J. Numer. Methods Biomed. Eng. 28:626–641, 2012.CrossRefGoogle Scholar
  23. 23.
    Pahlevan, N. M., D. G. Rinderknecht, P. Tavallali, M. Razavi, T. T. Tran, M. W. Fong, R. A. Kloner, M. Csete, and M. Gharib. Noninvasive iPhone Measurement of Left Ventricular Ejection Fraction Using Intrinsic Frequency Methodology. Crit. Care Med. 45:1115–1120, 2017.CrossRefGoogle Scholar
  24. 24.
    Pak, P. H., W. L. Maughan, K. L. Baughman, R. S. Kieval, and D. A. Kass. Mechanism of acute mechanical benefit from VDD pacing in hypertrophied heart: similarity of responses in hypertrophic cardiomyopathy and hypertensive heart disease. Circulation 98:242–248, 1998.CrossRefGoogle Scholar
  25. 25.
    Quiñones, M. A., C. M. Otto, M. Stoddard, A. Waggoner, W. A. Zoghbi, and Doppler Quantification Task Force of the Nomenclature and Standards Committee of the American Society of Echocardiography. Recommendations for quantification of Doppler echocardiography: a report from the Doppler Quantification Task Force of the Nomenclature and Standards Committee of the American Society of Echocardiography. J. Am. Soc. Echocardiogr. 15:167–184, 2002.Google Scholar
  26. 26.
    Reymond, P., Y. Bohraus, F. Perren, F. Lazeyras, and N. Stergiopulos. Validation of a patient-specific one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. Physiol. 301:H1173–H1182, 2011.CrossRefGoogle Scholar
  27. 27.
    Reymond, P., F. Merenda, F. Perren, D. Rüfenacht, and N. Stergiopulos. Validation of a one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. Physiol. 297:H208–H222, 2009.CrossRefGoogle Scholar
  28. 28.
    Sagawa, K. The end-systolic pressure–volume relation of the ventricle: definition, modifications and clinical use. Circulation 63:1223–1227, 1981.CrossRefGoogle Scholar
  29. 29.
    Sagawa, K., H. Suga, A. A. Shoukas, and K. M. Bakalar. End-systolic pressure/volume ratio: a new index of ventricular contractility. Am. J. Cardiol. 40:748–753, 1977.CrossRefGoogle Scholar
  30. 30.
    Senzaki, H., C. H. Chen, and D. A. Kass. Single-beat estimation of end-systolic pressure–volume relation in humans. A new method with the potential for noninvasive application. Circulation 94:2497–2506, 1996.CrossRefGoogle Scholar
  31. 31.
    Shishido, T., K. Hayashi, K. Shigemi, T. Sato, M. Sugimachi, and K. Sunagawa. Single-beat estimation of end-systolic elastance using bilinearly approximated time-varying elastance curve. Circulation 102:1983–1989, 2000.CrossRefGoogle Scholar
  32. 32.
    Starling, M. R., R. A. Walsh, L. J. Dell’Italia, G. B. Mancini, J. C. Lasher, and J. L. Lancaster. The relationship of various measures of end-systole to left ventricular maximum time-varying elastance in man. Circulation 76:32–43, 1987.CrossRefGoogle Scholar
  33. 33.
    Stergiopulos, N., J. J. Meister, and N. Westerhof. Determinants of stroke volume and systolic and diastolic aortic pressure. Am. J. Physiol. 270:H2050–H2059, 1996.Google Scholar
  34. 34.
    Suga, H., and K. Sagawa. Instantaneous pressure–volume relationships and their ratio in the excised, supported canine left ventricle. Circ. Res. 35:117–126, 1974.CrossRefGoogle Scholar
  35. 35.
    Suga, H., K. Sagawa, and A. A. Shoukas. Load independence of the instantaneous pressure–volume ratio of the canine left ventricle and effects of epinephrine and heart rate on the ratio. Circ. Res. 32:314–322, 1973.CrossRefGoogle Scholar
  36. 36.
    Sunagawa, K., W. L. Maughan, and K. Sagawa. Effect of regional ischemia on the left ventricular end-systolic pressure–volume relationship of isolated canine hearts. Circ. Res. 52:170–178, 1983.CrossRefGoogle Scholar
  37. 37.
    Swamy, G., J. Kuiper, M. S. R. Gudur, N. B. Olivier, and R. Mukkamala. Continuous left ventricular ejection fraction monitoring by aortic pressure waveform analysis. Ann. Biomed. Eng. 37:1055–1068, 2009.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2018

Authors and Affiliations

  1. 1.Laboratory of Hemodynamics and Cardiovascular Technology (LHTC), Institute of BioengineeringEcole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland

Personalised recommendations