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An ensemble Kalman filter approach to parameter estimation for patient-specific cardiovascular flow modeling

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Abstract

Many previous studies have shown that the fidelity of three-dimensional cardiovascular flow simulations depends strongly on inflow and outflow boundary conditions that accurately describe the characteristics of the larger vascular network. These boundary conditions are generally based on lower-dimensional models that represent the upstream or downstream flow behavior in some aggregated fashion. However, the parameters of these models are patient-specific, and no clear technique exists for determining them. In this work, an ensemble Kalman filter (EnKF) is implemented for the purpose of estimating parameters in cardiovascular models through the assimilation of specific patients’ clinical measurements. Two types of models are studied: a fully zero-dimensional model of the right heart and pulmonary circulation, and a coupled 0D–1D model of the lower leg. Model parameters are estimated using measurements from both healthy and hypertensive patients, and demonstrate that the EnKF is able to generate distinct parameter sets whose model predictions produce features unique to each measurement set. Attention is also given toward the quality of model predictions made in the absence of direct clinical counterparts, as well as techniques to improve filter robustness against shrinking ensemble covariance.

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References

  1. Aguero, J., Ishikawa, K., Hadri, L., Santos-Gallego, C., Fish, K., Hammoudi, N., Chaanine, A., Torquato, S., Naim, C., Ibanez, B., Pereda, D., Garcia-Alvarez, A., Fuster, V., Sengupta, P.P., Leopold, J.A., Hajjar, R.J.: Characterization of right ventricular remodeling and failure in a chronic pulmonary hypertension model. Am. J. Physiol. Heart Circ. Physiol. 307(8), H1204–H1215 (2014)

    Google Scholar 

  2. Alastruey, J.: Numerical modelling of pulse wave propagation in the cardiovascular system: development, validation, and clinical applications. Ph.D. thesis, Imperial College London (2006)

  3. Alastruey, J., Parker, K.H., Peiró, J., Sherwin, S.J.: Lumped parameter outflow models for 1-d blood flow simulations: effect on pulse waves and parameter estimation. Commun. Comput. Phys. 4(2), 317–336 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Batzel, J.J., Kappel, F., Schneditz, D., Tran, H.T.: Cardiovascular and Respiratory Systems: Modeling, Analysis, and Control. SIAM, Philadelphia (2007)

    MATH  Google Scholar 

  5. Blanco, P.J., Trenhago, P.R., Fernandes, L.G., Feijóo, R.A.: On the integration of the baroreflex control mechanism in a heterogeneous model of the cardiovascular system. Int. J. Num. Methods Biomed. Eng. 28, 412–433 (2012)

    MathSciNet  Google Scholar 

  6. Boyers, D., Cuthbertson, J.G., Luetscher, J.A.: Simulation of the human cardiovascular system: a model with normal responses to change of posture, blood loss, transfusion, and autonomic blockade. Simulation 18, 197–206 (1972)

    Google Scholar 

  7. Burgers, G., Jan van Leeuwen, P., Evensen, G.: Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724 (1998)

    Google Scholar 

  8. Canuto, D., Chong, K., Bowles, C., Dutson, E.P., Eldredge, J.D., Benharash, P.: A regulated mutliscale closed-loop cardiovascular model, with applications to hemorrhage and hypertension. Int. J. Numer. Methods Biomed. Eng. (2018). https://doi.org/10.1002/cnm.2975

    Article  Google Scholar 

  9. Danielsen, M.: Modeling of feedback mechanisms which control the heart function in view to an implementation in cardiovascular models. Ph.D. thesis, Roskilde University (1998)

  10. Di Carlo, A., Nardinocchi, P., Pontrelli, G., Teresi, L.: A heterogeneous approach for modelling blood flow in an arterial segment. Trans. Biomed. Health 6, 69–78 (2003)

    Google Scholar 

  11. Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)

    Google Scholar 

  12. Evensen, G.: The ensemble Kalman filter for combined state and parameter estimation: Monte Carlo techniques for data assimilation in large systems. IEEE Control Sys. Mag. 29(3), 83–104 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Formaggia, L., Lamponi, D., Tuveri, M., Veneziani, A.: Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Comput. Methods Biomech. Biomed. Eng. 9(5), 273–288 (2006)

    Google Scholar 

  14. Frank, O.: Die grundform des arterielen pulses erste abhandlung: mathematische analyse. Z. Biol. 37, 483–526 (1899)

    Google Scholar 

  15. Gohean, J.R.: A closed-loop multi-scale model of the cardiovascular system for evaluation of ventricule assist devices. Master’s thesis, The University of Texas at Austin (2007)

  16. Jafarpour, B., Tarrahi, M.: Assessing the performance of the ensemble Kalman filter for subsurface flow data integration under variogram uncertainty. Water Resour. Res. (2011). https://doi.org/10.1029/2010WR009090

    Article  Google Scholar 

  17. Jain, K., Maka, S.: Sensitivity analysis and parameter estimation of cardiovascular model. In: Proceedings of 2016 International Conference on Systems in Medicine and Biology. IEEE (2016)

  18. Karamolegkos, N., Vicario, F., Chbat, N.W.: Cardiovascular system identification: simulation study using arterial and central venous pressures. In: 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) (2015)

  19. Khoshdel, A.R., Carney, S.L., Nair, B.R., Gillies, A.: Better management of cardiovascular diseases by pulse wave velocity: combining clinical practice with clinical research using evidence-based medicine. Clin. Med. Res. 5(1), 45–52 (2007)

    Google Scholar 

  20. Kim, H.J., Vignon-Clementel, I.E., Figueroa, C.A., LaDisa, J.F., Jansen, K.E., Feinstein, J.A., Taylor, C.A.: On coupling a lumped parameter heart model and a three-dimensional finite element aorta model. Annal. Biomed. Eng. 37(11), 2153–2169 (2009)

    Google Scholar 

  21. Kind, T., Faes, T.J.C., Lankhaar, J.W., Vonk-Noordergraaf, A., Verhaegen, M.: Estimation of three- and four-element Windkessel parameters using subspace model identification. IEEE Trans. Biomed. Eng. 57(7), 1531–1538 (2010)

    Google Scholar 

  22. Klingensmith, M.E., Chen, L.E., Glasgow, S.C., Goers, T.A., Melby, S.J. (eds.): The Washington Manual of Surgery, 5th edn. Wolters Kluwer/Lippincott Williams & Wilkins, Alphen aan den Rijn (2008)

    Google Scholar 

  23. Koivistoinen, T., Kööbi, T., Jula, A., Hutri-Kähönen, N., Raitakari, O.T., Majahalme, S., Kukkonen-Harjula, K., Lehtimäki, T., Reunanen, A., Viikari, J., Turjanmaa, V., Nieminen, T., Kähönen, M.: Pulse wave velocity reference values in healthy adults aged 26–75 years. Clin. Physiol. Funct. Imaging 27(3), 191–196 (2007)

    Google Scholar 

  24. Leiva, J.S., Blanco, P.J., Buscaglia, G.C.: Partitioned analysis for dimensionally-heterogenous hydraulic networks. Multiscale Model. Simul. 9(2), 872–903 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Liang, F., Takagi, S., Himeno, R., Liu, H.: Multi-scale modeling of the human cardiovascular system with applications to aortic valvular and arterial stenoses. Med. Biol. Eng. Comput. 47, 743–755 (2009)

    Google Scholar 

  26. Liang, F.Y., Takagi, S., Himeno, R., Liu, H.: Biomechanical characterization of ventricular-arterial coupling during aging: a multi-scale model study. J. Biomech. 42, 692–704 (2009)

    Google Scholar 

  27. Moghadam, M.E., Vignon-Clementel, I.E., Figliola, R., Marsden, A.L.: A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations. J. Comput. Phys. 244, 63–79 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Moradkhani, H., Sorooshian, S., Gupta, H.V., Houser, P.R.: Dual state-parameter estimation of hydrological models using ensemble Kalman filter. Adv. Wat. Res. 28(2), 135–147 (2005)

    Google Scholar 

  29. Mullen, T.J., Appel, M.L., Mukkamala, R., Mathias, J.M., Cohen, R.J.: System identification of closed-loop cardiovascular control: effects of posture and autonomic blockade. Am. J. Physiol. 272, H448–H461 (1997)

    Google Scholar 

  30. Müller, L., Toro, E.: A global multiscale mathematical model for the human circulation with emphasis on the venous system. Int. J. Numer. Methods Biomed. Eng. 30, 681–725 (2014)

    MathSciNet  Google Scholar 

  31. Mynard, J.P., Davidson, M.R., Penny, D.J., Smolich, J.J.: 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)

    MathSciNet  Google Scholar 

  32. Naumann, A., Kolb, O., Semplice, M.: On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws. Appl. Math. Comput. 325, 252–270 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Nauser, T.D., Sittes, S.W.: Diagnosis and treatment of pulmonary hypertension. Am. Fam. Physician 63, 1789–1798 (2001)

    Google Scholar 

  34. Olufsen, M.S., Ottesen, J.T., Tran, H.T., Ellwein, L.M., Lipsitz, L.A., Novak, V.: Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation. J. Appl. Physiol. 99, 1523–1537 (2005)

    Google Scholar 

  35. Ottesen, J.T., Danielsen, M.: Modeling ventricule contraction with heart rate changes. J. Theo. Biol. 222, 337–346 (2003)

    MATH  Google Scholar 

  36. Quarteroni, A., Veneziani, A.: Analysis of a geometerial multiscale model based on the coupling of odes and pdes for blood flow simulations. Multiscale Model. Simul. 1(2), 173–195 (2003)

    MathSciNet  MATH  Google Scholar 

  37. Raines, J.K., Jaffrin, M.Y., Shapiro, A.H.: A computer simulation of arterial dynamics in the human leg. J. Biomech. 7, 77–91 (1974)

    Google Scholar 

  38. Reiter, G., Reiter, U., Kovacs, G., Kainz, B., Schmidt, K., Maier, R., Olschewski, H., Rienmueller, R.: Magnetic resonance-derived 3-dimensional blood flow patterns in the main pulmonary artery as a marker of pulmonary hypertension and a measure of elevated mean pulmonary arterial pressure. Circ. Cardiovasc. Imaging 1, 23–30 (2008)

    Google Scholar 

  39. Reiter, G., Reiter, U., Kovacs, G., Olschewski, H., Fuchsjäger, M.: Blood flow vortices along the main pulmonary artery measured with MR imaging for diagnosis of pulmonary hypertension. Radiology 275(1), 71–79 (2015)

    Google Scholar 

  40. Reymond, P., Merenda, F., Perren, F., Rüfenacht, D., Stergiopulos, N.: Validation of a one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. 297, 208–222 (2009)

    Google Scholar 

  41. Sherwin, S.J., Franke, V., Peiró, J., Parker, K.: One-dimensional modelling of a vascular network in space-time variables. J. Eng. Math. 47, 217–250 (2003)

    MathSciNet  MATH  Google Scholar 

  42. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    MathSciNet  MATH  Google Scholar 

  43. Stergiopulos, N., Meister, J.J., Westerhof, N.: Determinants of stroke volume and systolic and diastolic pressure. Am. J. Physiol. Heart Circ. 270, H2050–H2059 (1996)

    Google Scholar 

  44. Stergiopulos, N., Young, D.F., Rogge, T.R.: Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25, 1477–1488 (1992)

    Google Scholar 

  45. Suga, H., Sagawa, H., Shoukas, A.: 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)

    Google Scholar 

  46. Taylor, C.A., Figueroa, C.A.: Patient-specific modeling of cardiovascular mechanics. Annu. Rev. Biomed. Eng. 11, 109–134 (2009)

    Google Scholar 

  47. Tran, J.S., Schiavazzi, D.E., Ramachandra, A.B., Kahn, A.M., Marsden, A.L.: Automated tuning for parameter identification and uncertainty quantification in multi-scale coronary simulations. Comput. Fluids 142, 128–138 (2017)

    MathSciNet  MATH  Google Scholar 

  48. van Griensven, A., Meixner, T., Grunwald, S., Bishop, T., Diluzio, M., Srinivasan, R.: A global sensitivity analysis tool for the parameters of mutli-variable catchment models. J. Hydrol. 324, 10–23 (2006)

    Google Scholar 

  49. Vignon, I.E., Taylor, C.A.: Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries. Wave Motion 39, 361–374 (2004)

    MathSciNet  MATH  Google Scholar 

  50. Wang, D., Chen, Y., Cai, X.: State and parameter estimation of hydrologic models using the constrained ensemble Kalman filter. Water Resour. Res. 45, W06401 (2009)

    Google Scholar 

  51. West, M.: Mixture models, Monte Carlo, Bayesian updating and dynamic models. Comput. Sci. Stat. 24, 325–333 (1993)

    Google Scholar 

  52. Westerhof, N., Bosman, F., De Vries, C.J., Noordergraaf, A.: Analog studies of the human systemic arterial tree. J. Biomech. 2, 121–143 (1969)

    Google Scholar 

  53. Westerhof, N., Lankhaar, J.W., Westerhof, B.E.: The arterial Windkessel. Med. Biol. Eng. Comput. 47, 131–141 (2009)

    Google Scholar 

  54. Whitaker, J.S., Hamill, T.M.: Evaluating methods to account for system errors in ensemble data assimilation. Mon. Weather Rev. 140, 3078–3089 (2012)

    Google Scholar 

  55. Yeh, W.W.G.: Review of parameter identification procedures in groundwater hydrology: the inverse problem. Water Resour. Res. 22(2), 95–108 (1986)

    Google Scholar 

  56. Yu, Y.C., Boston, J.R., Simaan, M.A., Antaki, J.F.: Estimation of systemic vascular bed parameters for artificial heart control. IEEE Trans. Autom. Control (1998). https://doi.org/10.1109/CDC.2003.1271791

    Article  Google Scholar 

  57. Zhang, F., Snyder, C., Sun, J.: Impacts of initial estimate and observation availability on convective-scale data assimilation with an ensemble Kalman filter. Mon. Weather Rev. 132, 1238–1253 (2004)

    Google Scholar 

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Acknowledgements

This work has been supported, in part, by the US Army Medical Research Acquisition Activity (Grant No. W81XWH-15-1-0147) and the US Office of Naval Research (Grant No. N00014-13-C-0357).

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Correspondence to Jeff D. Eldredge.

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Communicated by Kunihiko Taira.

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Appendix: Numerical values of model parameters

Appendix: Numerical values of model parameters

See Tables 4, 5, 6, 7, 8 and 9.

Table 4 Normal distribution characteristics for 0D pulmonary model parameters
Table 5 Converged parameter values for the 0D model in the healthy case
Table 6 Converged parameter values for the 0D model in the hypertensive case
Table 7 Geometric data for the one-dimensional arterial network
Table 8 Normal distribution characteristics for coupled 0D–1D lower leg model parameters
Table 9 Converged ensemble mean parameter values for the coupled 0D–1D lower leg model

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Canuto, D., Pantoja, J.L., Han, J. et al. An ensemble Kalman filter approach to parameter estimation for patient-specific cardiovascular flow modeling. Theor. Comput. Fluid Dyn. 34, 521–544 (2020). https://doi.org/10.1007/s00162-020-00530-2

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