Quantifying the Uncertainty in Model Parameters Using Gaussian Process-Based Markov Chain Monte Carlo: An Application to Cardiac Electrophysiological Models

  • Jwala DhamalaEmail author
  • John L. Sapp
  • Milan Horacek
  • Linwei Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10265)


Estimation of patient-specific model parameters is important for personalized modeling, although sparse and noisy clinical data can introduce significant uncertainty in the estimated parameter values. This importance source of uncertainty, if left unquantified, will lead to unknown variability in model outputs that hinder their reliable adoptions. Probabilistic estimation model parameters, however, remains an unresolved challenge because standard Markov Chain Monte Carlo sampling requires repeated model simulations that are computationally infeasible. A common solution is to replace the simulation model with a computationally-efficient surrogate for a faster sampling. However, by sampling from an approximation of the exact posterior probability density function (pdf) of the parameters, the efficiency is gained at the expense of sampling accuracy. In this paper, we address this issue by integrating surrogate modeling into Metropolis Hasting (MH) sampling of the exact posterior pdfs to improve its acceptance rate. It is done by first quickly constructing a Gaussian process (GP) surrogate of the exact posterior pdfs using deterministic optimization. This efficient surrogate is then used to modify commonly-used proposal distributions in MH sampling such that only proposals accepted by the surrogate will be tested by the exact posterior pdf for acceptance/rejection, reducing unnecessary model simulations at unlikely candidates. Synthetic and real-data experiments using the presented method show a significant gain in computational efficiency without compromising the accuracy. In addition, insights into the non-identifiability and heterogeneity of tissue properties can be gained from the obtained posterior distributions.


Probabilistic parameter estimation Personalized modeling Markov chain Monte Carlo Gaussian process 



This work is supported by the National Science Foundation under CAREER Award ACI-1350374 and the National Institute of Heart, Lung, and Blood of the National Institutes of Health under Award R21Hl125998.


  1. 1.
    Adrieu, C., Freitas, N., Doucet, A., Jordan, M.: An introduction to Markov chain Monte Carlo for machine learning. Mach. Learn. 50, 5–43 (2003)CrossRefGoogle Scholar
  2. 2.
    Aliev, R.R., Panfilov, A.V.: A simple two-variable model of cardiac excitation. Chaos, Solitons Fractals 7(3), 293–301 (1996)CrossRefGoogle Scholar
  3. 3.
    Christen, J.A., Fox, C.: Markov chain Monte Carlo using an approximation. J. Comput. Graph. Stat. 14(4), 795–810 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dhamala, J., Sapp, J.L., Horacek, M., Wang, L.: Spatially-adaptive multi-scale optimization for local parameter estimation: application in cardiac electrophysiological models. In: Ourselin, S., Joskowicz, L., Sabuncu, M.R., Unal, G., Wells, W. (eds.) MICCAI 2016. LNCS, vol. 9902, pp. 282–290. Springer, Cham (2016). doi: 10.1007/978-3-319-46726-9_33 CrossRefGoogle Scholar
  5. 5.
    Konukoglu, E., et al.: Efficient probabilistic model personalization integrating uncertainty on data and parameters: application to eikonal-diffusion models in cardiac electrophysiology. Prog. Biophys. Mol. Biol. 107(1), 134–146 (2011)CrossRefGoogle Scholar
  6. 6.
    Lê, M., Delingette, H., Kalpathy-Cramer, J., Gerstner, E.R., Batchelor, T., Unkelbach, J., Ayache, N.: Bayesian personalization of brain tumor growth model. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9350, pp. 424–432. Springer, Cham (2015). doi: 10.1007/978-3-319-24571-3_51 CrossRefGoogle Scholar
  7. 7.
    Plonsey, R.: Bioelectric Phenomena. Wiley Online Library, Hoboken (1969)Google Scholar
  8. 8.
    Powell, M.J.: Developments of NEWUOA for minimization without derivatives. IMA J. Numer. Anal. 28(4), 649–664 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rasmussen, C.E.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  10. 10.
    Sapp, J., Dawoud, F., Clements, J., Horáček, M.: Inverse solution mapping of epicardial potentials: quantitative comparison to epicardial contact mapping. Circ. Arrhythmia Electrophysiol. 5(5), 1001–1009 (2012)CrossRefGoogle Scholar
  11. 11.
    Schiavazzi, D., Arbia, G., Baker, C., et al.: Uncertainty quantification in virtual surgery hemodynamics predictions for single ventricle palliation. Int. J. Numer. Methods Biomed. Eng. (2015)Google Scholar
  12. 12.
    Sermesant, M., Chabiniok, R., Chinchapatnam, P., et al.: Patient-specific electromechanical models of the heart for the prediction of pacing acute effects in CRT: a preliminary clinical validation. Med. Image Anal. 16(1), 201–215 (2012)CrossRefGoogle Scholar
  13. 13.
    Siekmann, I., Sneyd, J., Crampin, E.J.: MCMC can detect nonidentifiable models. Biophys. J. 103(11), 2275–2286 (2012)CrossRefGoogle Scholar
  14. 14.
    Wang, L., Zhang, H., Wong, K.C., Liu, H., Shi, P.: Physiological-model-constrained noninvasive reconstruction of volumetric myocardial transmembrane potentials. IEEE Trans. Biomed. Eng. 57(2), 296–315 (2010)CrossRefGoogle Scholar
  15. 15.
    Wong, K.C.L., Relan, J., Wang, L., Sermesant, M., Delingette, H., Ayache, N., Shi, P.: Strain-based regional nonlinear cardiac material properties estimation from medical images. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012. LNCS, vol. 7510, pp. 617–624. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33415-3_76 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jwala Dhamala
    • 1
    Email author
  • John L. Sapp
    • 2
  • Milan Horacek
    • 2
  • Linwei Wang
    • 1
  1. 1.Rochester Institute of TechnologyRochesterUSA
  2. 2.Dalhousie UniversityHalifaxCanada

Personalised recommendations