MCMC methods for inference in a mathematical model of pulmonary circulation

This study performs parameter inference in a partial differential equations system of pulmonary circulation. We use a fluid dynamics network model that takes selected parameter values and mimics the behaviour of the pulmonary haemodynamics under normal physiological and pathological conditions. This is of medical interest as it enables tracking the progression of pulmonary hypertension. We show how we make the fluids model tractable by reducing the parameter dimension from a 55D to a 5D problem. The Delayed Rejection Adaptive Metropolis (DRAM) algorithm, coupled with constraint nonlinear optimization is successfully used to learn the parameter values and quantify the uncertainty in the parameter estimates. To accommodate for different magnitudes of the parameter values, we introduce an improved parameter scaling technique in the DRAM algorithm. Formal convergence diagnostics are employed to check for convergence of the Markov chains. Additionally, we perform model selection using different information criteria, including Watanabe Akaike Information Criteria.

[1]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo in Practice: A Roundtable Discussion , 1998 .

[2]  P. Green,et al.  Delayed rejection in reversible jump Metropolis–Hastings , 2001 .

[3]  David R. Anderson,et al.  Model selection and multimodel inference : a practical information-theoretic approach , 2003 .

[4]  C. Fox,et al.  A general purpose sampling algorithm for continuous distributions (the t-walk) , 2010 .

[5]  L Tierney,et al.  Some adaptive monte carlo methods for Bayesian inference. , 1999, Statistics in medicine.

[6]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[7]  P. Boggs,et al.  Sequential quadratic programming for large-scale nonlinear optimization , 2000 .

[8]  Naomi C Chesler,et al.  Persistent vascular collagen accumulation alters hemodynamic recovery from chronic hypoxia. , 2012, Journal of biomechanics.

[9]  Heikki Haario,et al.  DRAM: Efficient adaptive MCMC , 2006, Stat. Comput..

[10]  Naomi C. Chesler,et al.  Pulmonary vascular wall stiffness: An important contributor to the increased right ventricular afterload with pulmonary hypertension , 2011, Pulmonary circulation.

[11]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[12]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[13]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[14]  Aki Vehtari,et al.  Understanding predictive information criteria for Bayesian models , 2013, Statistics and Computing.

[15]  A. Noordegraaf,et al.  The role of the right ventricle in pulmonary arterial hypertension , 2011, European Respiratory Review.

[16]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[17]  Ralph C. Smith,et al.  Uncertainty Quantification: Theory, Implementation, and Applications , 2013 .

[18]  Tai-yong Lee,et al.  Heuristic scaling method for efficient parameter estimation , 2010 .

[19]  Christopher A Dawson,et al.  Flow and pressure distributions in vascular networks consisting of distensible vessels. , 2003, American journal of physiology. Heart and circulatory physiology.

[20]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[21]  Yubing Shi,et al.  Review of Zero-D and 1-D Models of Blood Flow in the Cardiovascular System , 2011, Biomedical engineering online.

[22]  John Geweke,et al.  Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments , 1991 .

[23]  A. Mira.  On Metropolis-Hastings algorithms with delayed rejection , 2001 .

[24]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[25]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[26]  Sumio Watanabe,et al.  Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory , 2010, J. Mach. Learn. Res..

[27]  Dirk Husmeier,et al.  Parameter Inference in the Pulmonary Circulation of Mice , 2017 .

[28]  Dirk Husmeier,et al.  A computational study of pulmonary hemodynamics in healthy and hypoxic mice , 2017, 1712.01699.

[29]  Paul Bratley,et al.  Algorithm 659: Implementing Sobol's quasirandom sequence generator , 1988, TOMS.

[30]  P. Lax,et al.  Systems of conservation laws , 1960 .

[31]  Vinzenz Gregor Eck,et al.  A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications , 2016, International journal for numerical methods in biomedical engineering.