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Inference in a Partial Differential Equations Model of Pulmonary Arterial and Venous Blood Circulation Using Statistical Emulation

  • ️Tue Oct 17 2017

Abstract

The present article addresses the problem of inference in a multiscale computational model of pulmonary arterial and venous blood circulation. The model is a computationally expensive simulator which, given specific parameter values, solves a system of nonlinear partial differential equations and returns predicted pressure and flow values at different locations in the arterial and venous blood vessels. The standard approach in parameter calibration for computer code is to emulate the simulator using a Gaussian Process prior. In the present work, we take a different approach and emulate the objective function itself, i.e. the residual sum of squares between the simulations and the observed data. The Efficient Global Optimization (EGO) algorithm [2] is used to minimize the residual sum of squares. A generalization of the EGO algorithm that can handle hidden constraints is described. We demonstrate that this modified emulator achieves a reduction in the computational costs of inference by two orders of magnitude.

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Notes

  1. 1.

    Vascular rarefaction is an old finding in patients with hypertension, and represents the condition of having fewer blood vessels per tissue volume.

  2. 2.

    See Eq. (4.14) in Rasmussen and Williams [9] Sect. 4.2.

  3. 3.

    In this context, surrogate or metamodel are synonyms for GP posterior mean, and were generated by the engineering community.

  4. 4.

    The presented results were obtained on a CentOS 7 machine using MATLAB®. Our code used to perform inference depends on the GPML toolbox by Rasmussen [9] and the standard MATLAB® Statistics Toolbox.

References

  1. Gelbart, M.A., Snoek, J., Adams, R.P.: Bayesian optimization with unknown constraints. In: Uncertainty in Artificial Intelligence (UAI) (2014)

    Google Scholar 

  2. Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13, 455–492 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kuss, M.: Gaussian process models for robust regression, classification, and reinforcement learning. Ph.D. thesis, Technische Universität, Darmstadt (2006)

    Google Scholar 

  4. Mockus, J., Tiesis, V., Zilinskas, A.: The application of bayesian methods for seeking the extremum. Towards Global Optim. 2, 117–129 (1978)

    MATH  Google Scholar 

  5. Nickisch, H., Rasmussen, C.E.: Approximations for binary gaussian process classification. J. Mach. Learn. Res. 9, 2035–2078 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Olufsen, M.S.: Structured tree outflow condition for blood flow in larger systemic arteries. Am. J. Physiol. Heart Circulatory Physiol. 276, 257–268 (1999)

    Google Scholar 

  7. Perttunen, C.D., Jones, D.R., Stuckman, B.E.: Lipschitzian optimization without the lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Qureshi, M.U., Vaughan, G.D.A., Sainsbury, C., et al.: Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation. Biomech. Model. Mechanobiol. 13(5), 1137–1154 (2014)

    Article  Google Scholar 

  9. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2005)

    MATH  Google Scholar 

  10. Rosenkranz, S., Preston, I.R.: Right heart catheterisation: best practice and pitfalls in pulmonary hypertension. Eur. Respir. Rev. 24, 642–652 (2015)

    Article  Google Scholar 

  11. Sasena, M.J.: Optimization of Computer Simulations via Smoothing Splines and Kriging Metamodels. MSc Thesis, University of Michigan (1998)

    Google Scholar 

  12. Snoek, J., Larochelle, H., Adams, R.P.: Practical bayesian optimization of machine learning algorithms. In: Advances in Neural Information Processing Systems, pp. 2951–2959 (2012)

    Google Scholar 

  13. Snoek, J.: Bayesian Optimization and Semiparametric Models with Applications to Assistive Technology. Ph.D. thesis, University of Toronto, Toronto, Canada (2013)

    Google Scholar 

  14. Ugray, Z., et al.: Scatter search and local nlp solvers: a multistart framework for global optimization. INFORMS J. Comput. 19(3), 328–340 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vanhatalo, J., et al.: GPstuff: Bayesian modeling with Gaussian processes. J. Mach. Learn. Res. 14(1), 1175–1179 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Wang, H.M., et al.: Structure-based finite strain modelling of the human left ventricle in diastole. Int. J. Numer. Methods Biomed. Eng. 29, 83–103 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

UN is supported by a scholarship from the Biometrika Trust. SofTMech is a research centre for Multi-scale Modelling in Soft Tissue Mechanics, funded by EPSRC (grant no. EP/N014642/1).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8SQ, UK

    Umberto Noè, Nicholas Hill & Dirk Husmeier

  2. Research Center for Regenerative Medicine, Guangxi Medical University, Nanning, 530021, China

    Weiwei Chen

  3. Eurecom, Campus SophiaTech, 450 Route des Chappes, Biot, France

    Maurizio Filippone

Authors

  1. Umberto Noè

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  2. Weiwei Chen

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  3. Maurizio Filippone

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  4. Nicholas Hill

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  5. Dirk Husmeier

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Corresponding author

Correspondence to Umberto Noè .

Editor information

Editors and Affiliations

  1. Computing Science and Mathematics, University of Stirling, Stirling, United Kingdom

    Andrea Bracciali

  2. School of Informatics, University of Edinburgh, Edinburgh, United Kingdom

    Giulio Caravagna

  3. Department of Computer Science, Brunel University London, Uxbridge, Middlesex, United Kingdom

    David Gilbert

  4. Department of Management and Innovation Systems DISA-MIS, University of Salerno, Fisciano, Italy

    Roberto Tagliaferri

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Noè, U., Chen, W., Filippone, M., Hill, N., Husmeier, D. (2017). Inference in a Partial Differential Equations Model of Pulmonary Arterial and Venous Blood Circulation Using Statistical Emulation. In: Bracciali, A., Caravagna, G., Gilbert, D., Tagliaferri, R. (eds) Computational Intelligence Methods for Bioinformatics and Biostatistics. CIBB 2016. Lecture Notes in Computer Science(), vol 10477. Springer, Cham. https://doi.org/10.1007/978-3-319-67834-4_15

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  • DOI: https://doi.org/10.1007/978-3-319-67834-4_15

  • Published: 17 October 2017

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67833-7

  • Online ISBN: 978-3-319-67834-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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