Uncertainty propagation and sensitivity analysis: results from the Ocular Mathematical Virtual Simulator

Christophe Prud'homme; Lorenzo Sala; Marcela Szopos; Christophe Prud'homme; Lorenzo Sala; Marcela Szopos

doi:10.3934/mbe.2021105

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 3: 2010-2032. doi: 10.3934/mbe.2021105

Previous Article Next Article

Research article Special Issues

Uncertainty propagation and sensitivity analysis: results from the Ocular Mathematical Virtual Simulator

1.
IRMA UMR CNRS 7501, Université de Strasbourg, Strasbourg 67000, France
2.
Centre de recherche INRIA de Paris, Paris 75012, France
3.
MAP5 UMR CNRS 8145, Université de Paris, Paris 75006, France

Received: 30 November 2020 Accepted: 17 February 2021 Published: 02 March 2021

We propose an uncertainty propagation study and a sensitivity analysis with the Ocular Mathematical Virtual Simulator, a computational and mathematical model that predicts the hemodynamics and biomechanics within the human eye. In this contribution, we focus on the effect of intraocular pressure, retrolaminar tissue pressure and systemic blood pressure on the ocular posterior tissue vasculature. The combination of a physically-based model with experiments-based stochastic input allows us to gain a better understanding of the physiological system, accounting both for the driving mechanisms and the data variability.
- predictive ocular vascular dynamics,
- computational model,
- sensitivity analysis,
- ocular mathematical virtual simulator,
- Sobol index analysis,
- uncertainty quantification
Citation: Christophe Prud'homme, Lorenzo Sala, Marcela Szopos. Uncertainty propagation and sensitivity analysis: results from the Ocular Mathematical Virtual Simulator[J]. Mathematical Biosciences and Engineering, 2021, 18(3): 2010-2032. doi: 10.3934/mbe.2021105

Related Papers:

Abstract

We propose an uncertainty propagation study and a sensitivity analysis with the Ocular Mathematical Virtual Simulator, a computational and mathematical model that predicts the hemodynamics and biomechanics within the human eye. In this contribution, we focus on the effect of intraocular pressure, retrolaminar tissue pressure and systemic blood pressure on the ocular posterior tissue vasculature. The combination of a physically-based model with experiments-based stochastic input allows us to gain a better understanding of the physiological system, accounting both for the driving mechanisms and the data variability.

References

[1]	D. R. Hose, P. V. Lawford, W. Huberts, L. R. Hellevik, S. W. Omholt, F. N. van de Vosse, Cardiovascular models for personalised medicine: Where now and where next?, Med. Eng. Phys., 72 (2019), 38–48. doi: 10.1016/j.medengphy.2019.08.007
[2]	A. Brault, L. Dumas, D. Lucor, Uncertainty quantification of inflow boundary condition and proximal arterial stiffness–coupled effect on pulse wave propagation in a vascular network, Int. J. Numer. Meth. Bio., 33 (2017), e2859. doi: 10.1002/cnm.2859
[3]	P. Chen, A. Quarteroni, G. Rozza, Simulation-based uncertainty quantification of human arterial network hemodynamics, Int. J. Numer. Meth. Bio., 29 (2013), 698–721. doi: 10.1002/cnm.2554
[4]	V. Eck, J. Feinberg, H. Langtangen, L. Hellevik, Stochastic sensitivity analysis for timing and amplitude of pressure waves in the arterial system, Int. J. Numer. Meth. Bio., 31 (2015), e02711. doi: 10.1002/cnm.2711
[5]	C. Leguy, E. Bosboom, A. Belloum, A. Hoeks, F. Van De Vosse, Global sensitivity analysis of a wave propagation model for arm arteries, Med. Eng. Phys., 33 (2011), 1008–1016. doi: 10.1016/j.medengphy.2011.04.003
[6]	C. M. Fleeter, G. Geraci, D. E. Schiavazzi, A. M. Kahn, A. L. Marsden, Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics, Comput. Methods Appl. Mech. Eng., 365 (2020), 113030.
[7]	A. D. Marquis, A. Arnold, C. Dean-Bernhoft, B. E. Carlson, M. S. Olufsen, Practical identifiability and uncertainty quantification of a pulsatile cardiovascular model, Math. Biosci., 304 (2018), 9–24.
[8]	S. Sankaran, A. L. Marsden, A stochastic collocation method for uncertainty quantification and propagation in cardiovascular simulations, J. Biomech. Eng., 133 (2011).
[9]	A. Quaglino, S. Pezzuto, P. S. Koutsourelakis, A. Auricchio, R. Krause, Fast uncertainty quantification of activation sequences in patient-specific cardiac electrophysiology meeting clinical time constraints, Int. J. Numer. Meth. Bio., 34 (2018), e2985. doi: 10.1002/cnm.2985
[10]	D. E. Hurtado, S. Castro, P. Madrid, Uncertainty quantification of 2 models of cardiac electromechanics, Int. J. Numer. Meth. Bio., 33 (2017), e2894.
[11]	J. Campos, J. Sundnes, R. Dos Santos, B. Rocha, Uncertainty quantification and sensitivity analysis of left ventricular function during the full cardiac cycle, Philos. Trans. R. Soc. A., 378 (2020), 20190381. doi: 10.1098/rsta.2019.0381
[12]	G. Guidoboni, A. Harris, R. Sacco, Mathematical modeling of ocular fluid dynamics: From theory to clinical applications, in Modeling and simulation in science, engineering, and technology, Springer, 2019 (2019).
[13]	A. Harris, G. Guidoboni, B. Siesky, S. Mathew, A. C. V. Vercellin, L. Rowe, et al., Ocular blood flow as a clinical observation: Value, limitations and data analysis, Prog. Ret. Eye Res., 2020 (2020), 100841.
[14]	A. C. V. Vercellin, A. Harris, J. V. Cordell, T. Do, J. Moroney, A. Belamkar, et al., Mathematical modeling and glaucoma: the need for an individualized approach to risk assessment, J. Mod. Ophth., 1 (2016), 6–20.
[15]	G. Guidoboni, A. Harris, S. Cassani, J. Arciero, B. Siesky, A. Amireskandari, et al., Intraocular pressure, blood pressure, and retinal blood flow autoregulation: a mathematical model to clarify their relationship and clinical relevance, Invest. Ophth. Visual Sci., 55 (2014), 4105–4118. doi: 10.1167/iovs.13-13611
[16]	G. Guidoboni, A. Harris, L. Carichino, Y. Arieli, B. A. Siesky, Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model, Math. Biosci. Eng., 11 (2014), 523–546. doi: 10.3934/mbe.2014.11.523
[17]	L. Carichino, G. Guidoboni, B. Siesky, A. Amireskandari, I. Januleviciene, A. Harris, et al., Effect of intraocular pressure and cerebrospinal fluid pressure on the blood flow in the central retinal vessels, Integr. Multidiscip. Approaches Study Care Hum. Eye Kugler Publ., 2014 (2014), 59–66.
[18]	M. Szopos, S. Cassani, G. Guidoboni, C. Prud'homme, R. Sacco, B. Siesky, et al., Mathematical modeling of aqueous humor flow and intraocular pressure under uncertainty: towards individualized glaucoma management, J. Mod. Ophth., 1 (2016), 29–39.
[19]	R. Sacco, S. Cassani, G. Guidoboni, M. Szopos, C. Prud'homme, A. Harris, Modeling the coupled dynamics of ocular blood flow and production and drainage of aqueous humor, 4th International Conference on Computational and Mathematical Biomedical Engineering (CMBE 2015), 2015.
[20]	L. Sala, Mathematical modelling and simulation of ocular blood flows and their interactions., Ph.D thesis, Université de Strasbourg, 2019.
[21]	L. Sala, C. Prud'Homme, G. Guidoboni, M. Szopos, Ocular mathematical virtual simulator: A hemodynamical and biomechanical study towards clinical applications, J. Coupled Sys. Multi. Dyn., 6 (2018), 241–247. doi: 10.1166/jcsmd.2018.1165
[22]	L. Sala, C. Prud'Homme, D. Prada, F. Salerni, C. Trophime, V. Chabannes, et al., Patient-specific virtual simulator of tissue perfusion in the lamina cribrosa, Invest. Ophth. Visual Sci., 58 (2017), 727.
[23]	Y. C. Tham, S. H. Lim, P. Gupta, T. Aung, T. Y. Wong, C. Y. Cheng, Inter-relationship between ocular perfusion pressure, blood pressure, intraocular pressure profiles and primary open-angle glaucoma: the Singapore epidemiology of eye diseases study, Brit. J. Ophth., 102 (2018), 1402–1406. doi: 10.1136/bjophthalmol-2017-311359
[24]	L. Sala, C. Prud'homme, G. Guidoboni, M. Szopos, The ocular mathematical virtual simulator: towards uncertainty quantification, 6th International Conference on Computational and Mathematical Biomedical Engineering (CMBE 2019), 2019.
[25]	L. Formaggia, A. Quarteroni, A. Veneziani, Cardiovascular Mathematics: Modeling and simulation of the circulatory system, Springer Science & Business Media, 2010.
[26]	E. H. Starling, On the absorption of fluids from the connective tissue spaces, J. Physiol., 19 (1896), 312–326. doi: 10.1113/jphysiol.1896.sp000596
[27]	S. Cassani, Blood circulation and aqueous humor flow in the eye: multi-scale modeling and clinical applications, Ph.D thesis, Purdue University, 2016.
[28]	P. Fritzson, P. Aronsson, A. Pop, H. Lundvall, K. Nystrom, L. Saldamli, et al., Openmodelica-a free open-source environment for system modeling, simulation, and teaching, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006.
[29]	L. R. Petzold, Description of DASSL: a differential/algebraic system solver, 1982. Available from: https://www.osti.gov/biblio/5882821.
[30]	A. Harris, K. Joos, M. Kay, D. Evans, R. Shetty, W. E. Sponsel, et al., Acute IOP elevation with scleral suction: effects on retrobulbar haemodynamics., Brit. J. Ophth., 80 (1996), 1055–1059. doi: 10.1136/bjo.80.12.1055
[31]	R. Rodriguez-Cantano, H. N. Finsberg, S. T. Wall, J. Sundnes, A bayesian approach for parameter estimation in computational models of cardiac mechanics, 6th International Conference on Computational and Mathematical Biomedical Engineering (CMBE 2019), 2019.
[32]	I. M. Sobol, Sensitivity analysis for non-linear mathematical models, Math. Model. Comp. Exp., 1 (1993), 407–414.
[33]	C. Prieur, S. Tarantola, Variance-based sensitivity analysis: Theory and estimation algorithms, Handb. Uncertainty Quantif., (2017), 1217–1239.
[34]	A. Saltelli, Making best use of model evaluations to compute sensitivity indices, Comput. Phys. Comm., 145 (2002), 280–297. doi: 10.1016/S0010-4655(02)00280-1
[35]	A. Saltelli, S. Tarantola, K. S. Chan, A quantitative model-independent method for global sensitivity analysis of model output, Technometrics, 41 (1999), 39–56. doi: 10.1080/00401706.1999.10485594
[36]	M. Baudin, A. Dutfoy, B. Iooss, A. L. Popelin, Openturns: An industrial software for uncertainty quantification in simulation, preprint, arXiv: 1501.05242.
[37]	H. D. Sesso, M. J. Stampfer, B. Rosner, C. H. Hennekens, J. M. Gaziano, J. E. Manson, et al., Systolic and diastolic blood pressure, pulse pressure, and mean arterial pressure as predictors of cardiovascular disease risk in men, Hypertension, 36 (2000), 801–807. doi: 10.1161/01.HYP.36.5.801
[38]	B. Gavish, I. Z. Ben-Dov, M. Bursztyn, Linear relationship between systolic and diastolic blood pressure monitored over 24 h: assessment and correlates, J. Hypertension, 26 (2008), 199–209. doi: 10.1097/HJH.0b013e3282f25b5a
[39]	E. Limpert, W. A. Stahel, M. Abbt, Log-normal distributions across the sciences: keys and clues: on the charms of statistics, and how mechanical models resembling gambling machines offer a link to a handy way to characterize log-normal distributions, which can provide deeper insight into variability and probabilityormal or log-normal: that is the question, BioScience, 51 (2001), 341–352. doi: 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2
[40]	W. Suh, C. Kee, N. S. Group, K. G. Society, The distribution of intraocular pressure in urban and in rural populations: the Namil study in South Korea, Am. J. Ophth., 154 (2012), 99–106. doi: 10.1016/j.ajo.2012.01.009
[41]	N. L. Johnson, S. Kotz, N. Balakrishnan, Lognormal distributions, Cont. Univariate Dist., 1 (1994), 601–606.
[42]	D. Wang, W. Huang, Y. Li, Y. Zheng, P. J. Foster, N. Congdon, et al., Intraocular pressure, central corneal thickness, and glaucoma in Chinese adults: the Liwan eye study, Am. J. Ophth., 152 (2011), 454–462. doi: 10.1016/j.ajo.2011.03.005
[43]	R. Ren, J. B. Jonas, G. Tian, Y. Zhen, K. Ma, S. Li, et al., Cerebrospinal fluid pressure in glaucoma: a prospective study, Ophthalmology, 117 (2010), 259–266. doi: 10.1016/j.ophtha.2009.06.058
[44]	G. T. Dorner, E. Polska, G. Garhöfer, C. Zawinka, B. Frank, L. Schmetterer, Calculation of the diameter of the central retinal artery from noninvasive measurements in humans, Curr. Eye Res., 25 (2002), 341–345. doi: 10.1076/ceyr.25.6.341.14231
[45]	C. E. Riva, G. T. Feke, B. Eberli, V. Benary, Bidirectional ldv system for absolute measurement of blood speed in retinal vessels, Appl. Opt., 18 (1979), 2301–2306. doi: 10.1364/AO.18.002301
[46]	C. E. Riva, J. E. Grunwald, S. H. Sinclair, B. Petrig, Blood velocity and volumetric flow rate in human retinal vessels, Invest. Ophth. Vis. Sci., 26 (1985), 1124–1132.
[47]	K. E. Lee, B. E. K. Klein, R. Klein, S. M. Meuer, Association of retinal vessel caliber to optic disc and cup diameters, Invest. Ophth. Vis. Sci., 48 (2007), 63–67. doi: 10.1167/iovs.05-1203
[48]	J. P. Garcia Jr, P. T. Garcia, R. B. Rosen, Retinal blood flow in the normal human eye using the canon laser blood flowmeter, Ophth. Res., 34 (2002), 295–299. doi: 10.1159/000065600
[49]	G. T. Feke, C. E. Riva, Laser doppler measurements of blood velocity in human retinal vessels, JOSA, 68 (1978), 526–531. doi: 10.1364/JOSA.68.000526
[50]	L. Sala, C. Prud'homme, G. Guidoboni, M. Szopos, B. A. Siesky, A. Harris, Analysis of IOP and CSF alterations on ocular biomechanics and lamina cribrosa hemodynamics, Invest. Ophth. Vis. Sci., 59 (2018), 4475–4475.
[51]	C. Xu, G. Z. Gertner, Reliability of global sensitivity indices, J. Stat. Comp. Sim., 81 (2011), 1939–1969. doi: 10.1080/00949655.2010.509317
[52]	G. Guidoboni, R. Sacco, M. Szopos, L. Sala, A. C. Verticchio-Vercellin, B. Siesky, et al., Neurodegenerative disorders of the eye and of the brain: a perspective on their fluid-dynamical connections and the potential of mechanism-driven modeling, Front. Neurosci., 14 (2020), 1173.
[53]	B. M. Adams, M. S. Ebeida, M. S. Eldred, G. Geraci, J. D. Jakeman, K. A. Maupin, et al., DAKOTA, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: version 5.0 user's manual, Sandia Natl. Lab. Tech. Rep. SAND, 2009 (2009), 2010–2183.
[54]	C. Daversin, C. Prud'homme, Simultaneous empirical interpolation and reduced basis method for non-linear problems, C. R. Math., 353 (2015).
[55]	R. Hild, Optimization and control of high fields magnets, Ph.D Thesis, Université de Strasbourg, 2020.
[56]	C. Prud'homme, V. Chabannes, V. Doyeux, M. Ismail, A. Samake, G. Pena, et al., Advances in FEEL++ : A domain specific embedded language in C++ for partial differential equations, ECCOMAS 2012 - European Congress on Computational Methods in Applied Sciences and Engineering, e-Book Full Papers, 2012.
[57]	C. Prud'homme, V. Chabannes, T. Metivet, C. Daversin-Catty, R. Hild, G. Dollé, et al., feelpp/feelpp: Feel++ V108, Available from: https://doi.org/10.5281/zenodo.3784254.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)