Classification of cerebrovascular pathologies in real-time using nonlinear ODE-based surrogate model
-
Yuriy V. Bugai
,
Alexander A. Cherevko
und
Maxim A. Shishlenin
Abstract
In this paper we consider the coefficient inverse problem for a second-order nonlinear ODE surrogate model describing hemodynamic parameters during intraoperative neurosurgical measurements. This mathematical model of cerebral hemodynamics is based on the generalized Van der Pol–Duffing equation and described the local interaction of the velocity and pressure of blood flow in cerebral vessels. For each patient, the coefficients of this equation are individual and are determined from clinical data in real-time by solving the coefficient inverse problem. We apply the gradient method for optimization of the cost functional with the analytical finding of initial guess to get the coefficients by clinical data obtained during neurosurgical operation in the vicinity of arterial pathologies. A good initial guess is based on the analytical Fourier method. Statistical analysis of clinical data has shown that the surrogate model equation is sensitive to different types of pathology, which allows intraoperative monitoring of the patient’s condition and assessment of the type of pathology in real time. Numerical results are presented and it is shown that the proposed mathematical model and numerical method predict clinical data well.
Funding statement: This research was carried out during the authors’ visit to the Sirius Mathematics Center in the framework of the Program of Small Research Groups. Alexander A. Cherevko and Yuriy V. Bugai acknowledges the support of the state assignment of LIH SB RAS (Theme no. FWGG-2021-0009-2.3.1.2.10). Maxim A. Shishlenin acknowledges the support of the state assignment of IM SB RAS (Theme No. FWNF-2024-0001).
References
[1] J. Alastruey, S. M. Moore, K. H. Parker, T. David, J. Peiró and S. J. Sherwin, Reduced modelling of blood flow in the cerebral circulation: Coupling 1-D, 0-D and cerebral auto-regulation models, Internat. J. Numer. Methods Fluids 56 (2008), no. 8, 1061–1067. 10.1002/fld.1606Suche in Google Scholar
[2]
J. S. Azamatov and S. I. Kabanikhin,
Volterra operator equations.
[3] N. Butenin, Y. Nejmark and N. Fufaev, Introduction in Nonlinear Oscillation Theory, Nauka, Moscow, 1987. Suche in Google Scholar
[4] E. Butikov, Simulations of Oscillatory Systems, CRC Press, Boca Raton, 2015. 10.1201/b18110Suche in Google Scholar
[5] M. L. Cartwright and J. E. Littlewood, Some fixed point theorems, Ann. of Math. (2) 54 (1951), 1–37. 10.2307/1969308Suche in Google Scholar
[6] A. Cherevko, E. Bord, A. Khe, V. P. K. Orlov and A. Chupakhin, Using non-linear analogue of nyquist diagrams for analysis of the equation describing the hemodynamics in blood vessels near pathologies, J. Phys. 722 (2016), Article ID 012005. 10.1088/1742-6596/722/1/012005Suche in Google Scholar
[7] A. Cherevko, A. Mikhaylova, A. Chupakhin, I. Ufimtseva, A. Krivoshapkin and K. Orlov, Relaxation oscillation model of hemodynamic parameters in the cerebral vessels, J. Phys. 722 (2016), Article ID 012045. 10.1088/1742-6596/722/1/012045Suche in Google Scholar
[8] N. Chumakova and A. Madyarov, Parametric sensitivity of methanol oxidation process as solution of boundary-value problem with an unknown parameter, Chem. Eng. J. 91 (2003), 159–166. 10.1016/S1385-8947(02)00149-3Suche in Google Scholar
[9] A. Chupakhin, A. Cherevko, A. Khe, N. Telegina, A. Krivoshapkin, K. Orlov, V. Panarin and V. Baranov, Measurements and analysis of cerebral hemodynamic for patients with vascular cerebral malformations, Circulation Pathol. Cardiac Surgery 44 (2012), 27–31. Suche in Google Scholar
[10] L. Cveticanin, Strongly Nonlinear Oscillators, Undergrad. Lect. Notes Phys., Springer, Cham, 2014. 10.1007/978-3-319-05272-4Suche in Google Scholar
[11] G. Dimitriadis, Introduction to Nonlinear Aeroelasticity, Wiley, New York, 2017. 10.1002/9781118756478Suche in Google Scholar
[12] T. K. Dobroserdova and M. A. Olshanskii, A finite element solver and energy stable coupling for 3D and 1D fluid models, Comput. Methods Appl. Mech. Engrg. 259 (2013), 166–176. 10.1016/j.cma.2013.03.018Suche in Google Scholar
[13] L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries, J. Engrg. Math. 47 (2003), no. 3–4, 251–276. 10.1023/B:ENGI.0000007980.01347.29Suche in Google Scholar
[14] L. Formaggia, D. Lamponi, M. Tuveri and A. Veneziani, Numerical modeling of 1d arterial networks coupled with a lumped parameters description of the heart, Comp. Methods Biomech. Biomed. Eng. 9 (2006), no. 5, 273–288. 10.1080/10255840600857767Suche in Google Scholar PubMed
[15] S. Fresca, A. Manzoni, L. Dedè and A. Quarteroni, Deep learning-based reduced order models in cardiac electrophysiology, PLOS One. 15 (2020), no. 10, Article ID e0239416. 10.1371/journal.pone.0239416Suche in Google Scholar PubMed PubMed Central
[16] G. Guglielmi, Analysis of the hemodynamic characteristics of brain arteriovenous malformations using electrical models: baseline settings, surgical extirpation, endovascular embolization, and surgical bypass, Neurosurgery 63 (2008), no. 1, 1–11. 10.1227/01.NEU.0000315286.77538.57Suche in Google Scholar
[17] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math. 72 (1995), no. 1, 21–37. 10.1007/s002110050158Suche in Google Scholar
[18] N. S. Hoang and A. G. Ramm, Dynamical systems gradient method for solving nonlinear equations with monotone operators, Acta Appl. Math. 106 (2009), no. 3, 473–499. 10.1007/s10440-008-9308-1Suche in Google Scholar
[19] F. Huang and S. Ying, On-line parameter identification of the lumped arterial system model: A simulation study, PLOS ONE 15 (2020), no. 7, Article ID e0236012. 10.1371/journal.pone.0236012Suche in Google Scholar PubMed PubMed Central
[20] S. I. Kabanikhin, Definitions and examples of inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 4, 317–357. 10.1515/JIIP.2008.019Suche in Google Scholar
[21] S. I. Kabanikhin and M. A. Shishlenin, Quasi-solution in inverse coefficient problems, J. Inverse Ill-Posed Probl. 16 (2008), no. 7, 705–713. 10.1515/JIIP.2008.043Suche in Google Scholar
[22] S. I. Kabanikhin and M. A. Shishlenin, Recovering a time-dependent diffusion coefficient from nonlocal data, Numer. Anal. Appl. 11 (2018), no. 1, 38–44. 10.1134/S1995423918010056Suche in Google Scholar
[23] S. I. Kabanikhin and M. A. Shishlenin, Theory and numerical methods for solving inverse and ill-posed problems, J. Inverse Ill-Posed Probl. 27 (2019), no. 3, 453–456. 10.1515/jiip-2019-5001Suche in Google Scholar
[24] E. Karabelas, S. Longobardi and J. Fuchsberger, Global sensitivity analysis of four chamber heart hemodynamics using surrogate models, IEEE Trans. Biomed. Eng. 69 (2022), no. 10, 3216–3223. 10.1109/TBME.2022.3163428Suche in Google Scholar PubMed PubMed Central
[25] A. Khe, A. Cherevko, A. Chupakhin, A. Krivoshapkin and K. Orlov, Endovascular blood flow measurement system, J. Phys. 722 (2016), 10.1088/1742-6596/722/1/012041. 10.1088/1742-6596/722/1/012041Suche in Google Scholar
[26] A. K. Khe, A. P. Chupakhin, A. A. Cherevko, S. S. Eliava and Y. V. Pilipenko, Viscous dissipation energy as a risk factor in multiple cerebral aneurysms, Russian J. Numer. Anal. Math. Modelling 30 (2015), no. 5, 277–287. 10.1515/rnam-2015-0025Suche in Google Scholar
[27] S. Kindermann, Convergence of the gradient method for ill-posed problems, Inverse Probl. Imaging 11 (2017), no. 4, 703–720. 10.3934/ipi.2017033Suche in Google Scholar
[28] A. Klimov, V. Ryabuhin, K. Orlov and V. Krilov, Surgery of the Cerebral Aneurysms, Vol. 3, T. A. Alexeeva, Moscow, 2012. Suche in Google Scholar
[29] D. V. Klyuchinskiy, N. S. Novikov and M. A. Shishlenin, CPU-time and RAM memory optimization for solving dynamic inverse problems using gradient-based approach, J. Comput. Phys. 439 (2021), Article ID 110374. 10.1016/j.jcp.2021.110374Suche in Google Scholar
[30] I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour, John Wiley & Sons, Chichester, 2011. 10.1002/9780470977859Suche in Google Scholar
[31] V. Krylov, Surgery of the Cerebral Aneurysms, Vol. 1, T. A. Alexeeva, Moscow, 2011. Suche in Google Scholar
[32] B. Kryukov, Forced Oscillations of Essentially Nonlinear Systems, Mashinostroenie, Moscow, 1984. Suche in Google Scholar
[33] M. Litao, C. Pilar-Arceo and G. Legaspi, Avm compartments: Do they modulate trasnidal pressures? an electrical network analysis, Asian J. Neurosurgery 7 (2012), no. 4, 174–180. 10.4103/1793-5482.106649Suche in Google Scholar PubMed PubMed Central
[34] S. Nagasawa, M. Kawanishi, H. Tanaka, T. Ohta, S. Nagayasu and H. Kikuchi, Haemodynamic study of arteriovenous malformations using a hydraulic model, Neurological Res. 15 (1993), no. 6, 409–412. 10.1080/01616412.1993.11740174Suche in Google Scholar PubMed
[35] A. Neubauer, Some generalizations for Landweber iteration for nonlinear ill-posed problems in Hilbert scales, J. Inverse Ill-Posed Probl. 24 (2016), no. 4, 393–406. 10.1515/jiip-2015-0086Suche in Google Scholar
[36] A. Neubauer and O. Scherzer, A convergence rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, Z. Anal. Anwend. 14 (1995), no. 2, 369–377. 10.4171/zaa/679Suche in Google Scholar
[37] G. P. Panasenko and R. Stavre, Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl. (9) 85 (2006), no. 4, 558–579. 10.1016/j.matpur.2005.10.011Suche in Google Scholar
[38] G. P. Panasenko and R. Stavre, Asymptotic analysis of a viscous fluid-thin plate interaction: Periodic flow, C. R. Mech. 340 (2012), no. 8, 590–595. 10.1016/j.crme.2012.06.001Suche in Google Scholar
[39] T. Pedley, The Fluid Mechanics of Large Blood Vessels, Cambridge University, Cambridge, 1980. 10.1017/CBO9780511896996Suche in Google Scholar
[40] L. Pegolotti, M. R. Pfaller and N. L. Rubio, Learning reduced-order models for cardiovascular simulations with graph neural networks, Comput. Biol. Med. 168 (2024), Article ID 107676. 10.1016/j.compbiomed.2023.107676Suche in Google Scholar PubMed PubMed Central
[41] A. Prikhodko, M. Shishlenin and O. Stadnichenko, Comparative analysis of numerical methods for determining parameters of chemical reactions from experimental data, J. Phys. 2092 (2021), Article ID 012011. 10.1088/1742-6596/2092/1/012011Suche in Google Scholar
[42] M. A. Z. Raja, F. H. Shah and M. I. Syam, Intelligent computing approach to solve the nonlinear Van der Pol system for heartbeat model, Neural Comput. Appl. 30 (2018), 3651–3675. 10.1007/s00521-017-2949-0Suche in Google Scholar
[43] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl. 194 (1995), no. 3, 911–933. 10.1006/jmaa.1995.1335Suche in Google Scholar
[44] R. Schmidt and G. Thews, Human Physiology, Springer, Berlin, 1989. 10.1007/978-3-642-73831-9Suche in Google Scholar
[45] S. J. Sherwin, L. Formaggia, J. Peiró and V. Franke, Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, J. Numer. Methods Fluids 43 (2003), no. 6–7, 673–700. 10.1002/fld.543Suche in Google Scholar
[46] J. Stoker, Nonlinear Vibrations, Interscience, New York, 1950. Suche in Google Scholar
[47] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Westview Press, Boulder, 2001. Suche in Google Scholar
[48] B. van der Pol, The nonlinear theory of electric oscillations, Proc. Inst. Radio Engineers 22 (1934), no. 9, 1051–1086. 10.1109/JRPROC.1934.226781Suche in Google Scholar
[49] V. V. Vasin, Convergence of gradient-type methods for nonlinear equations, Dokl. Akad. Nauk 359 (1998), no. 1, 7–9. Suche in Google Scholar
[50] N. A. Vorobtsova, A. A. Yanchenko, A. A. Cherevko, A. P. Chupakhin, A. L. Krivoshapkin, K. Y. Orlov, V. A. Panarin and V. I. Baranov, Modelling of cerebral aneurysm parameters under stent installation, Russian J. Numer. Anal. Math. Modelling 28 (2013), no. 5, 505–516. 10.1515/rnam-2013-0028Suche in Google Scholar
[51] A. M. Yaglom, Hydrodynamic Instability and Transition to Turbulence, Fluid Mech. Appl. 100, Springer, Dordrecht, 2012. 10.1007/978-94-007-4237-6Suche in Google Scholar
[52] A. A. Yanchenko, A. A. Cherevko, A. P. Chupakhin, A. L. Krivoshapkin and K. Y. Orlov, Nonstationary hemodynamics modelling in a cerebral aneurysm of a blood vessel, Russian J. Numer. Anal. Math. Modelling 29 (2014), no. 5, 307–317. 10.1515/rnam-2014-0025Suche in Google Scholar
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