Advertisement

Journal of Mathematical Biology

, Volume 67, Issue 1, pp 39–68 | Cite as

A practical approach to parameter estimation applied to model predicting heart rate regulation

  • Mette S. Olufsen
  • Johnny T. Ottesen
Article

Abstract

Mathematical models have long been used for prediction of dynamics in biological systems. Recently, several efforts have been made to render these models patient specific. One way to do so is to employ techniques to estimate parameters that enable model based prediction of observed quantities. Knowledge of variation in parameters within and between groups of subjects have potential to provide insight into biological function. Often it is not possible to estimate all parameters in a given model, in particular if the model is complex and the data is sparse. However, it may be possible to estimate a subset of model parameters reducing the complexity of the problem. In this study, we compare three methods that allow identification of parameter subsets that can be estimated given a model and a set of data. These methods will be used to estimate patient specific parameters in a model predicting baroreceptor feedback regulation of heart rate during head-up tilt. The three methods include: structured analysis of the correlation matrix, analysis via singular value decomposition followed by QR factorization, and identification of the subspace closest to the one spanned by eigenvectors of the model Hessian. Results showed that all three methods facilitate identification of a parameter subset. The “best” subset was obtained using the structured correlation method, though this method was also the most computationally intensive. Subsets obtained using the other two methods were easier to compute, but analysis revealed that the final subsets contained correlated parameters. In conclusion, to avoid lengthy computations, these three methods may be combined for efficient identification of parameter subsets.

Keywords

Parameter estimation Inverse problems Subset selection Simulation and modeling Nonlinear heart rate model Medical applications Patient specific modeling 

Mathematics Subject Classification

90C31 34K29 34K60 92C30 92C50 93A30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson DA (1982) Structural properties of compartmental models. Math Biosci 58: 61–81MathSciNetzbMATHCrossRefGoogle Scholar
  2. Astrom KJ, Cykhoff P (1971) System identification: a survey. Automatica 7: 123–162CrossRefGoogle Scholar
  3. Banks HT, Davidian M, Samuels JR, Sutton KL (2009) An inverse problem statistical methodology summary. In: Chowell G, Hayman JM, Bettencourt LMA, Castillo-Chavez C (eds) Mathematical and statistical estimation approaches in epidemiology. Springer, New YorkGoogle Scholar
  4. Baker CTH, Paul CAH (1997) Pitfalls in parameter estimation for delay differential equations. SIAM J Sci Comput 18: 305–314MathSciNetzbMATHCrossRefGoogle Scholar
  5. Batzel J, Baselli G, Mukkamala R, Chon KH (2009) Modeling and disentangling physiological mechanisms: linear and nonlinear identification techniques for analysis of cardiovascular regulation. Philos Trans A Math Phys Eng Sci 367: 1377–1391zbMATHCrossRefGoogle Scholar
  6. Bellman R, Astrom KJ (1970) On structural identifiability. Math Biosci 7: 329–339CrossRefGoogle Scholar
  7. Berman M, Schoenfeld R (1956) Invariants in experimental data on linear kinetics and the formulation of models. J Appl Phys 27: 1361–1370CrossRefGoogle Scholar
  8. Borella B, Lucchi E, Nicosia S, Valigi P (2004) A Kalman filtering approach to estimation of maximum ventricle elastance. Conf Proc IEEE Eng Med Biol Soc 5: 3642–3645Google Scholar
  9. Burnham KP, Anderson DR (2002) Model selection and multimodel inference: a practical information-theoretic approach, 2nd edn. Springer, New YorkGoogle Scholar
  10. Burth M, Verghese GC, Valez-Reyes MV (1999) Subset selection for improved parameter estimation in on-line identification of a synchronous generator. IEEE Trans Power Syst 14: 218–225CrossRefGoogle Scholar
  11. Carson ER, Finkelstein L, Cobelli C (1985) Mathematical modeling of metabolic and endocrine systems: model formulation, identification, and validation. Wiley, New YorkGoogle Scholar
  12. Cintron-Arias A, Banks HT, Capaldi A, Lloyd AL (2009) A sensitivity matrix based methodology for inverse problem formulation. J Inv Ill-Posed Problems 17: 545–564MathSciNetzbMATHCrossRefGoogle Scholar
  13. Cobelli C, DiStefano JJ (1980) Parameter and structural identifiability concepts and ambiguities: a critical review and analysis. Am J Physiol 239: R7–R24Google Scholar
  14. Daun S, Rubin J, Vodovotz Y, Roy A, Parker R, Clermont G (2008) An ensemble of models of the acute inflammatory response to bacterial lipopolysaccharide in rats: results from parameter space reduction. J Theor Biol 253: 843–853CrossRefGoogle Scholar
  15. Dochain D, Vanrolleghem PA (2001) Dynamical modelling and estimation in wastewater treatment processes. IWA Publishing, LondonGoogle Scholar
  16. Ellwein LM, Tran HT, Zapata C, Novak V, Olufsen MS (2008) Sensitivity analysis and model assessment: Mathematical models for arterial blood flow and blood pressure. J Cardiovasc Eng 8: 94–108CrossRefGoogle Scholar
  17. Ellwein LM, Otake H, Gundert TJ, Koo B-K, Shinke T, Honda Y, Shite J, LaDisa JF (2011) Optical coherence tomography for patient-specific 3D artery reconstruction and evaluation of wall shear stress in a left circumflex coronary artery. Cardiovasc Eng Technol. doi: 10.1007/s13239-011-0047-5
  18. Eslami M (1994) Theory of sensitivity in dynamic systems: an introduction. Springer, BerlinGoogle Scholar
  19. Fadel PJ, Stromstad M, Wray DW, Smith SA, Raven PB, Secher NH (2003) New insights into differential baroreflex control of heart rate in humans. Am J Physiol 284: H735–H743Google Scholar
  20. Ferguson DW, Abboud FM, Mark AL (1985) Relative contribution of aortic and carotid baroreflexes to heart rate control in man during steady state and dynamic increases in arterial pressure. J Clin Invest 76: 2265–2274CrossRefGoogle Scholar
  21. Fisher FM (1959) Generalization of the rank and order conditions for identifiability. Econometrica 27: 431–437MathSciNetzbMATHCrossRefGoogle Scholar
  22. Frank P (1978) Introduction to sensitivity theory. Academic Press, New YorkzbMATHGoogle Scholar
  23. Golub GH, van Loan CF (1996) Matrix computations. Johns Hopkins Studies in Mathematical Sciences, 3rd edn. Baltimore, MDGoogle Scholar
  24. Golub GH, Klema VC, Stewart GW (1976) Rank degeneracy and least squares problems. Technical report, Department of Computer Science, Stanford University, Stanford, CAGoogle Scholar
  25. Godfrey K (1983) Compartmental models and their application. Academic Press, New YorkGoogle Scholar
  26. Guyton AC, Coleman TG, Granger HJ (1972) Overall regulation. Ann Rev Physiol 34: 13–44CrossRefGoogle Scholar
  27. Guglielmi N, Hairer E (2010) Radar5: a fortran-90 code for the numerical integration of stiff and implicit systems of delay differential equations. http://www.unige.ch/math/folks/hairer
  28. Gustafson P (2009) What are the limits of posterior distributions arising from nonidentified models, and why should we care?. J Am Stat Assoc Theor Method 104: 1682–1695MathSciNetzbMATHCrossRefGoogle Scholar
  29. Gutenkunst RN, Waterfall JJ, Casey FP, Brown KS, Myers CR, Sethna JP (2007a) Universally sloppy parameter sensitivities in systems biology models. PLoS Comput Biol 3: 1871–1878MathSciNetCrossRefGoogle Scholar
  30. Gutenkunst RN, Casey FP, Waterfall JJ, Myers CR, Sethna JP (2007b) Extracting falsifiable predictions from sloppy models. Ann NY Acad Sci 1115: 203–211CrossRefGoogle Scholar
  31. Hu X, Nenov V, Bergsneider M, Glenn TC, Vespa P, Martin N (2007) Estimation of hidden state variables of the intracranial system using constrained nonlinear Kalman filters. IEEE Trans Biomed Eng 54: 597–610CrossRefGoogle Scholar
  32. Ipsen ICF, Kelley CT, Pope SR (2011) Nonlinear least squares problems and subset selection. SIAM J Numerical Anal 49: 1244–1266MathSciNetzbMATHCrossRefGoogle Scholar
  33. Jacquez JA (1985) Compartmental analysis in biology and medicine, 2nd edn. The University of Michigan Press, Ann ArborGoogle Scholar
  34. Kelley CT (1999) Iterative methods for optimization. Frontiers in applied mathematics, vol 18. SIAM, PhiladelphiaCrossRefGoogle Scholar
  35. Kelley CT (2011) Implicit filtering, software environments and tools, vol 23. SIAM, PhiladelphiaCrossRefGoogle Scholar
  36. Khoo MC (2008) Modeling of autonomic control in sleep-disordered breathing. Cardiovasc Eng 8: 30–41MathSciNetCrossRefGoogle Scholar
  37. Kim HJ, Vignon-Clementel IE, Coogan JS, Figueroa CA, Jansen KE, Taylor CA (2010) Patient-specific modeling of blood flow and pressure in human coronary arteries. Ann Biomed Eng 38: 3195–3209CrossRefGoogle Scholar
  38. Kiparissides A, Kucherenko SS, Mantalaris A, Pistikopoulos EN (2009) Global sensitivity analysis challenges in biological systems modeling. Ind Eng Chem Res 48: 7168–7180CrossRefGoogle Scholar
  39. Koopmans TC, Reiersol O (1950) Identification of structural characteristics. Ann Math Statist 21: 165–181MathSciNetzbMATHCrossRefGoogle Scholar
  40. Liang H, Miao H, Wu H (2010) Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model. Ann Appl Stat 4: 460–483MathSciNetzbMATHCrossRefGoogle Scholar
  41. Lo M-T, Novak V, Peng C-K, Liu Y, Hu K (2009) Nonlinear phase interaction between nonstationary signals: a comparison study of methods based on Hilbert-Huang and Fourier transforms. Phys Rev E Stat Nonlinear Soft Matter Phys 79(6 Pt 1): 061924MathSciNetCrossRefGoogle Scholar
  42. Ljung L (1999) System Identification, theory for the user, 2nd edn. Prentice Hall Inc, Upper Saddle RiverGoogle Scholar
  43. Mehra RK (2008) Topics in stochastic control theory-identification in control and econometrics: similarities and differences. Ann Econ Soc Meas 3: 24–47Google Scholar
  44. Miao H, Dykes C, Demeter LM, Cavenaugh J, Parka SY, Perelson AS, Wu H (2008) Modeling and estimation of kinetic parameters and replicative fitness of HIV-1 from flow-cytometry-based growth competition experiments. Bull Math Biol 70: 1749–1771MathSciNetzbMATHCrossRefGoogle Scholar
  45. Miao H, Xia X, Perelson AS, Wu H (2011) On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Rev 53: 3–39MathSciNetzbMATHCrossRefGoogle Scholar
  46. Olufsen MS, Ottesen JT, Tran HT, Ellwein LM, Lipsitz LA, Novak V (2005) Blood pressure and flow variation during postural change from sitting to standing: model development and validation. J Appl Physiol 99: 1523–1537CrossRefGoogle Scholar
  47. Olufsen MS, Tran HT, Ottesen JT, Lipsitz LA, Novak V (2006) Modeling baroreflex regulation of heart rate during orthostatic stress. Am J Physiol 291: R1355–R1368CrossRefGoogle Scholar
  48. Olufsen MS, Alston AV, Tran HT, Ottesen JT, Novak V (2008) Modeling heart rate regulation. Part I: sit-to-stand versus head-up tilt. J Cardiovasc Eng 8: 73–87CrossRefGoogle Scholar
  49. Ottesen JT (1997) Nonlinearity of baroreceptor nerves. Surv Math Ind 7: 187–201zbMATHGoogle Scholar
  50. Ottesen JT, Olufsen MS (2011) Functionality of the baroreceptor nerves in heart rate regulation. Comput Methods Prog Biomed 101: 208–219CrossRefGoogle Scholar
  51. Olgac U, Poulikakos D, Saur SC, Alkadhi H, Kurtcuoglu V (2010) Patient-specific three-dimensional simulation of LDL accumulation in a human left coronary artery in its healthy and atherosclerotic states. Am J Physiol 296: H1969–H1982Google Scholar
  52. Pennati G, Socci L, Rigano S, Boito S, Ferrazzi E (2008) Computational patient-specific models based on 3-D ultrasound data to quantify uterine arterial flow during pregnancy. IEEE Trans Med Imaging 27: 1715–1722CrossRefGoogle Scholar
  53. Pope SR (2009) Parameter identification in lumped compartment cardiorespiratory models. PhD thesis, North Carolina State University, Raleigh, NCGoogle Scholar
  54. Pope SR, Ellwein LM, Zapata CL, Novak V, Kelley CT, Olufsen MS (2009) Estimation and identification of parameters in a lumped cerebrovascular model. Math Biosci Eng 6: 93–115MathSciNetzbMATHCrossRefGoogle Scholar
  55. Poyton AA, Varziri MS, McAuley KB, McLellan PJ, Ramsay JO (2005) Parameter estimation in continuous-time dynamic models using principal differential analysis. Comput Chem Eng 30: 698–708CrossRefGoogle Scholar
  56. Ramsay JO, Hooker G, Campbell D, Cao J (2007) Parameter estimation for differential equations: a generalized smoothing approach. J Roy Stat Soc: Ser B (Stat Methodol) 69: 741–796MathSciNetCrossRefGoogle Scholar
  57. Rao CR (1971) Unified theory of linear estimation. Sankhya: Indian J Stat Ser A 33: 371–394zbMATHGoogle Scholar
  58. Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmuller U, Timmer J (2009) Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25: 1923–1929CrossRefGoogle Scholar
  59. Rech G, Terasvirta T, Tschernig R (2001) A simple variable selection technique for nonlinear models. Commun Stat Theor Methods 30: 1227–1241MathSciNetzbMATHCrossRefGoogle Scholar
  60. Reymond P, Bohraus Y, Perren F, Lazeyras F, Stergiopulos N (2010) Validation of a patient specific 1D model of the systemic arterial tree. Am J Physiol. doi: 10.1152/ajpheart.00821.2010
  61. Rodriguez-Fernandez M, Egea JA, Banga JR (2006) Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems. BMC Bioinformatics 7: 483. doi: 10.1186/1471-2105-7-483 CrossRefGoogle Scholar
  62. Taylor CA, Figueroa CA (2009) Patient-specific modeling of cardiovascular mechanics. Ann Rev Biomed Eng 11: 109–134CrossRefGoogle Scholar
  63. Thompson DE, McAuley KB, McLellan PJ (2009) Parameter estimation in a simplified MWD model for HDPE produced by a Ziegler-Natta catalyst. Macromol React Eng 3: 160–177CrossRefGoogle Scholar
  64. Valdez-Jasso D, Haider MA, Banks HT, Bia D, Zocalo Y, Armentano R, Olufsen MS (2008) Viscoelastic mapping of the arterial ovine system using a Kelvin model. IEEE Trans Biomed Eng 56: 210–219CrossRefGoogle Scholar
  65. Valdez-Jasso D, Banks HT, Haider MA, Bia D, Zocalo Y, Armentano RL, Olufsen MS (2009) Viscoelastic models for passive arterial wall dynamics. Adv Appl Math Mech 1: 151–165MathSciNetGoogle Scholar
  66. Velez-Reyes MV (1992) Decomposed algorithms for parameter estimation. PhD thesis, Massachusetts Institute of Technology, Cambridge, MAGoogle Scholar
  67. Vilela M, Vinga S, Grivet Mattoso Maia MA, Voit EO, Almeida JS (2009) Identification of neutral biochemical network models from time series data. BMC Syst Biol 3: 47. doi: 10.1186/1752-0509-3-47 CrossRefGoogle Scholar
  68. Wan J, Steele BN, Spicer SA, Strohband S, Feijoo GR, Hughes TJR, Taylor CA (2002) A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease. Comput Methods Biomech Biomed Eng 5: 195–206CrossRefGoogle Scholar
  69. Wu H, Zhua H, Miao H, Perelson AS (2008) Parameter identifiability and estimation of HIV/AIDS dynamic models. Bull Math Biol 70: 785–799MathSciNetzbMATHCrossRefGoogle Scholar
  70. Yue H, Brown M, Knowles J, Wang H, Broomhead DS, Kell DB (2006) Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kB signalling pathway. Mol BioSyst 2: 640–649CrossRefGoogle Scholar
  71. Zenker S, Rubin J, Clermont G (2009) From inverse problems in mathematical physiology to quantitative differential diagnoses. PLoS Comput Biol 3: 2072–2086MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Science, Systems, and ModelsRoskilde UniversityRoskildeDenmark
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA

Personalised recommendations