Advertisement

Journal of Mathematical Biology

, Volume 67, Issue 1, pp 143–168 | Cite as

Inverse problems from biomedicine

Inference of putative disease mechanisms and robust therapeutic strategies
  • James LuEmail author
  • Elias August
  • Heinz Koeppl
Article

Abstract

Many complex diseases that are difficult to treat cannot be mapped onto a single cause, but arise from the interplay of multiple contributing factors. In the study of such diseases, it is becoming apparent that therapeutic strategies targeting a single protein or metabolite are often not efficacious. Rather, a systems perspective describing the interaction of physiological components is needed. In this paper, we demonstrate via examples of disease models the kind of inverse problems that arise from the need to infer disease mechanisms and/or therapeutic strategies. We identify the challenges that arise, in particular the need to devise strategies that are robust against variable physiological states and parametric uncertainties.

Keywords

Systems biology Disease modeling Inverse problems Dynamical systems 

Mathematis Subject Classification

92C42 97M60 65P30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Acikgoz SU, Diwekar UM (2010) Blood glucose regulation with stochastic optimal control for insulin-dependent diabetic patients. Chem Eng Sci 65(3): 1227–1236CrossRefGoogle Scholar
  2. Adiels M (2002) A compartmental model for kinetics of apolipoprotein B-100 and triglycerides in VLDL 1 and VLDL2 in normolipidemic subjects. Master’s thesis, Chalmers University of TechnologyGoogle Scholar
  3. August E, Papachristodoulou A (2009) Efficient, sparse biological network determination. BMC Syst Biol 3: 25CrossRefGoogle Scholar
  4. August E, Parker KH, Barahona M (2007) A dynamical model of lipoprotein metabolism. Bull Math Biol 69: 1233–1254zbMATHCrossRefGoogle Scholar
  5. Aylward EM, Parrilo PA, Slotine JJE (2008) Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming. Automatica 44(8): 2163–2170MathSciNetCrossRefGoogle Scholar
  6. Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton Series in applied mathematics. Princeton University Press, PrincetonGoogle Scholar
  7. Ben-Zvi A, Vernon SD, Broderick G (2009) Model-based therapeutic correction of hypothalamic–pituitary–adrenal axis dysfunction. PLoS Comput Biol 5: e1000273MathSciNetCrossRefGoogle Scholar
  8. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgezbMATHGoogle Scholar
  9. Cedersund G, Knudsen C (2005) Improved parameter estimation for systems with an experimentally located Hopf bifurcation. Syst Biol (Stevenage) 152: 161–168CrossRefGoogle Scholar
  10. Converse CA, Skinner ER (1992) Lipoprotein analysis: a practical approach. IRL Press at Oxford University Press, New YorkGoogle Scholar
  11. Cooper GM (2000) The cell (a molecular approach), 2nd edn. ASM Press, BostonGoogle Scholar
  12. d’Aspremont A, El Ghaoui L, Jordan MI, Lanckriet GRG (2007) A direct formulation for sparse PCA using semidefinite programming. SIAM Rev 49(3):434–448Google Scholar
  13. Davidson MH, Jacobson TA (2001) How statins work: the development of cardiovascular disease and its treatment with 3-hydroxy-3-methylglutarylcoenzyme A reductase inhibitor. Cardiol Clin. Medscape, IncGoogle Scholar
  14. Mol C, Mosci S, Traskine M, Verri A (2009) A regularized method for selecting nested groups of relevant genes from microarray data. J Comput Biol 16(5): 677–690MathSciNetCrossRefGoogle Scholar
  15. Dietschy, JM, Gotto, AM, Ontko , JA (eds) (1978) Disturbances in lipid and lipoprotein metabolism. American Physiological Society, BethesdaGoogle Scholar
  16. Dietschy JM, Turley SD, Spady DK (1993) Role of liver in the maintenance of cholesterol and low density lipoprotein homeostasis in different animal species, including humans. J Lipid Res 34: 1637–1659Google Scholar
  17. Donoho DL (2006) For most large underdetermined systems of equations, the minimal l 1-norm near-solution approximates the sparsest near-solution. Commun Pure Appl Math 59(7): 907–934MathSciNetCrossRefGoogle Scholar
  18. El Ghaoui L, Lebret H (1997) Robust solutions to least-squares problems with uncertain data. SIAM J Matrix Anal Appl 18(4): 1035–1064MathSciNetzbMATHCrossRefGoogle Scholar
  19. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Mathematics and its applications, vol 375. Kluwer, DordrechtCrossRefGoogle Scholar
  20. Engl HW, Flamm C, Kügler P, Lu J, Müller S, Schuster P (2009) Inverse problems in systems biology. Inverse Problems 25(12): 123014CrossRefGoogle Scholar
  21. Faratian D, Goltsov A, Lebedeva G, Sorokin A, Moodie S, Mullen P, Kay C, Um IH, Langdon S, Goryanin I, Harrison DJ (2009) Systems biology reveals new strategies for personalizing cancer medicine and confirms the role of PTEN in resistance to trastuzumab. Cancer Res 69: 6713–6720CrossRefGoogle Scholar
  22. Glass L, Mackey MC (1979) Pathological conditions resulting from instabilities in physiological control systems. Ann NY Acad Sci 316: 214–235CrossRefGoogle Scholar
  23. Glass CK, Witztum JL (2001) Atherosclerosis: the road ahead. Cell 104: 503–516CrossRefGoogle Scholar
  24. Goldstein JL, Brown MS (1977) The low-density lipoprotein pathway and its relation to atherosclerosis. Ann Rev Biochem 46: 897–930CrossRefGoogle Scholar
  25. Gupta S, Aslakson E, Gurbaxani BM, Vernon SD (2007) Inclusion of the glucocorticoid receptor in a hypothalamic pituitary adrenal axis model reveals bistability. Theor Biol Med Model 4: 8CrossRefGoogle Scholar
  26. Hafner M, Koeppl H, Hasler M, Wagner A (2009) ‘Glocal’ robustness analysis and model discrimination for circadian oscillators. PLoS Comput Biol 5: e1000534MathSciNetCrossRefGoogle Scholar
  27. Hafner M, Petrov T, Lu J, Koeppl H (2011) Rational design of robust biomolecular circuits: from specification to parameters. In: Koeppl H, Densmore D, Setti G, Bernardo M (eds) Design and analysis of biomolecular circuits. Springer, Berlin, pp 253–279CrossRefGoogle Scholar
  28. Iadevaia S, Lu Y, Morales FC, Mills GB, Ram PT (2010) Identification of optimal drug combinations targeting cellular networks: integrating phospho-proteomics and computational network analysis. Cancer Res 70: 6704–6714CrossRefGoogle Scholar
  29. Kuznetsov YA (2004) Elements of applied bifurcation theory. Applied mathematical sciences, vol 112, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  30. Lasserre JB, Putinar M (2010) Positivity and optimization for semi-algebraic functions. SIAM J Optim 20(6): 3364–3383MathSciNetzbMATHCrossRefGoogle Scholar
  31. Libby P (2002) Atherosclerosis: the new view. Scientific American pp 46–55Google Scholar
  32. Lim CC, Teo KL (1989) A stochastic optimal control approach to a mathematical drug administration model. Math Comput Model 12(8): 1009–1015MathSciNetzbMATHCrossRefGoogle Scholar
  33. Liu ET, Lauffenburger DA (2009) Systems biomedicine: concepts and perspectives. Elsevier, BurlingtonGoogle Scholar
  34. Lu J (2009) Inverse eigenvalue problems for exploring the dynamics of systems biology models. Adv Appl Math Mech 1(6): 711–728MathSciNetGoogle Scholar
  35. Lu J, Engl HW, Schuster P (2006) Inverse bifurcation analysis: application to simple gene systems. Algorithms Mol Biol 1: 11CrossRefGoogle Scholar
  36. Lu J, Müller S, Machné R, Flamm C (2008) SBML ODE solver library: extensions for inverse analysis. In: Proceedings of WCSB 2008, Leipzig, GermanyGoogle Scholar
  37. Ma PTS, Gil G, Südhof TC, Bilheimer DW, Goldstein JL, Brown MS (1986) Mevinolin, an inhibitor of cholesterol synthesis, induces mRNA for low density lipoprotein receptor in livers of hamsters and rabbits. PNAS 83: 8370–8374CrossRefGoogle Scholar
  38. Magombedze G, Garira W, Mwenje E, Bhunu CP (2011) Optimal control for HIV-1 multi-drug therapy. Int J Comput Math 88(2): 314–340MathSciNetzbMATHGoogle Scholar
  39. Mathews CK, van Holde KE, Ahern KG (2000) Biochemistry. Addison Wesley Longman, San FranciscoGoogle Scholar
  40. Miao H, Xia X, Perelson AS, Wu H (2011) On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Rev: Soc Ind Appl Math 53(1): 3–39. doi: 10.1137/090757009 MathSciNetzbMATHGoogle Scholar
  41. Murty KG, Kabadi SN (1987) Some NP-complete problems in quadratic and nonlinear programming. Math Program 39: 117–129MathSciNetzbMATHCrossRefGoogle Scholar
  42. Orton RJ, Adriaens ME, Gormand A, Sturm OE, Kolch W, Gilbert DR (2009) Computational modelling of cancerous mutations in the EGFR/ERK signalling pathway. BMC Syst Biol 3: 100CrossRefGoogle Scholar
  43. Packard CJ, Demant T, Stewart JP, Bedford D, Caslake MJ, Schwertfeger G, Bedynek A, Shepherd J, Seidel D (2000) Apolipoprotein B metabolism and the distribution of VLDL and LDL subfractions. J Lipid Res 41: 305–317Google Scholar
  44. Parrilo PA (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program Ser B 96: 293–320MathSciNetzbMATHCrossRefGoogle Scholar
  45. Parrilo PA (2005) Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, Pasadena, CaliforniaGoogle Scholar
  46. Prajna S, Papachristodoulou A, Parrilo PA (2002) SOSTOOLS—sum of squares optimization toolbox, user’s guide. http://www.cds.caltech.edu/sostools
  47. Rodríguez JEFB, Herrera JAC, Tusiente NT, Andino AB, Vilaú F (1999) Aterosclerosis, colesterol y pared arterial: Algunas reflexiones. Rev Cubana Invest Biomed 18(3): 169–175Google Scholar
  48. Shimamura T, Imoto S, Yamaguchi R, Fujita A, Nagasaki M, Miyano S (2009) Recursive regularization for inferring gene networks from time-course gene expression profiles. BMC Syst Biol 3: 41CrossRefGoogle Scholar
  49. Stengel RF (1986) Stochastic optimal control. A Wiley-Interscience Publication. John Wiley and Sons Inc., Wiley, New YorkzbMATHGoogle Scholar
  50. Sturm JF (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11–12:625–653. http://sedumi.ie.lehigh.edu
  51. Swan GW (1984) Applications of optimal control theory in biomedicine. Monographs and textbooks in pure and applied mathematics, vol 81. Marcel Dekker Inc, New YorkGoogle Scholar
  52. Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev 38(1): 49–95MathSciNetzbMATHCrossRefGoogle Scholar
  53. Velez DA, Mayberg MR, Ludlam WH (2007) Cyclic Cushing syndrome: definitions and treatment implications. Neurosurg Focus 23:E4; (discussion E4a)Google Scholar
  54. White DA, Baxter M (1984) Hormones and metabolic control. Edward Arnold Ltd, LondonGoogle Scholar
  55. Yang K, Bai H, Ouyang Q, Lai L, Tang C (2008) Finding multiple target optimal intervention in disease-related molecular network. Mol Syst Biol 4: 228CrossRefGoogle Scholar
  56. Zarzer CA (2009) On Tikhonov regularization with non-convex sparsity constraints. Inverse Problems 25(2):1–13Google Scholar
  57. Zenker S, Rubin J, Clermont G (2007) From inverse problems in mathematical physiology to quantitative differential diagnoses. PLoS Comput Biol 3(11): 2072–2086MathSciNetCrossRefGoogle Scholar
  58. Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc Ser B Stat Methodol 67(2): 301–320MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Biomolecular Signaling and Control Group, Automatic Control Laboratory, ETH ZurichZurichSwitzerland
  2. 2.Clinical Modeling & Simulation, Translational Research SciencesBaselSwitzerland

Personalised recommendations