Skip to main content
SpringerLink
Log in

Some Classes of Stochastic Differential Equations as an Alternative Modeling Approach to Biomedical Problems

  • Chapter
  • First Online:
Nonautonomous Dynamical Systems in the Life Sciences

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2102))

Abstract

Stochastic differential equations (SDEs) provide an appropriate framework for modeling biomedical problems, since they allow detailed a priori biochemical knowledge to be accounted for and at the same time are able to describe the noise in the systems under investigation and in the data without excessively complicating the settings. We present three application paradigms related to an intracellular signaling pathway, to radio-oncological treatments, and to cell dispersal.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Random ODEs are ODEs that include random variables or stochastic processes in their coefficients. Unlike SDEs they can be handled pathwise using deterministic rather than stochastic calculus, see e.g., [37].

  2. 2.

    All these models can actually be put in the framework of SDEs, some of which are however driven by jump processes and not by Brownian motions.

  3. 3.

    Taken from [75].

  4. 4.

    We used that x 3(0) = 0 and extended x 3 by this value on the interval [−δ, 0].

  5. 5.

    Due to the nonlinearity of the system, the large number of parameters to be estimated, and the rather small amount of available data, however, the practical handling of this issue is still not feasible.

  6. 6.

    Hereby the growth limiting is realized by the expectation of the stochastic process of the relevant concentration.

  7. 7.

    See [75].

  8. 8.

    Characterizing the evolution of the number of clonogens w.r.t. the standard reference population of N ref individuals (usually in literature N ref  = 106) after the moment of the illness detection. Hence t represents the time passed since the diagnosis of cancer has been set. A patient is considered to be cured at the first time τ c when \(C_{\gamma }(\tau _{c}) = 0\).

  9. 9.

    However, it can also have some drawbacks (see, e.g., [32, 41]).

  10. 10.

    For a comprehensive deduction of a corresponding model in a related, but much more complex framework (cancer cell migration through a tissue network) we refer to [38, 39].

  11. 11.

    Following the terminology in literature, we shall say that equation (9.37) characterizes the mesoscopic scale of cell dispersal.

  12. 12.

    Observe that this is a mixture of two simpler (Gaussian) kernels, whereby its weights may vary with the cell’s inner dynamics. Thus, the cell motion experiences a higher bias in the direction of the chemoattractant gradient \(\nabla S\) if the weight α 2(y) outperforms its unit conjugate α 1(y). In [38, 39] the same type of turning kernel has been used in a multiscale model for tumor cell migration through tissue network. Unlikely most of the turning kernels proposed so far in the literature (see e.g., [28]) it explicitly involves the gradient of the chemoattractant.

  13. 13.

    For instance, it could be defined as in [39] by ϕ(ξ):= ξ for \(s_{1} \leq \vert \xi \vert \leq s_{2}\) and \(\phi (\xi ):= s_{2} \frac{\xi } {\vert \xi \vert }\) for |ξ| > s 2 , respectively \(\phi (\xi ):= s_{1} \frac{\xi } {\vert \xi \vert }\) for |ξ| < s 1 , whereby the set V of velocities is assumed to be symmetric, of the form \(V = [s_{1},s_{2}] \times {\mathbb{S}}^{n-1}\).

  14. 14.

    Some parts of this subsection have been reproduced from [74] with permission.

  15. 15.

    Observe that the method outlined in Sect. 9.4.3 enables to directly compute the macroscopic cell density n(t, x), without needing to go through the intermediate step of assessing its mesoscopic (higher dimensional) counterpart f(t, x, v, y) as e.g., in the PDE approach. Hence this implies a dimension reduction from 2N + 2 to N.

  16. 16.

    Of course, the influence of the subcellular dynamics cannot be seen on this level of a random individual path.

  17. 17.

    Taken from [76].

  18. 18.

    More precisely their respective projections on the x 1 Ox 2 plane.

  19. 19.

    See [73].

  20. 20.

    Figures reproduced from [73].

  21. 21.

    See [73, 74].

  22. 22.

    Observe that for γ > 0 the bacterial population asymptotically goes over into an unbiased regime.

  23. 23.

    Though not in an explicit way.

  24. 24.

    Related multiscale models in the much more complex framework of tumor cell migration through a tissue network were addressed in [38, 39, 51].

References

  1. A. Bellen, M. Zenaro, Numerical Methods for Delay Differential Equations (Oxford University Press, Oxford, 2003)

    Book  MATH  Google Scholar 

  2. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Complexity and mathematical tools toward the modeling of multicellular growing systems. Math. Comput. Model. 51, 441–451 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. H.C. Berg, How bacteria swim. Sci. Am. 233, 36–44 (1975)

    Article  Google Scholar 

  4. H.C. Berg, D.A. Brown, Chemotaxis in Escherichia Coli analysed by three-dimensional tracking. Nature 232, 500–504 (1972)

    Article  Google Scholar 

  5. N. Bournaveas, V. Calvez, Global existence for the kinetic chemotaxis model without pointwise memory effects, and including internal variables (2008). arXiv:0802.2316v1 [math.AP]

    Google Scholar 

  6. D. Bray, Cell Movements. From Molecules to Motility (Garland, New York, 2001)

    Google Scholar 

  7. M.E. Burton, L.M. Shaw, J.J. Schentag, W.E. Evans, Applied Pharmacokinetics and Pharmacodynamics. Principles of Therapeutic Drug Monitoring (Lippincott Williams & Wilkins, Baltimore, 2006)

    Google Scholar 

  8. T. Cacoullos, Estimation of a multivariate density. Ann. Inst. Stat. Math. 18, 179–189 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Carlson, R. Stuart, X. Lin et al., Comparison of in vitro and in vivo α∕β ratios for prostate cancer. Phys. Med. Biol. 49, 4477–4491 (2004)

    Article  Google Scholar 

  10. C. Cercignani, The Boltzmann Equation and Its Application (Springer, Berlin, 1988)

    Book  Google Scholar 

  11. K.N. Chadwick, H.P. Leenhouts. The Molecular Theory of Radiation Biology (Springer, Berlin, 1981)

    Book  Google Scholar 

  12. E.A. Codling, N.A. Hill, Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters. J. Math. Biol. 51, 527–556 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Czirók, K. Schlett, E. Madarász, T. Vicsek, Exponential distribution of locomotion activity in cell cultures. Phys. Rev. Lett. 81, 3038–3041 (1998)

    Article  Google Scholar 

  14. S.B. Curtis, Lethal and potentially lethal lesions induced by radiation — A unified repair model. Radiat. Res. 106, 252–271 (1986)

    Article  Google Scholar 

  15. J.E. Darnell Jr., STATs and Gene Regulation. Science 277, 1630–1635 (1997)

    Google Scholar 

  16. A. Dawson, T. Hillen, Derivation of the tumour control prabability (TCP) from a cell cycle model. Comput. Math. Method. Med. 7, 121–142 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Deheuvels, Estimation non paramétrique de la densité par histogrames généralisés (II). Publications de l’Institut Statistique de l’Université de Paris 22, 1–23 (1977)

    MATH  MathSciNet  Google Scholar 

  18. L. Devroye, L. Györfi, Nonparametric Density Estimation: The L 1 View (Wiley, New York, 1985)

    Google Scholar 

  19. L. Devroye, Universal smoothing factor selection in density estimation: theory and practice. Test 6, 223–320 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. O. Diekmann, S. van Gils, S. Verduyn Lunel, H.-O. Walter, Delay Equations, Functional-, Complex-, and Nonlinear Analysis (Springer, New York, 1995)

    MATH  Google Scholar 

  21. S. Donnet, J.-L. Foulley, A. Samson, Bayesian analysis of growth curves using mixed models defined by stochastic differential equations. Biometrics 66, 733–741 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Downward, The ins and outs of signalling. Nature 411, 759–762 (2001)

    Article  Google Scholar 

  23. M. Eisenbach, A. Wolf, M. Welch, S.R. Caplan, I.R. Lapidus, R.M. Macnab, H. Aloni, O. Asher, Pausing, switching and speed fluctuation of the bacterial flagellar motor and their relation to motility and chemotaxis. J. Mol. Biol. 211, 551–563 (1990)

    Article  Google Scholar 

  24. M. Eisenbach, J.W. Lengeler, M. Varon, D. Gutnick, R. Meili, R.A. Firtel, J.E. Segall, G.M. Omann, A. Tamada, F. Murakami, Chemotaxis (Imperial College Press, London, 2004)

    Google Scholar 

  25. L.E. Elsgol’s, S.B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Academic, New York, 1973)

    Google Scholar 

  26. R. Erban, H.G. Othmer, From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math. 65, 361–391 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Erban, H.G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology. Multiscale Model. Simul. 3, 362–394 (2005)

    Google Scholar 

  28. R. Erban, H.J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete Continuous Dyn. Syst. Ser. B 6, 1239–1260 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. B. Firmani, L. Guerri, L. Preziosi Tumor immune system competition with medically induced activation disactivation. Math. Models Methods Appl. Sci. 9, 491–512 (1999)

    Google Scholar 

  30. E. Greenberg, E. Canale-Parola, Chemotaxis in Spirocheta aurantia. J. Bacteriol. 130, 485–494 (1977)

    Google Scholar 

  31. J.K. Hale, S. Verduyn Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)

    Book  MATH  Google Scholar 

  32. L. Hanin, M. Zaider, Cell-survival probability at large doses: an alternative to the linear quadratic model. Phys. Med. Biol. 55, 4687–4702 (2010)

    Article  Google Scholar 

  33. T. Hillen, Transport equations with resting phases. Eur. J. Appl. Math. 14, 613–636 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. T. Hillen, G. De Vries, J. Gong, C. Finlay, From cell population models to tumor control probability: including cell cycle effects. Acta Oncologica 49, 1315–1323 (2010)

    Article  Google Scholar 

  35. T. Hillen, H.G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61, 751–775 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. L. Holmström, J. Klemelä, Asymptotic bounds for the expected L 1 error of a multivariate kernel density estimator. J. Multivar. Anal. 42, 245–266 (1992)

    Article  MATH  Google Scholar 

  37. A. Jentzen, P.E. Kloeden, in Taylor Approximations of Stochastic Partial Differential Equations. CBMS Lecture Series (SIAM, Philadelphia, 2011)

    Google Scholar 

  38. J. Kelkel, C. Surulescu, On some models for cancer cell migration through tissue. J. Math. Biosci. Eng. 8, 575–589 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. J. Kelkel, C. Surulescu, A multiscale approach to cancer cell migration through network tissue. Math. Models Methods Appl. Sci. 22, 1150017.1–1150017.25 (2012)

    Article  MathSciNet  Google Scholar 

  40. E.F. Keller, L.A. Segel, Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)

    MATH  Google Scholar 

  41. J.P. Kirkpatrick, J.J. Meyer, L.B. Marks, The linear-quadratic model is inappropriate to model high dose per fraction effects in radiosurgery. Semin. Radiat. Oncol. 18, 240–243 (2008)

    Article  Google Scholar 

  42. P.E. Kloeden, T. Lorenz, Stochastic differential equations with nonlocal sample dependence. Stoch. Anal. Appl. 28, 937–945 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  43. P.E. Kloeden, T. Lorenz, A Peano-like theorem for stochastic differential equations with nonlocal sample dependence. Stoch. Anal. Appl. 31, 19–30 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  44. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992)

    Book  MATH  Google Scholar 

  45. F. Kozusko, Z. Bajzer, Combining Gompertzian growth and cell population dynamics. Math. Biosci. 185, 153–167 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  46. F. Kozusko, M. Bourdeau, A unified model for sigmoid tumor growth based on cell proliferation and quiescence. Cell Prolif. 40, 824–834 (2007)

    Article  Google Scholar 

  47. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic, Boston, 1993)

    MATH  Google Scholar 

  48. R. Lapidus, R. Schiller, A mathematical model for bacterial chemotaxis. Biophys. J. 14, 825–834 (1974)

    Article  Google Scholar 

  49. D.F. Lea, Actions of Radiations on Living Cells (Cambridge University Press, New York, 1955)

    Google Scholar 

  50. T. Lorenz, in Mutational Analysis: A Joint Framework for Cauchy Problems in and Beyond Vector Spaces. Lecture Notes in Mathematics, vol. 1996 (Springer, New York, 2010)

    Google Scholar 

  51. T. Lorenz, C. Surulescu, On a class of multiscale cancer cell migration models: Well-posedness in less regular function spaces, pp. 1–67, TU Kaiserslautern. Preprint (2013) in reviewing at Math. Models Methods Appl. Sci. (submitted)

    Google Scholar 

  52. X. Mao, Stochastic Differential Equations and Applications (Harwood, Chichester, 1997)

    MATH  Google Scholar 

  53. G.I. Marchuk, Mathematical Modeling of Immune Response in Infectious Diseases (Kluwer, Dordrecht, 1997)

    Book  Google Scholar 

  54. J.S. Marron, D. Nolan, Canonical Kernels for Density Estimation. Stat. Probab. Lett. 7, 195–199 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  55. O.V. Matvii, I.M. Cherevko, Approximation of systems with delay and their stability. Nonlinear Oscillations 7, 207–215 (2004)

    Article  MathSciNet  Google Scholar 

  56. T.C. Meng, S. Somani, P. Dhar, Modelling and simulation of biological systems with stochasticity. Silico Biol. 4, 293–309 (2004)

    Google Scholar 

  57. J. Murray, Mathematical Biology I (Springer, Berlin, 2002)

    Google Scholar 

  58. R.M. Nisbet, W.S.C. Gurney, Modeling Fluctuating Populations (Wiley, Chichester, 1982)

    Google Scholar 

  59. S.F.C. O’ Rourke, H. McAneney, T. Hillen, Linear quadratic and tumour control probability modelling in external beam radiotherapy. J. Math. Biol. 58, 799–817 (2009)

    Google Scholar 

  60. B. Øksendal, Stochastic Differential Equations. An Introduction with Applications (Springer, New York, 2003)

    Google Scholar 

  61. D. Ölz, C. Schmeiser, A. Soreff, Multistep navigation of leukocytes: a stochastic model with memory effects. Math. Med. Biol. 22, 291–303 (2005)

    Article  MATH  Google Scholar 

  62. H.G. Othmer, S.R. Dunbar, W. Alt, Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  63. H.G. Othmer, T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  64. A.R. Pagan, A. Ullah, Nonparametric Econometrics (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  65. S. Pellegrini, I. Dusanter-Fourt, The structure, regulation and function of the Janus kinase (JAK) and the signal transducers and activators of transcription (STATs). Eur. J. Biochem. 248, 615–633 (1997)

    Article  Google Scholar 

  66. A.D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC, London/Boca Raton, 2001)

    Book  Google Scholar 

  67. H. Qian, Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems — An analytical theory. Nonlinearity 24, R19–R49 (2011)

    Article  MATH  Google Scholar 

  68. Yu.M. Repin, On the approximation of systems with delay by ordinary differential equations. Prikl. Mat. Mekh. 29(2), 226–245 (1965)

    MathSciNet  Google Scholar 

  69. A.M. Reuther, T.R. Willoughby, P. Kupelian, Toxicity after hypofractionated external beam radiotherapy (70 Gy at 2.5 Gy per fraction) versus standard fractionation radiotherapy (78 Gy at 2 Gy per fraction) for localized prostate cancer. Int. J. Radiat. Oncol. Biol. Phys. 54, suppl. 1, 187–188 (2002)

    Google Scholar 

  70. D.W. Scott, Multivariate Density Estimation: Theory, Practice and Visualization (Wiley, New York, 1992)

    Book  MATH  Google Scholar 

  71. B.W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman & Hall, London, 1986)

    Book  MATH  Google Scholar 

  72. S.K. Srinivasan, R. Vasudevan, Introduction to Random Differential Equations and Their Applications (Elsevier, Amsterdam, 1971)

    MATH  Google Scholar 

  73. C. Surulescu, N. Surulescu, A nonparametric approach to cell dispersal. Int. J. Biomath. Biostat. 1, 109–128 (2010)

    Google Scholar 

  74. C. Surulescu, N. Surulescu, Modeling and simulation of some cell dispersion problems by a nonparametric method. Math. Biosci. Eng. 8, 263–277 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  75. C. Surulescu, N. Surulescu, On some stochastic differential equation models with applications to biological problems. ECMTB 14, 106–117 (2011)

    Google Scholar 

  76. C. Surulescu, N. Surulescu, On two approaches to a multiscale model for bacterial chemotaxis. Preprint, Institute for Numerical and Applied Mathematics, University of Münster (2011). http://wwwmath.uni-muenster.de/num/publications/2011/SS11a/

  77. N. Surulescu, On Some Continuous Time Series Models and Their Use in Financial Economics, Ph.D. thesis, University of Heidelberg, 2010

    Google Scholar 

  78. I. Swameye, T.G. Müller, J. Timmer, O. Sandra, U. Klingmüller, Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. PNAS 100(3), 1028–1033 (2003)

    Article  Google Scholar 

  79. H. Takagi, M.J. Sato, T. Yanagida, M. Ueda, Functional analysis of spontaneous cell movement under different physiological conditions. PLoS One 3(7), e2648 (2008)

    Google Scholar 

  80. C.A. Tobias, E.A. Blakeley, F.Q.H. Ngo, T.C.H. Yang, The repair-misrepair model of cell survival, in Radiation Biology and Cancer Research, ed. by R.E. Meyn, H.R. Withers (Raven Press, New York, 1980), pp. 195–230

    Google Scholar 

  81. J. Touboul, G. Hermann, O. Faugeras, Noise-induced behaviors in neural mean field dynamics (2011) [ArXiv: 1104.5425v1]

    Google Scholar 

  82. A. Upadhyaya, J.-P. Rieu, J.A. Glazier, Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates. Phys. A Stat. Mech. Appl. 293, 549–558 (2001)

    Article  Google Scholar 

  83. P. Vieu, Quadratic errors for nonparametric estimates under dependence. J. Multivar. Anal. 39, 324–347 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  84. M. Zaider, G.N. Minerbo, Tumour control probability: A formulation applicable to any temporal protocol of dose delivery. Phys. Med. Biol. 45, 279–293 (2000)

    Article  Google Scholar 

Download references

Acknowledgement

Christina Surulescu was partially supported by the Baden-Württemberg Foundation.

Author information

Authors and Affiliations

  1. Felix Klein Center of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 31, D-67663, Kaiserslautern, Deutschland

    Christina Surulescu

  2. Institute for Mathematical Statistics, University of Münster, Einsteinstr. 62, D-48149, Münster, Deutschland

    Nicolae Surulescu

Authors
  1. Christina Surulescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolae Surulescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christina Surulescu .

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Peter E. Kloeden

  2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Klagenfurt am Wörthersee, Austria

    Christian Pötzsche

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Surulescu, C., Surulescu, N. (2013). Some Classes of Stochastic Differential Equations as an Alternative Modeling Approach to Biomedical Problems. In: Kloeden, P., Pötzsche, C. (eds) Nonautonomous Dynamical Systems in the Life Sciences. Lecture Notes in Mathematics(), vol 2102. Springer, Cham. https://doi.org/10.1007/978-3-319-03080-7_9

Download citation

Publish with us

Policies and ethics

Search

Navigation