# American Institute of Mathematical Sciences

2011, 8(2): 263-277. doi: 10.3934/mbe.2011.8.263

## Modeling and simulation of some cell dispersion problems by a nonparametric method

 1 ICAM, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany, Germany

Received  February 2010 Revised  December 2010 Published  April 2011

Starting from the classical descriptions of cell motion we propose some ways to enhance the realism of modeling and to account for interesting features like allowing for a random switching between biased and unbiased motion or avoiding a set of obstacles. For this complex behavior of the cell population we propose new models and also provide a way to numerically assess the macroscopic densities of interest upon using a nonparametric estimation technique. Up to our knowledge, this is the only method able to numerically handle the entire complexity of such settings.
Citation: Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263
##### References:
 [1] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, Journal of Mathematical Biology, 9 (1980), 147-177. doi: 10.1007/BF00275919. [2] S. Asmussen and P. W. Glynn, "Stochastic Simulation. Algorithms and Analysis," Springer, 2007. [3] T. Cacoullos, Estimation of a multivariate density, Annals of the Institute of Statistical Mathematics, 18 (1966), 179-189. doi: 10.1007/BF02869528. [4] F. A. C. C. Chalub, Y. Dolak-Struss, P. Markowich, D. Oelz, C. Schmeiser and A. Soreff, Model hierarchies for cell aggregation by chemotaxis, Mathematical Models and Methods in the Applied Sciences, 16 (2006), 1173-1197. doi: 10.1142/S0218202506001509. [5] F. A. C. C. Chalub, P. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatshefte für Mathematik, 142 (2004), 123-141. [6] E. A. Codling and N. A. Hill, Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters, Journal of Mathematical Biology, 51 (2005), 527-556. doi: 10.1007/s00285-005-0317-7. [7] A. Czirók, K. Schlett, E. Madarász and T. Vicsek, Exponential distribution of locomotion activity in cell cultures, Physical Review Letters, 81 (1998), 3038-3041. doi: 10.1103/PhysRevLett.81.3038. [8] P. Deheuvels, Estimation non paramétrique de la densité par histogrames généralisés (II), Publications de l'Institut Statistique de l'Université de Paris, 22 (1977), 1-23. [9] L. Devroye and L. Györfi, "Nonparametric Density Estimation: The $L_1$ View," John Wiley, New York 1985. [10] L. Devroye, Universal smoothing factor selection in density estimation: Theory and practice, Test, 6 (1997), 223-320. doi: 10.1007/BF02564701. [11] R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in E. Coli: A paradigm for multiscale modeling in biology, Multiscale Modeling and Simulation, 3 (2005), 362-394. doi: 10.1137/040603565. [12] F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, Journal of Mathematical Biology, 50 (2005), 189-207. doi: 10.1007/s00285-004-0286-2. [13] C. W. Gear, J. Li and I. G. Kevrekidis, The gap-tooth method in particle simulations, Physics Letters A, 316 (2003), 190-195. doi: 10.1016/j.physleta.2003.07.004. [14] T. Hillen, "Transport Equations and Chemosensitive Movement," Habilitation Thesis, University of Tübingen, 2001. [15] T. Hillen, Hyperbolic models for chemosensitive movement, Mathematical Models and Methods in the Applied Sciences, 12 (2002), 1-28. doi: 10.1142/S0218202502002008. [16] T. Hillen, Transport equations with resting phases, European Journal of Applied Mathematics, 14 (2003), 613-636. doi: 10.1017/S0956792503005291. [17] T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM Journal of Applied Mathematics, 61 (2000), 751-775. doi: 10.1137/S0036139999358167. [18] J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, preprint IANS, University of Stuttgart, 2010, submitted. [19] L. Holmström and J. Klemelä, Asymptotic bounds for the expected $L^1$ error of a multivariate kernel density estimator, Journal of Multivariate Analysis, 42 (1992), 245-266. doi: 10.1016/0047-259X(92)90046-I. [20] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer, 2000. [21] K. V. Mardia, P. E. Jupp, "Directional Statistics," Wiley, 2000. doi: 10.1016/0167-7152(88)90050-8. [22] J. S. Marron and D. Nolan, Canonical kernels for density estimation, Statistics and Probability Letters, 7 (1988), 195-199. doi: 10.1214/aos/1176348653. [23] J. S. Marron and M. P. Wand, Exact mean integrated squared error, Annals of Statistics, 20 (1992), 712-736. doi: 10.1007/BF00277392. [24] D. Ölz, C. Schmeiser and A. Soreff, Multistep navigation of leukocytes: A stochastic model with memory effects, preprint, TU Vienna, 2004. doi: 10.1137/S0036139900382772. [25] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298. [26] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM Journal of Applied Mathematics, 62 (2002), 1222-1250. [27] A. R. Pagan and A. Ullah, "Nonparametric Econometrics," Cambridge University Press, 1999. [28] D. W. Scott, "Multivariate Density Estimation: Theory, Practice and Visualization," John Wiley & Sons, 1992. [29] B. W. Silverman, "Density Estimation for Statistics and Data Analysis," Chapman & Hall, 1986. [30] D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 28 (1974), 305-315. [31] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal, preprint IANS 14/2007, University of Stuttgart. [32] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal, International Journal of Biomathematics and Biostatistics, 1 (2010), 109-128. [33] C. Surulescu and N. Surulescu, On two approahes to a multiscale system modeling bacterial chemotaxis, preprint IANS, University of Stuttgart, 2/2010. [34] H. Takagi, M. J. Sato, T. Yanagida and M. Ueda, Functional analysis of spontaneous cell movement under different physiological conditions, PLoS ONE, 3 (2008), e2648. doi: 10.1371/journal.pone.0002648. [35] A. Upadhyaya, J. P. Rieu, J. A. Glazier and Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates, Physica A: Statistical Mechanics and its Applications, 293 (2001), 549-558. [36] P. Vieu, Quadratic errors for nonparametric estimates under dependence, Journal of Multivariate Analysis, 39 (1991), 324-347. doi: 10.1016/0047-259X(91)90105-B.

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##### References:
 [1] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, Journal of Mathematical Biology, 9 (1980), 147-177. doi: 10.1007/BF00275919. [2] S. Asmussen and P. W. Glynn, "Stochastic Simulation. Algorithms and Analysis," Springer, 2007. [3] T. Cacoullos, Estimation of a multivariate density, Annals of the Institute of Statistical Mathematics, 18 (1966), 179-189. doi: 10.1007/BF02869528. [4] F. A. C. C. Chalub, Y. Dolak-Struss, P. Markowich, D. Oelz, C. Schmeiser and A. Soreff, Model hierarchies for cell aggregation by chemotaxis, Mathematical Models and Methods in the Applied Sciences, 16 (2006), 1173-1197. doi: 10.1142/S0218202506001509. [5] F. A. C. C. Chalub, P. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatshefte für Mathematik, 142 (2004), 123-141. [6] E. A. Codling and N. A. Hill, Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters, Journal of Mathematical Biology, 51 (2005), 527-556. doi: 10.1007/s00285-005-0317-7. [7] A. Czirók, K. Schlett, E. Madarász and T. Vicsek, Exponential distribution of locomotion activity in cell cultures, Physical Review Letters, 81 (1998), 3038-3041. doi: 10.1103/PhysRevLett.81.3038. [8] P. Deheuvels, Estimation non paramétrique de la densité par histogrames généralisés (II), Publications de l'Institut Statistique de l'Université de Paris, 22 (1977), 1-23. [9] L. Devroye and L. Györfi, "Nonparametric Density Estimation: The $L_1$ View," John Wiley, New York 1985. [10] L. Devroye, Universal smoothing factor selection in density estimation: Theory and practice, Test, 6 (1997), 223-320. doi: 10.1007/BF02564701. [11] R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in E. Coli: A paradigm for multiscale modeling in biology, Multiscale Modeling and Simulation, 3 (2005), 362-394. doi: 10.1137/040603565. [12] F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, Journal of Mathematical Biology, 50 (2005), 189-207. doi: 10.1007/s00285-004-0286-2. [13] C. W. Gear, J. Li and I. G. Kevrekidis, The gap-tooth method in particle simulations, Physics Letters A, 316 (2003), 190-195. doi: 10.1016/j.physleta.2003.07.004. [14] T. Hillen, "Transport Equations and Chemosensitive Movement," Habilitation Thesis, University of Tübingen, 2001. [15] T. Hillen, Hyperbolic models for chemosensitive movement, Mathematical Models and Methods in the Applied Sciences, 12 (2002), 1-28. doi: 10.1142/S0218202502002008. [16] T. Hillen, Transport equations with resting phases, European Journal of Applied Mathematics, 14 (2003), 613-636. doi: 10.1017/S0956792503005291. [17] T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM Journal of Applied Mathematics, 61 (2000), 751-775. doi: 10.1137/S0036139999358167. [18] J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, preprint IANS, University of Stuttgart, 2010, submitted. [19] L. Holmström and J. Klemelä, Asymptotic bounds for the expected $L^1$ error of a multivariate kernel density estimator, Journal of Multivariate Analysis, 42 (1992), 245-266. doi: 10.1016/0047-259X(92)90046-I. [20] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations," Springer, 2000. [21] K. V. Mardia, P. E. Jupp, "Directional Statistics," Wiley, 2000. doi: 10.1016/0167-7152(88)90050-8. [22] J. S. Marron and D. Nolan, Canonical kernels for density estimation, Statistics and Probability Letters, 7 (1988), 195-199. doi: 10.1214/aos/1176348653. [23] J. S. Marron and M. P. Wand, Exact mean integrated squared error, Annals of Statistics, 20 (1992), 712-736. doi: 10.1007/BF00277392. [24] D. Ölz, C. Schmeiser and A. Soreff, Multistep navigation of leukocytes: A stochastic model with memory effects, preprint, TU Vienna, 2004. doi: 10.1137/S0036139900382772. [25] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298. [26] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM Journal of Applied Mathematics, 62 (2002), 1222-1250. [27] A. R. Pagan and A. Ullah, "Nonparametric Econometrics," Cambridge University Press, 1999. [28] D. W. Scott, "Multivariate Density Estimation: Theory, Practice and Visualization," John Wiley & Sons, 1992. [29] B. W. Silverman, "Density Estimation for Statistics and Data Analysis," Chapman & Hall, 1986. [30] D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 28 (1974), 305-315. [31] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal, preprint IANS 14/2007, University of Stuttgart. [32] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal, International Journal of Biomathematics and Biostatistics, 1 (2010), 109-128. [33] C. Surulescu and N. Surulescu, On two approahes to a multiscale system modeling bacterial chemotaxis, preprint IANS, University of Stuttgart, 2/2010. [34] H. Takagi, M. J. Sato, T. Yanagida and M. Ueda, Functional analysis of spontaneous cell movement under different physiological conditions, PLoS ONE, 3 (2008), e2648. doi: 10.1371/journal.pone.0002648. [35] A. Upadhyaya, J. P. Rieu, J. A. Glazier and Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates, Physica A: Statistical Mechanics and its Applications, 293 (2001), 549-558. [36] P. Vieu, Quadratic errors for nonparametric estimates under dependence, Journal of Multivariate Analysis, 39 (1991), 324-347. doi: 10.1016/0047-259X(91)90105-B.
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