# 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. doi: 10.1007/BF00275919. Google Scholar [2] S. Asmussen and P. W. Glynn, "Stochastic Simulation. Algorithms and Analysis,", Springer, (2007). Google Scholar [3] T. Cacoullos, Estimation of a multivariate density,, Annals of the Institute of Statistical Mathematics, 18 (1966), 179. doi: 10.1007/BF02869528. Google Scholar [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. doi: 10.1142/S0218202506001509. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00285-005-0317-7. Google Scholar [7] A. Czirók, K. Schlett, E. Madarász and T. Vicsek, Exponential distribution of locomotion activity in cell cultures,, Physical Review Letters, 81 (1998), 3038. doi: 10.1103/PhysRevLett.81.3038. Google Scholar [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. Google Scholar [9] L. Devroye and L. Györfi, "Nonparametric Density Estimation: The $L_1$ View,", John Wiley, (1985). Google Scholar [10] L. Devroye, Universal smoothing factor selection in density estimation: Theory and practice,, Test, 6 (1997), 223. doi: 10.1007/BF02564701. Google Scholar [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. doi: 10.1137/040603565. Google Scholar [12] F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, Journal of Mathematical Biology, 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar [13] C. W. Gear, J. Li and I. G. Kevrekidis, The gap-tooth method in particle simulations,, Physics Letters A, 316 (2003), 190. doi: 10.1016/j.physleta.2003.07.004. Google Scholar [14] T. Hillen, "Transport Equations and Chemosensitive Movement,", Habilitation Thesis, (2001). Google Scholar [15] T. Hillen, Hyperbolic models for chemosensitive movement,, Mathematical Models and Methods in the Applied Sciences, 12 (2002), 1. doi: 10.1142/S0218202502002008. Google Scholar [16] T. Hillen, Transport equations with resting phases,, European Journal of Applied Mathematics, 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar [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. doi: 10.1137/S0036139999358167. Google Scholar [18] J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks,, preprint IANS, (2010). Google Scholar [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. doi: 10.1016/0047-259X(92)90046-I. Google Scholar [20] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", Springer, (2000). Google Scholar [21] K. V. Mardia, P. E. Jupp, "Directional Statistics,", Wiley, (2000). doi: 10.1016/0167-7152(88)90050-8. Google Scholar [22] J. S. Marron and D. Nolan, Canonical kernels for density estimation,, Statistics and Probability Letters, 7 (1988), 195. doi: 10.1214/aos/1176348653. Google Scholar [23] J. S. Marron and M. P. Wand, Exact mean integrated squared error,, Annals of Statistics, 20 (1992), 712. doi: 10.1007/BF00277392. Google Scholar [24] D. Ölz, C. Schmeiser and A. Soreff, Multistep navigation of leukocytes: A stochastic model with memory effects,, preprint, (2004). doi: 10.1137/S0036139900382772. Google Scholar [25] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, Journal of Mathematical Biology, 26 (1988), 263. Google Scholar [26] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM Journal of Applied Mathematics, 62 (2002), 1222. Google Scholar [27] A. R. Pagan and A. Ullah, "Nonparametric Econometrics,", Cambridge University Press, (1999). Google Scholar [28] D. W. Scott, "Multivariate Density Estimation: Theory, Practice and Visualization,", John Wiley & Sons, (1992). Google Scholar [29] B. W. Silverman, "Density Estimation for Statistics and Data Analysis,", Chapman & Hall, (1986). Google Scholar [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. Google Scholar [31] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal,, preprint IANS 14/2007, (). Google Scholar [32] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal,, International Journal of Biomathematics and Biostatistics, 1 (2010), 109. Google Scholar [33] C. Surulescu and N. Surulescu, On two approahes to a multiscale system modeling bacterial chemotaxis,, preprint IANS, (). Google Scholar [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). doi: 10.1371/journal.pone.0002648. Google Scholar [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. Google Scholar [36] P. Vieu, Quadratic errors for nonparametric estimates under dependence,, Journal of Multivariate Analysis, 39 (1991), 324. doi: 10.1016/0047-259X(91)90105-B. Google Scholar

show all references

##### References:
 [1] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations,, Journal of Mathematical Biology, 9 (1980), 147. doi: 10.1007/BF00275919. Google Scholar [2] S. Asmussen and P. W. Glynn, "Stochastic Simulation. Algorithms and Analysis,", Springer, (2007). Google Scholar [3] T. Cacoullos, Estimation of a multivariate density,, Annals of the Institute of Statistical Mathematics, 18 (1966), 179. doi: 10.1007/BF02869528. Google Scholar [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. doi: 10.1142/S0218202506001509. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00285-005-0317-7. Google Scholar [7] A. Czirók, K. Schlett, E. Madarász and T. Vicsek, Exponential distribution of locomotion activity in cell cultures,, Physical Review Letters, 81 (1998), 3038. doi: 10.1103/PhysRevLett.81.3038. Google Scholar [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. Google Scholar [9] L. Devroye and L. Györfi, "Nonparametric Density Estimation: The $L_1$ View,", John Wiley, (1985). Google Scholar [10] L. Devroye, Universal smoothing factor selection in density estimation: Theory and practice,, Test, 6 (1997), 223. doi: 10.1007/BF02564701. Google Scholar [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. doi: 10.1137/040603565. Google Scholar [12] F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, Journal of Mathematical Biology, 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar [13] C. W. Gear, J. Li and I. G. Kevrekidis, The gap-tooth method in particle simulations,, Physics Letters A, 316 (2003), 190. doi: 10.1016/j.physleta.2003.07.004. Google Scholar [14] T. Hillen, "Transport Equations and Chemosensitive Movement,", Habilitation Thesis, (2001). Google Scholar [15] T. Hillen, Hyperbolic models for chemosensitive movement,, Mathematical Models and Methods in the Applied Sciences, 12 (2002), 1. doi: 10.1142/S0218202502002008. Google Scholar [16] T. Hillen, Transport equations with resting phases,, European Journal of Applied Mathematics, 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar [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. doi: 10.1137/S0036139999358167. Google Scholar [18] J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks,, preprint IANS, (2010). Google Scholar [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. doi: 10.1016/0047-259X(92)90046-I. Google Scholar [20] P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,", Springer, (2000). Google Scholar [21] K. V. Mardia, P. E. Jupp, "Directional Statistics,", Wiley, (2000). doi: 10.1016/0167-7152(88)90050-8. Google Scholar [22] J. S. Marron and D. Nolan, Canonical kernels for density estimation,, Statistics and Probability Letters, 7 (1988), 195. doi: 10.1214/aos/1176348653. Google Scholar [23] J. S. Marron and M. P. Wand, Exact mean integrated squared error,, Annals of Statistics, 20 (1992), 712. doi: 10.1007/BF00277392. Google Scholar [24] D. Ölz, C. Schmeiser and A. Soreff, Multistep navigation of leukocytes: A stochastic model with memory effects,, preprint, (2004). doi: 10.1137/S0036139900382772. Google Scholar [25] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, Journal of Mathematical Biology, 26 (1988), 263. Google Scholar [26] H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM Journal of Applied Mathematics, 62 (2002), 1222. Google Scholar [27] A. R. Pagan and A. Ullah, "Nonparametric Econometrics,", Cambridge University Press, (1999). Google Scholar [28] D. W. Scott, "Multivariate Density Estimation: Theory, Practice and Visualization,", John Wiley & Sons, (1992). Google Scholar [29] B. W. Silverman, "Density Estimation for Statistics and Data Analysis,", Chapman & Hall, (1986). Google Scholar [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. Google Scholar [31] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal,, preprint IANS 14/2007, (). Google Scholar [32] C. Surulescu and N. Surulescu, A nonparametric approach to cell dispersal,, International Journal of Biomathematics and Biostatistics, 1 (2010), 109. Google Scholar [33] C. Surulescu and N. Surulescu, On two approahes to a multiscale system modeling bacterial chemotaxis,, preprint IANS, (). Google Scholar [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). doi: 10.1371/journal.pone.0002648. Google Scholar [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. Google Scholar [36] P. Vieu, Quadratic errors for nonparametric estimates under dependence,, Journal of Multivariate Analysis, 39 (1991), 324. doi: 10.1016/0047-259X(91)90105-B. Google Scholar
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