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

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

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

Christina Surulescu ^1, and Nicolae Surulescu ^1,

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

Received February 2010 Revised December 2010 Published April 2011

Abstract
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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.

Keywords: velocity jump processes, nonparametric density estimation., Cell movement.

Mathematics Subject Classification: Primary: 92C17, 82C31, 60H10; Secondary: 60K40, 65C20, 62G0.

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.

show all references

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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