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Germinal center dynamics during acute and chronic infection
Moments of von mises and fisher distributions and applications
1. | University of Alberta, Centre for Mathematical Biology, Edmonton, Alberta, T6G2G1, Canada |
2. | Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK |
3. | Cross Cancer Institute, 11560-University Ave NW, Edmonton, Alberta, T6G 1Z2, Canada |
The von Mises and Fisher distributions are spherical analogues to theNormal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth.
References:
[1] |
wikipedia. com, https://en.wikipedia.org/wiki/VonMises%E2%80%93Fisher_distribution, last accessed on 11/22/2016. |
[2] |
A. Banerjee, I. S. Dhillon, J. Ghosh and S. Sra,
Clustering on the unit hypersphere using von Mises-Fisher distributions, Journal of Machine Learning Research, 6 (2005), 1345-1382.
|
[3] |
E. Batschelet, Circular Statistics in Biology, Academic Press, London, 1981.
![]() ![]() |
[4] |
C. Beaulieu,
The basis of anisotropic water diffusion in the nervous system -a technical review, NMR Biomed., 15 (2002), 435-455.
|
[5] |
R. Bleck,
An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates, Ocean Mod., 4 (2002), 55-88.
|
[6] |
E. A. Codling, N. Bearon and G. J. Thorn,
Thorn, Diffusion about the mean drift location in a
biased random walk, Ecology, 91 (2010), 3106-3113.
|
[7] |
E. A. Codling, M. J. Plank and S. Benhamou,
Random walk models in biology, J. Roy. Soc. Interface, 5 (2008), 813-834.
|
[8] |
C. Darwin,
Perception in the lower animals, Nature, 7 (1873), 360.
|
[9] |
C. Engwer, T. Hillen, M. Knappitsch and C. Surulescu,
Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015), 551-582.
doi: 10.1007/s00285-014-0822-7. |
[10] |
N. I. Fisher, Statistical Analysis of Circular Data, Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511564345.![]() ![]() ![]() |
[11] |
A. Giese, L. Kluwe, B. Laube, H. Meissner, M. E. Berens and M. Westphal,
Migration of human glioma cells on myelin, Neurosurgery, 38 (1996), 755-764.
|
[12] |
P. G. Gritsenko, O. Ilina and P. Friedl,
Interstitial guidance of cancer invasion, Journal or Pathology, 226 (2012), 185-199.
|
[13] |
H. Hatzikirou, A. Deutsch, C. Schaller, M. Simaon and K. Swanson,
Mathematical modelling of glioblastoma tumour development: A review, Math. Models Meth. Appl. Sci., 15 (2005), 1779-1794.
doi: 10.1142/S0218202505000960. |
[14] |
T. Hillen,
${M}^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.
doi: 10.1007/s00285-006-0017-y. |
[15] |
T. Hillen, E. Leonard and H. van Roessel, Partial Differential Equations; Theory and Completely Solved Problems, Wiley, Hoboken, NJ, 2012. |
[16] |
T. Hillen and K. Painter, Transport and anisotropic diffusion models for movement in oriented habitats, In: Lewis, M., Maini, P., Petrovskii, S. (Eds.), Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective, Springer, Heidelberg, 2071 (2013), 177-222.
doi: 10.1007/978-3-642-35497-7_7. |
[17] |
T. Hillen,
On the $L^2$-moment closure of transport equations: The general case, Discr.Cont.Dyn.Systems, Series B, 5 (2005), 299-318.
doi: 10.3934/dcdsb.2005.5.299. |
[18] |
A. R. C. James and A. K. Stuart-Smith,
Distribution of caribou and wolves in relation to linear corridors, The Journal of Wildlife Management, 64 (2000), 154-159.
|
[19] |
A. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. Swanson, M. Pelegrini-Issac, R. Guillevin and H. Benali,
Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Mang. Res. Med., 54 (2005), 616-624.
|
[20] |
J. Kent, The Fisher-Bingham Distribution on the Sphere, J. Royal. Stat. Soc., 1982. |
[21] |
E. Konukoglu, O. Clatz, P. Bondiau, H. Delignette and N. Ayache,
Extrapolation glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins, Medical Image Analysis, 14 (2010), 111-125.
|
[22] |
K. V. Mardia and P. E. Jupp, Directional Statistics, Wiley, New York, 2000. |
[23] |
H. McKenzie, E. Merrill, R. Spiteri and M. Lewis,
How linear features alter predator movement and the functional response, Royal Society Interface, 2 (2012), 205-216.
|
[24] |
P. Moorcroft and M. A. Lewis, Mechanistic Home Range Analysis, Princeton University Press, USA, 2006.
![]() |
[25] |
J. A. Mortimer and A. Carr,
Reproduction and migrations of the Ascension Island green turtle (chelonia mydas), Copeia, 1987 (1987), 103-113.
|
[26] |
P. Mosayebi, D. Cobzas, A. Murtha and M. Jagersand,
Tumor invasion margin on the Riemannian space of brain fibers, Medical Image Analysis, 16 (2011), 361-373.
|
[27] |
A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition. Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[28] |
H. Othmer, S. Dunbar and W. Alt,
Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[29] |
K. Painter,
Modelling migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.
doi: 10.1007/s00285-008-0217-8. |
[30] |
K. Painter and T. Hillen,
Mathematical modelling of glioma growth: The use of diffusion tensor imaging DTI data to predict the anisotropic pathways of cancer invasion, J. Theor. Biol., 323 (2013), 25-39.
doi: 10.1016/j.jtbi.2013.01.014. |
[31] |
K. Painter and T. Hillen,
Navigating the flow: Individual and continuum models for homing
in flowing environments, Royal Society Interface, 12 (2015), 20150,647.
|
[32] |
K. J. Painter,
Multiscale models for movement in oriented environments and their application to hilltopping in butterflies, Theor. Ecol., 7 (2014), 53-75.
|
[33] |
J. Rao,
Molecular mechanisms of glioma invasiveness: The role of proteases, Nature Reviews Cancer, 3 (2003), 489-501.
|
[34] |
A. Swan, T. Hillen, J. Bowman and A. Murtha, A patient-specific anisotropic diffusion model for brain tumor spread, Bull. Math. Biol., to appear (2017). |
[35] |
K. Swanson, E. A. Jr and J. Murray,
A quantitative model for differential motility of gliomas in grey and white matter, Cell Proliferation, 33 (2000), 317-329.
|
[36] |
K. Swanson, R. Rostomily and E. Alvord Jr,
Predicting survival of patients with glioblastoma by combining a mathematical model and pre-operative MR imaging characteristics: A proof of principle, British Journal of Cancer, 98 (2008), 113-119.
|
[37] |
P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, 1998. |
[38] |
C. Xue and H. Othmer,
Multiscale models of taxis-driven patterning in bacterial populations, SIAM Journal for Applied Mathematics, 70 (2009), 133-167.
doi: 10.1137/070711505. |
show all references
References:
[1] |
wikipedia. com, https://en.wikipedia.org/wiki/VonMises%E2%80%93Fisher_distribution, last accessed on 11/22/2016. |
[2] |
A. Banerjee, I. S. Dhillon, J. Ghosh and S. Sra,
Clustering on the unit hypersphere using von Mises-Fisher distributions, Journal of Machine Learning Research, 6 (2005), 1345-1382.
|
[3] |
E. Batschelet, Circular Statistics in Biology, Academic Press, London, 1981.
![]() ![]() |
[4] |
C. Beaulieu,
The basis of anisotropic water diffusion in the nervous system -a technical review, NMR Biomed., 15 (2002), 435-455.
|
[5] |
R. Bleck,
An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates, Ocean Mod., 4 (2002), 55-88.
|
[6] |
E. A. Codling, N. Bearon and G. J. Thorn,
Thorn, Diffusion about the mean drift location in a
biased random walk, Ecology, 91 (2010), 3106-3113.
|
[7] |
E. A. Codling, M. J. Plank and S. Benhamou,
Random walk models in biology, J. Roy. Soc. Interface, 5 (2008), 813-834.
|
[8] |
C. Darwin,
Perception in the lower animals, Nature, 7 (1873), 360.
|
[9] |
C. Engwer, T. Hillen, M. Knappitsch and C. Surulescu,
Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015), 551-582.
doi: 10.1007/s00285-014-0822-7. |
[10] |
N. I. Fisher, Statistical Analysis of Circular Data, Cambridge University Press, Cambridge, 1993.
doi: 10.1017/CBO9780511564345.![]() ![]() ![]() |
[11] |
A. Giese, L. Kluwe, B. Laube, H. Meissner, M. E. Berens and M. Westphal,
Migration of human glioma cells on myelin, Neurosurgery, 38 (1996), 755-764.
|
[12] |
P. G. Gritsenko, O. Ilina and P. Friedl,
Interstitial guidance of cancer invasion, Journal or Pathology, 226 (2012), 185-199.
|
[13] |
H. Hatzikirou, A. Deutsch, C. Schaller, M. Simaon and K. Swanson,
Mathematical modelling of glioblastoma tumour development: A review, Math. Models Meth. Appl. Sci., 15 (2005), 1779-1794.
doi: 10.1142/S0218202505000960. |
[14] |
T. Hillen,
${M}^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.
doi: 10.1007/s00285-006-0017-y. |
[15] |
T. Hillen, E. Leonard and H. van Roessel, Partial Differential Equations; Theory and Completely Solved Problems, Wiley, Hoboken, NJ, 2012. |
[16] |
T. Hillen and K. Painter, Transport and anisotropic diffusion models for movement in oriented habitats, In: Lewis, M., Maini, P., Petrovskii, S. (Eds.), Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective, Springer, Heidelberg, 2071 (2013), 177-222.
doi: 10.1007/978-3-642-35497-7_7. |
[17] |
T. Hillen,
On the $L^2$-moment closure of transport equations: The general case, Discr.Cont.Dyn.Systems, Series B, 5 (2005), 299-318.
doi: 10.3934/dcdsb.2005.5.299. |
[18] |
A. R. C. James and A. K. Stuart-Smith,
Distribution of caribou and wolves in relation to linear corridors, The Journal of Wildlife Management, 64 (2000), 154-159.
|
[19] |
A. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. Swanson, M. Pelegrini-Issac, R. Guillevin and H. Benali,
Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Mang. Res. Med., 54 (2005), 616-624.
|
[20] |
J. Kent, The Fisher-Bingham Distribution on the Sphere, J. Royal. Stat. Soc., 1982. |
[21] |
E. Konukoglu, O. Clatz, P. Bondiau, H. Delignette and N. Ayache,
Extrapolation glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins, Medical Image Analysis, 14 (2010), 111-125.
|
[22] |
K. V. Mardia and P. E. Jupp, Directional Statistics, Wiley, New York, 2000. |
[23] |
H. McKenzie, E. Merrill, R. Spiteri and M. Lewis,
How linear features alter predator movement and the functional response, Royal Society Interface, 2 (2012), 205-216.
|
[24] |
P. Moorcroft and M. A. Lewis, Mechanistic Home Range Analysis, Princeton University Press, USA, 2006.
![]() |
[25] |
J. A. Mortimer and A. Carr,
Reproduction and migrations of the Ascension Island green turtle (chelonia mydas), Copeia, 1987 (1987), 103-113.
|
[26] |
P. Mosayebi, D. Cobzas, A. Murtha and M. Jagersand,
Tumor invasion margin on the Riemannian space of brain fibers, Medical Image Analysis, 16 (2011), 361-373.
|
[27] |
A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition. Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[28] |
H. Othmer, S. Dunbar and W. Alt,
Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[29] |
K. Painter,
Modelling migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.
doi: 10.1007/s00285-008-0217-8. |
[30] |
K. Painter and T. Hillen,
Mathematical modelling of glioma growth: The use of diffusion tensor imaging DTI data to predict the anisotropic pathways of cancer invasion, J. Theor. Biol., 323 (2013), 25-39.
doi: 10.1016/j.jtbi.2013.01.014. |
[31] |
K. Painter and T. Hillen,
Navigating the flow: Individual and continuum models for homing
in flowing environments, Royal Society Interface, 12 (2015), 20150,647.
|
[32] |
K. J. Painter,
Multiscale models for movement in oriented environments and their application to hilltopping in butterflies, Theor. Ecol., 7 (2014), 53-75.
|
[33] |
J. Rao,
Molecular mechanisms of glioma invasiveness: The role of proteases, Nature Reviews Cancer, 3 (2003), 489-501.
|
[34] |
A. Swan, T. Hillen, J. Bowman and A. Murtha, A patient-specific anisotropic diffusion model for brain tumor spread, Bull. Math. Biol., to appear (2017). |
[35] |
K. Swanson, E. A. Jr and J. Murray,
A quantitative model for differential motility of gliomas in grey and white matter, Cell Proliferation, 33 (2000), 317-329.
|
[36] |
K. Swanson, R. Rostomily and E. Alvord Jr,
Predicting survival of patients with glioblastoma by combining a mathematical model and pre-operative MR imaging characteristics: A proof of principle, British Journal of Cancer, 98 (2008), 113-119.
|
[37] |
P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, 1998. |
[38] |
C. Xue and H. Othmer,
Multiscale models of taxis-driven patterning in bacterial populations, SIAM Journal for Applied Mathematics, 70 (2009), 133-167.
doi: 10.1137/070711505. |







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