Abstract

To gain insight into brain tumor invasion, experiments are conducted on multicellular tumor spheroids grown in collagen gel. Typically, a radius of invasion is reported, which is obtained by human measurement. We present a simple, heuristic algorithm for automated invasive radii estimation (AIRE) that uses local fluctuations of the image intensity. We then derive an analytical expression relating the image graininess to the cell density for a model imaging system. The result agrees with the experiment up to a multiplicative constant and thus describes a novel method for estimating the cell density from bright-field images.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
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    [CrossRef] [PubMed]
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  7. M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. S. Osher and R. Fedkiw, "Snakes, active contours, and segmentation," in Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).
  10. A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).
  11. C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
    [CrossRef] [PubMed]
  12. S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
    [CrossRef] [PubMed]
  13. R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
    [CrossRef] [PubMed]
  14. D. Del Duca, T. Werbowetski, and R. Del Maestro, "Spheroid preparation from hanging drops: characterization of a model of brain tumor invasion," J. Neuro-Oncol. 67, 295-303 (2004).
    [CrossRef]
  15. J. C. Crocker and D. G. Grier, "Methods of digital video microscopy for colloidal studies," J. Colloid Interface Sci. 179, 298-310 (1996).
    [CrossRef]
  16. S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).
  17. H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
    [CrossRef]
  18. E. Khain, L. M. Sander, and A. M. Stein, "A model for glioma growth," Complexity 11, 53-57 (2005).
    [CrossRef]

2007 (1)

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

2006 (1)

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

2005 (5)

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

E. Khain, L. M. Sander, and A. M. Stein, "A model for glioma growth," Complexity 11, 53-57 (2005).
[CrossRef]

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

2004 (2)

T. E. Werbowetski, R. Bjerkvig, and R. F. Del Maestro, "Evidence for a secreted chemorepellent that directs glioma cell invasion. J. Neurobiol. 60, 71-88 (2004).
[CrossRef] [PubMed]

D. Del Duca, T. Werbowetski, and R. Del Maestro, "Spheroid preparation from hanging drops: characterization of a model of brain tumor invasion," J. Neuro-Oncol. 67, 295-303 (2004).
[CrossRef]

2003 (2)

S. Osher and R. Fedkiw, "Snakes, active contours, and segmentation," in Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

2002 (1)

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

2001 (1)

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

1997 (1)

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

1996 (1)

J. C. Crocker and D. G. Grier, "Methods of digital video microscopy for colloidal studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

1992 (1)

S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).

1989 (1)

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

Amberger, V. R.

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

Andor-Ardo, D.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Baird, G. S.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Berens, M. E.

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Bjerkvig, R.

T. E. Werbowetski, R. Bjerkvig, and R. F. Del Maestro, "Evidence for a secreted chemorepellent that directs glioma cell invasion. J. Neurobiol. 60, 71-88 (2004).
[CrossRef] [PubMed]

Bogdanov, A.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Brangwynne, C. P.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

Campbell, R. E.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Carlson, B. L.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Chan, A. Y.

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

Chiocca, E. A.

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Crocker, J. C.

J. C. Crocker and D. G. Grier, "Methods of digital video microscopy for colloidal studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).

Deckter, L. A.

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

Deisboeck, T. S.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Del Duca, D.

D. Del Duca, T. Werbowetski, and R. Del Maestro, "Spheroid preparation from hanging drops: characterization of a model of brain tumor invasion," J. Neuro-Oncol. 67, 295-303 (2004).
[CrossRef]

Del Maestro, R.

D. Del Duca, T. Werbowetski, and R. Del Maestro, "Spheroid preparation from hanging drops: characterization of a model of brain tumor invasion," J. Neuro-Oncol. 67, 295-303 (2004).
[CrossRef]

Del Maestro, R. F.

T. E. Werbowetski, R. Bjerkvig, and R. F. Del Maestro, "Evidence for a secreted chemorepellent that directs glioma cell invasion. J. Neurobiol. 60, 71-88 (2004).
[CrossRef] [PubMed]

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

Demuth, T.

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

Dennison, S.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Deutsch, A.

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

Fedkiw, R.

S. Osher and R. Fedkiw, "Snakes, active contours, and segmentation," in Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

Filippidi, E.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

Galanis, E.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Gardel, M.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Giannini, C.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Glassner, A. S.

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

Gordon, V.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Gordon, V. D.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

Grier, D. G.

J. C. Crocker and D. G. Grier, "Methods of digital video microscopy for colloidal studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

Hatzikirou, H.

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

James, C. D.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Kansal, A. R.

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Kasai, K.

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

Kasza, K. E.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

Kaufman, L. J.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

Khain, E.

E. Khain, L. M. Sander, and A. M. Stein, "A model for glioma growth," Complexity 11, 53-57 (2005).
[CrossRef]

McDonald, W.

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

Mobley, D.

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

Moore, E.

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

Nakada, M.

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

Osher, S.

S. Osher and R. Fedkiw, "Snakes, active contours, and segmentation," in Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

Palmer, A. E.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Saeki, Y.

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

Saito, A.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Sander, L. M.

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

E. Khain, L. M. Sander, and A. M. Stein, "A model for glioma growth," Complexity 11, 53-57 (2005).
[CrossRef]

Sarkaria, J. N.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Schaller, C.

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

Schroeder, M. A.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Simon, M.

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

Stein, A. M.

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

E. Khain, L. M. Sander, and A. M. Stein, "A model for glioma growth," Complexity 11, 53-57 (2005).
[CrossRef]

Steinbach, P. A.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Stemmer-Rachaminov, A. O.

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Swanson, K. R.

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

Symons, M.

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

Tamaki, M.

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

Torquato, S.

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Tour, O.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Tran, N. L.

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

Tsien, R. Y.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Uhm, J. H.

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Valentine, M.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Valster, A.

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

Weitz, D.

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Weitz, D. A.

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

Werbowetski, T.

D. Del Duca, T. Werbowetski, and R. Del Maestro, "Spheroid preparation from hanging drops: characterization of a model of brain tumor invasion," J. Neuro-Oncol. 67, 295-303 (2004).
[CrossRef]

Werbowetski, T. E.

T. E. Werbowetski, R. Bjerkvig, and R. F. Del Maestro, "Evidence for a secreted chemorepellent that directs glioma cell invasion. J. Neurobiol. 60, 71-88 (2004).
[CrossRef] [PubMed]

Yamamoto, S.

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

Zacharias, D. A.

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Biophys. J. (2)

L. J. Kaufman, C. P. Brangwynne, K. E. Kasza, E. Filippidi, V. D. Gordon, T. S. Deisboeck, and D. A. Weitz, "Glioma expansion in collagen i matrices: analyzing collagen concentration-dependent growth and motility patterns," Biophys. J. 89, 635-650 (2005).
[CrossRef] [PubMed]

A. M. Stein, T. Demuth, D. Mobley, M. E. Berens, and L. M. Sander, "Describing glioblastoma invasion in a 3D in vitro experiment using a mathematical model of motility, shedding, and proliferation," Biophys. J. 92, 356-365 (2007).
[CrossRef]

Cell Prolif. (1)

T. S. Deisboeck, M. E. Berens, A. R. Kansal, S. Torquato, A. O. Stemmer-Rachaminov, and E. A. Chiocca, "Pattern of self-organization in tumour systems: complex growth dynamics in a novel brain tumor spheroid model," Cell Prolif. 34, 115-134 (2001).
[CrossRef] [PubMed]

Complexity (1)

E. Khain, L. M. Sander, and A. M. Stein, "A model for glioma growth," Complexity 11, 53-57 (2005).
[CrossRef]

Exper. Cell Res. (1)

V. Gordon, M. Valentine, M. Gardel, D. Andor-Ardo, S. Dennison, A. Bogdanov, D. Weitz, and T. S. Deisboeck, "Measuring the mechanical stress induced by an expanding multicellular tumor system: a case study," Exper. Cell Res. 289, 58-66 (2003).
[CrossRef]

Gene Therapy (1)

S. Yamamoto, L. A. Deckter, K. Kasai, E. A. Chiocca, and Y. Saeki, "Imaging immediate-early and strict-late promoter activity during oncolytic herpes simplex virus type 1 infection and replication in tumors," Gene Therapy 13, 1731-1736 (2006).
[CrossRef] [PubMed]

J. Colloid Interface Sci. (1)

J. C. Crocker and D. G. Grier, "Methods of digital video microscopy for colloidal studies," J. Colloid Interface Sci. 179, 298-310 (1996).
[CrossRef]

J. Neuro-Oncol. (1)

D. Del Duca, T. Werbowetski, and R. Del Maestro, "Spheroid preparation from hanging drops: characterization of a model of brain tumor invasion," J. Neuro-Oncol. 67, 295-303 (2004).
[CrossRef]

J. Neurobiol. (1)

T. E. Werbowetski, R. Bjerkvig, and R. F. Del Maestro, "Evidence for a secreted chemorepellent that directs glioma cell invasion. J. Neurobiol. 60, 71-88 (2004).
[CrossRef] [PubMed]

J. Neurosurg. (1)

M. Tamaki, W. McDonald, V. R. Amberger, E. Moore, and R. F. Del Maestro, "Implantation of C6 astrocytoma spheroid into collagen type I gels: invasive, proliferative, and enzymatic characterizations. J. Neurosurg. 87, 602-609 (1997).
[CrossRef] [PubMed]

Math. Mod. Meth. Appl. Sci. (1)

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon, and K. R. Swanson, "Mathematical modelling of glioblastoma tumour development: a review," Math. Mod. Meth. Appl. Sci. 15, 1779-1794 (2005).
[CrossRef]

Methods (1)

A. Valster, N. L. Tran, M. Nakada, M. E. Berens, A. Y. Chan, and M. Symons, "Cell migration and invasion assays," Methods 37, 208-215 (2005).
[CrossRef] [PubMed]

Neuro-Oncology (1)

C. Giannini, J. N. Sarkaria, A. Saito, J. H. Uhm, E. Galanis, B. L. Carlson, M. A. Schroeder, and C. D. James, "Patient tumor egfr and pdgfra gene amplifications retained in an invasive intracranial xenograft model of glioblastoma multiforme," Neuro-Oncology 7, 164-176 (2005).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

R. E. Campbell, O. Tour, A. E. Palmer, P. A. Steinbach, G. S. Baird, D. A. Zacharias, and R. Y. Tsien, "A monomeric red fluorescent protein," Proc. Natl. Acad. Sci. USA 99, 7877-7882 (2002).
[CrossRef] [PubMed]

Other (4)

S. R. Deans, The Radon Transform and Some of Its Applications (Krieger, 1992).

Central Brain Tumor Registry of the United States, "Primary brain tumors in the United States: statistical report, 1995-1999" (CBTRUS, 2002-2003).

S. Osher and R. Fedkiw, "Snakes, active contours, and segmentation," in Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003).

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

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Figures (6)

Fig. 1
Fig. 1

Bright-field images of four different cell lines on two different days. The circle indicates the automatically identified invasive radii. Each image is approximately 3   mm across. Note that the algorithm is robust to different lighting conditions, different cell lines, and different degrees of invasion.

Fig. 2
Fig. 2

Notice how the graininess, G, of the image is large in the invasive zone but small elsewhere. The images are 2.88   mm across. The outer circle in the experimental image and the dot in the graph of G ( r ) illustrate the invasive boundary when the threshold is θ inv = 0.5 .

Fig. 3
Fig. 3

Model imaging system.

Fig. 4
Fig. 4

Wide-field confocal microscopy image of the invasive tumor spheroid. The image dimensions are 3.7   mm × 3.7   mm . The focal plane is near the center of the tumor spheroid.

Fig. 5
Fig. 5

Estimates for the cell density (in cells / mm 3 ) made by counting cells (dots) and by using the image graininess (line), averaged across eight experiments. The vertical bars indicate the standard error. Each column is for a different lithium concentration, and each row is for a measurement at a different time.

Fig. 6
Fig. 6

Image of the invasive zone ( 360   μm × 360   μm ) . The cells are far more structured and complex than can be described by a Gaussian blur.

Equations (30)

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G ( I i j ) = ( I i + 1 , j I i j ) 2 + ( I i 1 , j I i j ) 2 + ( I i , j + 1 I i j ) 2 + ( I i , j 1 I i j ) 2 .
( x 0 , y 0 ) = min x 0 , y 0 { F = 1 ( I ˜ ( x , y ) c 1 ) 2 d x d y + F = 0 ( I ˜ ( x , y ) c 0 ) 2 d x d y | c 0 > c 1 + θ cent } ,
c 1 = F = 1 I ˜ ( x , y ) d x d y ,
c 0 = F = 0 I ˜ ( x , y ) d x d y .
R inv = max r { r : G ( r ) > θ inv × max r ( G ( r ) ) } ; θ inv ( 0 , 1 ) .
( log I i + 1 , j log I i j ) 2 = [ ( log I i + 1 , j 0 log I i j 0 ) ( A i + 1 , j A i j ) ] 2 ( A i + 1 , j A i j ) 2 .
I ( x , y ) = I 0 ( x , y ) exp [ μ l ( x , y , z ) d z ] = I 0 ( x , y ) exp [ α ( 2 π σ 0 2 ) 3 / 2 e ( x 2 + y 2 + z 2 ) / 2 σ 0 2 d z ] = I 0 ( x , y ) exp [ α 2 π σ 0 2 e ( x 2 + y 2 ) / 2 σ 0 2 ] ,
b ( a M / 2 s ) | z k | ,
A i j cell = α 1 2 π σ k 2 e [ ( x i x l ) 2 + ( y j y m ) 2 ] / 2 σ k 2 .
A i j = α l m k C l m k 1 2 π σ k 2 e [ ( x i x l ) 2 + ( y j y m ) 2 ] / 2 σ k 2 .
E [ G ( A i j ) ] 3 D = 1 3 π α 2 ρ ( 1 p ) h d 2 b σ 0 3 .
K α 2 h d 2 3 π b σ 0 2 .
e r r ( z ) = ρ 0 exp [ λ x 2 + y 2 ] ρ 0 exp [ λ x 2 + y 2 + z 2 ] ρ 0 exp [ λ x 2 + y 2 ] = 1 exp [ λ x 2 + y 2 + z 2 x 2 + y 2 ] λ z .
E [ G ( A i ) ] = E [ ( A i + 1 A i ) 2 + ( A i 1 A i ) 2 ] = E [ ( A i + 1 A i ) 2 ] + E [ ( A i 1 A i ) 2 ] .
E [ G ( A i ) ] = 2 E [ ( A i + 1 A i ) 2 ] = 2 E [ ( A i + 1 μ A i + μ ) 2 ] = 2 E [ ( A i + 1 μ ) 2 ] 4 E [ ( A i + 1 μ ) ( A i μ ) ] + 2 E [ ( A i μ ) 2 ] = 4 var [ A i ] 4 cov [ A i + 1 , A i ] .
E [ A i ] = E [ α l C l 1 2 π σ 0 2 e ( x l x i ) 2 / 2 σ 0 2 ] = α l E [ C l ] 1 2 π σ 0 2 e ( x l x i ) 2 / 2 σ 0 2 = α p h l 1 2 π σ 0 2 e ( x l x i ) 2 / 2 σ 0 2 h α ρ .
E [ A i 2 ] = E [ ( α l C l 1 2 π σ 0 2 e x l 2 / 2 σ 0 2 ) 2 ] = E [ ( α l C l 1 2 π σ 0 2 e x l 2 / 2 σ 0 2 ) × ( α m C m 1 2 π σ 0 2 e x m 2 / 2 σ 0 2 ) ] = E [ α 2 l m C l C m 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 ] = α 2 m l m E [ C l C m ] 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 + l E [ C l 2 ] 1 2 π σ 0 2 e 2 x l 2 / 2 σ 0 2 = α 2 m l m p 2 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 + α 2 l p 1 2 π σ 0 2 e 2 x l 2 / 2 σ 0 2 .
α 2 l p 2 1 2 π σ 0 2 e 2 x l 2 / 2 σ 0 2 ,
E [ A i 2 ] = α 2 m l p 2 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 + α 2 l ( p p 2 ) 1 2 π σ 0 2 e 2 x l 2 / 2 σ 0 2 α 2 ρ 2 + 1 4 π α 2 ρ ( 1 p ) σ 0 .
var [ A i ] = 1 4 π α 2 ρ ( 1 p ) σ 0 .
E [ A i A i + 1 ] = E [ ( α l C l 1 2 π σ 0 2 e ( x l ) 2 / 2 σ 0 2 ) × ( α m C m 1 2 π σ 0 2 e ( x m h d ) 2 / 2 σ 0 2 ) ] = E [ ( α l C l 1 2 π σ 0 2 e ( x l ) 2 / 2 σ 0 2 ) × ( α m C m + 1 1 2 π σ 0 2 e ( x m + 1 h d ) 2 / 2 σ 0 2 ) ] = E [ α 2 l m C l C m + 1 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 ] = α 2 m l m + 1 E [ C l C m + 1 ] 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 + l E [ C l 2 ] 1 2 π σ 0 2 e ( x l 2 + x l 1 2 ) / 2 σ 0 2 = α 2 m l m + 1 p 2 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 + α 2 l p 1 2 π σ 0 2 e ( x l 2 + ( x l h d ) 2 ) / 2 σ 0 2 = α 2 m l p 2 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 + α 2 l ( p p 2 ) 1 2 π σ 0 2 e ( x l 2 + ( x l h d ) 2 ) / 2 σ 0 2 .
x l 2 + ( x l h d ) 2 = 2 x l 2 2 x l h d + h d 2 = 2 ( x l 2 x l h d + h d 2 / 4 ) + h d 2 / 2 = 2 ( x l h d / 2 ) 2 + h d 2 / 2 .
E [ A i A i + 1 ] = α 2 ρ 2 m l 1 2 π σ 0 2 e ( x l 2 + x m 2 ) / 2 σ 0 2 h 2 + α 2 ρ ( 1 p ) e h d 2 / 4 σ 0 2 4 π σ 0 2 l 1 π σ 0 2 e ( x l h d / 2 ) 2 / σ 0 2 h α 2 ρ 2 + 1 4 π e h d 2 / 4 σ 0 2 σ 0 α 2 ρ ( 1 p ) .
cov [ A i , A i + 1 ] = 1 4 π e h d 2 / 4 σ 0 2 σ 0 α 2 ρ ( 1 p ) .
E [ G ( A i ) ] 1 D = 2 π α 2 ρ ( 1 p ) σ 0 ( 1 e h d 2 / 4 σ 0 2 ) .
E [ G ( A i j ) ] 2 D = 2 π α 2 ρ ( 1 p ) σ 0 2 ( 1 e h d 2 / 4 σ 0 2 ) .
A i j = k ( α l m C l m k 1 2 π σ k 2 e ( x l x i ) 2 ( y m y j ) 2 / 2 σ k 2 ) .
E [ G ( A i j ) ] 3 D = 2 π α 2 ρ ( 1 p ) k 1 σ k 2 ( 1 e h d 2 / 4 σ k 2 ) h .
E [ G ( A i j ) ] 3 D 2 π α 2 ρ ( 1 p ) h d 2 4 σ k 4 d z = 2 π α 2 ρ ( 1 p ) h d 2 4 ( σ 0 + b | z | ) 4 d z = 2 π α 2 ρ ( 1 p ) h d 2 4 ( σ 0 + b | z | ) 4 d z = 1 π α 2 ρ ( 1 p ) h d 2 0 ( σ 0 + b z ) 4 d z = 1 π α 2 ρ ( 1 p ) h d 2 / b σ 0 u 4 d u ,
E [ G ( A i j ) ] 3 D = 1 3 π α 2 ρ ( 1 p ) h d 2 b σ 0 3 .

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