January  2017, 14(1): 289-304. doi: 10.3934/mbe.2017019

On a mathematical model of bone marrow metastatic niche

1. 

Departamento de Matemática Aplicada, Ciencia e Ingeniería de Materiales y Tecnología Electrónica, ESCET, Universidad Rey Juan Carlos, E28933, Móstoles, Madrid, Spain

2. 

Depto. Matemática Aplicada a las T.I.C. ETSI Sistemas Informáticos, Universidad Politécnica de Madrid. Madrid 28031, Spain

3. 

Centro de Simulación Computacional, Universidad Politécnica de Madrid. Boadilla del Monte, Madrid 28660, Spain

* Corresponding author: J. Ignacio Tello

Received  November 23, 2015 Accepted  June 14, 2016 Published  October 2016

Fund Project: The first author is supported by Project TEC2012-39095-C03-02, the second is supported by Project MTM2013-42907-P.

We propose a mathematical model to describe tumor cells movement towards a metastasis location into the bone marrow considering the influence of chemotaxis inhibition due to the action of a drug. The model considers the evolution of the signaling molecules CXCL-12 secreted by osteoblasts (bone cells responsible of the mineralization of the bone) and PTHrP (secreted by tumor cells) which activates osteoblast growth. The model consists of a coupled system of second order PDEs describing the evolution of CXCL-12 and PTHrP, an ODE of logistic type to model the Osteoblasts density and an extra equation for each cancer cell. We also simulate the system to illustrate the qualitative behavior of the solutions. The numerical method of resolution is also presented in detail.

Citation: Ana Isabel Muñoz, J. Ignacio Tello. On a mathematical model of bone marrow metastatic niche. Mathematical Biosciences & Engineering, 2017, 14 (1) : 289-304. doi: 10.3934/mbe.2017019
References:
[1]

R. A. Adams, Sobolev Spaces Academic Press, New York-London, 1975.  Google Scholar

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A. A. BrydenS. IslamA. J. FreemontJ. H. ShanksN. J. George and N. W. Clarke, Parathyroid hormone-related peptide: Expression in prostate cancer bone metastases, Prostate Cancer Prostatic Dis, 5 (2002), 59-62.  doi: 10.1038/sj.pcan.4500553.  Google Scholar

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L. M. CalviG. B. AdamsK. W. WeibrechtJ. M. WeberD. P. OlsonM. C. KnightR. P. MartinE. SchipaniP. DivietiF. R. BringhurstL. A. MilnerH. M. Kronenberg and D. T. Scadden, Osteoblastic cells regulate the haematopoietic stem cell niche, Nature, 425 (2003), 841-846.  doi: 10.1038/nature02040.  Google Scholar

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N. L. CogginsD. TrakimasS. L. ChangA. EhrlichP. RayK. E. LukerJ. J. Linderman and G. D. Luker, CXCR7 controls competition for recruitment of β-arrestin 2 in cells expressing both CXCR4 and CXCR7, PLoS One, 9 (2014), 841-846.  doi: 10.1371/journal.pone.0098328.  Google Scholar

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K. GolanO. Kollet and T. Lapidot, Dynamic cross talk between S1P and CXCL12 regulates hematopoietic stem cells migration, development and bone remodeling, Pharmaceuticals, 6 (2013), 1145-1169.  doi: 10.3390/ph6091145.  Google Scholar

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G. InnamoratiM. T. ValentiF. GiovinazzoL. Dalle CarbonareM. Parenti and C. Bassi, Molecular approaches to target gpcrs in cancer therapy, Pharmaceuticals, 4 (2011), 567-589.  doi: 10.3390/ph4040567.  Google Scholar

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A. J. Lilly, W. E. Johnson and C. M. Bunce, The haematopoietic stem cell niche: New insights into the mechanisms regulating haematopoietic stem cell behaviour Stem Cells International, 2011 (2011), ID 274564. doi: 10.4061/2011/274564.  Google Scholar

[10]

A. I. Muñoz, Numerical resolution of a model of tumor growth, Mathematical Medicine and Biology, 33 (2016), 1-29.  doi: 10.1093/imammb/dqv004.  Google Scholar

[11]

G. O'Boyle, I. Swidenbank, H. Marshall, C. E. Barker, J. Armstrong, S. A. White, S. P. Fricker, R. Plummer, M. Wright and P. E. Lovat, Inhibition of CXCR4/CXCL12 chemotaxis in melanoma by AMD11070, Br J Cancer., 108 (2013), 1634--1640, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3668477/ doi: 10.1038/bjc.2013.124.  Google Scholar

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T. OskarssonE. Batlle and J. Massague, Metastatic stem cells: Sources, niches, and vital pathways, Cell Stem Cell, 14 (2014), 306-321.  doi: 10.1016/j.stem.2014.02.002.  Google Scholar

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A. A. Rose and P. M. Siegel, Emerging therapeutic targets in breast cancer bone metastasis, Future Oncol., 6 (2010), 55-74.  doi: 10.2217/fon.09.138.  Google Scholar

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M. D. RyserN. Nigam and S. V. Komarova, Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit, J. of Bone and Mineral Research, 24 (2009), 860-870.  doi: 10.1359/jbmr.081229.  Google Scholar

[15]

J. SceneayM. J. Smyth and A. Möller, The pre-metastatic niche: Finding common ground, Cancer Metastasis Rev., 32 (2013), 449-464.  doi: 10.1007/s10555-013-9420-1.  Google Scholar

[16]

Y. X. SunJ. WangC. E. ShelburneD. E. LopatinA. M. ChinnaiyanM. A. RubinK. J. Pienta and R. S. Taichman, Expression of CXCR4 and CXCL12 (SDF-1) in human prostate cancers (PCa) in vivo, Journal of Cellular Biochemistry, 89 (2003), 462-473.  doi: 10.1002/jcb.10522.  Google Scholar

[17]

R. S. TaichmanC. CooperE. T. KellerK. J. PientaN. S. Taichman and L. K. McCauley, Use of the stromal cell-derived factor-1/CXCR4 pathway in prostate cancer metastasis to bone, Cancer Research, 62 (2002), 1832-1837.   Google Scholar

[18]

J. I. Tello, On a mathematical model of tumor growth based on cancer stem cells, Math. Biosc. Eng., 10 (2013), 263-278.   Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces Academic Press, New York-London, 1975.  Google Scholar

[2]

A. A. BrydenS. IslamA. J. FreemontJ. H. ShanksN. J. George and N. W. Clarke, Parathyroid hormone-related peptide: Expression in prostate cancer bone metastases, Prostate Cancer Prostatic Dis, 5 (2002), 59-62.  doi: 10.1038/sj.pcan.4500553.  Google Scholar

[3]

L. M. CalviG. B. AdamsK. W. WeibrechtJ. M. WeberD. P. OlsonM. C. KnightR. P. MartinE. SchipaniP. DivietiF. R. BringhurstL. A. MilnerH. M. Kronenberg and D. T. Scadden, Osteoblastic cells regulate the haematopoietic stem cell niche, Nature, 425 (2003), 841-846.  doi: 10.1038/nature02040.  Google Scholar

[4]

S. L. Chang, S. P. Cavnar, S. Takayama, G. D. Luker and J. J. Linderman, Cell, isoform, and environment factors shape gradients and modulate chemotaxis PLoS One 10 (2015), e0123450. doi: 10.1371/journal.pone.0123450.  Google Scholar

[5]

N. L. CogginsD. TrakimasS. L. ChangA. EhrlichP. RayK. E. LukerJ. J. Linderman and G. D. Luker, CXCR7 controls competition for recruitment of β-arrestin 2 in cells expressing both CXCR4 and CXCR7, PLoS One, 9 (2014), 841-846.  doi: 10.1371/journal.pone.0098328.  Google Scholar

[6]

K. GolanO. Kollet and T. Lapidot, Dynamic cross talk between S1P and CXCL12 regulates hematopoietic stem cells migration, development and bone remodeling, Pharmaceuticals, 6 (2013), 1145-1169.  doi: 10.3390/ph6091145.  Google Scholar

[7]

G. InnamoratiM. T. ValentiF. GiovinazzoL. Dalle CarbonareM. Parenti and C. Bassi, Molecular approaches to target gpcrs in cancer therapy, Pharmaceuticals, 4 (2011), 567-589.  doi: 10.3390/ph4040567.  Google Scholar

[8]

S. V. KomarovaR. J. SmithS. J. DixonS. M. Sims and L. M. Wahlb, Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling, Bone, 33 (2003), 206-215.  doi: 10.1016/S8756-3282(03)00157-1.  Google Scholar

[9]

A. J. Lilly, W. E. Johnson and C. M. Bunce, The haematopoietic stem cell niche: New insights into the mechanisms regulating haematopoietic stem cell behaviour Stem Cells International, 2011 (2011), ID 274564. doi: 10.4061/2011/274564.  Google Scholar

[10]

A. I. Muñoz, Numerical resolution of a model of tumor growth, Mathematical Medicine and Biology, 33 (2016), 1-29.  doi: 10.1093/imammb/dqv004.  Google Scholar

[11]

G. O'Boyle, I. Swidenbank, H. Marshall, C. E. Barker, J. Armstrong, S. A. White, S. P. Fricker, R. Plummer, M. Wright and P. E. Lovat, Inhibition of CXCR4/CXCL12 chemotaxis in melanoma by AMD11070, Br J Cancer., 108 (2013), 1634--1640, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3668477/ doi: 10.1038/bjc.2013.124.  Google Scholar

[12]

T. OskarssonE. Batlle and J. Massague, Metastatic stem cells: Sources, niches, and vital pathways, Cell Stem Cell, 14 (2014), 306-321.  doi: 10.1016/j.stem.2014.02.002.  Google Scholar

[13]

A. A. Rose and P. M. Siegel, Emerging therapeutic targets in breast cancer bone metastasis, Future Oncol., 6 (2010), 55-74.  doi: 10.2217/fon.09.138.  Google Scholar

[14]

M. D. RyserN. Nigam and S. V. Komarova, Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit, J. of Bone and Mineral Research, 24 (2009), 860-870.  doi: 10.1359/jbmr.081229.  Google Scholar

[15]

J. SceneayM. J. Smyth and A. Möller, The pre-metastatic niche: Finding common ground, Cancer Metastasis Rev., 32 (2013), 449-464.  doi: 10.1007/s10555-013-9420-1.  Google Scholar

[16]

Y. X. SunJ. WangC. E. ShelburneD. E. LopatinA. M. ChinnaiyanM. A. RubinK. J. Pienta and R. S. Taichman, Expression of CXCR4 and CXCL12 (SDF-1) in human prostate cancers (PCa) in vivo, Journal of Cellular Biochemistry, 89 (2003), 462-473.  doi: 10.1002/jcb.10522.  Google Scholar

[17]

R. S. TaichmanC. CooperE. T. KellerK. J. PientaN. S. Taichman and L. K. McCauley, Use of the stromal cell-derived factor-1/CXCR4 pathway in prostate cancer metastasis to bone, Cancer Research, 62 (2002), 1832-1837.   Google Scholar

[18]

J. I. Tello, On a mathematical model of tumor growth based on cancer stem cells, Math. Biosc. Eng., 10 (2013), 263-278.   Google Scholar

Figure 1.  Geometry of the problem
Figure 2.  Results of the first scenario simulation, where therapeutic treatment had not been applied: Above we show the trajectory followed by the cell in the period of time [0, 0.04]. At the bottom, there appear the results obtained for the osteoblast density at different times
Figure 3.  Results of the second scenario simulation, where a therapeutic treatment inhibiting the chemotactic movement of the cell is applied: Above it is depicted the trajectory of the cell in the period of time [0, 0.2]. At the bottom, we illustrate the corresponding results for the osteoblast density at different times
Figure 4.  Results of the first scenario simulation regarding the evolution of the variable $u$ , tumor factors. We show the corresponding the level curves of $u$ for $t=0.02, $ $0.025$ and $0.04$
Figure 5.  Results of the second scenario simulation for the variable $u$ . We present the corresponding level curves at times $t=0.02$ and $t=0.2$
Figure 6.  Results of the first scenario for the variable $w$ , the chemoattractant CXCL-12 concentration at $t=0.02$ , when the cell has not reached the niche yet, and at time $t=0.08$ , when the cell is already well established in the niche
Figure 7.  Results of the second scenario for the variable $w$ , the chemoattractant CXCL-12 concentration at $t=0.02$ and $t=0.2$
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