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Osteoprotegerin in bone metastases: mathematical solution to the puzzle.

Ryser MD, Qu Y, Komarova SV - PLoS Comput. Biol. (2012)

Bottom Line: Consistently, systemic application of OPG decreases metastatic tumor burden in bone.However, OPG produced locally by cancer cells was shown to enhance osteolysis and tumor growth.The proposed mechanism highlights the importance of the spatial distribution of receptors, decoys and ligands, and can be applied to other systems involving regulation of spatially anisotropic processes.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada.

ABSTRACT
Bone is a common site for cancer metastasis. To create space for their growth, cancer cells stimulate bone resorbing osteoclasts. Cytokine RANKL is a key osteoclast activator, while osteoprotegerin (OPG) is a RANKL decoy receptor and an inhibitor of osteoclastogenesis. Consistently, systemic application of OPG decreases metastatic tumor burden in bone. However, OPG produced locally by cancer cells was shown to enhance osteolysis and tumor growth. We propose that OPG produced by cancer cells causes a local reduction in RANKL levels, inducing a steeper RANKL gradient away from the tumor and towards the bone tissue, resulting in faster resorption and tumor expansion. We tested this hypothesis using a mathematical model of nonlinear partial differential equations describing the spatial dynamics of OPG, RANKL, PTHrP, osteoclasts, tumor and bone mass. We demonstrate that at lower expression rates, tumor-derived OPG enhances the chemotactic RANKL gradient and osteolysis, whereas at higher expression rates OPG broadly inhibits RANKL and decreases osteolysis and tumor burden. Moreover, tumor expression of a soluble mediator inducing RANKL in the host tissue, such as PTHrP, is important for correct orientation of the RANKL gradient. A meta-analysis of OPG, RANKL and PTHrP expression in normal prostate, carcinoma and metastatic tissues demonstrated an increase in expression of OPG, but not RANKL, in metastatic prostate cancer, and positive correlation between OPG and PTHrP in metastatic prostate cancer. The proposed mechanism highlights the importance of the spatial distribution of receptors, decoys and ligands, and can be applied to other systems involving regulation of spatially anisotropic processes.

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Host tissue RANKL and systemic OPG.A Starting from the initial conditions described in Figure 3, the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) are shown at 30, 60 and 90 days, respectively. The outcomes for three different values of the host-RANKL level  are shown. The computational domain is 15 mm long, but since the fields are symmetric, only the right half is shown. The y-axes have the following units: RANKL in pmol/mm; OC in cells/mm; tumor density is normalized between 0, when there is no tumor per unit length, and 1, when the unit space is fully occupied by tumor. The resorption fronts of osteoclasts either reach wave-like propagation () or die out (). B Starting from the initial conditions described in Figure 3, the evolution of the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) is shown after 45 and 90 days, respectively. The initial host-RANKL level is , and between 20 and 90 days, a uniform source of OPG is administered at  (green) and  (blue), respectively. Compared to the control at  (red), the respective tumor burdens are reduced.
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pcbi-1002703-g004: Host tissue RANKL and systemic OPG.A Starting from the initial conditions described in Figure 3, the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) are shown at 30, 60 and 90 days, respectively. The outcomes for three different values of the host-RANKL level are shown. The computational domain is 15 mm long, but since the fields are symmetric, only the right half is shown. The y-axes have the following units: RANKL in pmol/mm; OC in cells/mm; tumor density is normalized between 0, when there is no tumor per unit length, and 1, when the unit space is fully occupied by tumor. The resorption fronts of osteoclasts either reach wave-like propagation () or die out (). B Starting from the initial conditions described in Figure 3, the evolution of the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) is shown after 45 and 90 days, respectively. The initial host-RANKL level is , and between 20 and 90 days, a uniform source of OPG is administered at (green) and (blue), respectively. Compared to the control at (red), the respective tumor burdens are reduced.

Mentions: The results presented in Figures 4–8 are based on numerical solutions of different versions of system (6), together with periodic boundary conditions and initial conditions as specified above. The parameter values are matched to in vivo observations where available, and a tuning method is applied to the set of unmatched parameters as explained in Text S1. The time stepping is performed with a fractional step method as described in [38]. Thereby, adaptive Runge-Kutta solvers are used for the advection and reaction parts, and a TR-BDF2 solver for the diffusion parts. Spatial discretisations are performed by means of finite differences (chemotactic term) and spectral collocation (diffusion terms). See Text S1 for details.


Osteoprotegerin in bone metastases: mathematical solution to the puzzle.

Ryser MD, Qu Y, Komarova SV - PLoS Comput. Biol. (2012)

Host tissue RANKL and systemic OPG.A Starting from the initial conditions described in Figure 3, the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) are shown at 30, 60 and 90 days, respectively. The outcomes for three different values of the host-RANKL level  are shown. The computational domain is 15 mm long, but since the fields are symmetric, only the right half is shown. The y-axes have the following units: RANKL in pmol/mm; OC in cells/mm; tumor density is normalized between 0, when there is no tumor per unit length, and 1, when the unit space is fully occupied by tumor. The resorption fronts of osteoclasts either reach wave-like propagation () or die out (). B Starting from the initial conditions described in Figure 3, the evolution of the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) is shown after 45 and 90 days, respectively. The initial host-RANKL level is , and between 20 and 90 days, a uniform source of OPG is administered at  (green) and  (blue), respectively. Compared to the control at  (red), the respective tumor burdens are reduced.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3475686&req=5

pcbi-1002703-g004: Host tissue RANKL and systemic OPG.A Starting from the initial conditions described in Figure 3, the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) are shown at 30, 60 and 90 days, respectively. The outcomes for three different values of the host-RANKL level are shown. The computational domain is 15 mm long, but since the fields are symmetric, only the right half is shown. The y-axes have the following units: RANKL in pmol/mm; OC in cells/mm; tumor density is normalized between 0, when there is no tumor per unit length, and 1, when the unit space is fully occupied by tumor. The resorption fronts of osteoclasts either reach wave-like propagation () or die out (). B Starting from the initial conditions described in Figure 3, the evolution of the RANKL concentration, osteoclast population density (OC) and tumor density (Tumor) is shown after 45 and 90 days, respectively. The initial host-RANKL level is , and between 20 and 90 days, a uniform source of OPG is administered at (green) and (blue), respectively. Compared to the control at (red), the respective tumor burdens are reduced.
Mentions: The results presented in Figures 4–8 are based on numerical solutions of different versions of system (6), together with periodic boundary conditions and initial conditions as specified above. The parameter values are matched to in vivo observations where available, and a tuning method is applied to the set of unmatched parameters as explained in Text S1. The time stepping is performed with a fractional step method as described in [38]. Thereby, adaptive Runge-Kutta solvers are used for the advection and reaction parts, and a TR-BDF2 solver for the diffusion parts. Spatial discretisations are performed by means of finite differences (chemotactic term) and spectral collocation (diffusion terms). See Text S1 for details.

Bottom Line: Consistently, systemic application of OPG decreases metastatic tumor burden in bone.However, OPG produced locally by cancer cells was shown to enhance osteolysis and tumor growth.The proposed mechanism highlights the importance of the spatial distribution of receptors, decoys and ligands, and can be applied to other systems involving regulation of spatially anisotropic processes.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada.

ABSTRACT
Bone is a common site for cancer metastasis. To create space for their growth, cancer cells stimulate bone resorbing osteoclasts. Cytokine RANKL is a key osteoclast activator, while osteoprotegerin (OPG) is a RANKL decoy receptor and an inhibitor of osteoclastogenesis. Consistently, systemic application of OPG decreases metastatic tumor burden in bone. However, OPG produced locally by cancer cells was shown to enhance osteolysis and tumor growth. We propose that OPG produced by cancer cells causes a local reduction in RANKL levels, inducing a steeper RANKL gradient away from the tumor and towards the bone tissue, resulting in faster resorption and tumor expansion. We tested this hypothesis using a mathematical model of nonlinear partial differential equations describing the spatial dynamics of OPG, RANKL, PTHrP, osteoclasts, tumor and bone mass. We demonstrate that at lower expression rates, tumor-derived OPG enhances the chemotactic RANKL gradient and osteolysis, whereas at higher expression rates OPG broadly inhibits RANKL and decreases osteolysis and tumor burden. Moreover, tumor expression of a soluble mediator inducing RANKL in the host tissue, such as PTHrP, is important for correct orientation of the RANKL gradient. A meta-analysis of OPG, RANKL and PTHrP expression in normal prostate, carcinoma and metastatic tissues demonstrated an increase in expression of OPG, but not RANKL, in metastatic prostate cancer, and positive correlation between OPG and PTHrP in metastatic prostate cancer. The proposed mechanism highlights the importance of the spatial distribution of receptors, decoys and ligands, and can be applied to other systems involving regulation of spatially anisotropic processes.

Show MeSH
Related in: MedlinePlus