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Modeling invasion of metastasizing cancer cells to bone marrow utilizing ecological principles.

Chen KW, Pienta KJ - Theor Biol Med Model (2011)

Bottom Line: These modified equations allow a more flexible way to model the space competition between the two cell species.The ability to model initial density, metastatic seeding into the bone marrow and growth once the cells are present, and movement of cells out of the bone marrow niche and apoptosis of cells are all aspects of the adapted equations.These equations are currently being applied to clinical data sets for verification and further refinement of the models.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Internal Medicine, The University of Michigan, 7308 CCC, 1500 E, Medical Center Drive, Ann Arbor, MI 48109, USA. kpienta@umich.edu

ABSTRACT

Background: The invasion of a new species into an established ecosystem can be directly compared to the steps involved in cancer metastasis. Cancer must grow in a primary site, extravasate and survive in the circulation to then intravasate into target organ (invasive species survival in transport). Cancer cells often lay dormant at their metastatic site for a long period of time (lag period for invasive species) before proliferating (invasive spread). Proliferation in the new site has an impact on the target organ microenvironment (ecological impact) and eventually the human host (biosphere impact).

Results: Tilman has described mathematical equations for the competition between invasive species in a structured habitat. These equations were adapted to study the invasion of cancer cells into the bone marrow microenvironment as a structured habitat. A large proportion of solid tumor metastases are bone metastases, known to usurp hematopoietic stem cells (HSC) homing pathways to establish footholds in the bone marrow. This required accounting for the fact that this is the natural home of hematopoietic stem cells and that they already occupy this structured space. The adapted Tilman model of invasion dynamics is especially valuable for modeling the lag period or dormancy of cancer cells.

Conclusions: The Tilman equations for modeling the invasion of two species into a defined space have been modified to study the invasion of cancer cells into the bone marrow microenvironment. These modified equations allow a more flexible way to model the space competition between the two cell species. The ability to model initial density, metastatic seeding into the bone marrow and growth once the cells are present, and movement of cells out of the bone marrow niche and apoptosis of cells are all aspects of the adapted equations. These equations are currently being applied to clinical data sets for verification and further refinement of the models.

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Competition among two species: Cancer cells as the superior species. The red (dash) line indicates the superior species (cancer cells); the green (solid) line indicates the inferior species (HSC). (A) Superior species (cancer cells) has a colonization rate of β1 = 0.2, mortality rate μ1 = 0.1 time-1, and initial density = 0.0001. Inferior species (HSC) has a colonization rate β2 = 0.8, mortality rate μ2 = 0.1 time-1, and initial density = 0.5. The * of the density curve of the inferior species is obtained from Eq. 2. (B) The superior species (cancer cells) has initial density = 0.01, and the inferior species (HSC) has initial density = 0.85; all other conditions remained the same as in (A). (C) The inferior species (HSC) has initial density = 0.84 and colonization rate β2 = 0.6 time-1 ; all other conditions remained the same as in (A).
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Figure 3: Competition among two species: Cancer cells as the superior species. The red (dash) line indicates the superior species (cancer cells); the green (solid) line indicates the inferior species (HSC). (A) Superior species (cancer cells) has a colonization rate of β1 = 0.2, mortality rate μ1 = 0.1 time-1, and initial density = 0.0001. Inferior species (HSC) has a colonization rate β2 = 0.8, mortality rate μ2 = 0.1 time-1, and initial density = 0.5. The * of the density curve of the inferior species is obtained from Eq. 2. (B) The superior species (cancer cells) has initial density = 0.01, and the inferior species (HSC) has initial density = 0.85; all other conditions remained the same as in (A). (C) The inferior species (HSC) has initial density = 0.84 and colonization rate β2 = 0.6 time-1 ; all other conditions remained the same as in (A).

Mentions: Alternatively, it can be assumed that cancer cells are more avid for the bone marrow niche and are the superior species (Figure 3A, B and 3C). Notably, a high initial density is set for the inferior species (HSC) instead of the low initial density for the inferior species above. The observations found in Figure 2 were again demonstrated. First, although the superior species (cancer cells) has very low initial density, the superior species (cancer cells) is not affected by inferior HSC cells. Second, the initial density of either superior species (cancer cells) or inferior species (HSC) does not affect the equilibrium density of both species, but the initial density of the superior species (cancer cells) affects the time it takes to reach the equilibrium density. Third, compared to Figure 3(A) and 3(B), 3(C) shows a lower equilibrium density for the inferior species (HSC) because of a lower birth rate of species 2. The results are consistent with Eq. 4.


Modeling invasion of metastasizing cancer cells to bone marrow utilizing ecological principles.

Chen KW, Pienta KJ - Theor Biol Med Model (2011)

Competition among two species: Cancer cells as the superior species. The red (dash) line indicates the superior species (cancer cells); the green (solid) line indicates the inferior species (HSC). (A) Superior species (cancer cells) has a colonization rate of β1 = 0.2, mortality rate μ1 = 0.1 time-1, and initial density = 0.0001. Inferior species (HSC) has a colonization rate β2 = 0.8, mortality rate μ2 = 0.1 time-1, and initial density = 0.5. The * of the density curve of the inferior species is obtained from Eq. 2. (B) The superior species (cancer cells) has initial density = 0.01, and the inferior species (HSC) has initial density = 0.85; all other conditions remained the same as in (A). (C) The inferior species (HSC) has initial density = 0.84 and colonization rate β2 = 0.6 time-1 ; all other conditions remained the same as in (A).
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC3198921&req=5

Figure 3: Competition among two species: Cancer cells as the superior species. The red (dash) line indicates the superior species (cancer cells); the green (solid) line indicates the inferior species (HSC). (A) Superior species (cancer cells) has a colonization rate of β1 = 0.2, mortality rate μ1 = 0.1 time-1, and initial density = 0.0001. Inferior species (HSC) has a colonization rate β2 = 0.8, mortality rate μ2 = 0.1 time-1, and initial density = 0.5. The * of the density curve of the inferior species is obtained from Eq. 2. (B) The superior species (cancer cells) has initial density = 0.01, and the inferior species (HSC) has initial density = 0.85; all other conditions remained the same as in (A). (C) The inferior species (HSC) has initial density = 0.84 and colonization rate β2 = 0.6 time-1 ; all other conditions remained the same as in (A).
Mentions: Alternatively, it can be assumed that cancer cells are more avid for the bone marrow niche and are the superior species (Figure 3A, B and 3C). Notably, a high initial density is set for the inferior species (HSC) instead of the low initial density for the inferior species above. The observations found in Figure 2 were again demonstrated. First, although the superior species (cancer cells) has very low initial density, the superior species (cancer cells) is not affected by inferior HSC cells. Second, the initial density of either superior species (cancer cells) or inferior species (HSC) does not affect the equilibrium density of both species, but the initial density of the superior species (cancer cells) affects the time it takes to reach the equilibrium density. Third, compared to Figure 3(A) and 3(B), 3(C) shows a lower equilibrium density for the inferior species (HSC) because of a lower birth rate of species 2. The results are consistent with Eq. 4.

Bottom Line: These modified equations allow a more flexible way to model the space competition between the two cell species.The ability to model initial density, metastatic seeding into the bone marrow and growth once the cells are present, and movement of cells out of the bone marrow niche and apoptosis of cells are all aspects of the adapted equations.These equations are currently being applied to clinical data sets for verification and further refinement of the models.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Internal Medicine, The University of Michigan, 7308 CCC, 1500 E, Medical Center Drive, Ann Arbor, MI 48109, USA. kpienta@umich.edu

ABSTRACT

Background: The invasion of a new species into an established ecosystem can be directly compared to the steps involved in cancer metastasis. Cancer must grow in a primary site, extravasate and survive in the circulation to then intravasate into target organ (invasive species survival in transport). Cancer cells often lay dormant at their metastatic site for a long period of time (lag period for invasive species) before proliferating (invasive spread). Proliferation in the new site has an impact on the target organ microenvironment (ecological impact) and eventually the human host (biosphere impact).

Results: Tilman has described mathematical equations for the competition between invasive species in a structured habitat. These equations were adapted to study the invasion of cancer cells into the bone marrow microenvironment as a structured habitat. A large proportion of solid tumor metastases are bone metastases, known to usurp hematopoietic stem cells (HSC) homing pathways to establish footholds in the bone marrow. This required accounting for the fact that this is the natural home of hematopoietic stem cells and that they already occupy this structured space. The adapted Tilman model of invasion dynamics is especially valuable for modeling the lag period or dormancy of cancer cells.

Conclusions: The Tilman equations for modeling the invasion of two species into a defined space have been modified to study the invasion of cancer cells into the bone marrow microenvironment. These modified equations allow a more flexible way to model the space competition between the two cell species. The ability to model initial density, metastatic seeding into the bone marrow and growth once the cells are present, and movement of cells out of the bone marrow niche and apoptosis of cells are all aspects of the adapted equations. These equations are currently being applied to clinical data sets for verification and further refinement of the models.

Show MeSH
Related in: MedlinePlus