Modeling invasion of metastasizing cancer cells to bone marrow utilizing ecological principles
© Chen and Pienta; licensee BioMed Central Ltd. 2011
Received: 10 August 2011
Accepted: 3 October 2011
Published: 3 October 2011
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).
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.
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.
A patient with prostate cancer, Mr. S. presents to clinic for consideration of further therapy. He is a 65 year old man that was diagnosed 12 years ago with a poorly differentiated, localized prostate cancer (PCa) when he presented for a routine physical exam and was found to have an elevated prostate specific antigen (PSA) blood test. Digital rectal exam revealed no abnormalities but prostate ultrasound and biopsy revealed a Gleason 5+4 = 9 cancer (clinical stage T1cNxMx). Because Mr. S. was in otherwise excellent health, he chose to undergo a radical retropubic prostatectomy and his prostate was removed. All of his lymph nodes were negative for cancer. He was considered to be cured of his disease.
Five years later, Mr. S's PSA became detectable and he now has 3 lesions present on bone scan. He has metastatic prostate cancer, now incurable. Each year, approximately 40,000 men who "should" have been cured of their prostate cancer by surgery or radiation therapy present with incurable metastatic disease that will manifest itself as metastatic lesions in the bone, usually years after primary treatment. The only explanation for this is that disseminated tumor cells (DTCs) are present in the bone microenvironment before surgery or radiation eradicated the primary tumor. How these cells traffic to the bone, become dormant, and then ultimately begin to proliferate are subjects of great interest to the cancer field.
Mr. S's cancer experience is not unique. PCa remains the most common cancer and the second leading cause of cancer-related death in American men today. Approximately 72% of patients who undergo radical prostatectomy had DTCs in their marrow prior to surgery suggesting that marrow dissemination is an early event in the progression of PCa disease. Clearly the ability of DTCs to proliferate, undergo apoptosis or become dormant must occur soon after the initial arrest of circulating tumor cells (CTCs) in the marrow. Unquestionably, a greater understanding of the molecular events that regulate a DTC's ability to become, and remain dormant over long periods is crucial to define new therapeutic strategies to combat disease progression. New mathematical models to understand these events may help further define strategies for understanding mechanisms of cancer cell trafficking and subsequent dormancy.
Eugene Odum, one of the founders of the science of ecology, defined an ecosystem as: "Any unit that includes all of the organisms (the biotic community) in a given area interacting with the physical environment so that a flow of energy leads to clearly defined biotic structures and cycling of materials between living and nonliving components is an ecological system" . The ecosystem is the first unit of the ecological hierarchy that is complete, i.e., that has all of the necessary components for survival. Ecosystems do not exist independently, but interact in a complex web of relationships that connect all ecosystems to make up the biosphere . Using the ecosystem paradigm, cancer cells, growing in an organ, can be considered to be a species co-existing in a complex habitat with other host cells. Together, the cancer cells and host cells, interacting within their habitat, create an ecosystem. This ecosystem, in turn, exists within a larger environment (the host patient as biosphere). The study of ecology, then, has the potential to offer insights into tumor biology. This is especially true for the study of metastasis [2, 3].
Ecologists have studied the population biology of invasive species for decades and have documented their impact on local environments as well as the global ecosystem as a whole [4–6]. Invasive species start as a native population within a defined community and are then transported by some means to a new environment . In this new environment, the invader either then dies off or enters a period of time during which it establishes itself (lag period). It then begins to spread and have impact on the local environment, disrupting the ecosystem as a whole. This disruption has broad implications for the native species and the broader ecosystem . Biologic traits that result in a robust invasive species include rapid proliferative capacity, adaptation to environmental stress (phenotypic plasticity) and high tolerance to environmental heterogeneity [7–10].
Tilman's model of the dynamics of two species invasion into a structured habitat
Symbols utilized in the equations
Density of species 1
Density of species 2
Birth rate of species 1
Birth rate of species 2
Death rate of species 1
Death rate of species 2
The ability (proportion) that species 1 can displace species 2 by the growth (birth) of species 1
The ability (proportion) that species 2 can displace species 1 by the growth (birth) of species 2
Eq. 1 describes that the dynamics of the superior species is only dependent on the colonization and its own death rate. Colonization, β1 ρ1(1-ρ1), indicates that when one member of the superior species grows, the member is not affected by any member of the inferior species. In other words, any member of superior species can displace the member of the inferior species. The part of the equation μ1ρ1 describes the density-independent mortality of superior species itself. Eq. 2 describes the dynamics of the inferior species is dependent on the colonization, mortality and the competitive displacement. Colonization, β2ρ2(1-ρ1-ρ2), indicates that the inferior species can colonize a lattice if it is unoccupied. A member of the inferior species, therefore, cannot displace any member of superior species. The competitive displacement, -β2 ρ1 ρ2, indicates that the inferior species is displaced by superior species growth (birth).
The initial density of the two different species does not affect the final equilibrium density. A very small initial density, however, requires more time to reach the equilibrium density.
Adapting the invasion model to describe the invasion of cancer cells into the hematopoietic stem cell niche of the bone marrow microenvironment
As noted, the Tilman's model describes the coexisting dynamics between superior and inferior species. However, the superior species is not affected by the inferior species. When modeling the real dynamics between HSC and cancer cells, it is doubtful that HSC or cancer cells behave in this absolute manner as a superior or inferior species. Thus, Tilman's model was amended to allow more interaction between the two species.
Coefficients, k(0 ≤ k ≤ 1), expresses the ability (proportion) that species 1 can displace species 2 by the growth (birth) of species 1. Therefore, (1-k) expresses the proportion of species 2 that is not displaced by species 1. The colonization of the species 1, β1ρ1[1-ρ1-(1-k) ρ2], indicates that when the species 1 grows, the species 1 is affected by the (1-k) proportion of species 2 (Eq. 5). Species 1 can displace the k proportion of the species 2, but not all of species 2. Similarly, coefficients, v (0 ≤ v ≤ 1), expresses the ability (proportion) that species 2 can displace species 1 by the growth (birth) of species 2. If k is larger than v, then species 1 has a stronger ability to displace species 2 (species 1 is the relative superior competitor).
The competitive displacement in Eq. 5, -β2ρ2ρ1v, indicates that the v proportion of species 1 is displaced by species 2's invasion or growth (birth in the equation). The competitive displacement in Eq. 6, -β1ρ1ρ2k , indicates that the k proportion of species 2 is displaced by species 1's birth. Both k and v express the displacement ability from species 1 and 2's birth, and are independent of each other. It is assumes that species 1 and species 2 do not change the lattice (occupancy of the niche) once they are given a birth until they are displaced. It is also assumes that when either species 1 or species 2 grow (birth rate), the lattice that they are going to occupy is chosen randomly. In addition, once either member of species 1 or species 2 is displaced, they will abandon the fixed and homogeneous space. Thus, the only reason that they may compete for the same lattice is because of the birth (movement in of a new cell or growth).
This results in cancer cells displacing most (90%) of the HSCs. However, the HSCs can also displace a small portion (10%) of the cancer cells. Comparing the conditions in Figure 3 versus Figure 4, the equilibrium densities of the cancer cells are lower in Figure 4 and it takes longer time for both species to reach their equilibrium densities in Figure 4.
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. This required accounting for the fact that this is the natural home of hematopoietic stem cells and that they already occupy this structured space. These modified equations allow a more flexible way to model the space competition between the two cell species. The ability to model initial density, birth rate (metastatic seeding into the bone marrow and growth once the cells are present) and death rate (movement of cells out of the bone marrow niche and apoptosis of cells), are key components of this equations. The equations allow modeling of metastasis and the lag period of cancer cells. These equations are currently being applied to clinical data sets for verification and further refinement of the models. First, it is likely that during the progression from primary tumors to metastasis, genomic alterations occur, resulting in genetic and phenotypic changes. This article treated the cancer cells as a single species when in fact, it may be reasonable to treat them as multiple species based on their genetic heterogeneity (different genomes = different species) (18,19). The model can also be adapted to take into account mutation over time, i.e., the cancer cell population is relatively weak following initial metastasis, and becomes strong over time and dominates during the late stages. The fields of ecology and population biology are rich sources for understanding the biology of metastasis.
Conflict of interest
The authors declare that they have no competing interests.
This work was directly supported by National Cancer Institute grant CA093900 and U01CA143055, the Department of Defense; the Prostate Cancer Foundation. K.J. Pienta receives support as an American Cancer Society Clinical Research Professor and a Taubman Scholar of the University of Michigan, NIH SPORE in prostate cancer grant P50 CA69568, and Cancer Center support grant P30 CA46592.
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