Theoretical Approach
Scientific Paper
This analysis was based on the scientific paper “Modeling invasion of metastasizing cancer cells to bone marrow using ecological principles” by Kun-Wan Chen and Kenneth J. Pienta [1] and in a work exploration of the same, done by Catarina Sá Silva and Tiago Curado [2] in Computer Models of Physiological Processes, a subject of the Integrated Masters in Biomedical Engineering of the University of Coimbra. The last one is available for downloading, in the “Download” tab.
A .pdf version of “Modeling invasion of metastasizing cancer cells to bone marrow using ecological principles” by Kun-Wan Chen and Kenneth J. Pienta [1] is available in: http://tbiomed.biomedcentral.com/articles/10.1186/1742-4682-8-36 [3].
Glossary
Ecosystem: This describes a system of interconnected elements within a bounded area, and the dynamics that are created among them. In biological terms, it includes a biological community of living beings (the biotic component) and all the non-living (physical and chemical) elements that are presented in their surrounding environment (the abiotic component) [4] [5]. The network of interactions that exist between the several parts of these components (dynamic over space and time) are the ones that establish the “system”. The boundaries of an ecosystem are specified simply for practical reasons. The study of an ecosystem is mainly focused on the study of the mentioned processes, which usually comprise energy transformations and biogeochemical cycles [5].
Population biology: It covers the study of populations of organisms and the existing relationships, both among themselves and between them and the environment. This concerns the tracking of population changes (related, for example, to population size, historical events, diseases, overcrowding, extinction, etc.) and the underlying processes that lead to that [6], dealing with genetic, physiological, demographic and age heterogeneities [7].
Ecological niche: This describes the position of a population in its environment or habitat, showing the way it responds to external factors, as well as the way it is affected by them. These factors may be abiotic components, food and shelter resources, competitors or predators, for example [8] [9]. This leads to the evolution of the population in the niche, since all these factors affect the survival and reproduction of the individuals [8].
Intraspecific competition: The individuals of a population have similar needs in terms of survival, growth and reproduction, which may lead to demands for resources that exceed the existing supplies. In this case, the individuals of the same species will compete for the resources among them (intraspecific competition). This scenario may influence innumerous events, such as: decreased rates of resource intake per individual, decreased rates of individual growth or development, decreases in the amounts of stored reserves, increased risks of predation, decreases in survivorship and decreases in fecundity [10].
Interspecific competition: It may cause, in a population, the same fecundity, growth and survivorship consequences of the last concept, but, in this case, the resource exploitation or interference is made by individuals of another species. As seen, this phenomenon affects the populations’ dynamics, and the dynamics, in their turn, can influence, over time, the species’ distributions and their evolution in the environment [11].
Hematopoietic stem cell niche: Hematopoietic stem cells are a subpopulation of hematopoietic cells capable of differentiating to all blood cell lineages. Their characteristic of having a long-lived self-renewal ensures the generation of blood cells throughout life. These stem cells are mostly located in the bone marrow, surrounded by a specific microenvironment, called the hematopoietic stem cell niche. This niche provides protection and plays a role in signaling the start of the stem cells’ differentiation [12].
Bone marrow: Located in the hollow spaces in the inside of bones, bone marrow is a soft, spongy and gelatinous tissue that consists of the two main types of stem (undifferentiated) cells, supported by a fibrous tissue called stroma. The bone marrow may, then, be red or yellow, depending on whether it consists of mainly red-colored hematopoietic tissue or yellow-colored fatty tissue. The first one is involved in producing blood cells, whereas the other is involved in producing stromal cells [13].
Metastasis: Cancer cells from a metastatic cancer have the ability of breaking away from a tumor in the primary site and spreading to other parts of the body, both by moving into nearby normal tissues and by moving through the bloodstream and lymph system to more distant sites [14] [15]. When some of these cells are able to settle in the new location and to grow and form new tumors, it is said that a metastasis occurred [15].
Content Review
A niche can be considered a space occupied by an organism in an ecological community. Similar species tend to reside in similar niches [16]. When the niches of two different species overlap, an interspecific competition occurs, which is a competition of two species for the same resources [17] [18].
An invasive species can be defined as an organism that causes ecological damage in an ecosystem where it is not native [19]. The invasive species can either die or settle in, starting to propagate and creating impact on the new environment, causing consequences to the native species [1].
It is possible to make an analogy between invasive species and cancer metastasis, once cancer cells grow in a place, escape and survive until they reach a new tissue (i.e. new environment) [1]. Thus, in this context, the following analogy may be made:
Figure 1: Analogy between invasive species and metastasizing cancer. Adapted from [1] [2].
Computational Simulation
Algorithm Analysis
The following analysis is based on what is referred and reported in [1]. The symbols used in the equations and their meaning are in Table 1, which was transcribed from [1].
Table 1: Symbols used in the equations. Transcribed from [1].
| Symbol | Definition |
|---|---|
| p1 | Density of species 1 |
| p1 | Density of species 2 |
| b1 | Birth rate of species 1 |
| b2 | Birth rate of species 2 |
| u1 | Death rate of species 1 |
| u2 | Death rate of species 2 |
| k | The ability (proportion) that species 1 can displace species 2 by the growth (birth) of species 1 |
| v | The ability (proportion) that species 2 can displace species 1 by the growth (birth) of species 2 |
| t | Time |
In Tilman’s model [20], a superior species can occupy the niche of an inferior species. However, an inferior species can only colonize an environment that is not occupied neither by itself nor by the superior species [1] [2].
dp1/dt = b1p1 (1 - p1) - u1p1 (1)
dp2/dt = b2p2 (1 - p1 - p2) - u2p2 - b1p1p2 (2)
In Equation 1, it is seen that the proliferation of the superior species (species 1) depends only on its birth and death rates. Observing the Equation 2, it is seen that the birth and death rates of the inferior species (species 2) are dependent on the competitive interactions between the two species [1] [2].
Kun-Wan Chen and Kenneth J. Pienta adapted the Tilman’s model of invasion dynamics, considering that the invaders are the tumor cells and that the Hematopoietic Stem Cells (HSC) are already in the bone marrow [1] [2]. As seen in Tilman’s unmodified model, the superior species is not affected by the inferior species. However, when modeling the real dynamics between HSC and tumor cells, it is difficult to understand which is the superior and inferior species and it shall be taken into account that they both have the capability to displace each other [1] [2].
dp1/dt = b1p1 [1 - p1 - (1 - k)p2] - u1p1 - b2p2p1v (3)
dp2/dt = b2p2 [1 - (1 - v)p1 - p2] - u2p2 - b1p1p2k (4)
It was considered that k= 0.9 and v = 0.1, which indicates that the tumor cells displace 90% of the HSCs and the HSCs displace 10% of the tumor cells [1].
In the simulations performed in the next section, it was considered the following values for the graphic 1, graphic 2 and graphic 3, according to what was simulated in [1]: b1 = 0.2, b2 = 0.8, u1 = 0.1 time-1, u2 = 0.1 time-1. The initial values may be changed in the simulation.
Computational Simulation
SELECT YOUR DATA:
Graphic A:Competition among two species: HSCs as the superior species
0.6
0.4
0.0001
0.1
Competition among two species: Cancer Cells as the superior species
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0.01
0.5
0.85
Competition among two species: Cancer Cells as the superior species
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Results Interpretation
The graphics obtained are identical to those presented in [1].
The first graphic analyzes the Tilman’s unmodified equations, to the dynamics between tumor cells and HSCs. HSCs are considered to be the superior species. As expected, the colonization of the superior species (HSCs) is completely independent of the behavior of the inferior species (tumor cells). After the local density of the HSCs stabilize, the density of the tumor cells also tends to a constant value. The lower is the initial local density of HSCs, the sooner will the local density of tumor cells tend to stabilization. In this case, the initial values for local density were always higher for HSCs and near zero to tumor cells.
The second graphic presents the Tilman’s unmodified equations, to the dynamics between tumor cells and HSCs, in which tumor cells are considered the superior species. Again, the behavior of the superior species (tumor cells) is independent of the inferior species (HSCs). The local densities of both species tend to stabilize. Although the initial values for local density are higher for HSCs (inferior species) and nearby zero to tumor cells (superior species), the stability values for the local density are higher to tumor cells than to HSCs cells.
In the third graphic, the Kun-Wan Chen and Kenneth J. Pienta adapted Tilman’s model of invasion dynamics is analyzed. In this case, it is taken into account that both species have influence on each other. Tumor cells are considered to be the superior species. In this perspective, as expected, the graphical behavior of one influences the other’s behavior. Although tumor cells are still the superior species and the initial values are identical, unlike the previous case, the local density of tumor cells does not surpass the density of HSCs, although the density of the superior species increases and the inferior species decreases. Again, the density tends to stabilize.
Experimental Procedure
This virtual laboratory studies how the competition affects the growth of a population. It was created by Glencoe/McGraw-Hill [21] and can be found in [22].
Download
The work exploration of [1], done by Catarina Sá Silva and Tiago Curado [2] in Computer Models of Physiological Processes, a subject of the Integrated Masters in Biomedical Engineering of the University of Coimbra, can be downloaded through the following link.
References
[INTRO IMAGE] “Bone Marrow Cancer,” Cancer Box - By Dr John Morris on February 13, 2015, [Online]. Available: http://www.cancerbox.org/bone-marrow-cancer. [Acedido em 2016 August 24].
[1] Kun-Wan Chen, Kenneth J. Pienta, “Modeling invasion of metastasizing cancer cells to bone marrow using ecological principles,” Theoretical Biology and Medical Modelling, vol. 8, p. 36, 2011.
[2] Catarina Sá Silva, Tiago Curado, “Projecto de Modelos Computacionais de Processos Fisiológicos acerca de: Modelo Matemático para a Representação da Evolução de Metástases de Células Cancerosas da Medula Óssea recorrendo a Princípios Ecológicos,” Coimbra, 2013/2014.
[3] Kun-Wan Chen, Kenneth J Pienta, “Modeling invasion of metastasizing cancer cells to bone marrow utilizing ecological principles,” BioMed Central - The Open Access Publisher, [Online]. Available: https://tbiomed.biomedcentral.com/articles/10.1186/1742-4682-8-36. [Acedido em 26 August 2016].
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[11] Michael Begon, Colin R. Townsend, John L. Harper, “Chapter 8: Interspecific Competition,” em Ecology: From Individuals to Ecosystems, Blackwell Publishing, Fourth Edition, 2006.
[12] Alexander Birbrair, Paul S. Frenette, “Niche heterogeneity in the bone marrow,” Annals of the New York Academy of Sciences, vol. 1370, nº Hematopoietic Stem Cells IX, pp. 82-96, 2016.
[13] Vinod K Panchbhavi, “Bone Marrow Anatomy,” WebMD LLC, [Online]. Available: http://emedicine.medscape.com/article/1968326-overview#showall. [Accessed 24 August 2016].
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[16] “Interdependence,” BBC, [Online]. Available: http://www.bbc.co.uk/schools/gcsebitesize/science/ocr_gateway/understanding_environment/interdependencerev1.shtml. [Accessed 2016 August 2016].
[17] unesp, “Nicho Ecológico,” SlideShare - LinkedIn Corporation, 31 January 2013. [Online]. Available: http://www.slideshare.net/popecologia/nicho-ecolgico. [Accessed 25 August 2016].
[18] “Interspecific Competition, Competitive Exclusion & Niche Differentiation,” Study.com, [Online]. Available: http://study.com/academy/lesson/interspecific-competition-competitive-exclusion-niche-differentiation.html. [Accessed 25 August 2016].
[19] “What is an invasive species?,” National Ocean Service, [Online]. Available: http://oceanservice.noaa.gov/facts/invasive.html. [Accessed 25 August 2016].
[20] David Tilman, “Competition and biodiversity in spatially structured habitats,” Ecology, vol. 75, nº 1, pp. 2-16, 1994.
[21] “McGraw-Hill Education,” [Online]. Available: http://www.mheducation.com/highered/home-guest.html. [Accessed 24 August 2016].
[22] “Population Biology,” Glencoe/McGraw-Hill, [Online]. Available: http://glencoe.mheducation.com/sites/dl/free/0078802849/383928/BL_04.html. [Accessed 25 August 2016].
