(675d) Mathematical Modeling of Metastatic Cancer Migration through a Remodeling Extracellular Matrix
Materials & Methods: For the setting of the model, it is assumed that the primary tumor is approaching the metastatic stage and that the cancer cells have already penetrated the basement membrane surrounding the tumor. This indicates that the malignant cancer cells are ready to detach away from the tumor mass and invade the ECM.
The model developed here is a continuous system of five coupled partial differential equations (PDEs) targets to unravel the spatiotemporal dynamics and interconnection of the following components: cancer cells population density, ECM collagen fibers concentrations, and the concentrations of enzymes MMP and LOX. The first PDE models the dynamics of cancer cell population density governed by three main factors: random diffusion, cell proliferation, and haptotactic motility toward the remodeling ECM collagen fibers of cancer cells. The center of the primary tumor mass is modeled to reside at the left edge of the system domain where the spatial unit x = 0. Initially, a fixed cluster of cancer cells is assumed to already exist in the system domain in the range of [0, 0.25]. We incorporate a logistic growth factor in all the terms of the equation that involves motility. This is to ensure the migration of cancer cells only in a physical space that is not too congested and occupied by the dense network of cancer cells and ECM collagen fibers.
Two separate PDEs for two population of collagen fibers concentrations are considered: those that are oriented randomly and those that have been crosslinked. Since ECM collagen fibers do not diffuse, no motility-related term is incorporated in the two PDEs for populations of ECM fibers. The dynamics of ECM hence is expressed primarily via the remodeling activity of its collagen fibers. Initially, at time zero, ECM is a meshwork of only randomly oriented collagen fibers with no cross-linked fibers.
The reaction-diffusion PDEs for mass transport are utilized to model the evolution of the two chemicals, MMP and LOX, in the system. Both PDEs account for the chemical diffusion, its natural decay, and the secretion of the chemical by cancer cells. We assumed that there is zero concentration of MMP and LOX presented in the system initially.
Results and Discussion: Nondimensionalization was performed before solving the model. We collected and analyzed the in silico experimental results in MATLAB for three case studies of the model as follows: (i) no LOX is present in the system; (ii) LOX is present in the system, but has no effect on the haptotactic migration of cancer cells toward the cross-linked ECM fibers; (iii) LOX is present in the system and has all the modelled effects on ECM remodeling and haptotactic migration of cancer cells. Simulation results in all three cases are snapshots of the system dynamics in one dimension with respect to the position at four chosen simulation dimensionless times t = 0, 1, 10 and 20.
The purpose of excluding LOX from the model in case (i) study is to validate our results with ones obtained from a prominent cancer invasion model previously established by Anderson et al. (2000). The model done by Anderson included three PDEs for cancer cell population density, ECM concentration, and MMP concentration. The results for the three same components in case (i) agree with ones obtained in the Anderson model.
Some of the similarities with case (i) results are repeated in case (ii) including the slow detachment of cancer cells away from the center of the primary tumor mass and the effect of ECM degradation by MMP. The only difference between the results of two cases is the existence of the cross-linked ECM fibers and their dynamics.
In case (iii) simulation results, again, besides the recurrence of expected phenomena occurring also in cases (i) and (ii), there are a total of four distinct peaks that appear in the simulation. Cancer cell population density has two peaks represent corresponding to the locations of highest concentrations of cross-linked ECM fibers. This suggests clusters of cancer cells that have invaded the ECM, and the migration of cancer cells are clustered in the area where there is a high density of cross-linked ECM collagen fibers. Such behavior is the consequence of haptotaxis effect caused by cancer cells directional motility toward the cross-linked ECM collagen fibers.
Conclusion: The model in its base, case (i), has been verified and validated via comparison with previous work done by Anderson et al. (2000). The extended features of the enzyme LOX and its effect on ECM and cancer migration are successfully implemented in our new model. Simulation result of case (iii) mentioned above has confirmed the capability of our new model to capture the cross-linking effect LOX performs on ECM and how cross-linked fibers enhance the overall migration of cancer cells. The model provides a fundamental understanding that could possibly facilitate predictions of new therapeutic development, for example, for inhibiting the LOX term or altering or slowing the remodeling rate of ECM to slow down or prevent metastasis.
 Anderson, A.R.A.; Chaplain, M.A.J.; Newman, E.L.; Steele, R.J.C.; Thompson, A.M. Mathematical modelling of tumour invasion and metastasis. J Theor Med 2000, 2, 129â154