Multiscale Modeling of Cancer Dynamics

Tumor emergence and progression is a complex phenomenon that assumes special molecular and cellular interactions. The hierarchical structuring and communication via feedback signaling of different cell types, which are categorized as the stem, progenitor, and differentiated cells in dependence of their maturity level, plays an important role. Under healthy conditions, these cells build a dynamical system that is responsible for facilitating the homeostatic regulation of the tissue via feedback signaling. Generally, in this hierarchical setting, stem and progenitor cells are yet likely to undergo a mutation, when a cell divides into two daughter cells. This may lead to the development of abnormal characteristics, i.e. mutation in the cell, yielding an unrestrained growth in number of cells.

• A maturity based mathematical model with feedback regulation is formulated for healthy and mutated cell lineages and it is comprised of coupled ordinary and partial differential equations.
• The variable C₀(t) and C₁(t) refer to the number of stem cells with zero and one mutation. P₀(x,t) and P₁(x,t) correspond to the progenitor cells with zero and one mutation, respectively, while M₀(t) and M₁(t) refer to the number of fully differentiated healthy and mutated cell populations.
• The terms α_S, α_A and α_D depict the probability of symmetric/asymmetric self-renewal and differentiation in stem cells, respectively.
• The functions g₀(x) and g₁(x) stand for the differentiation rate of progenitor cells, whereas β₀(x,s) and β₁(x,s) represent their proliferation rates.
• The feedback signal s=1/(1+kM) derived from total mature cell population M(t)=M₀(t)+M₁(t), represents growth and anti-growth factors which are invoked into the stem and progenitor cell populations.
• The focus is laid on the dynamical effects resulting from acquiring a mutation in the hierarchical structure of stem, progenitor and fully differentiated cells.
• Here m and m' represent the mutations rates in stem and progenitor cells and x* is maximum maturity level.

•  The model has been validated with the data set derived by TUBO Cancer cell linet generated in vitro experiments on TUBO cancer cells and is reported in the literature.
• TUBO cancer cells are a cloned line derived in vitro from a BALB-neuT mouse mammary carcinoma. The data set consists of mean ± standard deviation for total cell count.
• The blue line is the model fit to the experimental measurements in the right figure. The model fits to the mean values for more than 65 percent of the data set. The initial conditions for healthy cell line are equal to their respective steady-state values.
• Our proposed model predicts an establishment of the steady-state in grey shaded area where no measurements of cell count were available.
•  The initial condition of mutated stem cells is equal to 2x10⁴m/L and for mutated progenitor and mature cells are equal to zero.
• In the figure below, model validation has been further performed using the tumor volume measurements for three different cancers, namely prostate, breast and colon. These data sets are obtained by establishing human tumor xenografts in mice.