Lehrstuhl für Mechatronik in Maschinenbau und Fahrzeugtechnik (MEC)

Multiscale Modeling of Cancer Dynamics


M.Sc. Iqra Batool
Gottlieb-Daimler-Str. 42
67663, Kaiserslautern
Phone: +49 (0)631/205-3702
Fax: +49 (0)631/205-4201


State of Rhineland-Palatinate

Maturity-structured Modeling of Normal and Mutated Cell Lineage

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.

Mathematical Modeling

  • 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 αS, αA and αD 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.

Model Validation

  •  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.


Positive steady-states

Behavior of healthy and mutated cell lineages. (a) Healthy stem cells with initial value of 18000 m/L grow exponentially and converge to a steady-state. (b) Distribution of healthy progenitor cells P0(t,x). (c) Healthy mature cells depict a similar behavior with initial condition equal to zero. (d) Mutated stem cells with initial condition equal to zero, increase exponentially and attain a steady-state at relatively large value. (e) Distribution of mutated progenitor cells P1(t,x). It continues to grow towards a higher number and approaches a steady-state. (f) Mutated mature cells depict a similar behavior with initial condition equal to zero.


A Multiscale Model of Proliferating and Quiescent Cell Populations Coupled with Cell Cycle Dynamics.
Computer Aided Chemical Engineering, 2022. DOI
I. Batool, N. Bajcinca.

Evolution of Cancer Stem Cell lineage involving Feedback Regulation.
I. Batool, N. Bajcinca.
Well-posedness of a coupled PDE-ODE Model of Stem Cell Lineage Involving Homeostatic Regulation.
Results in Applied Mathematics, 2020. DOI
I. Batool, N. Bajcinca.

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