A panel showing a comparison for the measurable TCP curves when <i>k</i> = 3, for scheme 1 (conventional therapy), scheme 2 (hyper-fractionated therapy), and scheme 3 (accelerated hyper-fractionated therapy), noting that there is qualitative agreement with the theoretical TCP curves.

<p>The left graph is for TCP as a function of time in days, and the right is a for TCP as a function of dose of radiaton administered.</p

Tumour Control Probability in Cancer Stem Cells Hypothesis

<div><p>The tumour control probability (TCP) is a formalism derived to compare various treatment regimens of radiation therapy, defined as the probability that given a prescribed dose of radiation, a tumour has been eradicated or controlled. In the traditional view of cancer, all cells share the ability to divide without limit and thus have the potential to generate a malignant tumour. However, an emerging notion is that only a sub-population of cells, the so-called cancer stem cells (CSCs), are responsible for the initiation and maintenance of the tumour. A key implication of the CSC hypothesis is that these cells must be eradicated to achieve cures, thus we define TCP_S as the probability of eradicating CSCs for a given dose of radiation. A cell surface protein expression profile, such as CD44high/CD24low for breast cancer or CD133 for glioma, is often used as a biomarker to monitor CSCs enrichment. However, it is increasingly recognized that not all cells bearing this expression profile are necessarily CSCs, and in particular early generations of progenitor cells may share the same phenotype. Thus, due to the lack of a perfect biomarker for CSCs, we also define a novel measurable TCP_CD+, that is the probability of eliminating or controlling biomarker positive cells. Based on these definitions, we use stochastic methods and numerical simulations parameterized for the case of gliomas, to compare the theoretical TCP_S and the measurable TCP_CD+. We also use the measurable TCP to compare the effect of various radiation protocols.</p></div

The images, adapted from [20], show that CD133+ tumor cells can proliferate in culture as non-adherent spheres, whereas CD133− tumor cells are not able to proliferate and form spheres.

<p>We assume that CSCs and early progenitors (from non-CSC compartment) share the CD133 biomarker. Here , (), and denote stem, progenitors and mature cells, respectively.</p

The fixation probability of mutants in the absence of plasticity.

<p>We assume <i>N</i>_S = <i>N</i>_D = 10, , and <i>η</i>₁ = <i>η</i>₂ = 0 in simulations (points) and exact calculations (solid lines) for an initial mutant in the SC compartment. Each error bar shown at each point is the standard error of the mean. In (a) changing parameters <i>u</i>₁, <i>u</i>₂, that are the differentiation rates of normal and tumor SCs respectively, the trends for the fixation probability of mutants is given as a function of relative fitness of mutants, referred to as . In (b) and (c) the fixation probability variation is given in terms of asymmetric differentiation rates <i>u</i>₁ = <i>u</i>₂ = <i>u</i> and various values of <i>r</i> and the ratio of the differentiation rates of normal SCs <i>ϵ</i> = <i>u</i>₂/<i>u</i>₁. In (b) <i>ϵ</i> = 0.5 and in (c) <i>ϵ</i> = 1.5.</p

Phase diagram of plastic mutant SCs.

<p>The phase boundary for advantageous and disadvantageous mutant populations are given as differentiation and plasticity rates change. We assume that , , <i>u</i>₁ = <i>u</i>₂ = <i>u</i>, and <i>η</i>₁ = 0. Different regions for advantageous and disadvantageous mutant SCs are given in (a) as <i>u</i> changes. A similar analysis has been carried out in (b) as <i>η</i> varies. In (a) <i>η</i> = 0.1, 0.3, 0.7, here the alteration in the plasticity rate of DCs results in a tendency to approach various regions of fixation for mutant SCs, while the extinction domain shrinks with increasing <i>η</i>. In (b) <i>u</i> = 0.1, 0.3, 0.7. Increasing the asymmetric division rate <i>u</i>, the region for advantageous mutants expands to provide a higher survival chance for mutant SCs. In both cases, advantageous criteria relate to either fixation of mutants or coexistence of mutants and WT individuals.</p

A graph showing a comparison of the survival fraction, as a function of a single dose for biomarker-positive and biomarker-negative cells, assuming a three-fold increase in the radio-sensitivity parameters <i>α</i> and <i>β</i> for biomarker-negative cells as compared to positive cells.

<p>A graph showing a comparison of the survival fraction, as a function of a single dose for biomarker-positive and biomarker-negative cells, assuming a three-fold increase in the radio-sensitivity parameters <i>α</i> and <i>β</i> for biomarker-negative cells as compared to positive cells.</p

Phenotypic heterogeneity in modeling cancer evolution

<div><p>The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and non-stem cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exactly calculated results for the fixation probability of a mutant arising in each of the subpopulations. The exactly calculated results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and non-stem cells’ turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. The derived results are novel and general with potential applications in nature; we discuss our model in the context of colorectal/intestinal cancer (at the epithelium). However, the model clearly needs to be validated through appropriate experimental data. This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.</p></div

A panel showing a comparison for the measurable and theoretical TCP curves for <i>k</i> = 0,1,2,3, for the conventional therapy (scheme 1), noting that as <i>k</i> is increased, a slight shift occurs in the TCP curve.

<p>The left graph is for TCP as a function of time in days, and the right is a for TCP as a function of dose of radiaton administered.</p

Phenotypic—Genotypic changes in individuals within a four–compartmental structure.

<p>We consider constant population sizes <i>N</i>_S and <i>N</i>_D for SCs and DCs respectively. With respect to the finite Markov chain, we consider a generalized model to take into account the competition between normal and malignant individuals in each of the SC and DC subpopulations. Differentiation and dedifferentiation events connect the selection dynamics between the two niches. In (a), all possible differentiation, dedifferentiation, and death events with their corresponding rates are represented. The SC-DC compartmental structure is depicted in (b) with the associated self—renewal and differentiation/plasticity possibilities.</p

Cellular interactions in the colonic crypt as a newborn mutant arises within the stem or non-stem cell compartments.

<p>Within this schematic cylindrical model, we represent how our model is structured through the four compartments of host and mutant stem and non-stem cells. In contrast to the circular model of five SCs considered in [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0187000#pone.0187000.ref063" target="_blank">63</a>], we assume a cylindrical model of two circles, one on the top of the other. SCs are located at the bottom circle while the circle on the top is full of partially DCs.</p