Stochastic Model for Tumor Control Probability: Effects of Cell Cycle and (A)symmetric Proliferation

Estimating the required dose in radiotherapy is of crucial importance since the administrated dose should be sufficient to eradicate the tumor and at the same time should inflict minimal damage on normal cells. The probability that a given dose and schedule of ionizing radiation eradicates all the tumor cells in a given tissue is called the tumor control probability (TCP), and is often used to compare various treatment strategies used in radiation therapy. In this paper, we aim to investigate the effects of including cell-cycle phase on the TCP by analyzing a stochastic model of a tumor comprised of actively dividing cells and quiescent cells with different radiation sensitivities. We derive an exact phase-diagram for the steady-state TCP of the model and show that at high, clinically-relevant doses of radiation, the distinction between active and quiescent tumor cells (i.e. accounting for cell-cycle effects) becomes of negligible importance in terms of its effect on the TCP curve. However, for very low doses of radiation, these proportions become significant determinants of the TCP. Moreover, we use a novel numerical approach based on the method of characteristics for partial differential equations, validated by the Gillespie algorithm, to compute the TCP as a function of time. We observe that our results differ from the results in the literature using similar existing models, even though similar parameters values are used, and the reasons for this are discussed.Comment: 12 pages, 5 figure

Mathematical Model of the Effect of Interstitial Fluid Pressure on Angiogenic Behavior in Solid Tumors

We present a mathematical model for the concentrations of proangiogenic and antiangiogenic growth factors, and their resulting balance/imbalance, in host and tumor tissue. In addition to production, diffusion, and degradation of these angiogenic growth factors (AGFs), we include interstitial convection to study the locally destabilizing effects of interstitial fluid pressure (IFP) on the activity of these factors. The molecular sizes of representative AGFs and the outward flow of interstitial fluid in tumors suggest that convection is a significant mode of transport for these molecules. The results of our modeling approach suggest that changes in the physiological parameters that determine interstitial fluid pressure have as profound an impact on tumor angiogenesis as those parameters controlling production, diffusion, and degradation of AGFs. This model has predictive potential for determining the angiogenic behavior of solid tumors and the effects of cytotoxic and antiangiogenic therapies on tumor angiogenesis

Phenotypic heterogeneity in modeling cancer evolution

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 differentiated 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 exact analytic results for the fixation probability of a mutant arising in each of the subpopulations. The analytic 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 differentiated cells turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. We discuss our model in the context of colorectal/intestinal cancer (at the epithelium). 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.Comment: 28 pages, 7 figures, 2 table

A PINN Approach to Symbolic Differential Operator Discovery with Sparse Data

Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime

Modeling the Spatial Distribution of Chronic Tumor Hypoxia: Implications for Experimental and Clinical Studies

Tumor oxygenation status is considered one of the important prognostic markers in cancer since it strongly influences the response of cancer cells to various treatments; in particular, to radiation therapy. Thus, a proper and accurate assessment of tumor oxygen distribution before the treatment may highly affect the outcome of the treatment. The heterogeneous nature of tumor hypoxia, mainly influenced by the complex tumor microenvironment, often makes its quantification very difficult. The usual methods used to measure tumor hypoxia are biomarkers and the polarographic needle electrode. Although these techniques may provide an acceptable assessment of hypoxia, they are invasive and may not always give a spatial distribution of hypoxia, which is very useful for treatment planning. An alternative method to quantify the tumor hypoxia is to use theoretical simulations with the knowledge of tumor vasculature. The purpose of this paper is to model tumor hypoxia using a known spatial distribution of tumor vasculature obtained from image data, to analyze the accuracy of polarographic needle electrode measurements in quantifying hypoxia, to quantify the optimum number of measurements required to satisfactorily evaluate the tumor oxygenation status, and to study the effects of hypoxia on radiation response. Our results indicate that the model successfully generated an accurate oxygenation map for tumor cross-sections with known vascular distribution. The method developed here provides a way to estimate tumor hypoxia and provides guidance in planning accurate and effective therapeutic strategies and invasive estimation techniques. Our results agree with the previous findings that the needle electrode technique gives a good estimate of tumor hypoxia if the sampling is done in a uniform way with 5-6 tracks of 20–30 measurements each. Moreover, the analysis indicates that the accurate measurement of oxygen profile can be very useful in determining right radiation doses to the patients

Instability and Fluctuations of Flux Lines with Point Impurities in a Parallel Current

A parallel current can destabilize a single flux line (FL), or an array of FLs. We consider the effects of pinning by point impurities on this instability. The presence of impurities destroys the long-range order of a flux lattice, leading to the so called Bragg glass (BrG) phase. We first show that the long-range topological order of the BrG is also destroyed by a parallel current. Nonetheless, some degree of short-range order should remain, whose destruction by thermal and impurity fluctuations, as well as the current, is studied here. To this end, we employ a cage model for a single FL in the presence of impurities and current, and study it analytically (by replica variational methods), and numerically (using a transfer matrix technique). The results are in good agreement, and in conjunction with a Lindemann criterion, provide the boundary in the magnetic field--temperature plane for destruction of short-range order. In all cases, we find that the addition of impurities or current (singly or in combination) leads to further increase in equilibrium FL fluctuations. Thus pinning to point impurities does not stabilize FLs in a parallel current $j_z$, although the onset of this instability is much delayed due to large potential barriers that diverge as $j_z^{-\mu}$.Comment: 10 pages, 6 figure