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We developed a computational platform including machine learning and a mechanistic mathematical model to find the optimal protocol for administration of platinum-doublet chemotherapy in a palliative setting. The platform has been applied to advanced metastatic non-small cell lung cancer (NSCLC). The 42 NSCLC patients treated with palliative intent at Maria Sklodowska-Curie National Research Institute of Oncology, Gliwice Branch, were collected from a retrospective cohort of patients diagnosed in 2004-2014. Patients were followed-up, for three years. Clinical data collected include complete information about the clinical course of the patients including treatment schedule, response according to RECIST classification, and survival. The core of the platform is the mathematical model, in the form of a system of ordinary differential equations, describing dynamics of platinum-sensitive and platinum-resistant cancer cells and interactions reflecting competition for space and resources. The model is simulated stochastically by sampling the parameter values from a joint probability distribution function. The machine learning model is applied to calibrate the mathematical model and to fit it to the overall survival curve. The model simulations faithfully reproduce the clinical cohort at three levels long-term response (OS), the initial response (according to RECIST criteria), and the relationship between the number of chemotherapy cycles and time between two consecutive chemotherapy cycles. In addition, we investigated the relationship between initial and long-term response. We showed that those two variables do not correlate which means that we cannot predict patient survival solely based on the initial response. We also tested several chemotherapy schedules to find the best one for patients treated with palliative intent. We found that the optimal treatment schedule depends, among others, on the strength of competition among various subclones in a tumor. The computational platform developed allows optimizing chemotherapy protocols, within admissible limits of toxicity, for palliative treatment of metastatic NSCLC. The simplicity of the method allows its application to chemotherapy optimization in different cancers.

Introduction: The application of magnetic resonance imaging (MRI) to acquire detailed descriptions of the brain morphology in vivo is a driving force in brain mapping research. Most atlases are based on parametric statistics, however, the empirical results indicate that the population brain tissue distributions do not exhibit exactly a Gaussian shape. Our aim was to verify the population voxel-wise distribution of three main tissue classes: gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF), and to construct the brain templates for the Polish (Upper Silesian) healthy population with the associated non-parametric tissue probability maps (TPMs) taking into account the sex and age influence.Material and Methods: The voxel-wise distributions of these tissues were analyzed using the Shapiro-Wilk test. The non-parametric atlases were generated from 96 brains of the ethnically homogeneous, neurologically healthy, and radiologically verified group examined in a 3-Tesla MRI system. The standard parametric tissue proportion maps were also calculated for the sake of comparison. The maps were compared using the Wilcoxon signed-rank test and Kolmogorov-Smirnov test. The volumetric results segmented with the parametric and non-parametric templates were also analyzed.Results: The results confirmed that in each brain structure (regardless of the studied sub-population) the data distribution is skewed and apparently not Gaussian. The determined non-parametric and parametric templates were statistically compared, and significant differences were found between the maps obtained using both measures (the maps of GM, WM, and CSF). The impacts of applying the parametric and non-parametric TPMs on the segmentation process were also compared. The GM volumes are significantly greater when using the non-parametric atlas in the segmentation procedure, while the CSF volumes are smaller.Discussion and Conclusion: To determine the population atlases the parametric measures are uncritically and widely used. However, our findings suggest that the mean and parametric measures of such skewed distribution may not be the most appropriate summary statistic to find the best spatial representations of the structures in a standard space. The non-parametric methodology is more relevant and universal than the parametric approach in constructing the MRI brain atlases.

Our goal was to determine the influence of sex, age and the head/brain size on the compartmental brain volumes in the radiologically verified healthy population (96 subjects; 54 women and 42 men) from the Upper Silesia region in Poland. The MRI examinations were done using 3T Philips Achieva with the same T1-weighted and T2-weighted protocols. The image segmentation procedures were performed with SPM (Statistical Parameter Mapping) and FSL-FIRST software. The volumes of 14 subcortical structures for the left and right hemispheres and 4 overall volumes were calculated. The General Linear Models (GLM) analysis was used with and without the Total Brain Volume (TBV) and Intracranial Volume (ICV) parameters as the covariates to study the regional vs. global brain atrophy. After the ICV/TBV adjustments, the majority of sex differences in the specific volumes of interest (VOIs) revealed to be linked to the difference in the head/brain size parameters. The analysis also confirmed the significant effect of the aging process on the brain loss. After the TBV adjustment, the age- and sex-related volumetric trends for the gray and white matter volumes were observed: the negative age dependence of the gray matter volume is more pronounced in the males, while in case of the white matter the positive age-related trend in the female group is weaker. The local losses of the left caudate nucleus and the right thalamus are more advanced than the global brain atrophy. Different head-size correction strategies are not interchangeable and may yield various volumetric results, but when used together, facilitate studies on the regional dependencies inherent to a healthy, but aging, brain.

Abstract Background Examination of physiological processes and the influences of the drugs on them can be efficiently supported by mathematical modeling. One of the biggest problems is related to the exact fitting of the parameters of a model. Conditions inside the organism change dynamically, so the rates of processes are very difficult to estimate. Perturbations in the model parameters influence the steady state so a desired therapeutic goal may not be reached. Here we investigate the effect of parameter deviation on the steady state in three simple models of the influence of a therapeutic drug on its target protein. Two types of changes in the model parameters are taken into account: small perturbations in the system parameter values, and changes in the switching time of a specific parameter. Additionally, we examine the systems response in case of a drug concentration decreasing with time. Results The models which we analyze are simplified, because we want to avoid influences of complex dynamics on the results. A system with a negative feedback loop is the most robust and the most rapid, so it requires the largest drug dose but the effects are observed very quickly. On the other hand a system with positive feedback is very sensitive to changes, so small drug doses are sufficient to reach a therapeutic target. In systems without feedback or with positive feedback, perturbations in the model parameters have a bigger influence on the reachability of the therapeutic target than in systems with negative feedback. Drug degradation or inactivation in biological systems enforces multiple drug applications to maintain the level of a drug’s target under the desired threshold. The frequency of drug application should be fitted to the system dynamics, because the response velocity is tightly related to the therapeutic effectiveness and the time for achieving the goal. Conclusions Systems with different types of regulation vary in their dynamics and characteristic features. Depending on the feedback loop, different types of therapy may be the most appropriate, and deviations in the model parameters have different influences on the reachability of the therapeutic target.

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.

We review a quite large volume of literature concerning mathematical modelling of processes related to carcinogenesis and the growth of cancer cell populations based on the theory of evolutionary games. This review, although partly idiosyncratic, covers such major areas of cancer-related phenomena as production of cytotoxins, avoidance of apoptosis, production of growth factors, motility and invasion, and intra- and extracellular signaling. We discuss the results of other authors and append to them some additional results of our own simulations dealing with the possible dynamics and/or spatial distribution of the processes discussed.

This paper presents analysis and biomedical implications of a certain class of bilinear systems that can be applied in modeling of cancer chemotherapy. It combines models that so far have been studied separately, taking into account both the phenomenon of gene amplification and drug specificity in chemotherapy in their different aspects. The methodology of analysis of such models, based on system decomposition, is discussed. The mathematical description is given by an infinite dimensional state equation with a system matrix, the form of which allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the second one is infinite-dimensional and tridiagonal. Then the optimal control problem is defined in $l^1$ space. To derive necessary conditions for optimal control, the model description is transformed into an integrodifferential one.

This paper concerns the problem of fitting of mathematical modelsof cell signaling pathways. Such models frequently take the form of a set ofnonlinear ordinary differential equations. While the model is continuous-time,the performance index, used in the fitting procedure, involves measurementstaken only at discrete-time moments. Adjoint sensitivity analysis is a toolthat can be used for finding a gradient of a performance index in the space ofthe model’s parameters. The paper uses a structural formulation of sensitivityanalysis, especially dedicated for hybrid, continuous/discrete-time systems. Anumerical example of fitting of the mathematical model of the NF-kB regulatorymodule is presented.

The idea for this volume came from the session ''Control and Dynamics ofBiomedical Systems,'' which we organized at the Fourth World Congress of NonlinearAnalysts held in Orlando, Florida, on June 30–July 7, 2004. We inviteda number of speakers, and although not all could attend and give a talk in person,many expressed interest in contributing a paper. Following an invitation fromYang Kuang to put together a volume on the topic as a special issue of MathematicalBiosciences and Engineering, some papers presented at the session weresubmitted immediately, and others were contributed after the Congress.For the full paper, please click the 'Full Text' button above.

In the paper ''Spatially Distributed Morphogen Production and Morphogen Gradient Formation' by Arthur D. Lander, Qing Nie, and Frederic Y. M. Wan (Mathematical Biosciences and Engineering, Volume 2, Number 2, April 2005), the following correction should be made:In the statement of Lemma 10, '(0,1)' should hav been '($-x_m$, 1).'