This October, after amazing two years as a Postdoctoral fellow at H. Lee Moffitt Cancer Center (Enderling Lab), I moved back to Poland and joined my old team in Laboratory of Mathematical Modelling of Biomedical Systems
at Nalecz Institute of Biocybernetics and Biomedical Engineering
Polish Academy of Sciences. On this occasion I decided to create a new website, which (hopefully) will be maintained better than the previous one…
Animal studies and clinical trials identified a synergy of fractionated irradiation and immunotherapy. This synergy stems from the fact that radiation induces cell stress and immunogenic cell death, thereby exposing a wealth of previously hidden tumor-associated antigens, stress proteins and dangerassociated molecular patterns (HSPs, DAMPs), which are endogenous immune adjuvants that can initiate and stimulate an immune response. Dendritic cells recognize these antigens, and present them to activate naïve T cells in the tumor draining lymph node. Activated T cells then travel through the lymphatic system, nter the blood circulation, and travel in cycles with the blood through the system of rteries, capillaries and veins. Post-irradiation immunotherapy stimulates rapid growth, proliferation, and differentiation of dendritic cells or T cells that are capable of specific imination of antigen-presenting tumor cell, thereby facilitating a second wave of cell kill nd overall tumor regression. Different radiation fractionation protocols in combination ith various immunotherapeutic approaches are currently being explored in more than 10 active clinical trials (7).
The goal of the project is to develop a quantitative mathematical framework that predicts systemic response of metastatic tumors to focal radiotherapy – either alone or in combination with immunotherapy. The ultimate aim of this project will be to propose such framework as a clinical decision support system to derive optimal radiation fractionation protocols and irradiation site for immune activation on a per patient basis.
Intermittent hemodialysis (HD) is the main method of renal replacement therapy in patients with end stage renal disease. Cardiovascular complications related to atherosclerosis and vascular calcification are the most frequent cause of death in this group of patients. A relatively new non-invasive tool for the assessment of the status of arteries is applanation tonometry that allows for precise measurements of the pulse wave shape and its velocity in peripheral (e. g. radial) arteries. Importantly, by performing pulse-wave analysis on a peripheral artery, one can estimate the central aortic pulse shape that is crucial for the stratification of CV disease risk. The required permanent blood access, such as arteriovenous fistula, induces substantial changes in systemic and peripheral blood circulation and has substantial impact on the measured pulse wave. There are also other important factors that have to be taken into account, such as the changes in fluid overload and blood volume during hemodialysis session and the changes in vascular resistance due to modifications of vasodilatation of the capillary bed. These factors make the standard interpretation of pulse wave analysis problematic if applied to patients on hemodialysis and there is a need for a theoretical evaluation that can combine all important factors. On the other hand, applanation tonometry may help to better understand and assess the status of the arteriovenous fistula and other possible clinical complications that occur in HD patients. We hypothesize that the presence of the fistula and other abnormalities in vascular system of HD patients may influence the standard assumptions necessary for pulse wave analysis and its interpretation.
Our main objective is to investigate the role of various abnormalities in HD patients on the propagation of pulse wave using mathematical modeling of the whole arterial tree with arteriovenous fistula and compare these predictions to the respective clinical measurements.
We propose the application of a mathematical model of pulse wave propagation in the whole arterial system modified and extended by the arteriovenous fistula for the analysis of the role of various possible (patho-)physiological conditions that may have an impact on the measured pulse wave. The models of pulse wave propagation describe 1D flow in compliant arterial vessels using two variables: blood flow velocity and blood pressure (or equivalently the cross-sectional area of compliant vessel), and were previously shown to be in agreement with rich set of physiological data. The mathematical modeling will be applied for the analysis of the results of a new clinical study that involves non-invasive measurements of pulse wave at different places of arteriovenous system before, during and after hemodialysis session together with non-invasive assessment of the heart activity (by impedance cardiography) and the changes in fluid status of the patient (intracellular and extracellular water by bioimpedance, blood volume by online measurement of changes in hematocrit) that strongly influence the pulse wave propagation.
The lecture is devoted to the widely understood mathematical modeling in biology and medicine. We mainly focus on ecological models which are built using differential and difference equations. We also consider models of immune reactions and those of classical genetics (Mendel theory) based on Markov chains.
News:
project2014
The lecture presents a wide area of applications of nonlinear mathematical modeling in biology, medical science and clinical medicine.
Computer laboratory classes are based on MATLAB® software and are aimed at the presentation of computer methods for simulation and analysis of solutions of ordinary differential equations and equations with discrete time. Students have an opportunity to investigate the models themselves.
Slides from computer labs:
compLab1 – Introduction to MATLAB
compLab2 – Migrations in logistic equation, Alee effect, Holling response
compLab3 – Discrete logistic equation (solutions, bifurcations), two dimensional discrete models
compLab4 – Classic L-V (solutions, average value, harvesting), L-V with logistic term
compLab5 and data – Interactions between populations (competition/mutualism), May model
compLab6 – Lorenz attractor (solutions, behavior)
compLab7 – Kinetics of single enzyme and product, kinetics of multiple enzymes/prodicts/substrates
compLab8 – Zeeman model for the heartbeat
compLab9 – Models with various types of antigens
compLab10 – Kuznetsov model (solutions, behavior)
compLab11 – TC model of hemodialysis, data fitting
compLab12 – Basic gene expression models, part of p53 signaling pathway
compLab13 – Logistic model with vaccinations, L-V with harvesting
compLab14 – Model of self-repressing gene, Mackey-Glass model
compLab15 – Projects presentations
Below is the code that was used in one of my projects – results published in:
Poleszczuk J, Enderling H, A High-Performance Cellular Automaton Model of Tumor Growth with Dynamically Growing Domains, Applied Mathematics 2014 (5): 144-152.
Abstract
Tumor growth from a single transformed cancer cell up to a clinically apparent mass spans many spatial and temporal orders of magnitude. Implementation of cellular automata simulations of such tumor growth can be straightforward but computing performance often counterbalances simplicity. Computationally convenient simulation times can be achieved by choosing appropriate data structures, memory and cell handling as well as domain setup. We propose a cellular automaton model of tumor growth with a domain that expands dynamically as the tumor population increases. We discuss memory access, data structures and implementation techniques that yield high-performance multi-scale Monte Carlo simulations of tumor growth. We discuss tumor properties that favor the proposed high-performance design and present simulation results of the tumor growth model. We estimate to which parameters the model is the most sensitive, and show that tumor volume depends on a number of parameters in a non-monotonic manner.
Download code:
naive_code
improved_code
Below is the code that was used in one of my projects – results published in:
Poleszczuk J, Piotrowska MJ, Forys U, Optimal protocols for the anti-VEGF tumor treatment, Mathematical Modelling of Natural Phenomena, 2014 (9): 204-215.
(Abstract below).
Abstract
Anti-angiogenic cancer treatments target the tumor vasculature that delivers oxygen and nutrients to tumors and thus induces the tumor starvation and regression. Mathematical models provide valuable tools to study the proof-of-concept, efficacy and underlying mechanisms of such treatment approach. In the paper we investigate optimal anti-angiogenic treatment protocols for the recently proposed modification of the well established family of tumor angiogenesis models. This modification made the models valid in the case of treatment which is focused on blocking angiogenic signaling. Moreover, we propose a~new mathematical description of the anti-angiogenic treatment goal which assumes that the main purpose of applying the anti-angiogenic treatment, despite the minimization of the tumor volume, is to maintain relatively high volume of vessels that support the tumor. The reason of such approach is related to novel combined therapies, namely anti-angiogenic and chemotherapy, because effectiveness of chemotherapy depends on the amount and quality of the tumor vessels.
We derive the structure of the optimal anti-angiogenic treatment protocol and investigate numerically its efficacy when used together with chemotherapy infusions. We show that the proposed optimal protocol might be more efficient when combined with chemotherapy, comparing to the standard full-dose protocol. Besides the higher reduction of the tumor volume it might also decrease the level of the therapy induced hypoxia. The last effect is important, as hypoxia can induce anti-angiogenic drug resistance of tumor cells.
Download code:
optimal control code_1