Advocating the need of a systems biology approach for personalised prognosis and treatment of B-CLL patients

The clinical course of B-CLL is heterogeneous. This heterogeneity leads to a clinical dilemma: can we identify those patients who will benefit from early treatment and predict the survival? In recent years, mathematical modelling has contributed significantly in understanding the complexity of diseases. In order to build a mathematical model for determining prognosis of B-CLL one has to identify, characterise and quantify key molecules involved in the disease. Here we discuss the need and role of mathematical modelling in predicting B-CLL disease pathogenesis and suggest a new systems biology approach for a personalised therapy of B-CLL patients

Mathematical models of Leukaemia and its treatment: A review

Leukaemia accounts for around 3% of all cancer types diagnosed in adults, and is the most common type of cancer in children of paediatric age. There is increasing interest in the use of mathematical models in oncology to draw inferences and make predictions, providing a complementary picture to experimental biomedical models. In this paper we recapitulate the state of the art of mathematical modelling of leukaemia growth dynamics, in time and response to treatment. We intend to describe the mathematical methodologies, the biological aspects taken into account in the modelling, and the conclusions of each study. This review is intended to provide researchers in the field with solid background material, in order to achieve further breakthroughs in the promising field of mathematical biology

Mass concentration in a nonlocal model of clonal selection

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To understand better this impact, we propose a mathematical model describing dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling, which describes regulatory feedback loops in cell proliferation and differentiation process. We show that such coupling leads to mass concentration in points corresponding to maximum of the self-renewal potential and the model solutions tend asymptotically to a linear combination of Dirac measures. Furthermore, using a Lyapunov function constructed for a finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in space of positive Radon measures equipped with flat metric. The analytical results are illustrated by numerical simulations

The first international workshop on the role and impact of mathematics in medicine: a collective account

The First International Workshop on The Role and Impact of Mathematics in Medicine (RIMM) convened in Paris in June 2010. A broad range of researchers discussed the difficulties, challenges and opportunities faced by those wishing to see mathematical methods contribute to improved medical outcomes. Finding mechanisms for inter- disciplinary meetings, developing a common language, staying focused on the medical problem at hand, deriving realistic mathematical solutions, obtainin

CAR T cell therapy in B-cell acute lymphoblastic leukaemia: Insights from mathematical models

Immunotherapies use components of the patient immune system to selectively target cancer cells. The use of CAR T cells to treat B-cell malignancies --leukaemias and lymphomas-- is one of the most successful examples, with many patients experiencing long-lasting complete responses to this therapy. This treatment works by extracting the patient's T cells and adding them the CAR group, which enables them to recognize and target cells carrying the antigen CD19+, that is expressed in these haematological tumors. Here we put forward a mathematical model describing the time response of leukaemias to the injection of CAR T-cells. The model accounts for mature and progenitor B-cells, tumor cells, CAR T cells and side effects by incorporating the main biological processes involved. The model explains the early post-injection dynamics of the different compartments and the fact that the number of CAR T cells injected does not critically affect the treatment outcome. An explicit formula is found that provides the maximum CAR T cell expansion in-vivo and the severity of side effects. Our mathematical model captures other known features of the response to this immunotherapy. It also predicts that CD19+ tumor relapses could be the result of the competition between tumor and CAR T cells analogous to predator-prey dynamics. We discuss this fact on the light of available evidences and the possibility of controlling relapses by early re-challenging of the tumor with stored CAR T cells

A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

We model a general, hierarchically organized tissue by a multi compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters.Comment: 4 figures, to appear in Royal Society Interfac

The nonlinear nature of biology

In this thesis, we explore the stability and the breakdown of stability of biological systems. The main examples are the blood system and invasion of cancer. However, the models presented in the thesis apply to several other examples. Biological systems are characterised by both competition and cooperation. Cooperation is based on an unsolvable dilemma: Even though mutual cooperation leads to higher payoff than mutual defection, a defector has higher payoff than a co-operator when they meet. It is not possible to represent this dilemma with a linear and deterministic model. Hence, the dilemma of cooperation must have a nonlinear and/or stochastic representation. More general, by using a linearised model to describe a biological system, one might lose dimensions inherent in the complexity of the system. In this thesis, we illustrate that a nonlinear description of a biological system is potentially more accurate and might provide new information. We show that even though a new type of individual is in general not advantageous when it appears in stable population, the newcomers can grow in number due to stochasticity. Moreover, the new type can only become advantageous if it manages to change the environment in such a way that it increases its fitness. We also propose a model that links self-organisation with symmetric and asymmetric cell division, and we illustrate that if symmetric stem cell division is regulated by differentiated cells, then the fitness of the stem cells can be affected by modifying the death rate of the mature cells. This result is interesting because stem cells are less sensitive than mature cells to medical therapy, and our results imply that stem cells can be manipulated indirectly by medical treatments that target the mature cells

Application of mathematical modelling to describe and predict treatment dynamics in patients with NPM1-mutated Acute Myeloid Leukaemia (AML)

Background: Acute myeloid leukaemia (AML) is a severe form of blood cancer, which in many cases can not be cured. Although chemotherapeutic treatment is effective in most cases, often the disease relapses. To monitor the course of disease, as well as to early identify a relapse, the leukaemic cell burden in the bone marrow is measured. In the genome of these cells certain mutations can be found, which lead to the occurrence of leukaemia. One of those mutations is in the neucleophosmin 1 (NPM1) gene. This mutation is found in about one third of all AML patients. The burden of leukaemic cells can be derived from the proportion of NPM1 transcripts carrying this mutation in a bone marrow sample. These values are measured routinely at specific time points during treatment and are then used to categorise the patients into defined risk groups. In the studies, the data for this work originates from, the NPM1 burden was measured beyond the treatment period. That leads to a more comprehensive picture of the molecular course of disease of the patients. Hypothesis: My hypothesis is that the risk group categorisation can be improved by taking into account the dynamic time course information of the patients. Another hypothesis of this work is that with the help of statistical methods and computer models the time course data can be used to describe the course of disease of AML patients and assess whether they will experience a relapse or not. Materials and Methods: For these investigations I was provided with a dataset consisting of quantitative NPM1 time course measurements of 340 AML patients (with a median of 6 mea- surements per patient). To analyse this data I used statistical methods, such as correlation, logistic regression and survival time analysis. For a better understanding of the course of disease I developed a mechanistic model describing the dynamics of the cell numbers in the bone marrow of an AML patient. This model can be fitted to the measurements of a patient by adjusting two parameters, which represent the individual severity of disease. To predict a possible relapse within 2 years after beginning of treatment, I used data that was generated using the mechanistic model (synthetic data). For the prediction three different methods were compared: the mechanistic model, a recurrent neural network (RNN) and a generalised linear model (GLM). Both, the RNN and the GLM were trained and tuned on part of the synthetic data. Afterwards all three methods were tested using the so far unseen part of the data set (test data). Results: Following the analysis of the data I found that the decreasing slope of NPM1 burden during primary treatment as well as the absolute burden after the treatment harbour information about the further course of disease. Specifically, I found that a faster decrease of NPM1 burden and a lower final burden lead to a better prognosis. Further, I could show that the developed simple mechanistic model is able to describe the course of disease of most patients. When I divided the patients into two different risk groups using the fitted parameters from the model I could show that the patients in those groups show distinct relapse-free survival times. The categorisation using the parameters lead to a better distinction of groups than using current categorisation by the WHO. Further, I tried to predict a 2-year relapse using synthetic data and three different prediction methods. I could show that it had nearly no impact at all which method I used. Much more important, however, was the quality of data. Especially the sparseness of data, which we find in the time courses of AML patients, has a considerable negative effect on the predictability of relapse. Using a synthetic data set with measurement times oriented on the times of chemotherapy I could show that a sophisticated measurement scheme could improve the relapse predictability. Conclusions: In conclusion, I suggest to include the dynamic molecular course of the NPM1 burden of AML patients in clinical routine, as this harbours additional information about the course of disease. The involvement of a mechanistic model to asses the risk of AML patients can help to make more accurate predictions about their general prognosis. An accurate prediction of the time of relapse is not possible. All three used methods (mechanistic model, statistical model and neural network) are in general suitable to predict relapse of AML patients. For reliable predictions, however, the quality of the data needs to be drastically improved

Dynamics of Resistance Development to Imatinib under Increasing Selection Pressure: A Combination of Mathematical Models and In Vitro Data

In the last decade, cancer research has been a highly active and rapidly evolving scientific area. The ultimate goal of all efforts is a better understanding of the mechanisms that discriminate malignant from normal cell biology in order to allow the design of molecular targeted treatment strategies. In individual cases of malignant model diseases addicted to a specific, ideally single oncogene, e.g. Chronic myeloid leukemia (CML), specific tyrosine kinase inhibitors (TKI) have indeed been able to convert the disease from a ultimately life-threatening into a chronic disease with individual patients staying in remission even without treatment suggestive of operational cure. These developments have been raising hopes to transfer this concept to other cancer types. Unfortunately, cancer cells tend to develop both primary and secondary resistance to targeted drugs in a substantially higher frequency often leading to a failure of treatment clinically. Therefore, a detailed understanding of how cells can bypass targeted inhibition of signaling cascades crucial for malignant growths is necessary. Here, we have performed an in vitro experiment that investigates kinetics and mechanisms underlying resistance development in former drug sensitive cancer cells over time in vitro. We show that the dynamics observed in these experiments can be described by a simple mathematical model. By comparing these experimental data with the mathematical model, important parameters such as mutation rates, cellular fitness and the impact of individual drugs on these processes can be assessed. Excitingly, the experiment and the model suggest two fundamentally different ways of resistance evolution, i.e. acquisition of mutations and phenotype switching, each subject to different parameters. Most importantly, this complementary approach allows to assess the risk of resistance development in the different phases of treatment and thus helps to identify the critical periods where resistance development is most likely to occur

Effect of Dedifferentiation on Time to Mutation Acquisition in Stem Cell-Driven Cancers

Accumulating evidence suggests that many tumors have a hierarchical organization, with the bulk of the tumor composed of relatively differentiated short-lived progenitor cells that are maintained by a small population of undifferentiated long-lived cancer stem cells. It is unclear, however, whether cancer stem cells originate from normal stem cells or from dedifferentiated progenitor cells. To address this, we mathematically modeled the effect of dedifferentiation on carcinogenesis. We considered a hybrid stochastic-deterministic model of mutation accumulation in both stem cells and progenitors, including dedifferentiation of progenitor cells to a stem cell-like state. We performed exact computer simulations of the emergence of tumor subpopulations with two mutations, and we derived semi-analytical estimates for the waiting time distribution to fixation. Our results suggest that dedifferentiation may play an important role in carcinogenesis, depending on how stem cell homeostasis is maintained. If the stem cell population size is held strictly constant (due to all divisions being asymmetric), we found that dedifferentiation acts like a positive selective force in the stem cell population and thus speeds carcinogenesis. If the stem cell population size is allowed to vary stochastically with density-dependent reproduction rates (allowing both symmetric and asymmetric divisions), we found that dedifferentiation beyond a critical threshold leads to exponential growth of the stem cell population. Thus, dedifferentiation may play a crucial role, the common modeling assumption of constant stem cell population size may not be adequate, and further progress in understanding carcinogenesis demands a more detailed mechanistic understanding of stem cell homeostasis