Stability Analysis of a Model of Interaction Between the Immune System and Cancer Cells in Chronic Myelogenous Leukemia

Dynamics and bifurcations in a model of chronic myeloid leukemia with optimal immune response windows

Chronic Myeloid Leukemia is a blood cancer for which standard therapy with Tyrosine-Kinase Inhibitors is successful in the majority of patients. After discontinuation of treatment half of the well-responding patients either present undetectable levels of tumor cells for a long time or exhibit sustained fluctuations of tumor load oscillating at very low levels. Motivated by the consequent question of whether the observed kinetics reflect periodic oscillations emerging from tumor-immune interactions, in this work, we analyze a system of ordinary differential equations describing the immune response to CML where both the functional response against leukemia and the immune recruitment exhibit optimal activation windows. Besides investigating the stability of the equilibrium points, we provide rigorous proofs that the model exhibits at least two types of bifurcations: a transcritical bifurcation around the tumor-free equilibrium point and a Hopf bifurcation around a biologically plausible equilibrium point, providing an affirmative answer to our initial question. Focusing our attention on the Hopf bifurcation, we examine the emergence of limit cycles and analyze their stability through the calculation of Lyapunov coefficients. Then we illustrate our theoretical results with numerical simulations based on clinically relevant parameters. Besides the mathematical interest, our results suggest that the fluctuating levels of low tumor load observed in CML patients may be a consequence of periodic orbits arising from predator-prey-like interactions.

Mathematical modeling of combined therapies for treating tumor drug resistance

Drug resistance is one of the most intractable issues to the targeted therapy for cancer diseases. To explore effective combination therapy schemes, we propose a mathematical model to study the effects of different treatment schemes on the dynamics of cancer cells. Then we characterize the dynamical behavior of the model by finding the equilibrium points and exploring their local stability. Lyapunov functions are constructed to investigate the global asymptotic stability of the model equilibria. Numerical simulations are carried out to verify the stability of equilibria and treatment outcomes using a set of collected model parameters and experimental data on murine colon carcinoma. Simulation results suggest that immunotherapy combined with chemotherapy contributes significantly to the control of tumor growth compared to monotherapy. Sensitivity analysis is performed to identify the importance of model parameters on the variations of model outcomes.

Optimizing chemotherapy treatment outcomes using metaheuristic optimization algorithms: A case study

BACKGROUND

This study explores the dynamics of a mathematical model, utilizing ordinary differential equations (ODE), to depict the interplay between cancer cells and effector cells under chemotherapy. The stability of the equilibrium points in the model is analysed using the Jacobian matrix and eigenvalues. Additionally, bifurcation analysis is conducted to determine the optimal values for the control parameters.

OBJECTIVE

To evaluate the performance of the model and control strategies, benchmarking simulations are performed using the PlatEMO platform.

METHODS

The Pure Multi-objective Optimal Control Problem (PMOCP) and the Hybrid Multi-objective Optimal Control Problem (HMOCP) are two different forms of optimal control problems that are solved using revolutionary metaheuristic optimisation algorithms. The utilization of the Hypervolume (HV) performance indicator allows for the comparison of various metaheuristic optimization algorithms in their efficacy for solving the PMOCP and HMOCP.

RESULTS

Results indicate that the MOPSO algorithm excels in solving the HMOCP, with M-MOPSO outperforming for PMOCP in HV analysis.

CONCLUSION

Despite not directly addressing immediate clinical concerns, these findings indicates that the stability shifts at critical thresholds may impact treatment efficacy.

Intratumor heterogeneity: models of malignancy emergence and evolution

Cancer is a complex and heterogeneous disease characterized by the accumulation of genetic alterations that drive uncontrolled cell growth and proliferation. Evolutionary dynamics plays a crucial role in the emergence and development of tumors, shaping the heterogeneity and adaptability of cancer cells. From the perspective of evolutionary theory, tumors are complex ecosystems that evolve through a process of microevolution influenced by genetic mutations, epigenetic changes, tumor microenvironment factors, and therapy-induced changes. This dynamic nature of tumors poses significant challenges for effective cancer treatment, and understanding it is essential for developing effective and personalized therapies. By uncovering the mechanisms that determine tumor heterogeneity, researchers can identify key genetic and epigenetic changes that contribute to tumor progression and resistance to treatment. This knowledge enables the development of innovative strategies for targeting specific tumor clones, minimizing the risk of recurrence and improving patient outcomes. To investigate the evolutionary dynamics of cancer, researchers employ a wide range of experimental and computational approaches. Traditional experimental methods involve genomic profiling techniques such as next-generation sequencing and fluorescence in situ hybridization. These techniques enable the identification of somatic mutations, copy number alterations, and structural rearrangements within cancer genomes. Furthermore, single-cell sequencing methods have emerged as powerful tools for dissecting intratumoral heterogeneity and tracing clonal evolution. In parallel, computational models and algorithms have been developed to simulate and analyze cancer evolution. These models integrate data from multiple sources to predict tumor growth patterns, identify driver mutations, and infer evolutionary trajectories. In this paper, we set out to describe the current approaches to address this evolutionary complexity and theories of its occurrence.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

Parameter perturbations in a post-treatment chronic myeloid leukemia model capture the essence of pre-diagnosis A-bomb survivor mysteries

A model of post-diagnosis chronic myeloid leukemia (CML) dynamics across treatment cessations is applied here to pre-diagnosis scenarios of A-bomb survivors. The main result is that perturbing two parameters of a two-state simplification of this model captures the essence of two A-bomb survivor mysteries: (1) in those exposed to > 1 Sv in Hiroshima, four of six female onsets arose as a cluster in 1969-1974, well after 5-10-year latencies expected and observed in two of six female- and nine of ten male cases (about one background case was expected in this high-dose cohort); and (2) no Nagasaki adult cases exposed to > 0.2 Sv were observed though about nine were expected (~ 1.5 background +  ~ 7.5 radiation-induced). Overall, it is concluded that: (1) whole-body radiation co-creates malignant and benign BCR-ABL clones; (2) benign clones are more likely to act as anti-CML vaccines in females than in males; (3) the Hong Kong flu of 1968 (and H3N2 seasonal flu thereafter) exhausted anti-CML immunity, thereby releasing radiation-induced clones latent in high-dose Hiroshima females; and (4) benign cells of 1-2 are CD4⁺ as human T-cell leukemia-lymphoma virus-1 endemic to Nagasaki but not Hiroshima expands numbers of such cells. The next goal is to see if these conclusions can be substantiated using banked A-bomb survivor blood samples.

Angiogenic inhibition therapy, a sliding mode control adventure

BACKGROUND AND OBJECTIVE

A sliding mode based inhibitory agent injection law is derived using angiogenic inhibition model of cancer progression which describes the variation of tumor and supporting vasculature volumes in targeted molecular therapies.

METHODS

The closed loop injection laws are derived by applying sliding mode control method which is known as a robust control approach. It is beneficial especially when there are parametric uncertainties in the dynamical model of the plant. In this research plant is represented by angiogenic cancer progression model. Random uncertainties are introduced to the physiological rate constants and simulations are repeated several times to see the deviations in the states and inhibitory agent rates.

RESULTS

Smooth inhibitory agent injection laws are obtained from the developed approach. Several different control configurations reveal that, it is possible to decrease the setup time to 6.1 days. A few of those settings failed to generate a satisfactory result. It appeared also that the sliding surface parameters have a distinct effect on the closed loop performance. Appropriate choice of the sliding surface parameters allows one to have a robust closed loop treatment where the deviation from the nominal response is relatively lower.

DISCUSSION

The lowest setup time obtained in this research is 6.1 days. This appear shorter than other similar studies where the plant is represented by the same or similar models. In the cases where the setup time is relatively shorter, the inhibitory agent injection requirement is higher than the other cases. This result seems larger compared to similar studies however the inhibitory agent stays at high levels for a short duration. In addition, the existence of uncertainty may also lead to an increase in the inhibitory agent rate requirements. Nevertheless, the results of the study reveals that one can reduce the tumor volume in a finite time without the necessity of constant application of high dosage inhibitory agent.

Global dynamics of healthy and cancer cells competing in the hematopoietic system

Stem cells in the bone marrow differentiate to ultimately become mature, functioning blood cells through a tightly regulated process (hematopoiesis) including a stem cell niche interaction and feedback through the immune system. Mutations in a hematopoietic stem cell can create a cancer stem cell leading to a less controlled production of malfunctioning cells in the hematopoietic system. This was mathematically modelled by Andersen et al. (2017) including the dynamic variables: healthy and cancer stem cells and mature cells, dead cells and an immune system response. Here, we apply a quasi steady state approximation to this model to construct a two dimensional model with four algebraic equations denoted the simple cancitis model. The two dynamic variables are the clinically available quantities JAK2V617F allele burden and the number of white blood cells. The simple cancitis model represents the original model very well. Complete phase space analysis of the simple cancitis model is performed, including proving the existence and location of globally attracting steady states. Hence, parameter values from compartments of stem cells, mature cells and immune cells are directly linked to disease and treatment prognosis, showing the crucial importance of early intervention. The simple cancitis model allows for a complete analysis of the long term evolution of trajectories. In particular, the value of the self renewal of the hematopoietic stem cells divided by the self renewal of the cancer stem cells is found to be an important diagnostic marker and perturbing this parameter value at intervention allows the model to reproduce clinical data. Treatment at low cancer cell numbers allows returning to healthy blood production while the same intervention at a later disease stage can lead to eradication of healthy blood producing cells. Assuming the total number of white blood cells is constant in the early cancer phase while the allele burden increases, a one dimensional model is suggested and explicitly solved, including parameters from all original compartments. The solution explicitly shows that exogenous inflammation promotes blood cancer when cancer stem cells reproduce more efficiently than hematopoietic stem cells.

A stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumours

Spatial interactions between cancer and immune cells, as well as the recognition of tumour antigens by cells of the immune system, play a key role in the immune response against solid tumours. The existing mathematical models generally focus only on one of these key aspects. We present here a spatial stochastic individual-based model that explicitly captures antigen expression and recognition. In our model, each cancer cell is characterised by an antigen profile which can change over time due to either epimutations or mutations. The immune response against the cancer cells is initiated by the dendritic cells that recognise the tumour antigens and present them to the cytotoxic T cells. Consequently, T cells become activated against the tumour cells expressing such antigens. Moreover, the differences in movement between inactive and active immune cells are explicitly taken into account by the model. Computational simulations of our model clarify the conditions for the emergence of tumour clearance, dormancy or escape, and allow us to assess the impact of antigenic heterogeneity of cancer cells on the efficacy of immune action. Ultimately, our results highlight the complex interplay between spatial interactions and adaptive mechanisms that underpins the immune response against solid tumours, and suggest how this may be exploited to further develop cancer immunotherapies.

Bridging blood cancers and inflammation: The reduced Cancitis model

A novel mechanism-based model - the Cancitis model - describing the interaction of blood cancer and the inflammatory system is proposed, analyzed and validated. The immune response is divided into two components, one where the elimination rate of malignant stem cells is independent of the level of the blood cancer and one where the elimination rate depends on the level of the blood cancer. A dimensional analysis shows that the full 6-dimensional system of nonlinear ordinary differential equations may be reduced to a 2-dimensional system - the reduced Cancitis model - using Fenichel theory. The original 18 parameters appear in the reduced model in 8 groups of parameters. The reduced model is analyzed. Especially the steady states and their dependence on the exogenous inflammatory stimuli are analyzed. A semi-analytic investigation reveals the stability properties of the steady states. Finally, positivity of the system and the existence of an attracting trapping region in the positive octahedron guaranteeing global existence and uniqueness of solutions are proved. The possible topologies of the dynamical system are completely determined as having a Janus structure, where two qualitatively different topologies appear for different sets of parameters. To classify this Janus structure we propose a novel concept in blood cancer - a reproduction ratio R. It determines the topological structure depending on whether it is larger or smaller than a threshold value. Furthermore, it follows that inflammation, affected by the exogenous inflammatory stimulation, may determine the onset and development of blood cancers. The body may manage initial blood cancer as long as the self-renewal rate is not too high, but fails to manage it if an inflammation appears. Thus, inflammation may trigger and drive blood cancers. Finally, the mathematical analysis suggests novel treatment strategies and it is used to discuss the in silico effect of existing treatment, e.g. interferon-α or T-cell therapy, and the impact of malignant cells becoming resistant.