Bounded-noise-induced transitions in a tumor-immune system interplay

Semiconductor laser systems excited by multiplicative Ornstein-Uhlenbeck noise and additive sine-Wiener noise in relation to real and imaginary parts

In this paper we study a semiconductor laser system excited by multiplicative Ornstein-Uhlenbeck (OU) noise and additive sine-Wiener (SW) noise with a related real part and imaginary part. The two-dimensional Langevin equation (LE) with cross-correlating complex OU noise and complex SW noise is equivalent to the six-dimensional LE. Based on functional methods and spatial dimension extension methods, the six-dimensional Fokker-Planck equation (FPE) is achieved. Through Taylor expansion approximation and linear transformation, the dimensionality of FPE is reduced, and the exact expression of the probability distribution (SPD) in steady state is obtained. The impacts of noise parameters on laser intensity on SPD are further analyzed. Moreover, based on the reduced FPE, the information entropy of the system is obtained. This calculation is useful for us to understand further the influence of system parameters and noise correlation coefficient on the system. The main work lies in obtaining an alternative method for deriving a FPE corresponding to the two-dimensional stochastic semiconductor laser model, which is more applicable than methods in previous literature. An equivalent expression for cross-correlated OU noise and cross-correlated SW noise as well as a method for approximate solutions of a high-dimensional FPE are also provided.

A noble extended stochastic logistic model for cell proliferation with density-dependent parameters

Cell proliferation often experiences a density-dependent intrinsic proliferation rate (IPR) and negative feedback from growth-inhibiting molecules in culture media. The lack of flexible models with explanatory parameters fails to capture such a proliferation mechanism. We propose an extended logistic growth law with the density-dependent IPR and additional negative feedback. The extended parameters of the proposed model can be interpreted as density-dependent cell-cell cooperation and negative feedback on cell proliferation. Moreover, we incorporate further density regulation for flexibility in the model through environmental resistance on cells. The proposed growth law has similarities with the strong Allee model and harvesting phenomenon. We also develop the stochastic analog of the deterministic model by representing possible heterogeneity in growth-inhibiting molecules and environmental perturbation of the culture setup as correlated multiplicative and additive noises. The model provides a conditional maximum sustainable stable cell density (MSSCD) and a new fitness measure for proliferative cells. The proposed model shows superiority to the logistic law after fitting to real cell culture datasets. We illustrate both conditional MSSCD and the new cell fitness for a range of parameters. The cell density distributions reveal the chance of overproliferation, underproliferation, or decay for different parameter sets under the deterministic and stochastic setups.

Tumour neoantigen heterogeneity thresholds provide a time window for combination immunotherapy

Following the advent of cancer immunotherapy, increasing insight has been gained on the role of mutational load and neoantigens as key ingredients in T cell recognition of malignancies. However, not all highly mutational tumours react to immune therapies, and initial success is often followed by eventual relapse. Heterogeneity in the neoantigen landscape of a tumour might be key in the failure of immune surveillance. In this work, we present a mathematical framework to describe how neoantigen distributions shape the immune response. The model predicts the existence of an antigen diversity threshold level beyond which T cells fail at controlling heterogeneous tumours. Incorporating this diversity marker adds predictive value to antigen load for two cohorts of anti-CTLA-4 treated melanoma patients. Furthermore, our analytical approach indicates rapid increases in epitope heterogeneity in early malignancy growth following immune escape. We propose a combination therapy scheme that takes advantage of preexisting resistance to a targeted agent. The model indicates that the selective sweep for a resistant subclone reduces neoantigen heterogeneity, and we postulate the existence of a time window before tumour relapse where checkpoint blockade immunotherapy can become more effective.

Cancer-induced immunosuppression can enable effectiveness of immunotherapy through bistability generation: A mathematical and computational examination

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The presence of an immunological barrier in cancer- immune system interaction (CISI) is consistent with the bistability patterns in that system.

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In CISI models, bistability patterns are consistent with immunosuppressive effects dominating immunoproliferative effects.

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Bistability could be harnessed to devise effective combination immunotherapy approaches.

Cancer immunotherapies rely on how interactions between cancer and immune system cells are constituted. The more essential to the emergence of the dynamical behavior of cancer growth these interactions are, the more effectively they may be used as mechanisms for interventions. Mathematical modeling can help unearth such connections, and help explain how they shape the dynamics of cancer growth. Here, we explored whether there exist simple, consistent properties of cancer-immune system interaction (CISI) models that might be harnessed to devise effective immunotherapy approaches. We did this for a family of three related models of increasing complexity. To this end, we developed a base model of CISI, which captures some essential features of the more complex models built on it. We find that the base model and its derivates can plausibly reproduce biological behavior that is consistent with the notion of an immunological barrier. This behavior is also in accord with situations in which the suppressive effects exerted by cancer cells on immune cells dominate their proliferative effects. Under these circumstances, the model family may display a pattern of bistability, where two distinct, stable states (a cancer-free, and a full-grown cancer state) are possible. Increasing the effectiveness of immune-caused cancer cell killing may remove the basis for bistability, and abruptly tip the dynamics of the system into a cancer-free state. Additionally, in combination with the administration of immune effector cells, modifications in cancer cell killing may be harnessed for immunotherapy without the need for resolving the bistability. We use these ideas to test immunotherapeutic interventions in silico in a stochastic version of the base model. This bistability-reliant approach to cancer interventions might offer advantages over those that comprise gradual declines in cancer cell numbers.

A nonlinear mathematical model of cell-mediated immune response for tumor phenotypic heterogeneity

Human cancers display intra-tumor heterogeneity in many phenotypic features, such as expression of cell surface receptors, growth, and angiogenic, proliferative, and immunogenic factors, which represent obstacles to a successful immune response. In this paper, we propose a nonlinear mathematical model of cancer immunosurveillance that takes into account some of these features based on cell-mediated immune responses. The model describes phenomena that are seen in vivo, such as tumor dormancy, robustness, immunoselection over tumor heterogeneity (also called "cancer immunoediting") and strong sensitivity to initial conditions in the composition of tumor microenvironment. The results framework has as common element the tumor as an attractor for abnormal cells. Bifurcation analysis give us as tumor attractors fixed-points, limit cycles and chaotic attractors, the latter emerging from period-doubling cascade displaying Feigenbaum's universality. Finally, we simulated both elimination and escape tumor scenarios by means of a stochastic version of the model according to the Doob-Gillespie algorithm.

Mathematical analysis of a tumour-immune interaction model: A moving boundary problem

A spatio-temporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is developed. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour swells to a dormant level.

Impact of bounded noise on the formation and instability of spiral wave in a 2D Lattice of neurons

Spiral waves in the neocortex may provide a spatial framework to organize cortical oscillations, thus help signal communication. However, noise influences spiral wave. Many previous theoretical studies about noise mainly focus on unbounded Gaussian noise, which contradicts that a real physical quantity is always bounded. Furthermore, non-Gaussian noise is also important for dynamical behaviors of excitable media. Nevertheless, there are no results concerning the effect of bounded noise on spiral wave till now. Based on Hodgkin-Huxley neuron model subjected to bounded noise with the form of Asin[ωt + σW(t)], the influences of bounded noise on the formation and instability of spiral wave in a two-dimensional (2D) square lattice of neurons are investigated in detail by separately adjusting the intensity σ, amplitude A, and frequency f of bounded noise. It is found that the increased intensity σ can facilitate the formation of spiral wave while the increased amplitude A tends to destroy spiral wave. Furthermore, frequency of bounded noise has the effect of facilitation or inhibition on pattern synchronization. Interestingly, for the appropriate intensity, amplitude and frequency can separately induce resonance-like phenomenon.

A model for effects of adaptive immunity on tumor response to chemotherapy and chemoimmunotherapy

Complete clinical regressions of solid tumors in response to chemotherapy are difficult to explain by direct cytotoxicity alone, because of low growth fractions and obstacles to drug delivery. A plausible indirect mechanism that might reconcile this is the action of the immune system. A model for interaction between tumors and the adaptive immune system is presented here, and used to examine controllability of tumors through the interplay of cytotoxic, cytostatic and immunogenic effects of chemotherapy and the adaptive immune response. The model includes cytotoxic and helper T cells, T regulatory cells (Tregs), dendritic cells, memory cells, and several key cytokines. Nearly all parameter estimates are derived from experimental and clinical data. Individual tumors are characterized by two parameters: growth rate and antigenicity, and regions of tumor control are identified in this parameter space. The model predicts that inclusion of the immune response significantly expands the region of tumor control for both cytostatic and cytotoxic chemotherapies. Moreover, outside the control zone, tumor growth is delayed significantly. An optimal fractionation schedule is predicted, for a fixed cumulative dose. The model further predicts expanded regions of tumor control when several forms of immunotherapy (adoptive T cell transfer, Treg depletion, and dendritic cell vaccination) are combined with chemotherapy. Outcomes depend greatly on tumor characteristics, the schedule of administration, and the type of immunotherapy chosen, suggesting promising opportunities for personalized medicine. Overall, the model provides insight into the role of the adaptive immune system in chemotherapy, and how scheduling and immunotherapeutic interventions might improve efficacy.

Role of the interpretation of stochastic calculus in systems with cross-correlated Gaussian white noises

We derive the Fokker-Planck equation for multivariable Langevin equations with cross-correlated Gaussian white noises for an arbitrary interpretation of the stochastic differential equation. We formulate the conditions when the solution of the Fokker-Planck equation does not depend on which stochastic calculus is adopted. Further, we derive an equivalent multivariable Ito stochastic differential equation for each possible interpretation of the multivariable Langevin equation. To demonstrate the usefulness and significance of these general results, we consider the motion of Brownian particles. We study in detail the stability conditions for harmonic oscillators with two white noises, one of which is additive, random forcing, and the other, which accounts for fluctuations of either the damping or the spring coefficient, is multiplicative. We analyze the role of cross correlation in terms of the different noise interpretations and confirm the theoretical predictions via numerical simulations. We stress the interest of our results for numerical simulations of stochastic differential equations with an arbitrary interpretation of the stochastic integrals.

Mathematical models of immune-induced cancer dormancy and the emergence of immune evasion

Cancer dormancy, a state in which cancer cells persist in a host without significant growth, is a natural forestallment of progression to manifest disease and is thus of great clinical interest. Experimental work in mice suggests that in immune-induced dormancy, the longer a cancer remains dormant in a host, the more resistant the cancer cells become to cytotoxic T-cell-mediated killing. In this work, mathematical models are used to analyse the possible causative mechanisms of cancer escape from immune-induced dormancy. Using a data-driven approach, both decaying efficacy in immune predation and immune recruitment are analysed with results suggesting that decline in recruitment is a stronger determinant of escape than increased resistance to predation. Using a mechanistic approach, the existence of an immune-resistant cancer cell subpopulation is considered, and the effects on cancer dormancy and potential immunoediting mechanisms of cancer escape are analysed and discussed. The immunoediting mechanism assumes that the immune system selectively prunes the cancer of immune-sensitive cells, which is shown to cause an initially heterogeneous population to become a more homogeneous, and more resistant, population. The fact that this selection may result in the appearance of decreasing efficacy in T-cell cytotoxic effect with time in dormancy is also demonstrated. This work suggests that through actions that temporarily delay cancer growth through the targeted removal of immune-sensitive subpopulations, the immune response may actually progress the cancer to a more aggressive state.

Cellular polarization: interaction between extrinsic bounded noises and the wave-pinning mechanism

Cell polarization (cued or uncued) is a fundamental mechanism in cell biology. As an alternative to the classical Turing bifurcation, it has been proposed that the onset of cell polarity might arise by means of the well-known phenomenon of wave-pinning [Gamba et al., Proc. Natl. Acad. Sci. USA 102, 16927 (2005)]. A particularly simple and elegant deterministic model of cell polarization based on the wave-pinning mechanism has been proposed by Edelstein-Keshet and coworkers [Biophys. J. 94, 3684 (2008)]. This model consists of a small biomolecular network where an active membrane-bound factor interconverts into its inactive form that freely diffuses in the cell cytosol. However, biomolecular networks do communicate with other networks as well as with the external world. Thus, their dynamics must be considered as perturbed by extrinsic noises. These noises may have both a spatial and a temporal correlation, and in any case they must be bounded to preserve the biological meaningfulness of the perturbed parameters. Here we numerically show that the inclusion of external spatiotemporal bounded parametric perturbations in the above wave-pinning-based model of cellular polarization may sometimes destroy the polarized state. The polarization loss depends on both the extent of temporal and spatial correlations and on the kind of noise employed. For example, an increase of the spatial correlation of the noise induces an increase of the probability of cell polarization. However, if the noise is spatially homogeneous then the polarization is lost in the majority of cases. These phenomena are independent of the type of noise. Conversely, an increase of the temporal autocorrelation of the noise induces an effect that depends on the model of noise.

Tumor-immune dynamics regulated in the microenvironment inform the transient nature of immune-induced tumor dormancy

Cancer in a host induces responses that increase the ability of the microenvironment to sustain the growing mass, for example, angiogenesis, but cancer cells can have varying sensitivities to these sustainability signals. Here, we show that these sensitivities are significant determinants of ultimate tumor fate, especially in response to treatments and immune interactions. We present a mathematical model of cancer-immune interactions that modifies generalized logistic growth with both immune-predation and immune-recruitment. The role of a growing environmental carrying capacity is discussed as a possible regulatory mechanism for tumor growth, and this regulation is shown to modify cancer-immune interactions and the possibility of achieving immune-induced tumor dormancy. This mathematical model qualitatively matches experimental observations of immune-induced tumor dormancy as it predicts dormancy as a transient period of growth that necessarily ends in either tumor elimination or tumor escape. As dormant tumors may exist asymptomatically and may be easier to treat with conventional therapy, an understanding of the mechanisms behind tumor dormancy may lead to new treatments aimed at prolonging the dormant state or converting an aggressive cancer to the dormant state.

The interplay of intrinsic and extrinsic bounded noises in biomolecular networks

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a biomolecular network. The influence of intrinsic and extrinsic noises on biomolecular networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: (i) the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, (ii) a model of enzymatic futile cycle and (iii) a genetic toggle switch. In (ii) and (iii) we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possible functional role of bounded noises.

A review of mathematical models of cancer-immune interactions in the context of tumor dormancy

The role of the immune system in tumor dormancy is now well established. In an immune-induced dormant state, potentially lethal cancer cells persist in a state where growth is restricted, to little or no increase, by the host's immune response. To describe this state in the context of cancer progression and immune response, basic temporal (spatially homogeneous) quantitative predator-prey constructs are discussed, along with some current and proposed augmentations that incorporate potentially significant biological phenomena such as the cancer cell transition to a quiescent state or the time delay in T-cell activation. Advances in cancer-immune modeling that describe complex interactions underlying the ability of the immune system to both promote and inhibit tumor growth are emphasized. Finally, the review concludes by discussing future mathematical challenges and their biological significance.

Evasion of tumours from the control of the immune system: consequences of brief encounters

BACKGROUND

In this work a mathematical model describing the growth of a solid tumour in the presence of an immune system response is presented. Specifically, attention is focused on the interactions between cytotoxic T-lymphocytes (CTLs) and tumour cells in a small, avascular multicellular tumour. At this stage of the disease the CTLs and the tumour cells are considered to be in a state of dynamic equilibrium or cancer dormancy. The precise biochemical and cellular mechanisms by which CTLs can control a cancer and keep it in a dormant state are still not completely understood from a biological and immunological point of view. The mathematical model focuses on the spatio-temporal dynamics of tumour cells, immune cells, chemokines and "chemorepellents" in an immunogenic tumour. The CTLs and tumour cells are assumed to migrate and interact with each other in such a way that lymphocyte-tumour cell complexes are formed. These complexes result in either the death of the tumour cells (the normal situation) or the inactivation of the lymphocytes and consequently the survival of the tumour cells. In the latter case, we assume that each tumour cell that survives its "brief encounter" with the CTLs undergoes certain beneficial phenotypic changes.

RESULTS

We explore the dynamics of the model under these assumptions and show that the process of immuno-evasion can arise as a consequence of these encounters. We show that the proposed mechanism not only shape the dynamics of the total number of tumor cells and of CTLs, but also the dynamics of their spatial distribution. We also briefly discuss the evolutionary features of our model, by framing them in the recent quasi-Lamarckian theories.

CONCLUSIONS

Our findings might have some interesting implication of interest for clinical practice. Indeed, immuno-editing process can be seen as an "involuntary" antagonistic process acting against immunotherapies, which aim at maintaining a tumor in a dormant state, or at suppressing it.

Spatiotemporal bounded noises and transitions induced by them in solutions of the real Ginzburg-Landau model

In this work, we introduce two spatiotemporal colored bounded noises, based on the zero-dimensional Cai-Lin and Tsallis-Borland noises. Then we study and characterize the dependence of the defined stochastic processes on both a temporal correlation parameter τ and a spatial coupling parameter λ. In particular, we found that varying λ may induce a transition of the distribution of the noise from bimodality to unimodality. With the aim of investigating the role played by bounded noises in nonlinear dynamical systems, we analyze the behavior of the real Ginzburg-Landau time-varying model additively perturbed by such noises. The observed phase transition phenomenology is quite different from that observed when the perturbations are unbounded. In particular, we observed an inverse order-to-disorder transition and a reentrant transition, with dependence on the specific type of bounded noise.

Fine-tuning anti-tumor immunotherapies via stochastic simulations

BACKGROUND

Anti-tumor therapies aim at reducing to zero the number of tumor cells in a host within their end or, at least, aim at leaving the patient with a sufficiently small number of tumor cells so that the residual tumor can be eradicated by the immune system. Besides severe side-effects, a key problem of such therapies is finding a suitable scheduling of their administration to the patients. In this paper we study the effect of varying therapy-related parameters on the final outcome of the interplay between a tumor and the immune system.

RESULTS

This work generalizes our previous study on hybrid models of such an interplay where interleukins are modeled as a continuous variable, and the tumor and the immune system as a discrete-state continuous-time stochastic process. The hybrid model we use is obtained by modifying the corresponding deterministic model, originally proposed by Kirschner and Panetta. We consider Adoptive Cellular Immunotherapies and Interleukin-based therapies, as well as their combination. By asymptotic and transitory analyses of the corresponding deterministic model we find conditions guaranteeing tumor eradication, and we tune the parameters of the hybrid model accordingly. We then perform stochastic simulations of the hybrid model under various therapeutic settings: constant, piece-wise constant or impulsive infusion and daily or weekly delivery schedules.

CONCLUSIONS

Results suggest that, in some cases, the delivery schedule may deeply impact on the therapy-induced tumor eradication time. Indeed, our model suggests that Interleukin-based therapies may not be effective for every patient, and that the piece-wise constant is the most effective delivery to stimulate the immune-response. For Adoptive Cellular Immunotherapies a metronomic delivery seems more effective, as it happens for other anti-angiogenesis therapies and chemotherapies, and the impulsive delivery seems more effective than the piece-wise constant. The expected synergistic effects have been observed when the therapies are combined.

Simple biophysical model of tumor evasion from immune system control

The competitive nonlinear interplay between a tumor and the host's immune system is not only very complex but is also time-changing. A fundamental aspect of this issue is the ability of the tumor to slowly carry out processes that gradually allow it to become less harmed and less susceptible to recognition by the immune system effectors. Here we propose a simple epigenetic escape mechanism that adaptively depends on the interactions per time unit between cells of the two systems. From a biological point of view, our model is based on the concept that a tumor cell that has survived an encounter with a cytotoxic T-lymphocyte (CTL) has an information gain that it transmits to the other cells of the neoplasm. The consequence of this information increase is a decrease in both the probabilities of being killed and of being recognized by a CTL. We show that the mathematical model of this mechanism is formally equal to an evolutionary imitation game dynamics. Numerical simulations of transitory phases complement the theoretical analysis. Implications of the interplay between the above mechanisms and the delivery of immunotherapies are also illustrated.

Noise-assisted interactions of tumor and immune cells

We consider a three-state model comprising tumor cells, effector cells, and tumor-detecting cells under the influence of noises. It is demonstrated that inevitable stochastic forces existing in all three cell species are able to suppress tumor cell growth completely. Whereas the deterministic model does not reveal a stable tumor-free state, the auto-correlated noise combined with cross-correlation functions can either lead to tumor-dormant states, tumor progression, as well as to an elimination of tumor cells. The auto-correlation function exhibits a finite correlation time τ, while the cross-correlation functions shows a white-noise behavior. The evolution of each of the three kinds of cells leads to a multiplicative noise coupling. The model is investigated by means of a multivariate Fokker-Planck equation for small τ. The different behavior of the system is, above all, determined by the variation of the correlation time and the strength of the cross-correlation between tumor and tumor-detecting cells. The theoretical model is based on a biological background discussed in detail, and the results are tested using realistic parameters from experimental observations.

Resistance to antitumor chemotherapy due to bounded-noise-induced transitions

Tumor angiogenesis is a landmark of solid tumor development, but it is also directly relevant to chemotherapy. Indeed, the density and quality of neovessels may influence the effectiveness of therapies based on blood-born agents. In this paper, first we define a deterministic model of antiproliferative chemotherapy in which the drug efficacy is a unimodal function of vessel density, and then we show that under constant continuous infusion therapy the tumor-vessel system may be multistable. However, the actual drug concentration profiles are affected by bounded even if possibly large fluctuations. Through numerical simulations, we show that the tumor volume may undergo transitions to the higher equilibrium value induced by the bounded noise. In case of periodically delivered boli-based chemotherapy, we model the fluctuations due to time variability of both the drug clearance rate and the distribution volume, as well as those due to irregularities in drug delivery. We observed noise-induced transitions also in case of periodic delivering. By applying a time dense scheduling with constant average delivered drug (metronomic scheduling), we observed an easier suppression of the transitions. Finally, we propose to interpret the above phenomena as an unexpected non-genetic kind of resistance to chemotherapy.