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

Modeling correlated uncertainties in stochastic compartmental models

We consider compartmental models of communicable disease with uncertain contact rates. Stochastic fluctuations are often added to the contact rate to account for uncertainties. White noise, which is the typical choice for the fluctuations, leads to significant underestimation of the disease severity. Here, starting from reasonable assumptions on the social behavior of individuals, we model the contacts as a Markov process which takes into account the temporal correlations present in human social activities. Consequently, we show that the mean-reverting Ornstein-Uhlenbeck (OU) process is the correct model for the stochastic contact rate. We demonstrate the implication of our model on two examples: a Susceptibles-Infected-Susceptibles (SIS) model and a Susceptibles-Exposed-Infected-Removed (SEIR) model of the COVID-19 pandemic and compare the results to the available US data from the Johns Hopkins University database. In particular, we observe that both compartmental models with white noise uncertainties undergo transitions that lead to the systematic underestimation of the spread of the disease. In contrast, modeling the contact rate with the OU process significantly hinders such unrealistic noise-induced transitions. For the SIS model, we derive its stationary probability density analytically, for both white and correlated noise. This allows us to give a complete description of the model's asymptotic behavior as a function of its bifurcation parameters, i.e., the basic reproduction number, noise intensity, and correlation time. For the SEIR model, where the probability density is not available in closed form, we study the transitions using Monte Carlo simulations. Our modeling approach can be used to quantify uncertain parameters in a broad range of biological systems.

Angiogenesis and vessel co-option in a mathematical model of diffusive tumor growth: The role of chemotaxis

This work considers the propagation of a tumor from the stage of a small avascular sphere in a host tissue and the progressive onset of a tumor neovasculature stimulated by a pro-angiogenic factor secreted by hypoxic cells. The way new vessels are formed involves cell sprouting from pre-existing vessels and following a trail via a chemotactic mechanism (CM). Namely, it is first proposed a detailed general family of models of the CM, based on a statistical mechanics approach. The key hypothesis is that the CM is composed by two components: i) the well-known bias induced by the angiogenic factor gradient; ii) the presence of stochastic changes of the velocity direction, thus giving rise to a diffusive component. Then, some further assumptions and simplifications are applied in order to derive a specific model to be used in the simulations. The tumor progression is favored by its acidic aggression towards the healthy cells. The model includes the evolution of many biological and chemical species. Numerical simulations show the onset of a traveling wave eventually replacing the host tissue with a fully vascularized tumor. The results of simulations agree with experimental measures of the vasculature density in tumors, even in the case of particularly hypoxic tumors.

Modeling the effect of immunotherapies on human castration-resistant prostate cancer

In this paper we analyze the potential effect of immunotherapies on castration-resistant form of human Prostate Cancer (PCa). In particular, we examine the potential effect of the dendritic vaccine sipuleucel-T, the only currently available immunotherapy option for advanced PCa, and of ipilimumab, a drug targeting the Cytotoxic T-Lymphocyte Antigen 4 (CTLA4), exposed on the CTLs membrane, currently under Phase II clinical trial. The model, building on the one by Rutter and Kuang, includes different types of immune cells and interactions and is parameterized on available data. Our results show that the vaccine has only a very limited effect on PCa, while repeated treatments with ipilimumab appear potentially capable of controlling and even eradicating an androgen-independent prostate cancer. From a mathematical analysis of a simplified model, it seems likely that, under continuous administration of ipilimumab, the system lies in a bistable situation where both the no-tumor equilibrium and the high-tumor equilibrium are attractive. The schedule of periodic treatments could then determine the outcome, and mathematical models could help determine an optimal schedule.

Identifying and characterising the impact of excitability in a mathematical model of tumour-immune interactions

We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting - elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.

Modeling tumor growth inhibition and toxicity outcome after administration of anticancer agents in xenograft mice: A Dynamic Energy Budget (DEB) approach

Host features, such as cell proliferation rates, caloric intake, metabolism and energetic conditions, significantly influence tumor growth; at the same time, tumor growth may have a dramatic impact on the host conditions. For example, in clinics, at certain stages of the tumor growth, cachexia (body weight reduction) may become so relevant to be considered as responsible for around 20% of cancer deaths. Unfortunately, anticancer therapies may also contribute to the development of cachexia due to reduced food intake (anorexia), commonly observed during the treatment periods. For this reason, cachexia is considered one of the major toxicity findings to be evaluated also in preclinical studies. However, although various pharmacokinetic-pharmacodynamic (PK-PD) tumor growth inhibition (TGI) models are currently available, the mathematical modeling of cachexia onset and TGI after an anticancer administration in preclinical experiments is still an open issue. To cope with this, a new PK-PD model, based on a set of tumor-host interaction rules taken from Dynamic Energy Budget (DEB) theory and a set of drug tumor inhibition equations taken from the well-known Simeoni TGI model, was developed. The model is able to describe the body weight reduction, splitting the cachexia directly induced by tumor and that caused by the drug treatment under study. It was tested in typical preclinical studies, essentially designed for efficacy evaluation and routinely performed as a part of the industrial drug development plans. For the first time, both the dynamics of tumor and host growth could be predicted in xenograft mice untreated or treated with different anticancer agents and following different schedules. The model code is freely available for downloading at http://repository.ddmore.eu (model number DDMODEL00000274).

Influence of parameter perturbations on the reachability of therapeutic target in systems with switchings

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.

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.

Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif

Gene switching dynamics is a major source of randomness in genetic networks, also in the case of large concentrations of the transcription factors. In this work, we consider a common network motif - the positive feedback of a transcription factor on its own synthesis - and assess its response to extrinsic noises perturbing gene deactivation in a variety of settings where the network might operate. These settings are representative of distinct cellular types, abundance of transcription factors and ratio between gene switching and protein synthesis rates. By investigating noise-induced transitions among the different network operative states, our results suggest that gene switching rates are key parameters to shape network response to external perturbations, and that such response depends on the particular biological setting, i.e. the characteristic time scales and protein abundance. These results might have implications on our understanding of irreversible transitions for noise-related phenomena such as cellular differentiation. In addition these evidences suggest to adopt the appropriate mathematical model of the network in order to analyze the system consistently to the reference biological setting.

The pharmacodynamics of the p53-Mdm2 targeting drug Nutlin: the role of gene-switching noise

In this work we investigate, by means of a computational stochastic model, how tumor cells with wild-type p53 gene respond to the drug Nutlin, an agent that interferes with the Mdm2-mediated p53 regulation. In particular, we show how the stochastic gene-switching controlled by p53 can explain experimental dose-response curves, i.e., the observed inter-cell variability of the cell viability under Nutlin action. The proposed model describes in some detail the regulation network of p53, including the negative feedback loop mediated by Mdm2 and the positive loop mediated by PTEN, as well as the reversible inhibition of Mdm2 caused by Nutlin binding. The fate of the individual cell is assumed to be decided by the rising of nuclear-phosphorylated p53 over a certain threshold. We also performed in silico experiments to evaluate the dose-response curve after a single drug dose delivered in mice, or after its fractionated administration. Our results suggest that dose-splitting may be ineffective at low doses and effective at high doses. This complex behavior can be due to the interplay among the existence of a threshold on the p53 level for its cell activity, the nonlinearity of the relationship between the bolus dose and the peak of active p53, and the relatively fast elimination of the drug.

P53 is an antitumor gene regulating vital cellular functions such as repair of DNA damage, cellular suicide, and cell proliferation: in many tumors p53 is lowly expressed and/or mutated. Drugs targeting the biomolecular network of p53 are becoming important. The network includes the key proteins Mdm2 and PTEN, whose production is regulated by p53, and which, in turn, enact positive and negative feedbacks on p53. Drug Nutlin, inhibiting the p53 inhibitor Mdm2, might be important for tumors where p53 is underproduced but unmutated. We investigate the cellular mechanism of action of Nutlin. The basic concept of our mathematical model is that the experimentally observed cell-to-cell variability of Nutlin efficacy is caused by the randomness of gene activation/deactivation of Mdmd2 and PTEN. Indeed, the abundance/scarceness of p53 regulates the probability that the relative genes are active or inactive. The model reproduced the experimental cell-specific response to different doses of Nutlin (dose-response curves) in some types of tumor cells. Much clinical research focus on 'metronomic' drug delivery regimens, where instead of giving large doses with long intervals, smaller doses are frequently delivered. In our simulations, dose-splitting of Nutlin produced a response generally worse than the response to a single dose.

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.

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.

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.

Non-Gaussian stochastic dynamics of spins and oscillators: a continuous-time random walk approach

We consider separately a spin and an oscillator that are coupled to their environment. After a finite interval of random length, the state of the environment changes, and each change causes a random change in the resonance frequency of the spin or vibrational frequency of the oscillator. Mathematically, the evolution of these frequencies is described by a continuous-time random walk. Physically, the stochastic dynamics can be understood as non-Gaussian because the frequency of the system and state of the environment change on comparable time scales. These dynamics are also nonstationary, and so might apply to a nonequilibrium environment. The resonance and vibrational spectra of the spin and oscillator, as well as the ensemble-averaged displacement of the oscillator, are investigated in detail. We observe some distinct non-Gaussian features of the dynamics, such as the narrow, leptokurtic shape of the resonance spectrum of the spin and beating of the average oscillator displacement. The convergence to Gaussian dynamics as changes in the environment occur with increasing frequency is also considered. Among other results, we observe narrowing of the resonance and vibrational lines in the Gaussian limit due to a weakening of the system-environment interaction.