Fine-tuning anti-tumor immunotherapies via stochastic simulations
Abstract
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.
Keywords
Hybrid Model Therapy Session Stochastic Simulation Algorithm Propensity Function Adoptive Cellular ImmunotherapyList of abbreviations used
- ACI
Adoptive Cellular Immunotherapy. IL or IL-2: Interleukin-2. ODE: Ordinary Differential Equation. T-IS: Tumor-Immune system (interplay). KP: Kirschner-Panetta (model). SSA: Stochastic Simulation Algorithm.
Introduction
A key problem of anti-tumor therapies is finding a suitable scheduling of their administration to the patients. Of course a major problem in medical oncology is avoiding severe therapy-related side effects which, unfortunately, may cause the death of the patient. However, also in the ideal case of no side-effects, a therapy aims at reducing to zero the number of tumor cells in the host, within its end. Indeed, also if a single tumor cell remains the patient has a tumor. Actually, the requirement might theoretically be milder by accepting to leave the patient with a sufficiently small number of tumor cells so that the residual tumor can be eradicated by the immune system. In any case, both the duration and the scheduling of a therapy becomes of great relevance, as experimentally shown and theoretically studied [1]. In this paper we shall focus on a computational study of some kinds of immunotherapies, whose underlying key idea is to modify the natural interplay between tumor cells and immune system, by boosting the latter.
Tumor cells are characterized by a vast number of genetic and epigenetic events eventually leading to the appearance of specific tumor antigens, called neo-antigens. Such antigens trigger anti-tumor actions by the immune system [2], thus resulting in the tumor-immune system interaction taking place. These observations provided a theoretical basis to the hypothesis of immune surveillance, i.e. the immune system may act to control and, in some case, to eliminate tumors [3]. Only recently studies in molecular oncology and epidemiology accumulated evidences of this [4]. The competitive interaction between tumor cells and the immune system is extremely complex. As such, a neoplasm may very often escape from immune control. A dynamic equilibrium may also be established with the tumor surviving in a microscopic undetectable "dormant" steady state [5]. If this is the case, on the one hand a dormant tumor may induce metastases, on the other hand over a long period of time, a significant fraction of the average host life span, the neoplasm may develop strategies to circumvent the action of the immune system, thus restarting to grow [2, 4, 6, 7]. We stress that this evolutionary adaptation, termed "immunoediting" [4], may negatively impact on the effectiveness of immunotherapies, as shown in [8].
Regarding immunotherapies, although the basic idea of immunotherapy is simple and promising [9], the clinical results are controversial since a huge inter-subjects variability is observed [10, 11, 12]. Immunotherapies are divided into two broad classes: passive and active therapies [13]. Among the passive ones, the Adoptive Cellular Immunotherapy (ACI) consisting in injecting cultured activated immune effectors in the diseased host [14, 15] is probably the most important. Active immunotherapies aim at stimulating the immune response by expanding, for instance, the proliferation of cytotoxic T cells. Among these, a prominent role is played by Interleukin-based therapies [16, 17].
Regarding the mathematical modeling of tumorimmune system interactions and and related therapies, many papers have appeared using various approaches. For instance, ordinary differential equations (ODEs) are used in [5, 8, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], the theory of kinetic active particles is used by Bellomo and Forni in [29, 30] and hybrid agent-based models have been introduced by Motta and Lollini [31, 32]. In [14] Kirschner and Panetta proposed a largely influential ODE-based model of Tumor-Immune system (T-IS) interplay, whose variables are tumor cells, effector cells and the concentration of interleukins-2 (IL-2). This model is able to explain various kinds of experimentally observed tumor size oscillations [33, 34, 35, 36, 37, 38] as well as both macroscopic and microscopic constant equilibria. Although a vast array of behaviors is mimicked by the solutions of the Kirschner-Panetta (KP) model, the tumor-free equilibrium is unstable for all biologically meaningful values of the parameters. However, in [39] we have shown that resetting the model in a hybrid setting where the interleukins are modeled with a continuous variable and the tumor and the immune system are modeled with discrete-state continuous-time stochastic process, the eradication via immune surveillance can be correctly reproduced. Since eradication is a fundamental topic in the study of immunotherapies, here we extend our hybrid version of the KP model to investigate the effects of both interleukin-based therapies and ACIs. Although our hybrid version of KP model is highly idealized, we think that it can provide useful information on the design of the above mentioned therapies.
Methods
In the next sections we recall the KP model [14], its hybrid definition [39] and we extend the hybrid model with general immunotherapies.
The deterministic Kirschner-Panetta model
where T_{ * } (t), E_{*} (t) and I (t) denote, respectively, the densities of tumor cells, effectors of the immune system and interleukins. The tumor induces the recruitment of the effectors at a linear rate cT thus c may be seen as a measure of the immunogenicity of the tumor. In other words, according to [14]c is "a measure of how different the tumor is from self". The proliferation of effectors is stimulated by the interleukins. The average lifespan of effectors is ${\mu}_{E}^{-1}$ and the average degradation time for interleukin is ${\mu}_{I}^{-1}$. The source of interleukin is modeled as linearly depending on effectors, and it also depends on the tumor burden. Finally, continuous infusion immunotherapy may be delivered when effectors and interleukins are injected at constant rates σ_{ E }, σ_{ I } ≥ 0.
In the case of no therapy, i.e. σ_{ E }, = σ_{ I } = 0, the main results obtained in [14] are that (i) the tumor-free equilibrium point (0, 0, 0) is always unstable, (ii) it exists positive c_{ m } ≪ 1 such that for c ∈ (0, c_{ m }) there is only one is locally stable equilibrium, whose size is very large due to the low value of c, (iii) it exists a c_{ M } > 0 such that if c ∈ (c_{ m }, c_{ M } ) there is a unique periodic solution whose period and amplitude decrease if c increases and, finally, (iv) when c >c_{ M } there is a unique globally stable equilibrium, whose size is a decreasing function of c. Thus, we note that this model explicitly precludes the possibility of tumor suppression in absence of immunotherapies. If idealized infinitely long constant continuous infusion therapies are considered the behavior of the system is complex, but in all the possible meaningful combinations of the parameters it is possible to find regions where globally stable limit cycles exist, as well as regions where there is cancer suppression. Moreover, there is a threshold value such that, for higher values of σ_{ I }, there is an unbounded growth of effectors, leading to severe side-effects.
A hybrid model with constant therapies
As discussed in [39], the low-level oscillations predicted by model (1) make stochastic effects on the cell populations worth investigating. Unfortunately, it is possible to see that a purely stochastic model with discrete populations becomes computationally too hard to analyze, being the number of IL-2 huge. In this case a hybrid approach, despite being more costly than the deterministic counterpart, still permits a feasible analysis.
Note that the modified deterministic model (2) is obtained by the original Kirschener-Panetta deterministic model (1) by means of a nonsingular linear transformation. As it is very well-known linear transformations of the state variables do not change the topological properties of the solutions, and as a consequence these transformations do not change all the stability properties of equilibria [40, 41].
Birth and death reactions for the hybrid model
reaction | Propensity | reaction | propensity |
---|---|---|---|
R_{1} : T ↦ T + 1 | a_{1} = r_{2}T | R_{2} : T ↦ T - 1 | ${a}_{2}=\frac{{r}_{2}b}{V}{T}^{2}$ |
R_{3} : T ↦ T - 1 | ${a}_{3}=\frac{{p}_{T}TE}{{g}_{T}V+T}$ | R_{4} : E ↦ E + 1 | ${a}_{4}\left(t\right)=\frac{{p}_{E}EI\left(t\right)}{{g}_{E}+I\left(t\right)}$ |
R_{5} : E ↦ E - 1 | a_{5} = μ_{ E }E | R_{6} : E ↦ E + 1 | a_{6} = cT |
R_{7} : E ↦ E + 1 | a_{7} = Vσ_{ E } |
with χ a random number with distribution Exp(1). The solution of equation (3) has no analytical form, thus requiring iterative methods to find its solution.
We remark that the same equation in the case of only time-independent propensity functions yields the well known SSA strategy to generate exponential jumps. As far as model analysis is concerned, differently from the deterministic model (1), the stochastic simulations performed in [39] show that, at least in some cases, suppression of the neoplasm might be reached, thanks to the conjunction of the intrinsic tendency of the Tumor-Immune System to oscillate with the stochastic dynamics.
A hybrid model with general therapies
It is important to notice that in the realistic case of finite duration therapies the deterministic system always predicts tumor re-growth, being the tumor-free equilibrium (0, 0, 0) unstable. Practically, if at the end of the therapy the solution is very close to the tumor-free equilibrium, immediately after the end the tumor restarts growing. We remark that in the oncological context it is important the state in which the tumor is at the end of a therapy, e.g. the tumor shrinkage up to an undetectable size, but it is far more important what is observed during the follow-up visits, i.e. that the tumor has not re-grown. In this framework the deterministic model is unable to reproduce the reality and it structurally gives negative answers on the effectiveness of the therapy. These reasons, combined with the possible low-level oscillations of model (1) makes again stochastic effects worth investigating. Along the line of [39], this shall permit to have an estimation of the average value of the random therapy-induced eradication time t_{ er } better than the one obtainable in the deterministic framework, that is better than min{t | T (t) < 1} which would be actually very rough. As in [39], switching from the deterministic to the hybrid version of the model is furthermore justified by the fact that, at the best of our knowledge, there is no proof that the average of the state variables of a nonlinear birth and death stochastic processes follows the dynamics of the corresponding ODE system, when variables assume low values. In fact, the ODE system would be a good approximation of the purely stochastic model if the state variables were "sufficiently large". In any case, by using the above deterministic approximation of t_{ er } its standard deviation could never be computed.
where r is a random number U[0, 1] and a_{ i }(t + τ ) = a_{ i }(x) if $i\in \mathcal{S}$. It is important to notice that equations (5) and (6) would reduce to the standard equations used in the SSA if the system was entirely stochastic.
The Hybrid Simulation Algorithm (Algorithm 1)
Require: (T_{0}, E_{0}, I_{0}), t_{0}, t_{ stop }. |
---|
1: set the initial state to (T_{0}, E_{0}, I(t_{0})) and the initial time t to t_{0}; |
2: while t < t_{ stop } do |
3: let x be the current state, for $j\in \mathcal{S}$ evaluate a_{ j }(x), define ${a}_{0}^{\mathcal{S}}={\sum}_{j\in \mathcal{S}}{a}_{j}\left(\mathbf{x}\right)$; |
4: let χ be a random number with distribution Exp(1), solve the transcendental equation |
${A}_{4}\left(\tau \right)+{a}_{0}^{S}\tau +{A}_{7}\left(\tau \right)=\chi $ |
and then define ${a}_{0}={a}_{0}^{S}+{a}_{4}\left(t+\tau \right)+{a}_{7}\left(t+\tau \right)$; |
5: let r be a random number U[0, 1], for the next event to fire find j by solving |
$\sum _{i=1}^{j-1}{a}_{i}\left(t+\tau \right)<r\cdot {a}_{0}\le \sum _{i=1}^{j}{a}_{i}\left(t+\tau \right)$ |
where if $i\in \mathcal{S}$ then a_{ i }(t + τ ) = a_{ i }(x); |
6: update (T, E, I(t)) to (T + ν_{T,j}, E + ν_{E,j}, I(t + τ )) and change clock to t + τ ; |
7: end while |
Values of the parameters
Values of the parameters
Par. | Value | Unit | Description |
---|---|---|---|
r | 0.18 | days ^{-1} | baseline growth rate of the tumor |
b | 10^{-9} | ml ^{-1} | carrying capacity of the tumor |
a | 1 | ml/days | baseline strength of the killing rate by immune effectors |
c | 10^{-4} | days ^{-1} | tumor antigenicity |
V | 3.2 | Ml | blood and bone marrow volumes for leukemia |
g _{ T } | 10^{5} | ml ^{-1} | 50% reduction factor of the killing rate by immune effectors |
g _{ E } | 2 · 10^{7} | pg/l | 50% reduction factor of IL-stimulated growth rate of effectors |
g _{ I } | 10^{3} | ml ^{-1} | 50% reduction factor of production rate of interleukins |
p _{ E } | 0.1245 | days ^{-1} | baseline strength of the IL-stimulated growth rate of effectors |
p _{ I } | 5 | pg/days | baseline strength of production rate of interleukins |
μ _{ E } | 0.03 | days ^{-1} | inverse of average lifespan of effectors |
μ _{ I } | 10 | days ^{-1} | loss/degradation rate of IL_{2} |
Interleukin-based immunotherapies
where θ_{ i } ∈ [t_{ s }, t_{ e }] for i = 0, . . . , k. In the following, with a slight abuse of terminology, we refer to each θ_{ i } ∈ Θ as a therapy session.
Piece-wise constant therapy
if θ_{max+1}≤ t_{ n } + φ < θ_{max+1}+ A and 0 otherwise. This holds since in the uppermost case ${\int}_{{\theta}_{max}+A}^{{t}_{n}+\phi}\eta \left(x\right)dx={\int}_{{\theta}_{max+1}}^{{t}_{n}+\phi}\eta \left(x\right)dx$. We remark that all these combinations of cases are necessary because of all the possible combinations of the parameters t and t_{ n } with the set Θ.
Impulsive therapy
for i = 0, . . . , k.
Adoptive cellular immunotherapies
In the next paragraphs we discuss how to specialize model (4) with either a piece-wise constant or an impulsive ACI. As in the previous section we consider a set of k therapy sessions Θ = {θ_{ i } | i = 0, . . . , k}.
Piece-wise constant therapy
Impulsive therapy
The Hybrid Simulation Algorithm (Algorithm 2)
Require: (T_{0}, E_{0}, I_{0}), t_{0}, t_{ stop }. |
---|
1: initialize the simulation as for Algorithm 1; |
2: while t < t_{ stop } do |
3: pick a value for τ as in Algorithm 1; |
4: get θ_{ next } = min{θ_{ i } ∈ Θ | θ_{ i } >t} |
5: if τ < θ_{ next } then |
6: fire a reaction as in Algorithm 1; |
7: else |
8: update (T, E, I(t)) to (T, E + w_{ next }, I(t + θ_{ next })) and change clock to t + θ_{ next }; |
9: end if |
10: end while |
Combining IL-2 therapies and ACI
Combining therapies requires combining the results of the previous sections. To shorten the presentation we briefly discuss how to perform the simulations of the hybrid model with combined immunotherapies.
Whenever an impulsive ACI is considered, independently of the IL-2 therapy, the model can be simulated by Algorithm 2. In the other cases the model can be simulated by Algorithm 1.
Results
In the next sections we perform both asymptotic and transitory analysis of the solutions of deterministic system (4). We show the existence of deterministic conditions that guarantee the eradication of the tumor, and that will be used to tune the parameters of our hybrid model when performing its simulations, which are discussed in the forthcoming section.
Deterministic asymptotic analysis
With the aid of elementary dynamical systems theory [41] and by using some mathematical properties summarized in the additional file 1, here we briefly investigate how the therapies may influence the asymptotic behavior of the solutions of the deterministic system (4). We shall focus on the mathematically idealized case of infinite length of the therapy and on the deterministic conditions guaranteeing the eradication of the tumor.
IL-2 immunotherapy
implies that (T(t), E(t)) → (0, +∞). In other words, the deterministic model predicts that eradication is possible only at the price of killing the patient for the excess of stimulation of the immune system, that is E(t) → +∞. We remark that the above inference does not consider finite length therapies. In fact, if at the end of the therapy the tumor has been eradicated, the number of effectors and the interleukin density will both decay. Inequality (17) is the condition which we obtain by the deterministic model. We now refine such a condition to account for the three therapies that we mentioned so far: (i) constant, (ii) piece-wise constant and (iii) impulsive. It holds that:
and γ_{1} < 1.
ACI
As for the case of IL-2 mono-therapy we now focus on the therapies considered so far, we have that:
(i) if we consider constant σ_{ E }(t) then condition (22) is σ_{ E } >rg_{ T } V/a.
if 0 ≤ t < P and the eradication condition can be consequently computed.
Combined therapies
As for the mono-thrapeutic case in some of the scenarios that we mentioned it is possible to infer analytical local eradication conditions. In the additional file 1 we show the derivation of the eradication condition for combined impulsive therapies with synchronous delivery.
Global stability of the eradication
Finally, it follows that if ∀x(0) > 0. X(t) → 0 then also T(t) → 0, and the tumor eradication is globally asymptotically stable [41].
Deterministic transitory analysis
Results of the previous section refer to the highly idealized case of a infinite horizon therapy. However, real therapies have a finite duration and, more important, the host organism has a finite lifespan. Thus, in this as well as in other applications of computational biology and medicine it is natural to wonder whether such results can be used at all [8, 28]. This is critically related to the velocity at which the solutions of the equations studied in the previous section tend to their asymptotic solutions.
As far as the IL-2 mono-therapy is concerned, the velocity of growth of E(t) - which in turn determines the velocity of reduction of T(t) - is ruled by the difference p_{ E }〈J_{∞}(t)/(g_{ E } + J_{∞}(t))〉 - μ_{ E }. Moreover, independently of the initial conditions the function J(t) converges to J_{∞}(t) in some multiple of average degradation time of the interleukin, i.e. 1/μ_{ I }, which is small. Thus, this means that very soon the asymptotic solution is reached. Observe now that since J_{∞}(t)/(g_{ E } + J_{∞}(t)) < 1 it follows that if p_{ E } < μ_{ E } then the constraint (17) is never fulfilled. We stress that this is the case for the values listed in Table 3. In practice, since in general 〈J_{∞}(t)/(g_{ E } + J_{∞}(t))〉 << 1 and since μ_{ E } is small, it follows that p_{ E } must be far larger than μ_{ E }. This requires that, for some of the settings that we will simulate in the next sections, the value of p_{ E } could be different from the one given in Table 3. Biologically, this might substantially reduce the number of patients to whom the IL-2 mono-therapy might be effective.
Further discussions are worth. In the case of impulsive therapy unless IL-2 is injected every few hours J_{∞}(t) → 0 rapidly, so that the eradication is unlikely unless large doses are delivered. This is also mirrored in the corresponding local eradication condition (19). Furthermore, in equation (16) it is also very important parameter g_{ E }, whose value used in our simulations is very large. Thus, if μ_{ I }P and g_{ E } are large we may roughly say that - unless huge doses are delivered or p_{ E } is particularly large - the coefficient of E in inequality (16) is almost always comparable to -μ_{ E }, so that a large rate of injection is required to fulfill eradication condition (19).
In the case of piece-wise continuous delivery of IL-2 J_{∞}(t) takes few hours to get sufficiently closer its maximum plateau value, as given by equation (18). This suggests that for this kind of drug delivery the duration of each therapy session, i.e. A, should be a quite large fraction of the unity. This, of course, poses some practical problems since the patients should receive very long daily infusions. However, in some recent clinical trials on cyrcadian rythms-tuned delivery of chemotherapy some special 24-hours infusors have been experimented [58]. Roughly speaking, the above fact might be related to the "indirect effect" of IL-based therapies. In fact, they aim at triggering the expansion of the number of effector cells which, in turn, kill tumor cells.
Differently, as far as the ACI mono-therapy is concerned, the velocity of convergence of ε_{ E }(t) is some multiple of the average lifespan of the effector, i.e. 1/μ_{ E }. Such a value is generally quite big and, for instance, it is 33 days about in our simulations. This means that the convergence is very slow and that, unless the duration of the therapy is exceptionally long, the results of the asymptotic analysis cannot be used as a basis for the stochastic simulations. Similar considerations can be done for the combined therapy.
Finally, it is important to recall that the conditions that derived in the previous section are of local nature. In order to guarantee the eradication for generic non-small initial conditions (T(0), E(0), I(0)) the constraint specific to the simulated therapy has to be largely fulfilled.
Stochastic simulations
We performed stochastic simulations of the model under various therapeutic settings, whose results are now reported. We have mostly considered a single-month daily therapy, i.e. Θ = {1, . . . , 30}. When different schedules are considered the parameters are explicitly reported. In all the figures representing simulation days and number of cells are given on the x-axis and the y-axis, respectively. To perform simulations a JAVA implementation of model (4) and Algorithms 1 and 2 has been developed.
IL-2 immunotherapy
Notice that (i.e. compare with the transitory analysis) the piece-wise constant immunotherapy seems more efficient than the impulsive one. Indeed (i) in the piece-wise case the eradication of the tumor is reached at t_{ e } ≈ 20 days and the maximum tumor size is around 10^{6} = 10 · T(0). Differently, (ii) in the impulsive case the tumor is eradicated a few days later, i.e. t_{ e } ≈ 23 days, and the maximum tumor size is almost 20·T(0). At the eradication day the number of effector cells is around 4 · 10^{6} in both cases, whereas the density of IL-2 is of the order of 10^{7} in (left) and 10^{4} in (right).
Adoptive Cellular Immunotherapy
Note that the figures show no remarkable difference in the tumor response. In particular, in both the simulations the eradication is obtained at around day 15. In both cases, at the eradication day the number of effector cells is around 6 · 10^{5}, and the density of IL-2 is of the order of 10^{2}.
Notice that it seems that the immune response is slightly better stimulated with this therapy setting than the one in Figure 2 (right). In fact, in this case the eradication day is around 12. The number of effector cells and the density of IL-2 are similar to those in Figure 2 (right).
Combined therapies
As expected, this combined therapy eradicates even though the parameters are lower than those used in the scenarios where the single therapies are used (i.e. Figure 1 (right) and 2 (right)). In both cases the eradication is observed a few days after the end of the therapy. In the case of synchronous therapy t_{ er } ≈ 39, in the asynchronous case t_{ er } ≈ 37. In both cases at the day of eradication the number of effector cells is around 2 · 10^{5} and the density of IL-2 is around 10. In both therapies the maximum size of the tumor is almost equal, i.e. it is around 2.5·10^{5}. Finally, since in both cases the proliferation of effector cells is almost equal, it seems that no remarkable differences are observed with these therapy schedules.
Larger initial tumor and ACI
In both cases the eradication is observed close to the end of the therapy (i.e. day 25) with the number of effector cells being around 10^{6} (left) and 10^{8} (right), and the density of IL-2 of the order of 10^{2} (left) and 10^{7} (right). Notice that in (left) the line of the effectors is interrupted at the eradication time since the simulation is interrupted when T = 0. Moreover, Differently form the other figures here the scales on the y-axes are different since the maximum in (right) is 100 times bigger than the maximum in (left).
Probabilistic analysis of IL-2 therapy
meaning that t_{ er } is the eradication time for tumor cells before 70 days, more than twice the duration of the simulated therapies. We evaluated the empirical probability density function of t_{ er }, denoted ϱ(t_{ er }), by performing 10^{2} simulations for both the scenarios in Figure 1. In all simulations we used the same configuration used in such a figure, that is in (left) A = 0.2 and d_{ i } = 4 · 10^{7}/A for i = 1, . . . , 30 whereas in (right) u_{ i } = 4 · 10^{7} for i = 1, . . . , 30. In both cases T(0) = 10^{5}, E(0) = I(0) = 0 and p_{ E } = 20 · 0.1245. All the other parameters are as in Table 3.
In case of daily delivery of the IL-2 mono-therapy, Figure 6 shows the evaluation of ϱ(t_{ er }) for piece-wise constant (left) and impulsive therapies (right). It is remarkable that for all the simulations the eradication is always reached (i.e. 10^{2} times out of 10^{2} simulations).
Averages and standard deviation, daily IL-2 mono-therapy
〈t_{ er }〉 | σ | 〈t_{ er }〉 | σ |
---|---|---|---|
19.34 | 0.33 | 22.71 | 0.41 |
Probabilistic analysis of ACI
As for the case of IL-2 mono-therapy the results on ACIs in Figure 2 motivated us to to investigat the relationship between impulsive and piece-wise constant ACIs, as well as the influence of the period P between two consecutive therapy sessions. Again, we considered the same time-dependent property over a single simulation, i.e. t_{ er } = min{t ≤ 70 | T (t) = 0}. Also in this case 70 days is more than twice the duration of the therapies that we simulated. We evaluated the empirical probability density function ϱ(t_{ er }) by performing 10^{2} simulations for each of a set of parameter configurations. In all simulations we used the initial configuration T(0) = 10^{5} and E(0) = I(0) = 0.
Averages and standard deviation, daily ACI mono-therapy
w _{*} | 〈t_{ er }〉 | σ | 〈t_{ er }〉 | σ |
---|---|---|---|---|
5 | 15.5 | 1.1180 | 15.2 | 1.7204 |
2.5 | 24.0 | 2.0 | 23.5 | 1.7078 |
2 | 27.6 | 1.9720 | 27 | 2 |
1.5 | 35.6 | 3.0397 | 34.7 | 3.1638 |
1 | 62.0 | 4.3204 | 61.0 | 4.3204 |
Averages and standard deviation, weekly ACI mono-therapy
w _{*} | 〈t_{ er }〉 | σ | 〈t_{ er }〉 | σ |
---|---|---|---|---|
5 | 14.98 | 0.80 | 11.39 | 0.93 |
2.5 | 23.12 | 1.22 | 19.27 | 1.30 |
2 | 26.81 | 1.28 | 22.95 | 1.23 |
1.5 | 33.66 | 2 | 28.9 | 2.42 |
1 | 61.18 | 3.75 | 53.19 | 5.70 |
By means of the Wilcoxon statistical test we compared the observed realizations of t_{ er }, grouped by kind of delivery (i.e. impulsive or piece-wise constant) and by frequency (i.e. daily or weekly). We obtained that (i) if delivered daily the eradication times of the piece-wise constant and the impulsive ACIs are not statistically different (i.e. p > 0.05 for all doses). Also, (ii) if delivered daily the eradication time of the impulsive therapy is significantly smaller than the one with piece-wise constant therapy (i.e. p < 10^{-12}) and (iii) for impulsive ACI the eradication time of the weekly delivered therapy is significantly smaller than the one of the daily delivered therapy (i.e. p < 10^{-12}). Finally, (iv) for impulsive ACI the eradication time in the weekly delivery is not statistically different from the one of the daily delivered therapy (i.e. p > 0.05).
Conclusions
In this work we extended our hybrid model [39] with IL-based immunotherapies and Adoptive Cellular Immunotherapies (ACIs), both modeled as piecewise constant or impulsive functions. We performed analytical analysis of the corresponding deterministic model, inspired by earlier work by Panetta and Kirschner [14]. We analyzed our hybrid model via stochastic simulations which seem to suggest results of some interest, which we briefly summarize:
(i) by the transitory analysis it turns out that IL-based immunotherapies require very large values of the parameter p_{ E }, which might substantially reduce the number of patients to whom it may be used as monotherapy;
(ii) in IL-based immunotherapies the piece-wise constant delivery seems more effective for tumor eradication than the impulsive one although at the price of very long infusion sessions;
(iii) in a daily delivered ACI the piece-wise constant delivery seems more or less equivalent to the impulsive one;
(iv) in a ACI the impulsive delivery seems slightly more effective than the daily delivery: less frequent deliveries of larger doses ensure a slightly more rapid eradication than frequent deliveries of smaller doses. Note that the latter type of delivery is called metronomic delivery, and it is of great relevance for other anti-tumor therapies such as anti-angiogenesis therapies and chemotherapies [1, 59, 60]. Furthermore, for those therapies the metronomic delivery is often more effective;
(v) in a ACI the weekly impulsive delivery seems slightly more effective than the weekly piecewise constant delivery;
(vi) when combined impulsive therapies are considered both the synchronous and the asynchronous delivery seem to be effective and no remarkable differences are observable.
Other more predictable effects were observed such as the synergistic effects of combined therapies, or the dependence of the eradication on the initial values. Of course, these results are strongly linked to the specific model, to its ability in describing the dynamics of real tumors and to the chosen parameters.
As far as the model is concerned, we have previously stressed that maybe the hypothesis that the linear antigenic effect cT due to the tumor size should be corrected by assuming a saturating stimulation cT/(1+dT); here we also add that the assumption that E' linearly depend on E could be corrected, as there are cases where this dependence might be nonlinear (see [26] and references therein). Note also that, although computationally useful, representing the piece-wise constant delivery of ACI by means of a continuous input σ_{ E }(t) is only an approximation. Indeed, in reality the infusion should be more realistically represented as a series of injections of a group of cells each Δt ≪ 1 time units. The time interval Δt should be modeled as a Poisson random variable.
As far as the parameters are concerned, in order to obtain more general biological inferences an extensive and systematic exploration of the space of parameters is mandatory. Of course this will require the exploitation of intelligent algorithms (e.g. approximated stochastic simulations [61, 62]) to tackle the computational hardness of model analysis.
Finally, here we have only explored the effects of the intrinsic stochasticity on the dynamics of tumor-immune system interplay under therapy. However, it has been shown that without therapy the extrinsic stochasticity may play a significant role in shaping tumor evasion from the immune control [28]. Moreover, it has also been proposed that realistic bounded stochastic fluctuations affecting chemotherapy may deeply influence the outcome of chemotherapies of solid vascularized tumors [63].
Note that the inclusion of realistic extrinsic noise would require minor changes in the proposed hybrid simulation algorithms besides the inclusion of the stochastic nonlinear equations for correlated bounded noises [28, 63]. However, that would require extensive numerical simulations (e.g. a higher number of samples of the stochastic process underlying the hybrid system) when inferring heuristic probability densities of eradication times, for instance.
Notes
Acknowledgements
This work was conducted within the framework of the official agreement on "Computational Medicine" between the European Institute of Oncology and the University of Pisa. The research of AdO has been done in the framework of the Integrated Project "P-medicine - from data sharing and integration via VPH models to personalized medicine" (project ID: 270089), partially funded by the European Commission under the 7th framework program. The authors would like to thank the reviewers for their comments that helped to improve the manuscript.
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 4, 2012: Italian Society of Bioinformatics (BITS): Annual Meeting 2011. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/13/S4.
Supplementary material
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