Institute for Informatics and Telematics, National Research Council, Via. G. Moruzzi 1, Pisa, 56127, Italy

Dipartimento di Informatica, Università di Pisa, Largo Pontecorvo 3, Pisa, 56127, Italy

Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, Milano, 20141, Italy

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.

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

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

Regarding immunotherapies, although the basic idea of immunotherapy is simple and promising

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

Methods

In the next sections we recall the KP model

The deterministic Kirschner-Panetta model

In

where _{* }_{* }(_{E}_{I }

In the case of no therapy, i.e. _{E}_{I }_{m }_{m}_{M }_{m}_{M }_{M }_{I}

A hybrid model with constant therapies

As discussed in

We now recall the hybrid model of T-IS interplay that we defined in _{* }and _{* }in (1) converted into total number of cells by means of the volume _{* }= ^{-1 }and _{* }= ^{-1}. This leads to the ODE system

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

From (2) a bi-dimensional stochastic process ruling the dynamics of _{i }**x**) is associated. With the system state **x **at time _{i }**x**) _{4 }depends on the continuous part of the hybrid model, i.e. _{4 }(

Birth and death reactions for the hybrid model

**reaction**

**Propensity**

**reaction**

**propensity**

_{1 }:

_{1 }= _{2}

_{2 }:

_{3 }:

_{4 }:

_{5 }:

_{5 }= _{E}E

_{6 }:

_{6 }=

_{7 }:

_{7 }= _{E}

Reaction-based events and propensity functions

We remark that the original SSA is assumed to simulate only time-independent chemical reactions. The algorithm used to simulate this model takes inspiration from algorithms simulating hybrid systems with time-independent propensity functions _{n }

with

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

A hybrid model with general therapies

In this paper we consider more general immunotherapies than the constant ones considered in _{E }_{I }

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 _{er }_{er }

We now present an SSA-based algorithm to simulate the hybrid version of the above model. This new algorithm extends the algorithm which simulates model (2) in a natural way. The reactions in Table _{7 }is modified as _{7}(_{E }_{E }_{4 }(_{7 }(

Here the _{T }_{E}_{T,i }and _{E,i}, describe how the reaction _{i }_{n }

and

The exact simulation algorithm for the hybrid system with generic immunotherapies is Algorithm 1 and is defined in Table **x **and the propensity functions in

where ^{-8 }∧ |^{-6}. Once the jump is determined, given _{j }

where _{i}_{i}**x**) if

Some considerations about the ODEs constituting the model (4) are worth discussing. If the last stochastic event happened at time _{n }_{n+1 }the equation for

which, since in such intervals the other two state variables are constant, is a linear ODE with constant input and constant coefficients. Thus, given _{n}_{n }_{n}, _{n+1}) is

where

Notice that equation (8) is necessary to evaluate, at each step of Algorithm 1, the value of _{4}(

The Hybrid Simulation Algorithm (Algorithm 1)

**Require: **(_{0}, _{0}, _{0}), _{0}, _{stop}

1: set the initial state to (_{0}, _{0}, _{0})) and the initial time _{0};

2: **while **_{stop }**do**

3: let **x **be the current state, for _{j}**x**), define

4: let

and then define

5: let

where if _{i}_{i}**x**);

6: update (_{T,j}, _{E,j},

7: **end while**

Input: initial state (_{0}, _{0}, _{0}), start time _{0}, stop time _{stop}

Values of the parameters

We discuss now the values of the parameters used to simulate the model. In Table _{I}_{E}

Values of the parameters

**Par**.

**Value**

**Unit**

**Description**

0.18

^{-1}

baseline growth rate of the tumor

10^{-9}

^{-1}

carrying capacity of the tumor

1

baseline strength of the killing rate by immune effectors

10^{-4}

^{-1}

tumor antigenicity

3.2

blood and bone marrow volumes for leukemia

_{T}

10^{5}

^{-1}

50% reduction factor of the killing rate by immune effectors

_{E}

2 · 10^{7}

50% reduction factor of IL-stimulated growth rate of effectors

_{I}

10^{3}

^{-1}

50% reduction factor of production rate of interleukins

_{E}

0.1245

^{-1}

baseline strength of the IL-stimulated growth rate of effectors

_{I}

5

baseline strength of production rate of interleukins

_{E}

0.03

^{-1}

inverse of average lifespan of effectors

_{I}

10

^{-1}

loss/degradation rate of _{2}

Parameters of model (4), as given in

Interleukin-based immunotherapies

In the next paragraphs we discuss how to specialize model (4) with either a piece-wise constant or an impulsive interleukin-based immunotherapy. In both cases, we consider a therapy starting at time _{s}_{e }_{s}_{e}

where _{i }_{s}_{e}_{i }

Piece-wise constant therapy

In the case of a continuous infusion each therapy session has a duration of _{i }

where _{i+1 }- _{i }|

where _{n }

where _{n }_{I}^{μIx}_{0}, _{1}, . . . , _{k}_{n}_{n }+

Now, since Θ contains ordered time-points then also _{n}, _{j}_{n},

so that the indexes _{n}_{n }+ _{n}_{min}_{min}_{max }_{max }_{n }

The integral in [_{min}_{max }_{n}, _{j}_{j }_{j }_{n},

Notice that for any _{j }_{n}, _{I}_{j }_{j+1}) (i.e. when the therapy is not delivered) yielding

then the overall integral evaluates as

Notice that this quantity can be easily computed in a iterative fashion. The cases of the rightmost and leftmost integration intervals are similarly accounted. Let us consider the leftmost interval [_{n}_{min}_{n }

if _{min-1 }_{n }_{min-1 }+ _{max}_{n }_{n }_{max+1}

if _{max+1 }≤ _{n }_{max+1 }+ _{n }

Impulsive therapy

In many cases the infusions are very short implying that the in flux rate reaches very large values, so that one may approximate _{I}

Here, _{i }_{i}_{i}_{n}

where _{i }_{i}_{i }

for

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 _{i }

Piece-wise constant therapy

As before we assume

This scenario can be simulated by Algorithm 1 where, in this case, the function _{7}(

where

Impulsive therapy

We consider the case of an impulsive ACI where at each _{i }_{i }

Thus to the "natural" stochastic events external deterministic events are superimposed. As a consequence, at such times (_{n }_{j }_{n }_{n}_{3 }and _{5 }change after _{j}_{4 }and _{7 }are, by definition, potentially non-constant. We argue that there is a similarity between scheduling-based SSAs for systems with delays and this hybrid system (i.e. consider this ACI as a set of events scheduled at the preset times in Θ). Although this differs from such algorithms where the scheduling times are stochastically chosen during the simulation, this allows to modify Algorithm 1 into Algorithm 2 presented in Table

The Hybrid Simulation Algorithm (Algorithm 2)

**Require: **(_{0}, _{0}, _{0}), _{0}, _{stop}

1: initialize the simulation as for Algorithm 1;

2: **while **_{stop }**do**

3: pick a value for

4: get _{next }_{i }_{i }

5: **if **_{next }**then**

6: fire a reaction as in Algorithm 1;

7: **else**

8: update (_{next}_{next}_{next}

9: **end if**

10: **end while**

Input: initial state (_{0}, _{0}, _{0}), start time _{0}, stop time _{stop}

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

**Supplementary Materials (text)**. Results about scalar differential inequalities, scalar linear ODEs with periodic coefficients, impulsive ODEs and combined impulsive immunotherapies are given.

Click here for file

IL-2 immunotherapy

We start from the case of the delivery of a IL-based mono-therapy (i.e. _{E}_{∞}(_{∞}(

Thus, we found a tumor-free state

which is, however, unstable. In fact, this follows by setting

and linearizing since the equation for tumor cells reads as _{∞}(

it follows that _{∞}(

yields the inequality

Finally, from the properties of periodical linear differential equations recalled the additional file

implies that (

(_{I}_{I }_{∞ }= _{I }_{I }

(_{I}_{i }_{i }

where

In this case condition (17) reads as

where

and _{1 }

(_{I}_{i }_{i }

where

ACI

When only ACI is delivered (i.e. _{I}

where _{E}

which is the equation for

and hence the local eradication condition (17) is

However, by averaging both the sides of (20) yields

which implies that

Finally, in this case the local stability condition corresponding to equation (17) becomes

As for the case of IL-2 mono-therapy we now focus on the therapies considered so far, we have that:

(_{E}_{E }_{T }V/a

(_{E}_{e }_{i }_{i }_{i }

where

if 0 ≤

if 0 ≤

(_{E}_{i }

and hence the eradication condition (22) is

Combined therapies

Finally, we shortly consider when both the therapies are delivered. In this case there is the following tumor-free asymptotic solution

where _{∞}(

In the case of synchronous delivery with common period

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

Global stability of the eradication

We conclude by investing the global stability of the eradication. Since for sufficiently large _{∞}(

follows that, for large times, _{∞}(

it follows that

Finally, it follows that if ∀

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

As far as the IL-2 mono-therapy is concerned, the velocity of growth of _{E}_{∞}(_{E }_{∞}(_{E}_{∞}(_{I}_{∞}(_{E }_{∞}(_{E }< μ_{E }_{∞}(_{E }_{∞}(_{E }_{E }_{E}_{E }

Further discussions are worth. In the case of impulsive therapy unless IL-2 is injected every few hours _{∞}(_{E}_{I}P _{E }_{E }_{E}

In the case of piece-wise continuous delivery of IL-2 _{∞}(

Differently, as far as the ACI mono-therapy is concerned, the velocity of convergence of _{E}_{E}

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 (

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

IL-2 immunotherapy

In Figure ^{5 }and _{i }^{7}/_{i }_{i }^{7 }for _{E }_{E }

Single run, IL-2 mono-therapy

**Single run, IL-2 mono-therapy**. Single-run of piece-wise constant (left) and impulsive (right) IL-2 daily immunotherapy. In (left) _{i }^{7}/_{i }^{7 }for ^{5}, _{E }

Notice that (i.e. compare with the transitory analysis) the piece-wise constant immunotherapy seems more efficient than the impulsive one. Indeed (_{e }^{6 }= 10 · _{e }^{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

In Figure ^{5 }and _{i }^{4 }for ^{4 }effector cells are injected, i.e. _{i }^{4 }for

Single run, daily ACI mono-therapy

**Single run, daily ACI mono-therapy**. Single-run of piece-wise constant (left) and impulsive (right) daily ACI. In (left) _{i }^{4}/(_{i }^{4 }for ^{5 }and

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}.

Finally, to discover the relation between the frequency of the therapy sessions and the dosage of each session, in Figure _{i }_{i }^{4 }for

Single run, weekly ACI mono-therapy

**Single run, weekly ACI mono-therapy**. Single-run of piece-wise constant (left) and impulsive (right) weekly ACI, i.e. in both panels _{i }_{i }^{4}/_{i }^{4 }for

Notice that it seems that the immune response is slightly better stimulated with this therapy setting than the one in Figure

Combined therapies

In Figure ^{5}, _{i }_{i }^{6 }for ^{4 }effector cells are injected, i.e. _{i }^{4 }for _{E }_{E }

Single run, combined immunotherapies

**Single run, combined immunotherapies**. Single-run of synchronous (left) and asynchronous (right) combined impulsive IL-2 and ACI daily immunotherapies. The asynchronous delivery is a shift of 0.5 days. In (left) _{i }^{6 }for _{i }^{4 }for ^{5}, _{E }

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 _{er }_{er }^{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

We analyzed the effect of varying the initial number of tumor cells in a scenario with impulsive ACI. Again, we analyzed this scenario because it was the one which permitted a computationally easier analysis. In Figure ^{6 }(left) and ^{7 }(right) is shown. In both cases ^{4 }effector cells are injected, i.e. _{i }^{4 }for ^{7 }and the same schedule is applied the eradication was not observed and the tumor size reached, quite rapidly, a size of the order of 10^{9}. In order to obtain eradication also for ^{7}, as shown in the right panel of Figure _{i }^{4 }for

Single run, ACI mono-therapy and larger tumor

**Single run, ACI mono-therapy and larger tumor**. Single-run of impulsive daily ACI with ^{6 }(left) and ^{7 }(right). In both cases _{i }^{4 }for _{i }^{5 }for

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

Probabilistic analysis of IL-2 therapy

As we already said some of the simulations we performed are time-consuming, especially when the value of

Probability density function, IL-2 mono-therapy

**Probability density function, IL-2 mono-therapy**. Empirical evaluation of _{er}_{i }^{7}/_{i }^{7 }for ^{5}, _{E }^{2 }simulations for each configuration.

We defined the following time-dependent property over a single simulation: we want to evaluate

meaning that _{er }_{er}_{er}^{2 }simulations for both the scenarios in Figure _{i }^{7}/_{i }^{7 }for ^{5}, _{E }

In case of daily delivery of the IL-2 mono-therapy, Figure _{er}^{2 }times out of 10^{2 }simulations).

In Table _{er}_{er}_{er }_{er}^{-16 }for the Wilcoxon statistical test.

Averages and standard deviation, daily IL-2 mono-therapy

**〈 t**

**
σ
**

**〈 t**

**
σ
**

19.34

0.33

22.71

0.41

Daily delivered IL-2 immunotherapy. Average of _{er }

Probabilistic analysis of ACI

As for the case of IL-2 mono-therapy the results on ACIs in Figure _{er }_{er}^{2 }simulations for each of a set of parameter configurations. In all simulations we used the initial configuration ^{5 }and

Figure _{er}_{er}_{* }∈ {5, 2.5, 2, 1.5, 1}, in (left) each therapy session lasts _{i }_{* }· 10^{4}/(_{i }_{* }· 10^{4 }for

Probability density function, daily ACI mono-therapy

**Probability density function, daily ACI mono-therapy**. Empirical evaluation of _{er}_{i }_{* }· 10^{4}/(_{i }_{* }· 10^{4 }for _{* }∈ {5, 2.5, 2, 1.5, 1}. The value of _{* }is given in the figure, all the other parameters are as in Table 3. The densities are obtained by performing 10^{2 }simulations for each value of _{*}.

For any simulation (i.e. 10^{2 }times out of 10^{2 }simulations) the eradication was found if _{* }≠ 1. In the case of _{* }= 1 only half of the simulations predicted eradication before day 70. However, at around day 70 the tumor was small (i.e. always less than 100 cells) meaning that the eradication could have been reached immediately in the days after the 70-th. Interestingly, in both cases it seems that [1.5; 2] is a range for _{* }to have eradication before the end of the therapy, as often desired. In Table _{er}_{er}

Averages and standard deviation, daily ACI mono-therapy

**
w
_{*}
**

**〈 t**

**
σ
**

**〈 t**

**
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

Daily delivered ACI. Average of _{er }

In case of weekly delivered therapy, we considered the simulated therapies where the quantity of effectors are injected each week is equal to the one injected per week in the daily therapies above described. This implies _{k }_{* }_{k }_{* }· 10^{4}/(_{k }_{* }· 10^{4 }(right). The densities are plotted in Figure

Probability density function, weekly ACI mono-therapy

**Probability density function, weekly ACI mono-therapy**. Empirical evaluation of _{er}_{i }_{* }· 7 · 10^{4}/(_{i }_{* }· 7 · 10^{4 }for _{* }∈ {5, 2.5, 2, 1.5, 1}. The value of _{* }is given in the figure, all the other parameters are as in Table 3. The densities are obtained by performing 10^{2 }simulations for each value of _{*}.

Averages and standard deviation, weekly ACI mono-therapy

**
w
_{*}
**

**〈 t**

**
σ
**

**〈 t**

**
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

Weekly delivered ACI. Average of _{er }

By means of the Wilcoxon statistical test we compared the observed realizations of _{er}^{-12}) and (^{-12}). Finally, (

Conclusions

In this work we extended our hybrid model

(_{E}

(

(

(

(

(

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 _{E}

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

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

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

List 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.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

AdO conceived of the study and defined the model, GC implemented the model and performed the simulations, AdO, GC and RB analysed the model and wrote the manuscript. All authors read and approved the final version of the manuscript.

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.

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