Mathematical Modeling of the Role of Cytokines in Sindbis Virus Treatment of Glioblastoma

ABSTRACT

Oncolytic viruses, specifically Sindbis virus (SINV), combined with cytokines show promising results in slowing glioma progression, but a quantitative understanding of their effects remains limited. In this study, we use an ordinary differential equation (ODE) model to examine the effect of adding cytokines to oncolytic SINV therapy. We fit the mathematical model to data extracted from published tumor growth curves to estimate key model parameters. We find that there are statistically significant differences between the infection rates of SINV and cytokine‐bearing SINV, as well as differences in the cytokine's ability to reduce viral production. Model simulations show that the addition of cytokines causes an almost immediate reduction in the tumor size caused by the increased viral infection rate. The simultaneous reduction in viral production caused by the cytokines results in oscillations in virus, cytokines, and tumor volume. By providing parameter estimates for key biological processes, our model can help optimize treatment strategies and guide future research in oncolytic virotherapy.

Study Highlights

• What is the current knowledge on the topic?

  • ○
    Oncolytic viruses are a promising treatment for glioblastoma, showing the ability to slow or stall growth of the tumor with fewer adverse side effects than standard chemotherapy. However, the curative success of oncolytic viruses is still low, leading researchers to modify viruses in order to enhance their ability to kill tumor cells. One such modification is to arm the virus with cytokines, which have shown improved efficiency in eradicating glioblastomas in animal models. While experiments show improved outcomes with the cytokine‐armed viruses, we still lack a clear understanding of how the presence of cytokines alters virus‐tumor dynamics.
• What question did this study address?

  • ○
    This study applies a mathematical model to the experimental data in order to quantify how cytokines change infection dynamics.
• What does this study add to our knowledge?

  • ○
    We find that the addition of cytokines results in an increased infection rate and a reduction in viral production leading to long‐lasting viral infections that are better able to reduce tumor volume.
• How might this change drug discovery, development, and/or therapeutics?

  • ○
    Our study provides a mechanistic understanding of the role of cytokines in oncolytic virus treatment of glioblastomas and can be used to help guide researchers as they further modify viruses to improve their efficacy.

1. Introduction

Gliomas are brain tumors that are rare but of great medical importance because of their rapid progression [1, 2]. They have been the leading cause of cancer death among males ages 39 and under and females 19 and under, with 2020 having 23,890 cases diagnosed and 18,020 deaths [3]. One kind of glioma, known as glioblastoma multiform (GBM), has a global incidence of less than 0.01% and, despite progress in methods of treatment, it remains mostly incurable [4]. Given the difficulty in curing gliomas with current treatment modalities, new treatment methods are being explored. These include the use of nanoparticles [5], immunotherapy [6], and oncolytic viruses [7]. Recent studies suggest that use of viral oncolytic therapy in particular might help reduce the characteristics of a glioma that make it immunosuppressive and chemo‐resistive [8].

With our new‐found ability to manipulate viral genomes, there has been renewed interest in oncolytic viruses [9]. Currently, the FDA has only approved one oncolytic virus, T‐Vec, a herpes simplex virus type 1 with the granulocyte macrophage‐colony stimulating factor (GM‐CSF) cytokine, for use as an oncolytic virus therapy in the US [10]. While many other oncolytic viruses are being tested, including in clinical trials [11], their ability to effectively treat tumors, while promising, still falls short of other potential cancer therapies [11, 12, 13]. Oncolytic virus research has focused on improving viral tropism because it is viewed as the main mechanism that controls viral treatment efficacy [14, 15]. However, recent work has also started examining the possibility of enhancing the immune response stimulated by the oncolytic virus. Many oncolytic viruses are known to impact protein synthesis, which leads to a series of events that culminates in the release of cytokines [16]. Cytokines play a big part in regulating the innate and adaptive immune response, allowing the immune system to communicate over a short distance, either by cells sending signals to themselves or to other cells [17].

Several studies have examined the possibility of adding cytokines to oncolytic virus therapy to enhance the effectiveness of the virus. One strategy is to use a cytokine in addition to oncolytic virus as a form of combination therapy. One such study showed that pre‐conditioning B16 tumors with three daily doses of GM‐CSF followed by injecting two doses daily of oncolytic reovirus increased delivery of the virus to tumor cells [18]. Another study used a chemokine modulating cocktail that included interferon‐α along with a poxvirus to treat colon cancer tumors in mice, leading to longer survival in mice treated with the cocktail [19]. An alternative strategy is to modify the viruses themselves to express particular cytokines. One such study modified vesicular stomatitis virus (VSV), a rhabdovirus that can cause mild illness in humans and has been tested as an oncolytic virus, to express interleukin‐12 (IL‐12), which slowed tumor growth more than the wild‐type virus [20]. VSV was also modified to express IL‐15, increasing treatment efficacy in a mouse model [21]. IL‐15 has also been added to adenovirus along with a second cytokine, RANTES, which led to greater survival in mice with neuroblastomas [22]. Another interleukin, IL‐36γ, was added to vaccinia virus and showed increased therapeutic efficacy via increased T cell activity [23]. Finally, a study by Shi et al. [24] developed a strain of Sindbis virus (SINV) that expressed GM‐CSF, finding that the addition of GM‐CSF eradicated hepatocellular carcinoma. Building on this, a study by Sun et al. [25] created a number of SINV strains expressing IL‐12, IL‐7, GM‐CSF, or various combinations thereof, and tested them on U87‐MG tumors in mice, finding that viral strains expressing cytokines were more effective at reducing tumor volume.

Further understanding of the complex interactions between oncolytic virus treatment and cytokines can be gained through the use of mathematical models. Several mathematical modeling studies have examined the interplay of cytokines and oncolytic virus treatment. A study by Kim et al. [26] modeled the effect of T‐cells stimulated by IL‐12, finding that the presence of T cells could lead to a short‐term reduction of the tumor volume, but a long‐term relapse of cancer once the virus and immune response wane. A more detailed spatiotemporal model suggests that a strong immune response is needed to help eradicate the tumor, but the strong immune response also leads to broader stochasticity resulting in more variability in patient outcomes [27]. Mathematical modeling was also used to create an in silico clinical trial to study optimal treatment regimens for combinations of GM‐CSF and T‐VEC [28] and to determine optimal release times of an IL‐12 and GM‐CSF expressing adenovirus along with immature dendritic cells from a hydrogel treatment platform [29]. Mathematical modeling has proved to be a useful tool in improving our ability to effectively use oncolytic viruses and cytokines to treat cancer.

In this paper, we create a mathematical model to analyze the effects of an oncolytic virus armed with various cytokines when used to treat gliomas. We fit the model to previously published data of treated tumor volumes in order to quantify the effects of armed SINV on tumor growth rates, finding statistically significant differences in the infection rate and the viruses ability to reduce viral production when cytokines are added to the virus. These findings elucidate the mechanisms underlying differences in dynamics between SINV and SINV armed with cytokines.

2. Methods

2.1. Experimental Data

We used the data from an experiment studying SINV treatment of gliomas in mice [25]. In the study, researchers modified SINV to express, IL‐12, IL‐7, GM‐CSF, IL‐12 and IL‐7, IL‐12 and GM‐CSF, or IL‐7 and GM‐CSF. The various viral strains were tested on U‐87MG, a glioma cell line, implanted in mice. Briefly, 4‐week‐old nude mice were injected with 5 × 10⁶ U‐87MG‐Luc cells subcutaneously in the thigh. The tumor was allowed to grow for 7 days. On Days 7, 9, and 11, 10⁶ pfu (plaque forming units) SINV was injected intratumorally. Tumor size was monitored until Day 21. Tumor growth data for both treated and untreated tumors is shown in Figure 4E of Sun et al. [25]. We extracted data from this figure using WebPlotDigitizer (https://automeris.io) and it is shown in Figure 1.

FIGURE 4.

Estimated parameter distributions for λ (top left), β (top right), δ (center left), γ (center right), k (bottom left), e (bottom right).

FIGURE 1.

Tumor growth data taken from Figure 4E of [25]. Lines show mean growth of untreated tumors (PBS) and tumors treated with various recombinant strains of Sindbis virus infected on Days 7, 9, and 11.

2.2. Mathematical Model

We use a mathematical model of oncolytic virus therapy that includes both the antitumor and antiviral effects of the cytokines,

$$\begin{array}{c}\frac{\mathit{dT}}{\mathit{dt}}=\mathit{\lambda T}-\mathit{\beta TV}-\mathit{kCT}\\ \frac{\mathit{dI}}{\mathit{dt}}=\mathit{\beta TV}-\mathit{\delta I}\\ \frac{\mathit{dV}}{\mathit{dt}}=\frac{\mathit{pI}}{I+\mathit{eC}}-\mathit{cV}\\ \frac{\mathit{dC}}{\mathit{dt}}=V-\mathit{\gamma C}\end{array}$$ (1)

The model is depicted in Figure 2. In the model, the uninfected tumor cells (T) replicate at an exponential rate (λ). These cells then get infected by the virus (V) at infection rate β. Infected cells (I) produce virus at rate p. The cells that have been infected by virus die at rate δ and the virus is cleared at rate c. Virus stimulates the secretion of cytokines (C). The effectiveness of cytokines in reducing viral production is given by e. Cytokines remove tumor cells at rate k and are cleared from the system at rate γ. Note that we have assumed a cytokine growth rate of 1 [cytokine]/[virus]·d, where [] denotes unit of, so the value of this parameter depends on the units we are using to measure cytokines. Since we are not matching the model cytokine predictions to data, we can express the cytokines in whatever unit we choose, so we choose a unit that sets the cytokine growth rate to 1 [cytokine]/d·[virus], to reduce the number of free parameters when fitting. Setting this rate constant to 1 essentially scales the cytokines to be expressed in the same units as the virus.

FIGURE 2.

Model diagram. Oncolytic virus infects tumor cells and also stimulates the production of cytokines. The cytokines can slow the production of virus, but also help in eliminating tumor cells.

2.3. Fitting the Model to Data

The data extracted from Figure 4E of Sun et al. [25] (and shown in Figure 1 in this manuscript) is used to estimate model parameters. Parameter fitting is performed by minimizing the sum of squared residuals (SSR) between model predictions and experimental data. In order to create the lines of best fit using the mathematical model we created, we replicated experimental conditions as closely as possible, meaning that we initiated the tumor with the number of cells used in the experiment and injected the stated amount of virus on the days stated in the experimental protocol. The initial tumor size is 17 mm³, where we used the 5 × 10⁶ cells injected in the study and assumed that U87‐MG cells have a volume of 3.375 × 10⁻⁶ μm³ [30]. The initial amount of virus was assumed to be zero and the tumor was allowed to grow until we added 10⁶ pfu of SINV on days 7, 9, and 11. We fixed the value of p = 100 TCID₅₀/d since p is not identifiable for this model [31] and we also fixed the value of c = 10/d, since this parameter is also not identifiable [32]. The remaining parameters were estimated by fitting to the data. Note that we fit λ, the tumor growth rate, for each curve separately even though they should all have the same base growth rate. A careful examination of the data shows that several of the treated tumor growth curves grow faster than the control curve before treatment is applied (see Figure 1), so using a single common value of λ for all the curves does not correctly capture the data.

2.4. Statistical Analysis

Posterior parameter distributions are estimated through bootstrapping. Bootstrapping is performed by shuffling the residuals and adding them to the best fit curve to create new data sets with the same error as the original data set. The new data set is fit using the same procedure as the original data. This is repeated 1000 times to determine parameter distributions.

The bootstrap estimates are used to perform Mann–Whitney U tests for each parameter value in pairwise comparison between parameter values for SINV alone and parameters for each of the cytokine‐bearing viruses. Running the statistical test on the full set of 1000 parameters leads to an over‐powered test, so we need a smaller sample size to determine a meaningful statistical significance. We used a power analysis on the parameters (specifically, β and e) assuming a three order of magnitude change (roughly what is observed) in the parameter value between the original virus and the armed viral strains to determine a reasonable sample size. We assumed a confidence level of 95% and a power of 95% and used a power analysis calculator specific for the Mann–Whitney test. The infection rate needed a sample size of 11 while e needed a sample size of 8, so we decided to use a sample size of 10 since we were using the power analysis just to get a rough idea of a reasonable sample size. We draw 10 samples from each distribution and perform the Mann–Whitney test using the 10 samples. This is done 100 times and the average p value is used to determine statistical significance. This test is used to assess whether the addition of a particular cytokine leads to a different parameter value than SINV alone.

2.5. Sensitivity Analysis

We perform a sensitivity analysis on the stability criterion derived in Section 3.1. We use Sobol sensitivity indices [33], which give a measure of how much of the total variance in a particular outcome is due to each parameter. The Sobol indices range from 0 to 1 with 0 indicating that the parameter does not contribute to the variance in the outcome at all, and 1 indicating that all of the variance in the outcome is due to that parameter. Sobol indices were calculated using the sobol_indices function in the scipy.stats package of Python. This function uses a “pick and freeze” method for calculating the Sobol indices. In this method, the parameter for which the Sobol index is being calculated is frozen while the other parameters are sampled and the output function is calculated [34].

3. Results

3.1. Mathematical Analysis

We found the basic reproduction number of the model using the next‐generation matrix method,

$$R_0=\frac{\mathit{\beta p}{T}_0}{\mathit{\delta c}\left(1+{\mathit{eC}}_0\right)}$$ (2)

The presence of an initial amount of cytokines reduces the ability of the virus to initiate an infection.

We can also examine the fixed points of the system and their stability to determine conditions for eradication of the tumor. The model has two fixed points: a disease‐free equilibrium,

$$\left(T^{*}=0,I^{*}=0,V^{*}=0,C^{*}=0\right),$$ (3)

and an endemic equilibrium,

$$\left(T^{*}=\frac{\mathit{\delta c}\left(\mathit{e\lambda }+k+\mathit{\beta \gamma }\right)}{\left(k+\mathit{\beta \gamma }\right)\mathit{\beta p}},I^{*}=\frac{\mathit{c\lambda \gamma }\left(\mathit{e\lambda }+k+\mathit{\beta \gamma }\right)}{p{\left(k+\mathit{\beta \gamma }\right)}^2},V^{*}=\frac{\mathit{\gamma \lambda }}{k+\mathit{\beta \gamma }},C^{*}=\frac{\lambda }{k+\mathit{\beta \gamma }}\right).$$ (4)

Thus the model predicts either eradication of the tumor or a chronic infection with a static tumor size.

The stability of either fixed point is determined by the eigenvalues of the Jacobian evaluated at the fixed point. For the disease‐free equilibrium, the eigenvalues are

$$\left(\lambda ,-\delta ,-c,-\gamma \right).$$ (5)

These are negative (indicating stability) except for λ, which represents growth of the tumor. The eigenvalues for the endemic equilibrium are the roots of the characteristic equation,

$$\begin{array}{c}{x}^4+x^3\left(\gamma +\delta +c\right)+x^2\left(\frac{D^{2}e}{p}{I}^{*}+\mathit{\delta c}-\beta T^{*}D+\mathit{\gamma \delta }+\mathit{\gamma c}\right)\\ +x\left(\frac{D^{2}e}{p}\delta I^{*}+\mathit{\gamma \delta c}-\mathit{\gamma \beta }{T}^{*}D+{\beta }^2{V}^{*}{T}^{*}D\right)+\beta V^{*}k{T}^{*}D+\gamma {\beta }^2{V}^{*}{T}^{*}D,\end{array}$$ (6)

where

$$D=\frac{p\left(k+\mathit{\beta \gamma }\right)}{k+\mathit{\beta \gamma }+\mathit{e\lambda }}.$$

We use the Routh‐Hurwitz criteria to determine stability conditions for this equilibrium. This gives the following condition,

$$\frac{\gamma }{\gamma +\delta +c}\left(\frac{e}{\mathit{e\lambda }+k+\mathit{\beta \gamma }}+\frac{\beta }{k+\mathit{\beta \gamma }}\right)\left(\frac{\mathit{ec\lambda }}{\mathit{e\lambda }+k+\mathit{\beta \gamma }}+\delta +c\right)-\frac{\mathit{\text{cλγδ}}\left(\gamma {\beta }^2+\mathit{e\gamma \beta }+\mathit{e\lambda \beta }+\mathit{\beta k}+\mathit{ek}\right)}{{\left(\mathit{e\lambda }+k+\mathit{\beta \gamma }\right)}^2{\left(k+\mathit{\beta \gamma }\right)}^2{\left(\gamma +\delta +c\right)}^2}>1$$ (7)

When this condition is satisfied, we expect that viral treatment will not lead to a cure of the tumor, but will result in a chronic infection that maintains a constant tumor volume.

3.2. Model Fits to Data

We fit the model given by Equations (1) to data from Figure 4E of Sun et al. [25], a graph that depicts the volume of a tumor over the span of 21 days when treated by wild‐type SINV or by SINV armed with various cytokine combinations. Figure 3 shows that our model and parameters successfully capture the experimental results of the different viral treatments used. The values of the best‐fit parameters and the 95% confidence intervals are shown in Table 1.

FIGURE 3.

Experimental data and model best fit curves for tumor growth data taken from [25]. Graphs show tumors treated with wild‐type SINV (top left), SINV + GM‐CSF (top right), SINV + IL‐12 (second row left), SINV + IL‐7 (second row right), SINV + GM‐CSF + IL‐12 (third row left), SINV + GM‐CSF + IL‐7 (third row right), SINV + IL‐12 + IL‐7 (bottom).

TABLE 1.

Best fit parameter estimates for different SINV/cytokine combinations.

Cytokines	λ (/d)	β (/(d·pfu))	δ (/d)	γ (/d)	k (/(d·pfu))	e	SSR	Stability criterion
None	0.294	4.02 × 10−3	5.69 × 10−2	1.01	6.93 × 10−4	0.344	0.00456	7.75
95% CI	0.271–0.314	8.54 × 10−4–0.132	1.04 × 10−16–0.505	2.01 × 10−6–223	3.50 × 10−6–0.773	0.00446–4.21 × 103	0.00233–0.270
GM‐CSF	0.436	0.509	9.26 × 10−2	0.783	1.51 × 10−12	1.70 × 103	0.00391	5.13
95% CI	0.423–0.449	0.0177–0.838	0.0574–0.239	0.416–21.4	(1.51–1.51) × 10−12	1.77–2.40 × 104	0.00192–0.270
IL‐12	0.389	1.70	8.22 × 10−2	1.00	0.267	1.85 × 104	0.00704	6.19
95% CI	0.376–0.403	0.0638–94.3	0.0630–0.134	0.144–441	0.00429–80.2	201–1.02 × 106	0.00413–0.177
IL‐7	0.414	2.98	0.123	0.848	0.342	4.60 × 104	0.00504	5.33
95% CI	0.406–0.422	0.766–368	0.108–0.152	0.186–10.3	0.0260–1.30	1.16 × 104–1.65 × 106	0.00226–1.53
GM‐CSF/IL‐12	0.327	1.25	0.163	6.38	5.68 × 10−8	2.18 × 104	0.0239	24.9
95% CI	0.302–0.353	0.0167–39.7	0.107–0.390	4.25–608	(2.40–11.5) × 10−8	119–3.67 × 104	0.0216–0.484
IL‐7/IL‐12	0.309	1.62	0.213	24.3	0.0870	4.69 × 105	0.0117	12.3
95% CI	0.300–0.321	0.565–2.46	0.171–0.298	7.95–444	0.0394–0.158	2.68 × 104–4.15 × 106	0.00447–0.0632
GM‐CSF/IL‐7	0.303	0.390	0.203	1.96	1.29 × 10−4	4.23 × 102	0.0312	12.3
95% CI	0.276–0.340	0.00577–1.01	0.119–0.455	0.795–5.06 × 103	(1.05–1.46) × 10−4	8.36–9.58 × 103	0.0277–2.77

We see that some parameters vary greatly between different SINV strains, while other parameters are relatively similar throughout. The rate at which uninfected tumor cells (T) replicate (λ) doesn't show too much difference between each viral combination, with the range being 0.294–0.436/d, reflecting the variability in base tumor growth seen in the data. Other studies that have examined the growth of U‐87 cells found growth rates of 0.212/d [35] and 0.47/d [36], so the values found here are within that range. Several parameters show differences, particularly between SINV alone and SINV engineered to express various cytokines. For example, the rate at which cells infected by the virus die (δ) has the lowest rate for SINV alone followed by the SINV combined with one other cytokine. The highest δ values occur for viruses that have combinations of two cytokines. This confirms that cytokines aid the virus in infecting and eventually eradicating tumor cells. There is also a change in the value of e over several orders of magnitude when cytokines are added to SINV. Interestingly, e is not necessarily largest for SINV expressing two cytokines, suggesting that additional cytokines do not have an additive or synergistic effect in altering viral production. We see a similar jump in the value of β, which is lower for SINV alone than for SINV expressing various cytokines. This also leads us to conclude that adding any cytokine has an effect on the rate at which SINV infects tumor cells. The decay rate of cytokines is largest for viruses bearing two cytokines, but is lower than wild‐type virus when SINV is only expressing a single cytokine. The rate at which cytokines remove tumor cells (k) did not show a consistent trend.

We also calculated the value of the stability criterion (Equation (7)) for the chronic infection as derived for our model in Section 3.1. The value of the criterion is consistently above 1, so the chronic infection state is stable for all the viruses. This suggests that while the viral treatment is driving down the size of the tumor, it will not completely eradicate the tumor, but will end up with a chronic infection and a static tumor.

3.3. Comparison of Parameter Values

Since we noted some differences in some of the parameter values, we do a more quantitative comparison of the parameter values. Histograms of the different parameter estimates are shown in Figure 4. We can see two distinct clusters for tumor growth rate λ which reflects the distinct tumor growth rates seen before virus injection with the three tumors injected with SINV bearing two cytokines growing faster pre‐injection than the remaining tumors. Histograms for β and e show a similar pattern, with the distribution for SINV alone somewhat to the left of the distributions of the remaining viruses. The distributions for δ and γ are more clustered together, while those for k are quite separated.

For a more quantitative assessment of differences between parameters, we perform a Mann–Whitney test comparing parameter values for each viral strain against parameter values for SINV alone. This is meant to assess whether the addition of cytokines changes the value of that particular parameter. The p values for these tests are given in Table 2 with statistically significant (p < 0.5) values indicated in bold. Confirming our observations from previous sections, all cytokine‐bearing strains have statistically significant differences in β and e from SINV alone. There are also statistically significant differences in k between SINV alone and cytokine‐bearing SINV, although since there is no consistent trend of cytokine‐bearing viruses being higher or lower than the wild‐type, this might be due to a lack of identifiability of this parameter. We also see a statistically significant difference in δ between the wild‐type virus and two of the double cytokine‐bearing viruses.

TABLE 2.

Statistical analysis comparing SINV virus alone with each of the other viral strains.

Cytokines	β	δ	γ	k	e
GM‐CSF	0.00037	0.33	0.35	0.00027	0.0040
IL‐12	0.00039	0.35	0.47	0.0048	0.0007
IL‐7	0.00020	0.28	0.30	0.0020	0.00032
GM‐CSF/IL‐12	0.00050	0.10	0.093	0.0011	0.0010
IL‐7/IL‐12	0.00020	0.046	0.011	0.0083	0.00023
GM‐CSF/IL‐7	0.012	0.043	0.056	0.016	0.086

Note: p‐values less than 0.05 indicate statistical significance and are in bold.

3.4. Virus and Cytokine Dynamics

To better understand how the parameter differences we have found affect viral and cytokine dynamics, we simulate a single injection of 10⁶ pfu on Day 7 into a tumor started at 17 mm³. While the actual experiment used three injections, we simulate only one injection so that we can more clearly see the behavior of virus and cytokines upon injection. We use the parameters given in Table 1 to simulate the different viral strains. The resulting tumor, virus, and cytokine time courses are shown in Figure 5. Note that the y‐axis is on a log scale so straight lines on these graphs represent exponential growth or decay of the quantity.

FIGURE 5.

Model predictions of tumor (black line), virus (red line), and cytokines (green line) for the different strains of virus. We use estimated parameters given in Table 1, starting with a tumor size of 17 mm³, and simulate a single injection of 10⁷ pfu of virus at Day 7.

It is immediately clear that SINV alone shows different behavior than the cytokine‐bearing SINV strains. When SINV alone is injected, the tumor continues to grow for some time, while for the other viral strains there is an almost immediate decline in the tumor volume. Use of a mechanistic mathematical model allows us to understand the cause of these differences in dynamics. The observed difference in tumor response is due to the difference in infection rates between SINV alone and the cytokine‐bearing strains. SINV alone has a low infection rate, so it takes some time to infect enough tumor cells to bring down the overall tumor volume. The other viral strains have a higher infection rate, which leads to a more rapid decrease in the tumor volume. We also see that SINV alone maintains a high amount of virus, whereas the remaining strains have a rapid drop in the amount of virus. This is caused by the difference in e. SINV alone maintains a high production rate even in the presence of cytokines while the production rate for the other viruses drops quickly once cytokines appear. This also results in the biphasic decline of virus and cytokines, where the initial rapid drop in both is caused by the rapid drop in viral production rate, followed by a slower decline as the level of cytokines reduces and viral production recovers.

Generally, the time courses of viruses and cytokines are similar, which is not surprising since the production of cytokines is driven by virus. In some cases, particularly for viruses with a single cytokine (GM‐CSF, IL‐7, IL‐12), the decay rate of cytokines is notably slower than the decay of virus. We also note that when we go beyond the 21 days of the original experiment, we see rebound of the original tumor. This is consistent with the mathematical analysis, since all viruses satisfied the stability criteria for the chronic infection with a non‐zero tumor size.

3.5. Sensitivity Analysis

We use Sobol sensitivity indices to assess which parameters most affect the stability criterion. The Sobol sensitivity indices assess the proportion of variance in a particular outcome that is due to variance in each parameter [33]. When calculating Sobol indices, model parameters, with the exception of p since it does not appear in the stability criterion, are varied about the values presented in Table 1 for each viral strain. The left‐hand side of Equation (7) is the output function. Parameters with a high Sobol index contribute most to variance in the stability criterion and are good parameters to target if we are trying to modify strains of oncolytic virus so that they fully cure the tumor. Sobol indices for all strains are shown in Figure 6.

FIGURE 6.

Sobol sensitivity indices for the stability criterion given by Equation (7) for the parameter combinations describing different strains of Sindbis virus. Parameter values are taken from Table 1.

The Sobol indices are fairly consistent across the different strains of virus. Most parameters have very low values of the sensitivity index. The parameter that has the largest sensitivity index is the tumor growth rate, λ. It is not surprising that the tumor growth rate has a large influence on whether viral treatment will lead to tumor elimination, but this is not a variable we can target in designing new strains of virus. More useful from the standpoint of modifying viruses is the strong dependence of the stability criterion on cytokine clearance rate, γ, which could be modified by choosing appropriate cytokines to incorporate into the virus.

4. Discussion

Glioblastomas are the most aggressive form of brain cancer, and viral therapy is one of the few effective treatments. In this project, we analyzed data from a previous study on tumor growth in which U‐87MG tumors were injected into the subcutaneous tissue in mice. These tumors were then injected with unmodified SINV or SINV expressing various cytokines. We fit a mathematical model to this data to estimate parameter values, finding that the addition of cytokines seemed to primarily affect the infection rate and the cytokine's ability to suppress viral production. Specifically, the infection rate is lower for SINV alone than for armed SINV, suggesting that cytokines have an effect on the infection rate that was not explicitly incorporated into the mathematical model. The value of e increased when cytokines were added to SINV, although it did not necessarily increase more when there were two cytokines, suggesting that one cytokine is sufficient to facilitate a reduction in the production rate of virus. Finally, we noted an increase in the death rate of infected cells when cytokines were present, although it was mostly not statistically significant. This supports the notion that the presence of cytokines assists in the clearance of the tumor. We then show that these parameter changes lead to distinct differences in viral and tumor dynamics, most notably a more immediate decrease in the tumor and a lower level of virus in the tumor. Unfortunately, mathematical analysis of the model suggests that these decreases in tumor volume will not lead to full eradication in any of the viral strains since the parameter values indicate that the chronic infection with static tumor volume is a stable state for all the viral strains.

These findings are consistent with other studies that have added cytokines to oncolytic virus treatment. In the study that pre‐conditioned B16 tumors with three daily doses of GM‐CSF before injecting two doses daily of reovirus [18], the finding that GM‐CSF pre‐treatment increased delivery of the virus to tumor cells could be explained by our finding that GM‐CSF increases the infection rate of the virus. The results of another study, where VSV was modified to express interleukin‐12 (IL‐12) [20], can also be explained by our finding of an increased infection rate. In this study, the researchers found that the addition of IL‐12 slowed tumor growth more than wild‐type VSV, which is similar to the immediate reduction in tumor volume seen in simulations of the cytokine‐bearing strains in this study. Thus, our study provides a mechanistic basis for the promising results seen in experiments that add cytokines to oncolytic virus treatment.

Our findings also suggest that caution should be used in designing such viruses since we also found that viral production is strongly reduced by the presence of cytokines. As seen in model simulations, this greatly reduces the overall viral load once the cytokine appears and can lead to premature clearance of the virus [25]. Our study also indicates that none of the viral strains tested here will result in full eradication of the tumor since the chronic infection state is stable in all cases. Therefore, it is important to engineer a working combination of OVs and cytokines that can have a complete and lasting anti‐tumor response. Our sensitivity analysis suggests that targeting the cytokine clearance rate will lead to the largest changes in the stability criteria, possibly making the chronic infection state unstable and leading to a cure of the cancer. Data analysis and statistical modeling, as done in this study, as well as studies examining treatment optimization [37], can help find the ideal oncolytic virus treatment for cancer.

The findings of our study are limited by the available data. The original study by Sun et al. [25] only presents tumor growth curves for tumors treated with the different strains of virus. This limits our ability to identify some of the model parameters. This is likely the case for the parameter k, which varied widely and with no consistent trend across all the viruses. Additional data that would help us identify more of the parameters are measurements of viral load and cytokines over time, preferably within the tumor. Another limitation is that the tumor growth data is the average of several tumors grown in several animals. This is known to cause problems when parameter fitting since the estimated parameters based on average measurements are not the same as averaging parameters based on individual fits [38, 39]. This type of averaging also prevents us from examining between‐individual differences in parameter values and dynamics, potentially missing correlations between certain parameters and specific outcomes. We can see some of the problems with this averaging in the original data, where data for several oncolytic virus‐treated tumors grew faster before injection of the virus than the untreated tumor.

We also need to be careful in extrapolating our results to humans since these experiments were performed in mice. There are many physiologic differences between mice and humans, particularly when it comes to the immune response and the effect of cytokines [40]. When using mouse models, we also need to consider that not just the immune response to the virus, but the immune response to the implanted tumor also differs from the human immune response [41]. Humanized mouse models [42], particularly those that more closely reproduce the human immune response, can help improve our ability to translate findings in animal models to humans. In fact, humanized mice geared specifically for studies of immuno‐oncology have been developed [43]. Despite the use of a mouse model, the results of this study are promising for humans since SINV was armed with cytokines that are known to help induce glioblastoma clearance in humans. For example, GM‐CSF recruits eosinophils to the site of the tumor and the eosinophils inhibit glioblastoma metabolism and proliferation [44]. IL‐12 triggers interferon‐γ production and induces macrophages to convert to a tumoricidal phenotype [45], while IL‐7 is required for proliferation of lymphocytes [46]. Given the important roles these cytokines play in clearing human glioblastomas, the additional benefit of increasing the potency of SINV as an oncolytic virus is likely to only improve the efficacy of treatment in humans.

There are also limitations arising from the choice of mathematical model. First, the model being used is an ordinary differential equation (ODE) model, so it replicates the average behavior of the tumor cells, virus, and cytokines. It also assumes that these quantities are continuous and well‐mixed, which can lead to erroneous predictions. For example, the assumption of continuity allows for fractions of tumor cells to grow, sometimes leading to incorrect predictions of tumor relapse. Stochastic mathematical models that incorporate the discrete nature of cells and virus can more accurately capture these types of dynamics [47]. The well‐mixed assumption ignores the spatial structure of solid tumors. There are some mathematical models that have attempted to describe the movement of virus and the immune response through a solid spherical tumor using partial differential equations [48, 49]. Agent‐based models replicate both the spatial and discrete nature of these systems, but are computationally expensive [50]. All three of these model types are more difficult to fit to experimental data than ODE models and would require more extensive experimental data.

The mathematical model also included a highly simplified immune response and did not explicitly account for all immune effects. Cytokines do not directly kill cancer cells, but rather stimulate production of T‐cells, which then attack cancer cells [26]. This process is not explicitly represented in our model. Additionally, cytokines themselves have more effects than reducing production of the virus [51], but these are also not explicitly represented in our model. There is also a natural host immune response to the presence of the cancer that is not included in the mathematical model [52]. While mathematical modeling of all these complex responses is possible [53], properly parameterizing these models requires immune response data [54]. However, in order to begin translating these results to humans, some of these processes will need to be incorporated into the mathematical model to develop a digital twin of the human. We will also need data from human patients in order to determine appropriate parameter values to replicate the dynamics of the system in humans.

In spite of limitations in the data and the simplicity of the mathematical model, our study has detected differences in viral dynamics when oncolytic SINV is armed with cytokines. We find that the presence of cytokines decreases viral production and increases the viral infection rate leading to earlier slowing of growth or even decay of the tumor. Our study provides a framework for using mathematical models to assess the effects of making changes to oncolytic viruses and confirms the findings that the addition of cytokines leads to more effective oncolysis.

Author Contributions

S.M. and H.M.D. wrote the manuscript. H.M.D. designed the research. S.M. and H.M.D. performed the research. S.M. analyzed the data. S.M. and H.M.D. contributed new reagents/analytical tools.

Funding

The authors have nothing to report.

Conflicts of Interest

The authors declare no conflicts of interest.

Makam S. and Dobrovolny H. M., “Mathematical Modeling of the Role of Cytokines in Sindbis Virus Treatment of Glioblastoma,” CPT: Pharmacometrics & Systems Pharmacology 15, no. 2 (2026): e70205, 10.1002/psp4.70205.