 Research Article
 Open Access
CD8 ^{+} T cell response to adenovirus vaccination and subsequent suppression of tumor growth: modeling, simulation and analysis
 Qing Wang†^{1},
 David J KlinkeII†^{2, 3}Email author and
 Zhijun Wang^{1}
Received: 12 December 2014
Accepted: 15 May 2015
Published: 6 June 2015
Abstract
Background
Using immune checkpoint modulators in the clinic to increase the number and activity of cytotoxic T lymphocytes that recognize tumor antigens can prolong survival for metastatic melanoma. Yet, only a fraction of the patient population receives clinical benefit. In short, these clinical trials demonstrate proofofprinciple but optimizing the specific therapeutic strategies remains a challenge. In many fields, CAD (computeraided design) is a tool used to optimize integrated system behavior using a mechanistic model that is based upon knowledge of constitutive elements. The objective of this study was to develop a predictive simulation platform for optimizing antitumor immunity using different treatment strategies.
Methods
To better understand the therapeutic role that cytotoxic CD8 ^{+} T cells can play in controlling tumor growth, we developed a multiscale mechanistic model of the biology using impulsive differential equations and calibrated it to a selfconsistent data set.
Results
The multiscale model captures the activation and differentiation of naïve CD8 ^{+} T cells into effector cytotoxic T cells in the lymph node following adenovirusmediated vaccination against a tumor antigen, the trafficking of the resulting cytotoxic T cells into blood and tumor microenvironment, the production of cytokines within the tumor microenvironment, and the interactions between tumor cells, T cells and cytokines that control tumor growth. The calibrated model captures the modest suppression of tumor cell growth observed in the B16F10 model, a transplantable mouse model for metastatic melanoma, and was used to explore the impact of multiple vaccinations on controlling tumor growth.
Conclusions
Using the calibrated mechanistic model, we found that the cytotoxic CD8 ^{+} T cell response was prolonged by multiple adenovirus vaccinations. However, the strength of the immune response cannot be improved enough by multiple adenovirus vaccinations to reduce tumor burden if the cytotoxic activity or local proliferation of cytotoxic T cells in response to tumor antigens is not greatly enhanced. Overall, this study illustrates how mechanistic models can be used for in silico screening of the optimal therapeutic dosage and timing in cancer treatment.
Keywords
Background
Cytotoxic CD8 ^{+} T cells are important effectors in the adaptive immune response against intracellular pathogens and play an important role in immunosurveillance against malignancy [1, 2]. Modulating an immune checkpoint to increase cytotoxic T lymphocytes (CTLs) that target malignant cells can cure patients of metastatic melanoma [3, 4]. While this clinical success demonstrates proofofprinciple, the clinical response is limited to a subset of patients. Yet, these results encourage alternative approaches to direct host immunity against tumors, including adoptive transfer of autologous T cells extracted from a patient’s own tumor (e.g., [5]), engineering of T cell receptors to recognize tumor antigens (e.g., [6, 7]), or vaccination against tumor antigens (e.g., [811]). Cancer vaccines based on patientspecific material is attractive as it would enable personalized treatments that enhance CTL response to the specific antigens expressed by a patient’s tumor [12]. One approach is to use adenoviruses that were initially developed as vehicles for gene therapy. Attempts to replace missing or faulty genes by adenoviral gene transfer were largely unsuccessful in experimental animals and human volunteers alike due to innate and adaptive immune responses induced by the adenoviral antigens ([13]). Replicationdeficient adenovirus vectors have been pursued as vaccine carriers in the clinic as they showed high efficiency in some rodent and simian preclinical models [13, 14]. The profile of the immune response elicited by adenovirus vaccines against tumor antigens in murine models was investigated by some research groups (see [1517]). While the approach seems promising, the results are suboptimal as similarly observed for the immune checkpoint modulators. In exploring one treatment variation, sequential treatments involving adenovirus and oncolytic viruses may lead to improved antitumor response [18]. However a more systematic approach to explore treatment variants may be helpful to improve overall response.
As illustrated by [1922], a variety of mathematical models based on ordinary differential equations (ODEs) have been developed to better understand cancer progression and response to immunotherapy in the last couple of decades. Early work employed LotkaVolterra equations to describe the interactions between tumor and the immune system where effector cells acted as predators and tumor cells as prey ([21, 23]). The immune surveillance phenomena was described qualitatively in [23] where low doses of tumor cells can escape immune defenses and grow into a larger tumor whereas larger doses of tumor cells are eliminated. The simple predatorprey model was generalized by Kirschner [21, 24], de Pillis et al. [25], Eftimie et al. [26], Wilson and Levy [27], and other researchers where different components of the immune system, such as particular cytokines or natural killer cells, were introduced into the model depending on different cancer treatment strategies. The effect of time delay in the immune response was considered in [20, 28, 29] where authors found that impact of time delay on tumor growth is almost negligible.
Mathematical modeling and computer simulations can be powerful tools in optimizing therapeutic strategies. Mathematical modeling and simulations can be used to screen in silico parameter regions that seem most promising for optimal timing and dosage of therapy and clinical trials can be focused on those regions [3032]. In [33], the authors explore how the timing of oral insulin delivery and immunomodulatory drugs can be optimized for maximum effect. Moreover, an in silico approach can suggest targeted experiments and then minimize the number of needed experiments [34]. It can also be applied to combine in a virtual way different modes of actions that are well characterized in isolation, such as immunotherapy and chemotherapy, and see how they may be combined to maximum benefit. For instance, Eftimie et al. explored how vaccination using two different viruses that carry the same tumor antigen achieves a greater therapeutic response than if one virus is used alone [26]. In this paper, we use simulations to investigate the impact of multiple adenovirus vaccinations on CD8 ^{+} T cell proliferation and recruitment to the tumor microenvironment and to identify important parameter ranges that control tumor growth through vaccinationinduced antitumor immunity.
The structure of this paper is as follows. First, we present a multiscale mechanistic model of antitumor immunity and tumor growth based on a set of coupled impulsive ODEs. Second, we describe how we calibrated the parameters of the model against published experimental data using a genetic algorithm. Next we investigate the stability of tumorfree and high tumor equilibria based on the linearized system. Finally, we used the simulation platform to explore the impact of multiple adenovirus vaccinations on T cell proliferation and recruitment to the tumor microenvironment to control tumor growth.
Methods
Here, we developed a multiscale impulsive ODE model based on our mechanistic understanding of underlying biology and calibrated the model using existing experimental data. This multiscale mathematical model represents the cytotoxic T cell response to adenovirus vaccination against a tumor antigen and subsequent control of the growth of B16F10 tumors. For the reported experiments, the B16F10 cell line was purchased from American Tissue Culture Collection (ATCC, Bethesda, MD). Numerical solutions of the model were obtained using simulators generated by C Sharp. Simulations start on day 0, the time of tumor implantation and conclude on day 49. At the initial time point, we assume that there is no activated tumor specific effector T cells present in the blood and at the site of the tumor. A genetic algorithm was used to find parameter sets that closely match the experimental data [15, 16]. Each parameter set was modeled using an individual chromosome in order to apply the algorithm to search in the parameter space. For each generation, the impulsive ODE set was solved using the RungeKutta method of order four for each parameter set. The fitness function value, or variance, was calculated using a linear combination of sum of error squared and sum of differences between slopes of lines of experimental data and corresponding model predictions. The calibrated mechanistic model was then used to investigate the longterm behavior through stability analysis. Finally, we used the calibrated model to explore the impact of multiple vaccinations on tumor growth to improve antitumor immunity, a scenario that is difficult to test experimentally using preclinical mouse models but could be potentially used in the clinic to treat patients. Details of model development, parameter calibration, stability analysis, and numerical simulations of multiple vaccinations are described in the following sections.
Results
A multiscale model of CD8 ^{+} T cell control of tumor growth

Naïve CD8 ^{ + } T cells ( T _{ N } , units: cells per mm ^{ 3 } ). As the immunization protocol induces the clonal expansion of small subset of CD8 ^{+} T cell clones rather than globally changing T cell numbers, we assumed that naïve CD8 ^{+} T cells expressing the T cell receptor that recognizes the epitope derived from the immunized tumor antigen are produced at a constant rate c _{1} from thymus and die naturally at a rate k _{ d1} T _{ N }. Naïve CD8 ^{+} T cells are maintained at a constant level in the absence of adenovirus, i.e., c _{1}=k _{ d1} T _{ N }(0). Naïve CD8 ^{+} T cells are recruited to the lymph node and activated by adenovirus vaccination and become effector CD8 ^{+} T cells (T _{ E1}) when they encounter adenovirusinduced antigen expression (LV) at a rate proportional to T _{ N } and a saturable adenovirusinduced antigen (LV) term defined by \(\frac {\text {LV}}{\text {LV}+\gamma }\).

Effector CD8 ^{ + } T cells in lymph node ( T _{ E 1 } , units: cells per mm ^{ 3 } ). The increase in the rate of concentration of effector CD8 ^{+} T cells in the lymph node due to activation of naïve CD8 ^{+} T cells from the blood stream is given by \( c_{2} \frac {T_{N}Vol_{b}}{Vol_{\textit {ln}}}\frac {\text {LV}}{\text {LV}+\gamma }\), where V o l _{ b }=1.4∗10^{3} m m ^{3} is the volume of the blood compartment ([39]) and V o l _{ ln }=0.25 m m ^{3} is the volume of the lymph node compartment ([40]). We assume that the natural death of effector T cells in the lymph node is negligible. Effector CD8 ^{+} T cells in the lymph node proliferate at a rate proportional to T _{ E1}, a saturable adenovirusinduced antigen term defined by \(\frac {\text {LV}}{\text {LV}+\gamma }\), and an immune checkpoint term defined by \(\frac {\alpha }{\alpha +T^{2}_{E1}}\), where α is the square root of the saturation constant of T _{ E1}. We also assume that influx rate of effector CD8 ^{+} T cells from blood to lymph node is \(a_{21}\frac {T_{E2}Vol_{b}}{Vol_{\textit {ln}}}\) and a _{12}·T _{ E1} is the efflux rate.

Adenovirus in lymph node (LV, units: Relative Light Units (RLU) per mm ^{ 3 } ). Since the adenovirus used in the calibration experiments are replicatedefective and include a GFP expression plasmid, we assume an exponential decay model for LV. We also used a difference equation \(\Delta \text {LV}(t)=\text {LV}(t^{+})\text {LV}(t^{})=\text {LV}_{k}\) to reflect the abrupt change of the concentration of adenovirus during vaccination at time t _{ k }, where LV_{ k } represents the dosage of vaccination at t _{ k } with k=1,2,…,n.

Effector CD8 ^{ + } T cells in blood ( T _{ E 2 } , units: cells per mm ^{ 3 } ). We assume the effector CD8 ^{+} T cells die naturally at a rate k _{ d3} T _{ E2} in blood. The influx rate of effector CD8 ^{+} T cells from lymph node to blood is equal to \(a_{12}\frac {T_{E1}Vol_{\textit {ln}}}{Vol_{b}}\) and the efflux rate of effector CD8 ^{+} T cells from blood to lymph node is equal to a _{21} T _{ E2}. The influx rate of CD8 ^{+} T effectors from the tumor to blood is \(a_{32}\frac {C_{MHCI^{}}}{\epsilon +C(t)}\frac {T_{E3}Vol_{t}}{Vol_{b}}\) and the efflux rate of CD8 ^{+} T effectors from blood to tumor is a _{23} T _{ E2}, where \(C(t)=C_{MHCI^{}}+C_{MHCI^{+}}\phantom {\dot {i}\!}\) is the number of tumor cells, \(C_{MHCI^{+}}\phantom {\dot {i}\!}\) is the number of major histocompatibility complex (MHC) class I positive tumor cells, \(\phantom {\dot {i}\!}C_{MHCI^{}}\) is the number of MHC class I negative tumor cells, and ε is a small positive constant representing a small volume of tissue that excludes tumor and effector CD8 ^{+} T cells in the tumor compartment.

MHC class I positive tumor cells ( \(\phantom {\dot {i}\!}C_{MHCI^{+}}\) , units: cell number). MHC class I positive tumor cells are converted from MHC class I negative tumor cells (\(C_{MHCI^{}}\phantom {\dot {i}\!}\)) with the assistance of Interferon γ (IFNγ) at a rate \(c_{3}\frac {\text {IFN}{\gamma }}{k_{1} +\text {IFN}{\gamma }}C_{MHCI^{}}\) and the effector CD8 ^{+} T cellmediated MHC class I positive tumor cells death rate is \(c_{4} T_{E3}\frac {C_{MHCI^{+}}}{\epsilon +C(t)}\phantom {\dot {i}\!}\). We assume that the dilution rate of MHC class I positive tumor cells due to proliferation is \(k_{p2} C_{MHCI^{+}}\phantom {\dot {i}\!}\). The natural death rate of MHC class I positive tumor cells is assumed to be \(k_{d4}C_{MHCI^{+}}\phantom {\dot {i}\!}\).

MHC class I negative tumor cells ( \(\phantom {\dot {i}\!}C_{MHCI^{}}\) , units: cell number). MHC class I negative tumor cells are converted to MHC class I positive tumor cells with the assistance of Interferon gamma (IFN_{ γ }) at a rate \(c_{3}\frac {\text {IFN}{\gamma }}{k_{1} +\text {IFN}{\gamma }}C_{MHCI^{}}\). We assume that the proliferation rate of MHC class I positive tumor cells is equal to \(2k_{p2} C_{MHCI^{+}}\phantom {\dot {i}\!}\). As MHC class I positive tumor cells proliferate, they lose MHC class I expression and become MHC class I negative cells. A logistic growth pattern is assumed for the number of MHC class I negative tumor cells in the absence of vaccination treatment.

Effector CD8 ^{ + } T cells in tumor microenvironment ( T _{ E 3 } , units: cells per mm ^{ 3 } ). We assume that effector CD8 ^{+} T cells can proliferate locally upon recognition of the corresponding tumor antigen presented by MHCI positive tumor cells at a saturable rate equal to \(k_{p3}\frac {C_{MHCI^{+}}}{\epsilon +C(t)}T_{E3}\phantom {\dot {i}\!}\). Effector CD8 ^{+} T cells have a finite lifespan and die within the tumor microenvironment as a rate equal to k _{ d5}·T _{ E3}. The influx rate of effector CD8 ^{+} T cells from the blood to tumor is defined by \(a_{23}\frac {T_{E2}Vol_{b}}{Vol_{t}}\), where V o l _{ t }=ε+s _{ t } C(t)+V _{ i } T _{ E3} m m ^{3} is the volume of the tumor compartment, ε is a small positive constant representing a small volume of tissue that excludes tumor and effector CD8 ^{+} T cells in the tumor compartment, s _{ t }=6∗10^{−7} m m ^{3} is the average size of a B16F10 tumor cell ([41]), and V _{ i }=10^{−7} m m ^{3} is the average size of a T effector cell ([42]). The efflux rate of effector CD8 ^{+} T cells from the tumor to blood is \(a_{32}T_{E3}\frac {C_{MHCI^{}}}{\epsilon +C(t)}\).

Interferon gamma (IFN γ , units: moles per mm ^{ 3 } ). We assume that Interferon γ is secreted solely by effector CD8 ^{+} T cells within the tumor at a rate proportional to the concentration of effector CD8 ^{+} T cells within the tumor microenvironment and decays at a rate proportional to its concentration. While this assumption may not hold in all model systems, the presence of IFN γ in the tumor was dependent on CD8 ^{+} T cell activation [43].

Tumor Necrosis Factor α (TNF α , units: moles per mm ^{ 3 } ). We assume that Tumor Necrosis Factor α decays naturally at a rate proportional to its concentration and is secreted solely by effector CD8 ^{+} T cells in the tumor at a rate that includes both autocrine and constitutive production terms: \(\left (k_{c2}\frac {\text {TNF}{\alpha }}{k_{2}+\text {TNF}{\alpha }}+k_{3}\right)T_{E3}\). While this assumption may not hold in all model systems, the presence of TNF α in the tumor was also dependent on CD8 ^{+} T cell activation [43].
where ΔLV(t)=LV(t ^{+})−LV(t ^{−}) reflects the abrupt change of adenovirus at vaccination time t and LV_{ k } is the dosage of the adenovirus vaccination at the administration time t _{ k } with k=1,2,3,⋯,n.
Model calibration
Next, we calibrated model parameters against selfconsistent experimental data. These data were acquired from two papers. The first paper described the general dynamics of a CD8 ^{+} T cell response to vaccination with a recombinant human adenovirus serotype 5 (rHuAd5) vector that can be used as a general delivery vehicle to express human tumor antigens [16]. The second paper describes using this adenovirus vector to induce a CD8 ^{+} T cell response to the human dopachrome tautomerase antigen (hDCT; vector: rHuAd5hDCT) [15]. In contrast, the same adenovirus vector engineered to vaccinate against the glycoprotein gp100 (rHuAd5hgp100) was unable to control the growth of B16F10 in prophylactic and neoadjuvant settings. The B16F10 cell line exhibits a defect in the processing and presentation of peptides derived from gp100 through the Major Histocompatibility Complex class I pathway [44]. Together these results suggest that the control of tumor growth induced by rHuAd5hDCT is through tumorspecific CD8 ^{+} T cells.

CD8 ^{+} T cells in the secondary lymph nodes (T _{ E1}) and effector CD8 ^{+} T cells in the blood (T _{ E2}) are obtained from Figure 1(A) of Yang’s paper ([16]).

Antigen expression derived from adenovirus vaccination (LV) corresponds to data presented in Figure 3(B) of Yang’s paper ([16]).

Total volume of B16F10derived tumors was calibrated against data shown in Figure 1(B) of McGray’s paper ([15]).

The concentration of effector CD8 ^{+} T cells present within the tumor (T _{ E3}) are found in Figure 4(A) of McGray’s paper ([15]).

Expression of Interferon gamma (\(\overline {\text {IFN}} \gamma \)) and Tumor Necrosis Factor alpha (\(\overline {\text {TNF}}\alpha \)) genes within the tumor are obtained from Figure 1(E) of McGray’s paper ([15]).
As there are more data points (93) than parameters (27) parameters, the mechanistic model is identifiable in theory.
Parameter values determined by calibrating model against experimental data
Parameter  Units  Description  Calibrated values 

k _{ d1}  d a y ^{−1}  Naïve CD8 ^{+} T cell natural death rate constant  3.809×10^{−3} 
k _{ d2}  d a y ^{−1}  Adenovirus natural death rate constant  0.364 
k _{ d3}  d a y ^{−1}  Blood T effector natural death rate constant  1.80×10^{−2} 
k _{ d4}  d a y ^{−1}  Tumor cell natural death rate constant  2.08×10^{−6} 
k _{ d5}  d a y ^{−1}  Tumor T effector natural death rate constant  0.800 
k _{ d6}  d a y ^{−1}  Interferon γ natural degradation rate constant  0.082 
k _{ d7}  d a y ^{−1}  Tumor Necrosis Factor α natural degradation rate constant  3.10×10^{−6} 
k _{ p1}  d a y ^{−1}  Lymph node T effector proliferation rate constant due to adenovirus vaccination  12.017 
k _{ p2}  d a y ^{−1}  Tumor cell proliferation rate constant  0.5 
k _{ p3}  d a y ^{−1}  Tumor T effector proliferation rate constant due to tumor growth  5.73×10^{−6} 
a _{12}  d a y ^{−1}  Rate constant for T cell flow from lymph node to blood  5.706 
a _{21}  d a y ^{−1}  Rate constant for T cell flow from blood to lymph node  3.540×10^{−3} 
a _{23}  d a y ^{−1}  Rate constant for T cell flow from blood to tumor  5.546×10^{−2} 
a _{32}  d a y ^{−1}  Rate constant for T cell flow from tumor to blood  6.89×10^{−18} 
c _{1}  c e l l·m m ^{−3}·d a y ^{−1}  Naïve T cell natural production rate  2.719×10^{−4} 
c _{2}  d a y ^{−1}  T cell to lymph node T effector transfer rate constant  0.5263 
c _{3}  d a y ^{−1}  MHC class I negative to positive tumor cells transfer rate constant  0.8759 
c _{4}  m m ^{3}·d a y ^{−1}  MHCI positive tumor death rate due to T effector (in tumor) lysis  2.49×10^{−13} 
α  (c e l l·m m ^{−3})^{2}  Lymph node T effector saturation constant  6.520×10^{10} 
k _{1}  m o l e s·m m ^{−3}  Interferon γ saturation constant  3.69×10^{−9} 
k _{2}  m o l e s·m m ^{−3}  Tumor Necrosis Factor α saturation constant  6.924×10^{6} 
k _{3}  m o l e s·d a y ^{−1}·c e l l ^{−1}  Constitutive Tumor Necrosis Factor α production rate constant  2.634×10^{−4} 
k _{ c1}  m o l e s·d a y ^{−1}·c e l l ^{−1}  Cellular Interferon γ production rate constant  7.295×10^{8} 
k _{ c2}  m o l e s·d a y ^{−1}·c e l l ^{−1}  Autocrine Tumor Necrosis Factor α production rate constant  9.939×10^{8} 
γ  R L U·m m ^{−3}  Adenovirus saturation constant  2.905×10^{3} 
β _{1}  AU  Constant of proportionality in calculating T C R _{ α } gene expression  8.79×10^{−6} 
r _{2}  c e l l ^{−1}·d a y ^{−1}  Constant in tumor logistic growth (or MHC class I negative tumor growth rate divided by the carrying capacity K, i.e., \(\frac {k_{p2}k_{d4}}{K}\))  3.34×10^{−10} 
Stability analysis
We note that the tumorfree equilibrium has only one nonezero element: the naïve T cells T _{ N }, this occurs when there are no tumor cells present and no adenovirus immunization treatment is administered and also corresponds to tumorspecific effector CD8 ^{+} T cells and cytokines being equal to zero. The high tumor equilibrium \(\overrightarrow {X}_{1}\) has two nonzero elements: the naïve T cells T _{ N } and the MHC class I negative tumor cells \(\phantom {\dot {i}\!}C_{\textit {MHCI}}^{}\), which reflects the status of the steady state when the onetime adenovirus vaccination treatment failed to completely eradicate the tumor cells. This situation occurs when adenovirus LV decays to zero and the MHC class I positive tumor cells are all killed by tumor infiltrating lymphocytes, which causes exhaustion of effector CD8 ^{+} T cells in three compartments and cytokines decay to zero. The rest of the MHC class I negative tumor cells then approach the carrying capacity and the naïve T cells return to their original constant level.
Thus when k _{ p2}>k _{ d4}, the tumorfree equilibrium \(\overrightarrow {X}_{0}\) is unstable and when k _{ p2}<k _{ d4}, the tumorfree equilibrium \(\overrightarrow {X}_{0}\) is stable since all eigenvalues of the Jacobian matrix have negative real parts.
where a _{2}=a _{12}+a _{21}+a _{23}+a _{32}+k _{ d3}+k _{ d5}, a _{1}=a _{12}(k _{ d5}+a _{32}+k _{ d3}+a _{23})+k _{ d3}(k _{ d5}+a _{32})+a _{21}(k _{ d5}+a _{32})+k _{ d5} a _{23}, and a _{0}=a _{12}(k _{ d3} k _{ d5}+a _{32} k _{ d3}+a _{23} k _{ d5}). It is easy to see that a _{ i }>0 for i=0,1,2 since all parameters are positive. By simple calculation, we can obtain a _{2} a _{1}−a _{0}>0 which implies that roots of equation (13) (λ _{7}, λ _{8}, λ _{9}) all have negative real parts by the RouthHurwitz criterion. Thus when the proliferation rate of tumor cells is greater than the natural death rate of tumor cells (i.e., k _{ p2}>k _{ d4}), the high tumor equilibrium \(\overrightarrow {X}_{1}\) is stable and when the proliferation rate of tumor cells is less than the natural death rate of tumor cells (i.e., k _{ p2}<k _{ d4}), the hightumor equilibrium \(\overrightarrow {X}_{1}\) is unstable.
Therefore, using the parameter values obtained from model calibration (k _{ p2}>k _{ d4}), the tumorfree equilibrium \(\overrightarrow {X}_{0}\) is unstable and the high tumor equilibrium \(\overrightarrow {X}_{1}\) is stable. This implies that under the current status of the mouse immune system, a small tumor will keep growing to its carrying capacity because of the fast proliferation of tumor cells without adenovirus vaccination treatment. On the other hand, the onetime adenovirus immunization as applied in the experiment was not very successful in completely eliminating tumor cells due to limited effects on enhancing CTL immune response. It seems that the CTL response falls to zero before all MHC negative tumor cells are converted to MHC positive tumor cells and killed by the cytotoxic CD8 ^{+} T cells. Then, MHC positive tumor cells approach zero while MHC negative tumor cells approach the carrying capacity and all T cell effectors and cytokines drop back to zero soon after the one time vaccination treatment (see Fig. 2).
Effect of multiple vaccinations
Next, we investigated in silico the impact of multiple adenovirus vaccinations on T cell proliferation and recruitment, cytokine secretion, and tumor growth using the calibrated model in conjunction with an impulsive control mechanism where control laws are discrete in time, as represented by Eq. (10).
Impact of the single adenovirus vaccination with enhanced T cell cytotoxicity or proliferation
Discussion
Mathematical modeling and simulation are increasingly being used in the pharmaceutical industry to better understand the underlying biology targeted by a drug and to explore therapeutic scenarios that may be difficult to test experimentally [45]. Here, we developed a threecompartment mechanistic mathematical model to describe the clonal expansion of CD8 ^{+} T cells in a mouse model of metastatic melanoma in response to adenovirus vaccination against a defined tumor antigen. Based on the collective knowledge of this preclinical mouse model, the model represents the primary CD8 ^{+} T cell response to adenovirus immunization and the subsequent impact on the growth of a tumor derived from the B16F10 cell line. Using the mechanistic model as a framework to integrate different experimental studies, model parameters were calibrated against published experimental data that describes the primary response. As shown in Fig. 2, our model predictions of adenovirus concentration, tumor size, concentrations of CD8 ^{+} T effectors in blood and tumor, gene expression of IFNG and TNF α matched the experimental data. The proposed model structure reflects a tradeoff between biological realism, parameter identifiability, and a fitnessforpurpose. As mentioned above, an excess of data points (93) relative to the number of parameters (27) suggests that the model is identifiable in theory. Efforts are underway to identify the appropriate topology of the network, given the available data [46].
In terms of the tradeoff between biological realism and fitnessforpurpose, we settled on the proposed model structure to facilitate stability analysis. Stability analysis of tumorfree and high tumor equilibria was conducted based on the linearized system. Impulsive stabilization using the Lyapunov method will be considered in the future to provide conditions on parameters such that the hightumor equilibrium may be stabilized using impulsive control through manipulation of strength and frequency of the multiple vaccinations. However, the proposed model structure imposes some limitations in how the model represents the system and interpreting the model predictions. In particular, we note that the model predicted an earlier peak time for the concentration of effector CD8 ^{+} T cells in the lymph node compared to experimental data, which may suggest a more complicated model structure for the lymph node compartment than proposed here. While additional model structure may help in capturing the dynamics of T cells within the lymph node, the current structure is sufficient to capture the dynamics of CD8 ^{+} T cells within the blood, which is the pool that gets recruited to the tumor compartment. Additional lymph node structure would then have limited impact on our conclusions. We also note that the effective concentration of effector CD8 ^{+} T cells within the tumor compartment (e.g., number of effector CD8 ^{+} T cells per weight of tumor) peaked at 10 days despite the blood population of CD8 ^{+} T cells peaking at day 20. As CD8 ^{+} T cell recruitment from the blood into the tumor compartment was assumed to be independent of tumor size and the parameter values suggest that proliferation of CD8 ^{+} T cells within the tumor was negligible, this decline in CD8 ^{+} T cell concentration was due to dilution of recruited effector CD8 ^{+} T cells into an exponentially growing tumor mass. While a direct measure of tumor infiltrating CD8 ^{+} lymphocytes was reported at a single time point, expression of IFNG and TNF α are implicit surrogate markers for CD8 ^{+} T cell infiltration as these two cytokines are directly proportional to the concentration of CD8 ^{+} T cells within the tumor compartment. Measuring the number of tumor infiltrating lymphocytes in this mouse model at additional time points would help confirm these assumptions. This would be interesting, as tumors, like the B16 model, are known to develop immunosuppressive mechanisms that are proportional to tumor size that could alter the relationship between the presence of tumor infiltrating lymphocytes and cytokine production [47].
Given the rapid growth of the B16F10 model and ethical limitations of animal studies, studies using this preclinical mouse model is limited typically to a single round of therapy. Yet, the treatment of human cancers typically involves multiple rounds of therapy to control tumor growth. Here we used a calibrated mechanistic model coupled with computer simulation to explore clinically relevant treatment options in silico. In exploring the impact of multiple vaccinations, our model indicates that increasing the dose of adenovirus vaccination, the time period between successive adenovirus vaccinations, or the number of adenovirus vaccinations results in a prolonged lifespan of effector CD8 + T cells in all three compartments and extended length of secretion of the cytokines IFNG and TNF α within the tumor microenvironment. However these changes in multiple vaccinations have little impact on the magnitude of the clonal CD8 ^{+} T cell immune response and therefore have very little impact on reducing tumor growth. If technically and ethically feasible, additional animal experiments using multiple vaccinations may be helpful to confirm these predictions. As the adenovirus vector promotes clonal expansion of CD8 ^{+} T cells that recognize a small number of epitopes derived from tumor antigens, the number and diversity of effector CD8 ^{+} T cells might not be sufficient to eliminate tumor cells completely. The results are consistent with recent findings in literature, where Budhu et al. reported that CD8 ^{+} T cell concentration determines their efficiency in killing melanoma cells [48]. As reported in [49, 50], immunotherapy of patients with cancer requires the in vivo generation of large numbers of highly reactive antitumor lymphocytes that are not restrained by normal tolerance mechanisms and are capable of sustaining immunity against solid tumors. Immunization of melanoma patients with a broader array of cancer antigens can increase the number of circulating effector CD8 ^{+} T cells (eCTLs), but to date this has not correlated with clinical tumor regression, suggesting a defect in function of the eCTLs. In contrast, a clinical benefit has been observed in patients with metastatic melanoma using antibodies against CTLA4, which globally increase the number of circulating CD8 ^{+} T cells irrespective of antigen specificity [4]. The onetime adenovirus immunization experimental data [15, 16] and the computational simulations exploring the efficacy of multiple adenovirus vaccinations using our calibrated model all indicate the limited impact of a single antigenspecific therapy to eliminate tumors. Our simulation results also suggest that increasing the cytotoxic activity of effector CD8 ^{+} T cell or the local proliferation of CD8 ^{+} T cells within the tumor microenvironment, as observed following antiPD1 therapy [51], may completely eliminate tumor cells.
Conclusions
In summary, we present a multiscale mechanistic model of CD8 ^{+}mediated control of tumor growth in response to adenovirusvaccination mediated T cell stimulation using a system of impulsive ordinary differential equations. The model parameters were calibrated against experimental data employing a genetic algorithm whose fitness function is given by a linear combination of sum of normalized error squared and sum of normalized difference of slopes squared. With the calibrated parameter values, our model predictions match experimental data very well. Stability analysis via linearization implies that, in the case of no vaccination treatment, a small tumor will grow to its carrying capacity as a result of a stable tumorfree equilibrium and a unstable high tumor equilibrium. Using the calibrated model, numerical simulation of multiple adenovirus vaccinations suggest that this treatment strategy will significantly prolong T cell immune response but not necessarily enhance a cytotoxic CD8 ^{+} T cell response to a tumor antigen that noticeably reduces tumor size. A reduction in tumor size can be obtained if the cytotoxic activity or proliferation of effector CD8 ^{+} T cells present within the tumor microenvironment are enhanced. Along those lines, simulation results also show that a tumor may be completely eliminated by a single adenovirus vaccination that creates a highly enhanced cytotoxic T cell efficacy or with enhanced local proliferation of cytotoxic T cells within the tumor. Overall, the results illustrate how mechanistic models can be used to predict tumor growth response to antigenspecific immunotherapies and screen in silico for optimal therapeutic dosage and timing in treating patients with cancer.
Ethics
The authors declare that no animal or human experiments were performed as part of the research for this manuscript.
Declarations
Acknowledgements
This work was supported by grants from the National Science Foundation (CAREER 1053490 to DJK), the National Cancer Institute (R15CA132124 to DJK), and the National Institute of General Medical Sciences of the National Institutes of Health grant as part of the West Virginia IDeA Network of Biomedical Research Excellence (P20GM103434 to QW). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NSF, the NCI, or the National Institutes of Health. The authors thank Logan Lyda and Darryl Johnson for assistance with data analysis of the simulation results.
Authors’ Affiliations
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