Abstract
Recent research has indicated that viruses specifically infecting tumor cells could be used as an alternative therapeutic approach in cancer patients. A particular example is the adenovirus ONYX-015, which has entered clinical trials in the context of head and neck cancer. Successful therapy crucially requires an understanding about how viral and host parameters influence tumor load. The interactions between the growing tumor, the replicating virus, and possible immune responses are multifactorial and nonlinear. Hence, a complete understanding of how virus and host characteristics influence the outcome of therapy requires mathematical models. In this study, such mathematical models are presented and analyzed. The study investigates three possible scenarios that could be relevant for therapy: (a) viral cytotoxicity alone kills tumor cells; (b) a virus-specific lytic CTL response contributes to killing of infected tumor cells; (c) the virus elicits immunostimulatory signals within the tumor that promote the development of tumor-specific CTL. The models precisely define conditions required for successful therapy. They identify the parameters that need to be measured and modulated to evaluate and refine the existing therapy regimes.
INTRODUCTION
Traditional therapy of tumors and cancers is characterized by a relatively low efficacy and high toxicity for the patient. Although efforts are under way to design more efficient drugs that target genetic abnormalities only present in cancer cells, advances in genetic engineering have opened up possibilities to use replicating viruses as “biological control agents” to combat tumors (1). Several viruses have been altered to selectively infect cancer cells. Some examples are herpes simplex virus-1, NDV, and adenoviruses (1). A specific example that has drawn attention recently is ONYX-015, an attenuated adenovirus that selectively infects tumor cells with a defect in p53 (1, 2, 3, 4, 5, 6, 7). This virus has been shown to have significant antitumor activity and has proven relatively effective at reducing or eliminating tumors in clinical trials (8, 9, 10).
How do such viruses combat tumors? There are two modes of virus-mediated antitumor activity (1). First, the virus can be directly cytolytic to the tumor cells thereby contributing to tumor remission. Second, the presence of the virus might induce specific immune responses that lyse tumor cells. In individuals with solid cancers, tumor-specific CTL responses are characteristically absent (11, 12, 13). Although a fraction of patients bear tumors that have down-regulated major histocompatibility complex and, therefore, are resistant to CTL, many tumors are potentially immunogenic yet lack significant responses (11, 12, 13, 14). Several hypotheses can be put forward to account for this observation (11). One possibility is that uptake of tumor antigen by professional antigen-presenting cells is inefficient because the tumor cells are proliferating and long-lived cells. Related to this, absence of necrotic tumor cell death could result in the absence of so called danger signals that might be required for the efficient induction and activation of specific immunity (11, 12). According to these arguments, tumor-specific viruses can induce immune responses in two ways: (a) virus antigen displayed on tumors can lead to virus-specific responses directed against virus-infected tumor cells; and (b) the presence of the virus could alert tumor-specific immunity. This could be achieved by delivering the missing “danger signal” and by enhancing presentation of tumor antigens on antigen presenting cells after virus-mediated destruction of the tumor cells.
The central question in this context concerns the optimal characteristics of a virus required for combating tumors (1). Apart from the obvious requirement for selective cancer cell infection, it has been argued that viral replication and viral cytotoxicity, as well as the induction of virus- and tumor-specific CTL responses, might all be beneficial to the patient. On the other hand, detrimental immune responses, slowing down the replication rate of the virus, should be avoided, and the virus should not integrate into the human genome. For details, see Ref. (1). The interactions between the growing tumor, the replicating virus population, and antiviral immune responses are highly complex and nonlinear. Hence, to precisely define the conditions that are required for successful therapy by this approach, mathematical models are needed. In this study, mathematical models are constructed describing the interactions between the tumor, the virus, and the immune system. The models are used to define the viral characteristics required for tumor remission and to evaluate the efficacy of virus-mediated anticancer therapy. The report starts by examining the interactions between the tumor, the virus, and the virus-specific CTL response. The model is then extended to include not only virus-specific but also tumor-specific CTL.
MATERIALS AND METHODS
Dynamics between the Tumor, the Virus, and Virus-specific CTL.
The model consists of three variables: uninfected tumor cells (x), tumor cells infected by the virus (y), and virus-specific CTL (zv). It is given by the following set of differential equations:
The CTL response is modeled according to predator-prey dynamics. Upon exposure to antigen, the CTL proliferates and kills. There are many different functional responses that can be used to model CTL dynamics; however, the exact functional response that would be applicable is currently not known. The equation used here represents a relatively simple response. CTL proliferation is directly proportional both to the amount of antigen, y, and the number of CTL, z. Because no saturation term is involved, this represents a relatively strong and efficient response. The advantage of this model is its analytical simplicity. Alternative and more complicated functional responses have been tested analytically and numerically, and the results presented here remain robust. Different ways of modeling CTL dynamics have been compared by DeBoer and Perelson (15) and by Wodarz et al.(16).
Furthermore, the model is simplified in that it assumes mass action kinetics. The spatial structure of some solid tumors might require spatially explicit models. I have analyzed a simple cellular automaton in which an infected cell can only infect its nearest neighbors. Extensive simulations of such models gave rise to results that were qualitatively very similar to the ones obtained from the simple mass action model. Although the kinetics differed, the possible outcomes and the condition under which those outcomes were achieved were qualitatively the same. Hence, in this context, the models presented here are a valid first step to investigate the basic dynamics of tumor cell-infecting viruses. More complicated spatial models will be an additional step but would be beyond the scope of this paper.
In the absence of the virus, the trivial equilibrium is attained given by E0:
The virus can establish an infection in the tumor cell population if [βk (r − d) + sd]/r > a. First, consider virus infection in the absence of a CTL response. The virus can either attain 100% prevalence in the tumor cell population, or it may only infect a fraction of the tumor cells. One hundred percent virus prevalence is described by equilibrium E1:
Coexistence of infected and uninfected tumor cells is described by equilibrium E2:
Next, consider virus replication in the presence of the virus-specific CTL response. One hundred percent virus prevalence in the tumor cell population is described by equilibrium E3:
Coexistence of infected and uninfected cells is described by equilibrium E4:
The model described here assumes that upon division of infected cells, the virus is passed on to both daughter cells. Although this is the case for viruses that integrate into the tumor cell genome, this assumption should also be appropriate for nonintegrating viruses, because active virion production should result in a very high probability that the virus is transmitted to both daughter cells. A more detailed model would assume that with a probability q, the virus is passed on to both daughter cells, whereas with a probability (1 − q), the virus is lost in one of the daughter cells. Analysis of such a model (data not shown) demonstrates that conditions for tumor remission are qualitatively identical to the case examined here. The difference is that the expressions for invasion conditions and minimum tumor load are shifted by a constant factor.
Virus Infection and Tumor-specific CTL.
A model is described taking into account the interactions between the tumor, tumor-specific CTL, and the virus. It contains three variables: uninfected tumor cells (x), infected tumor cells (y), and tumor-specific CTL (zT). It is given by the following set of differential equations.
The focus is on the equilibrium expressions in the presence of the tumor-specific CTL. Again, the virus can attain 100% prevalence in the tumor cell population, or only a fraction of the cells can be infected. One hundred percent prevalence in the tumor population is described by equilibrium E1:
Coexistence of infected and uninfected tumor cells is described by equilibrium E2:
Interaction between Tumor-specific and Virus-specific CTL.
The two above described models are combined to study the interactions between the virus-specific and the tumor-specific CTL responses. The model is given by the following set of differential equations.
As explained in the text, coexistence of both CTL responses is only possible if the virus does not attain 100% prevalence in the tumor cell population. If this is the case, the system converges to the following equilibrium:
RESULTS
Basic Dynamics between Tumor and Virus.
This section investigates the basic dynamics between a growing tumor population, a replicating virus selective for the tumor cells, and a specific CTL response directed against viral antigen displayed on the surface of infected tumor cells. Various aspects of tumor growth and inhibition have been modeled in a variety of ways (17, 18, 19, 20). This study concentrates on a simple model, capturing the essential assumptions for analyzing virus-mediated therapy. The model contains three variables: uninfected tumor cells (x), tumor cells infected by the virus (y), and CTL specific for viral antigen (zv). It is explained schematically in Fig. 1. For mathematical details, see “Materials and Methods.” The tumor cells grow in a logistic fashion at a rate r and die at a rate d. The maximum size or space the tumor is allowed to occupy is given by its carrying capacity k. The virus spreads to tumor cells at a rate β (this parameter can be viewed as summarizing the replication rate of the virus). Infected tumor cells are killed by the virus at a rate a and grow in a logistic fashion at a rate s. This assumes that division of infected tumor cells results in both daughter cells carrying the virus. This would certainly be the case with a virus that integrates into the tumor cell genome, but with a nonintegrating virus, the chances of transmission upon cell division should be sufficiently high to justify this assumption (for a more detailed discussion, see “Materials and Methods”). The virus-specific CTL expands in response to antigen at a rate cv and decays at a rate b (for a detailed discussion of CTL dynamics, see “Materials and Methods”). The CTL kills infected tumor cells at a rate pv.
In the absence of the virus, the tumor eventually grows to its maximum size, given by
Viral Cytotoxicity.
This section investigates how the replicating virus by itself can alter the size of the tumor. The virus can establish an infection in the tumor cell population if [βk (r − d) + sd]/r > a. Then, the virus attains 100% prevalence in the tumor population, or only a fraction of the tumor cells are infected (Fig. 2 A). The virus infects all of the tumor cells if a < s(d + βk)/(r + βk).
With this result in mind, how does viral cytotoxicity influence the size of the overall tumor? The tumor size is defined as the sum of infected and uninfected tumor cells, x + y, at equilibrium. Viral cytotoxicity has an opposing influence on tumor load, depending on which equilibrium is attained (Fig. 2,A). If all of the tumor cells are infected, then x + y = k(s − a)/s. An increase in viral cytotoxicity results in a reduction in tumor load (Fig. 2,A). On the other hand, if not all of the tumor cells are infected, then x + y = k (r − s + a − d)/(βk + r − s). Now, an increase in the viral cytotoxicity increases tumor load (Fig. 2 A). The reason is that increased rates of tumor cell killing eliminate infected tumor cells before the virus had a chance to significantly spread. This, in turn, increases the pool of uninfected tumor cells and, therefore, the tumor load.
Hence, there is an optimal cytotoxicity, aopt, at which the tumor size reaches a minimum. This optimum is the degree of cytotoxicity at which the system jumps from the equilibrium describing 100% virus prevalence to the equilibrium where uninfected tumor cells are also present (Fig. 2 A). The optimal viral cytotoxicity is thus given by
There are a number of points worth noting about this result. The minimum tumor size this therapy regime can achieve is most strongly determined by the replication rate of the virus, β (Fig. 2 A). When the replication rate of the virus is higher, the minimum size of the tumor is smaller. To achieve this minimum, the viral cytotoxicity must be around its optimum value. A major determinant of the optimal viral cytotoxicity is the rate of growth of uninfected and infected tumor cells (r and s, respectively).
If the infected tumor cells grow at a significantly slower rate relative to uninfected cells (s ≪ r), the optimal cytotoxicity is low (Fig. 3,A). In the extreme case where the virus abolishes the ability of the tumor cell to divide, a noncytotoxic virus is required to achieve optimal treatment results. More cytotoxic viruses result in tumor persistence (Fig. 3 A).
On the other hand, if the growth rate of infected tumor cells is not significantly lower than that of uninfected tumor cells, a more cytotoxic virus is required to achieve minimum tumor size (Fig. 3,B). If viral cytotoxicity is too weak, the tumor persists. However, if the viral cytotoxicity is too high, the tumor also persists because infected cells die too fast for the virus to spread efficiently (Fig. 3 B). In general, when the replication rate of the virus is faster, the optimal level of cytotoxicity is higher.
Virus-specific CTL.
The effect of an antiviral CTL response is examined here. We assume that the CTL expand in response to viral antigen.
First, we define the conditions under which an antiviral CTL response is established. This condition is different depending on whether the virus attains 100% prevalence in the tumor cell population in the absence of the CTL. The strength of the CTL response, or CTL responsiveness, is denoted by cv. If the virus has attained 100% prevalence in the absence of CTL, the CTL become established if cv > bs/[k(s − a)]. On the other hand, if the virus is not 100% prevalent in the tumor cell population in the absence of CTL, the CTL invades if cv > bβ(βk + r − s)/[r(βk − a) − d(βk − s)].
Given that the CTL response becomes established, we investigated how the presence of the CTL influences the outcome of therapy. Again, two situations have to be distinguished.
If the virus has established 100% prevalence in the tumor cell population in the absence of the CTL response, the presence of CTL can both be beneficial and detrimental to the patient (Fig. 2,B). The virus can remain 100% prevalent in the tumor in the presence of CTL. In this case, overall tumor size is given by
Fig. 4 shows a simulation of therapy where an intermediate CTL responsiveness results in tumor remission, whereas a stronger CTL response can result in failure of therapy because virus spread is inhibited.
If the virus is not 100% prevalent already in the absence of the CTL response, a CTL-mediated increase in the death rate of infected cells can only be detrimental to the patient because it increases tumor load. The system converges to an equilibrium tumor size described by
Virus Infection and Tumor-specific CTL.
The above sections explored how virus infection and the virus-specific CTL response can influence tumor load. However, virus infection might not only induce a CTL response specific for viral antigen displayed on the surface of the tumor cells. In addition, active virus replication could induce a CTL response specific for tumor antigens (11, 12). The reason is that virus replication could result in the release of substances and signals alerting and stimulating the immune system. This could be induced by tumor antigens being released and taken up by professional antigen presenting cells and/or by danger signals released from the necrotic tumor cells. Here, such a tumor-specific CTL response is included in the model. It is assumed that the responsiveness of the tumor-specific CTL requires two signals: (a) the presence of the tumor antigen; and (b) the presence of infected tumor cells providing immunostimulatory signals. First, the interactions between the tumor, the virus, and the tumor-specific CTL are investigated. In a second step, the model is expanded to include both a tumor-specific and a virus-specific CTL response.
Tumor-specific CTL.
A model is constructed describing the interactions between the tumor population, the virus population, and a tumor-specific CTL response. It takes into account three variables: uninfected tumor cells (x), infected tumor cells (y), and tumor-specific CTL (zT). Mathematical details of the model are given in “Materials and Methods,” and Fig. 1 schematically shows the assumptions underlying the equations. The CTL expand in response to tumor antigen, which is displayed both on uninfected and infected cells (x + y) at a rate cT. However, it is assumed that the tumor-specific CTL response only has the potential to expand in the presence of the virus, y. In the model, virus load correlates with the ability of the tumor-specific response to expand, because high levels of viral replication result in stronger stimulatory signals. The tumor-specific CTL kills both uninfected and infected tumor cells at a rate pT.
If the virus has reached 100% prevalence in the absence of CTL, the tumor-specific CTL response becomes established if cT > bs2/[k(a − s)]2. If infected and uninfected tumor cells coexist in the absence of CTL, the tumor-specific CTL response becomes established if
We investigate how the responsiveness of the tumor-specific CTL, cT, influences the size of the tumor, x + y. The presence of the tumor-specific CTL can have the following effects. If the virus achieves 100% prevalence in the tumor cell population, then x + y = (b/cT)1/2. Thus, an increase in the responsiveness of the tumor-specific CTL results in a decrease in tumor load (Fig. 5,A). If cT > b(βk + r − s)2/[k (r − s + a − d)]2, the virus is not 100% prevalent in the tumor cell population. This switch is thus promoted by a high responsiveness of the tumor-specific CTL relative to the replication rate of the virus (Fig. 5,A). In this case, the size of the tumor is given by x + y = k(r − s + a − d)/(βk + r − s). This is the minimum tumor size that can be achieved. Thus, if the CTL responsiveness against the tumor lies above a threshold, tumor load reaches its minimum (Fig. 5,A). It also becomes independent of the strength of the CTL. Hence, a CTL responsiveness that lies above this threshold is not detrimental to the patient. In this situation, tumor size is determined by the replication rate and the cytotoxicity of the virus (Fig. 5,A). When the replication rate of the virus is higher and the degree of viral cytotoxicity is lower, the tumor is smaller. The reason is that fast viral replication and low cytotoxicity result in a higher virus load, which in turn results in stronger signals to induce the tumor-specific CTL. Fig. 5 B shows a simulation of treatment underscoring this result.
One note of caution is that the model assumes that the production of immunostimulatory signals induced by the virus is proportional to the amount of viral replication. If cellular debris after virus-mediated destruction of cells also contributes to these signals, then the effect of viral cytotoxicity could be more complex. However, the exact nature and concept of the so-called danger signals is still controversial. The model takes into account the simple observation that presence of signals typical of viral replication can enhance immunity to tumors.
Tumor-specific and Virus-specific CTL.
In this section, the two models explored thus far are combined. That is, both the virus- and the tumor-specific CTL responses are taken into consideration. The model is written out in “Materials and Methods,” and basic assumptions are depicted in Fig. 1. In this model, the virus- and the tumor-specific CTL responses are in competition with each other, because both can reduce tumor load and, hence, the strength of the stimulus required to induce CTL proliferation. In the following, these competition dynamics are examined.
If the virus has reached 100% prevalence in the tumor cell population in the absence of CTL, then virus- and tumor-specific CTL cannot coexist. If cv > (cTb)1/2, then the virus-specific CTL response is established. On the other hand, if cv < (cTb)1/2, then the tumor-specific CTL response becomes established.
If both infected and uninfected tumor cells are present in the absence of CTL, the situation is more complicated. Three outcomes are possible. The virus-specific response becomes established, the tumor-specific response becomes established, or both responses can coexist. The virus-specific response persists if cv > kcT(r − s + a − d)/(βk + r − s). The tumor-specific response persists if cT > cv2r/{k[cv(r − d) − bβ]}. Coexistence of both CTL responses is only observed if both of these conditions are fulfilled. The effect of either response alone has been explored above. If both responses coexist, then the size of the tumor is given by
DISCUSSION
This study has examined mathematical models to define viral and host characteristics necessary for successful cancer therapy using tumor cell-infecting viruses. Success of therapy was correlated with a low number of infected and uninfected tumor cells after therapy. Because we used deterministic models, the tumor can never go completely extinct but can be reduced to very low levels. The lower the equilibrium size of the tumor during therapy, the higher the chances that the tumor might be eliminated in practical terms (tumor size reduced below one tumor cell).
The outcome of therapy depends on a complex balance between host and viral parameters. An important variable is the death rate of infected tumor cells. To achieve maximum reduction of the tumor, the death rate of the infected cells must be around its optimum, defined by the mathematical models in this study. If the death rate of infected cells lies around its optimum, a fast replication rate of the virus and a slow growth rate of the tumor increase the chances of tumor eradication.
Three scenarios were investigated: (a) viral cytotoxicity alone kills tumor cells; (b) a CTL response against the virus contributes to killing infected tumor cells; and (c) the virus helps eliciting a tumor-specific CTL response after the release of immunostimulatory signals.
Viral Cytotoxicity.
The first and most basic question concerns the cytotoxicity of the virus required to eliminate the tumor. According to the literature (1), a desirable attribute for a tumor-specific virus should be that it causes lysis of infected cells, and it has been suggested that this could even be enhanced, e.g., by including toxin-encoding genes within the virus (1, 21). The models analyzed in this study suggest that the situation is more complicated than this. The optimal rate of virus-mediated cell killing depends on the growth rate of infected tumor cells relative to uninfected tumor cells.
If the infected cells grow at a significantly slower rate than uninfected cells, optimal treatment results are obtained with viruses characterized by a low degree of cytotoxicity. If the growth rate of infected tumor cells is low, the virus can only be maintained for a sufficiently long period of time if infected cells also die at a slow rate. If the virus is not maintained for a sufficiently long period of time, it fails to kill the tumor. Hence, one strategy could be to engineer a virus that interferes with the cell cycle of the tumor cells and is only weakly cytotoxic.
If the growth rate of infected tumor cells is not reduced relative to that of uninfected tumor cells, then a more cytotoxic virus is required for optimal treatment outcome. In general, when the replication rate of the virus is faster, the optimal level of viral cytotoxicity is higher.
Immune Responses.
Another important question concerns the role of immune responses for the outcome of therapy. Because a fast rate of viral replication promotes tumor extinction, any immune response that directly reduces the replication rate of the virus is detrimental to the patient. Some examples are antibody responses or other nonlytic effector mechanisms. However, in solid tumors this might be less of a problem because antibodies do not penetrate such tumors efficiently. On the other hand, antibodies could inhibit the virus to spread from a primary tumor to metastatic growths at other locations, and this could compromise the efficacy of therapy. It has been documented that removal of a primary tumor can lead to the outgrowth of micrometastatic patches (22, 23, 24). In this context, virus-mediated destruction of the primary tumor could be detrimental to the patient if the virus cannot spread to those metastatic patches and kill them.
The effect of specific lytic responses, such as CTL, is more complex. Two possibilities were examined. CTL could react against viral antigens, or the virus could deliver stimulatory signals that result in the development of CTL specific against tumor antigens.
An antiviral CTL response is only beneficial to the patient if the viral cytotoxicity lies below its optimum value. In this case, the CTL response is most effective if it pushes the death rate of infected cells toward the optimum. If the CTL response is stronger, it is detrimental to the patient, because the infected cells are killed too fast for the virus to spread efficiently. When the replication rate of the virus is faster, the CTL response has to be stronger to achieve optimum treatment results. It is important to note that the actual minimum size of the tumor that can be achieved in the model is the same both when viral cytotoxicity acts alone and when a CTL response is also present. The effect of a CTL response is to modulate the overall death rate of infected cells with the aim of pushing it toward its optimum value.
The situation is different with the tumor-specific CTL response. A strong tumor-specific CTL response is never detrimental to the patient. If the strength of the CTL lies below a threshold, an increase in the tumor-specific response reduces overall tumor load. If the strength of the CTL crosses a threshold, tumor load is maximally suppressed and becomes independent of the CTL response. In this case, the model suggests that a high rate of viral replication and a low degree of viral cytotoxicity can suppress tumor load, because this ensures sufficient levels of viral growth to provide the immunostimulatory signals. Because virus-specific CTL responses reduce virus load, they also reduce the stimulatory signals and, hence, weaken tumor-specific immunity. If the goal is to use the virus to deliver those signals required for a tumor-specific CTL response, it would be a good strategy to also vaccinate the patients with tumor-specific antigen. This increases the strength of the tumor-specific CTL.
Treatment Strategies.
The above discussion suggests that the most straightforward way to use viruses as anticancer weapons is in the absence of immunity. If the cytotoxicity of the virus is around its optimum value, minimum tumor size is achieved. The optimum cytotoxicity, in turn, depends on the replication rate of the virus as well as on the growth rate of infected and uninfected tumor cells.
If a virus-specific CTL response is induced, the best strategy would be to use a fast replicating and weakly cytotoxic virus. This is because the CTL will increase the death rate of infected cells. If the overall death rate of infected cells is too high, this is detrimental to the patient, because virus spread is prevented. In addition, a weakly cytotoxic and fast replicating virus also provides the strongest stimulatory signals for the establishment of tumor-specific immunity.
Because the model suggests that a fast growth rate of the tumor decreases the efficacy of treatment, success of therapy could be promoted by using a combination of virus therapy and conventional chemotherapy or radiotherapy. These suggestions are supported by recent experimental data (7, 25, 26, 27). A combination of treatment with the adenovirus ONYX-015 and chemotherapy or radiotherapy has been shown to be significantly more effective than treatment with either agent alone.
An issue that has been left open in the modeling thus far concerns the timing of therapy. In the model, the eventual outcome of therapy is independent of the size of the tumor when therapy is started, and, hence, it is independent of the timing of therapy. In the model, late start of therapy only results in a longer time period until the outcome of therapy is reached. This is because the analysis concentrated on equilibrium outcomes. Timing of therapy might become an important variable in a variety of circumstances. If therapy is initiated too late, the tumor might have grown to a size at which the virus cannot reduce the tumor before the death of the host. In addition, at later stages of the tumor, the spatial structure might limit the amount of virus spread (For a discussion of possible spatial extensions of the models, see “Materials and Methods”). Related to this, if metastatic patches have already been formed, the virus might successfully combat the primary tumor while allowing a burst of secondary tumors.
Conclusion.
In conclusion, the mathematical models have allowed us to precisely define the conditions under which treatment with tumor cell-infecting viruses is most likely to result in an optimal outcome. The analysis has demonstrated that success of this therapy regime depends on a complex balance between host and viral parameters. In particular, it depends on the fine-tuning between the rate of tumor growth, the rate of viral replication, the cytotoxicity of the virus, and the presence or absence of specific immune responses. Further experimental work should be coupled with mathematical modeling to estimate these parameters in specific systems. This would allow model predictions to be tested and could result in a refinement of this treatment regime.
Schematic representation of the assumptions underlying the mathematical models. In the text, the model is built up gradually. It starts with the interactions between the virus and the tumor cells. Then a virus-specific CTL response is added, followed by including a tumor-specific CTL response. For mathematical details, see “Materials and Methods.”
Schematic representation of the assumptions underlying the mathematical models. In the text, the model is built up gradually. It starts with the interactions between the virus and the tumor cells. Then a virus-specific CTL response is added, followed by including a tumor-specific CTL response. For mathematical details, see “Materials and Methods.”
A, dependence of overall tumor load on the cytotoxicity of the virus. There is an optimal cytotoxicity at which tumor load is smallest. This is also the point where the system switches from the equilibrium describing 100% virus prevalence in the tumor population to the equilibrium where infected and uninfected tumor cells coexist. The faster the rate of virus replication, the higher the optimal level of cytotoxicity and the smaller the minimum tumor load. Parameters were chosen as follows: k = 10; r = 0.2; s = 0.2; d = 0.1; for fast viral replication, β = 1; for slow viral replication, β = 0.1. B, dependence of overall tumor load on the strength of the virus-specific CTL response. There is an optimal CTL responsiveness at which tumor load is smallest. This is also the point where the system switches from the equilibrium describing 100% virus prevalence in the tumor population to the equilibrium where infected and uninfected tumor cells coexist. The faster the rate of virus replication, the higher the optimal strength of the CTL response and the smaller the minimum tumor load. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; d = 0.1; b = 0.1; P = 1; a = 0.2; for fast viral replication, β = 1; for slow viral replication, β = 0.1.
A, dependence of overall tumor load on the cytotoxicity of the virus. There is an optimal cytotoxicity at which tumor load is smallest. This is also the point where the system switches from the equilibrium describing 100% virus prevalence in the tumor population to the equilibrium where infected and uninfected tumor cells coexist. The faster the rate of virus replication, the higher the optimal level of cytotoxicity and the smaller the minimum tumor load. Parameters were chosen as follows: k = 10; r = 0.2; s = 0.2; d = 0.1; for fast viral replication, β = 1; for slow viral replication, β = 0.1. B, dependence of overall tumor load on the strength of the virus-specific CTL response. There is an optimal CTL responsiveness at which tumor load is smallest. This is also the point where the system switches from the equilibrium describing 100% virus prevalence in the tumor population to the equilibrium where infected and uninfected tumor cells coexist. The faster the rate of virus replication, the higher the optimal strength of the CTL response and the smaller the minimum tumor load. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; d = 0.1; b = 0.1; P = 1; a = 0.2; for fast viral replication, β = 1; for slow viral replication, β = 0.1.
Simulation of therapy using tumor cell-infecting viruses in the absence of immunity. A, the growth rate of infected tumor cells is significantly slower than that of uninfected tumor cells. A noncytotoxic virus now results in tumor eradiation. A more cytotoxic virus results in tumor persistence. Parameters were chosen as follows: k = 10; r = 0.5; s = 0; β = 1; d = 0.1; a = 0.1 for the noncytotoxic virus; and a = 0.5 for the more cytotoxic virus. B, the growth rate of infected tumor cells is not significantly reduced relative to that of uninfected cells. An intermediate level of cytotoxicity results in tumor eradication. Weaker or stronger levels of cytotoxicity result in tumor persistence. Parameters were chosen as follows: k = 10; r = 0.5; s = 0; β = 1; d = 0.1; a = 0.2 for the weakly cytotoxic virus; a = 0.55 for intermediate cytotoxicity; and a = 3 for strong cytotoxicity.
Simulation of therapy using tumor cell-infecting viruses in the absence of immunity. A, the growth rate of infected tumor cells is significantly slower than that of uninfected tumor cells. A noncytotoxic virus now results in tumor eradiation. A more cytotoxic virus results in tumor persistence. Parameters were chosen as follows: k = 10; r = 0.5; s = 0; β = 1; d = 0.1; a = 0.1 for the noncytotoxic virus; and a = 0.5 for the more cytotoxic virus. B, the growth rate of infected tumor cells is not significantly reduced relative to that of uninfected cells. An intermediate level of cytotoxicity results in tumor eradication. Weaker or stronger levels of cytotoxicity result in tumor persistence. Parameters were chosen as follows: k = 10; r = 0.5; s = 0; β = 1; d = 0.1; a = 0.2 for the weakly cytotoxic virus; a = 0.55 for intermediate cytotoxicity; and a = 3 for strong cytotoxicity.
Simulation of therapy using tumor cell-infecting viruses in the presence of virus-specific lytic CTL. An intermediate CTL responsiveness results in tumor eradication, whereas a stronger CTL response results in tumor persistence. With the stronger CTL response, the initial decay of the tumor is faster. However, subsequently the virus is removed from the tumor cell population before the tumor has been driven extinct. Therefore, the tumor cells can start to grow back again. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; β = 0.1; a = 0.2; P = 1; b = 0.1; d = 0.1. The intermediate CTL responsiveness is characterized by cv = 0.2625, whereas the stronger CTL response is characterized by cv = 2.
Simulation of therapy using tumor cell-infecting viruses in the presence of virus-specific lytic CTL. An intermediate CTL responsiveness results in tumor eradication, whereas a stronger CTL response results in tumor persistence. With the stronger CTL response, the initial decay of the tumor is faster. However, subsequently the virus is removed from the tumor cell population before the tumor has been driven extinct. Therefore, the tumor cells can start to grow back again. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; β = 0.1; a = 0.2; P = 1; b = 0.1; d = 0.1. The intermediate CTL responsiveness is characterized by cv = 0.2625, whereas the stronger CTL response is characterized by cv = 2.
A, dependence of overall tumor load on the strength of the tumor-specific CTL response. When the strength of the tumor-specific CTL is higher, the tumor load is lower. If the strength of the tumor-specific CTL crosses a threshold, tumor load becomes independent of CTL parameters. Instead it is determined by the replication rate and cytotoxicity of the virus. When the rate of virus replication is faster and the degree of viral cytotoxicity is smaller, the overall tumor load can be reduced further. The CTL responsiveness at which tumor load becomes independent of CTL parameters is also the point at which the system switches from the equilibrium describing 100% virus prevalence in the tumor population to the equilibrium where infected and uninfected tumor cells coexist. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; d = 0.1; b = 0.1. The fast replicating and weakly cytotoxic virus is characterized by β = 1 and a = 0.2. The slower replicating and more cytotoxic virus is characterized by β = 0.5 and a = 0.5. B, simulation of therapy using a tumor cell-infecting virus to stimulate a tumor-specific CTL response. If the virus replicates at a fast rate and is weakly cytotoxic, the level of immunostimulatory signals is high. Hence the tumor-specific response is strong and drives the tumor extinct. If the virus replicates slowly and is more cytotoxic, the level of stimulatory signals is lower. This compromises the efficacy of the tumor-specific CTL, which cannot drive the tumor into remission. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; d = 0.1; b = 0.1; cT = 0.2. The fast replicating and weakly cytotoxic virus is characterized by β = 0.5 and a = 0.2. The slower replicating and more cytotoxic virus is characterized by β = 0.1 and a = 0.6.
A, dependence of overall tumor load on the strength of the tumor-specific CTL response. When the strength of the tumor-specific CTL is higher, the tumor load is lower. If the strength of the tumor-specific CTL crosses a threshold, tumor load becomes independent of CTL parameters. Instead it is determined by the replication rate and cytotoxicity of the virus. When the rate of virus replication is faster and the degree of viral cytotoxicity is smaller, the overall tumor load can be reduced further. The CTL responsiveness at which tumor load becomes independent of CTL parameters is also the point at which the system switches from the equilibrium describing 100% virus prevalence in the tumor population to the equilibrium where infected and uninfected tumor cells coexist. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; d = 0.1; b = 0.1. The fast replicating and weakly cytotoxic virus is characterized by β = 1 and a = 0.2. The slower replicating and more cytotoxic virus is characterized by β = 0.5 and a = 0.5. B, simulation of therapy using a tumor cell-infecting virus to stimulate a tumor-specific CTL response. If the virus replicates at a fast rate and is weakly cytotoxic, the level of immunostimulatory signals is high. Hence the tumor-specific response is strong and drives the tumor extinct. If the virus replicates slowly and is more cytotoxic, the level of stimulatory signals is lower. This compromises the efficacy of the tumor-specific CTL, which cannot drive the tumor into remission. Parameters were chosen as follows: k = 10; r = 0.5; s = 0.5; d = 0.1; b = 0.1; cT = 0.2. The fast replicating and weakly cytotoxic virus is characterized by β = 0.5 and a = 0.2. The slower replicating and more cytotoxic virus is characterized by β = 0.1 and a = 0.6.
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