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Tumor Biology |
Institute for Advanced Study, Princeton, New Jersey 08540
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ABSTRACT |
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 Top ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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INTRODUCTION |
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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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.
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MATERIALS AND METHODS |
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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 | (1) |
 | (2) |
 | (3) |
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:
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 | (5) |
 | (6) |
 | (7) |
 | (8) |
 | (9) |
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.
 | (10) |
 | (11) |
 | (12) |
 | (13) |
 | (14) |
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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.
 | (16) |
 | (17) |
 | (18) |
 | (19) |
 | (20) |
 | (21) |
 | (22) |
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RESULTS |
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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. 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 ß (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.
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. The presence of the virus alters the size of the tumor. The system can converge to one of two qualitatively different equilibria. Either a given fraction of the tumor cell population is infected with the virus or the virus infects 100% of the tumor cells (Fig. 2)
. In the latter case, uninfected tumor cells are absent (x = 0). These outcomes can be observed both in the absence and in the presence of a CTL response. In the absence of CTL, viral replication and cytotoxicity alone are responsible for altering the size of the tumor. In the presence of CTL, lysis of infected cells contributes to the dynamics. These two scenarios will be examined in turn.
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. The virus infects all of the tumor cells if a < s(d + ßk)/(r + ßk).
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. 2A)
. 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. 2A)
. On the other hand, if not all of the tumor cells are infected, then x + y = k (r - s + a - d)/(ßk + r - s). Now, an increase in the viral cytotoxicity increases tumor load (Fig. 2A)
. 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. 2A)
. The optimal viral cytotoxicity is thus given by 
. At this optimal cytotoxicity, the tumor size is reduced maximally and is 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, ß (Fig. 2A)
. 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. 3A)
. 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. 3A)
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. 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. 3B)
. 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 > bß(ßk + r - s)/[r(ßk - a) - d(ßk - 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. 2B)
. The virus can remain 100% prevalent in the tumor in the presence of CTL. In this case, overall tumor size is given by 
. At this equilibrium, an increase in the CTL responsiveness against the virus decreases the tumor size (Fig. 2B)
. On the other hand, if the CTL responsiveness crosses a threshold given by cv > b(ßk + r)/[k(r - d)], the virus does not maintain 100% prevalence in the tumor cell population, and the overall tumor size is given by 
. In this case, an increase in the CTL responsiveness to the virus increases tumor load and is detrimental to the patient (Fig. 2B)
. This is because the CTL response kills the virus faster than it can spread. Hence, the optimal CTL responsiveness is given by 
. At this optimal CTL responsiveness, the tumor size is reduced maximally and is given by 
. When the replication rate of the virus is faster, the optimal CTL responsiveness is higher and the minimum size of the tumor that can be attained by therapy is lower (Fig. 2B)
. The minimum tumor size that can be achieved is the same as in the previous case where viral cytotoxicity alone was responsible for reducing the tumor. The effect of the CTL response is to modulate the overall death rate of infected cells with the aim of pushing it toward its optimum value.
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.
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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. 5A)
. If cT > b(ßk + 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. 5A)
. In this case, the size of the tumor is given by x + y = k(r - s + a - d)/(ßk + 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. 5A)
. 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. 5A)
. 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. 5B
shows a simulation of treatment underscoring this result.
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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)/(ßk + r - s). The tumor-specific response persists if cT > cv2r/{k[cv(r - d) - bß]}. 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 
. Thus, a strong tumor-specific response, cT, reduces tumor load. On the other hand, a strong virus-specific response, cv, increases tumor load. The reason is that a strong virus-specific response results in low virus load and, therefore, in low stimulatory signals promoting the induction of tumor-specific immunity. This last statement only applies to the parameter region where both types of CTL responses coexist. The effect of virus-specific and tumor-specific responses alone has been analyzed and discussed extensively above.
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DISCUSSION |
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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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.
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FOOTNOTES |
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1 To whom requests for reprints should be addressed, at Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540. Phone: (609) 734-8048; Fax: (609) 951-4438; E-mail: wodarz{at}ias.edu 
Received 10/20/00. Accepted 2/16/01.
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TopABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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