Abstract
Motivated by the rapid expansion in the development of replication-competent viral agents for the treatment of solid tumors, we formulated and analyzed a three-dimensional mathematical model of a tumor that is infected by a replication-competent virus. We initially considered three patterns of intratumoral injection in which a fixed fraction of cells are initially infected with the virus throughout (a) the entire tumor, (b) the tumor core, and (c) the tumor rim, respectively. For each injection pattern, an approximate analysis of the model provides a simple and accurate condition for whether the virus will eradicate the tumor. The model was then generalized to incorporate nutrient-limited necrosis and an innate immune response against virus-infected tumor cells. Recent preclinical and clinical data were used to validate the model and estimate key parameter values. Our analysis has the following implications: even in the absence of an immune response, tumor eradication requires widespread distribution of the virus within the tumor at the time of infection; core or rim injections alone may result in tumor escape, particularly in a well-vascularized tumor; the more rapidly a virus lyses infected cells the more effective it will be at controlling the tumor; and the innate immune response to the virus can potentially prevent the virus from controlling the tumor, even with repeat injections. Therefore, in addition to diffuse intratumoral infection, tumor eradication by oncolytic adenovirus will probably require potent suppression of innate immune clearance mechanisms (e.g., by replacement of adenovirus E3 genes), combinations with traditional (chemotherapy, radiotherapy) treatments, and/or concomitant therapeutic gene expression with resultant bystander effects.
INTRODUCTION
Nearly all of the approaches to cancer gene therapy suffer from low efficiency of gene transfer in vivo (1). Replication-competent agents may overcome this limitation by selectively replicating in and lysing tumor cells (2). Replication-selective agents have displayed antitumoral efficacy alone, with chemotherapy and with radiotherapy in murine tumor models, most of which were immunodeficient (3, 4, 5). Agents in, or nearing, clinical trials include the adenovirus dl1520 (otherwise known as ONYX-015; Refs. 6, 7), the herpes virus G207 (8), the prostate-specific adenoviruses CN706 and CN708 (now CG7060 and CG7080; Ref. 9), and a reovirus that attacks tumors with activated Ras pathways (10). Additional oncolytic virus species in development include vesicular stomatitis virus, polio virus, measles virus, Newcastle disease virus, vaccinia virus, and autonomous parvovirus (2). However, despite producing many thousands of infectious virions per infected cell within a tumor and causing necrosis, most viruses are unable to completely eradicate the majority of tumor nodules, even in immunodeficient mice with relatively small tumors. If larger tumors or immunocompetent mice are treated, the efficacy of such viruses can decrease even more (4, 10). In clinical trials with dl1520 adenovirus injected repeatedly into superficial squamous cell head and neck cancers, some antitumoral activity was demonstrated; however, durable complete responses were not demonstrated despite five consecutive daily treatments of four to eight needle tracts each, repeated every 3 weeks (6, 11). Similarly, durable complete responses were not reported from Phase I/II trials of dl1520 involving over 100 patients with other tumor types (including ovarian, colon, and pancreatic; Ref. 12). No objective tumor responses were reported from two Phase I clinical trials with different oncolytic herpes simplex viruses (13, 14). In summary, despite thousands of viruses being produced per infected cell, eradication of a solid three-dimensional tumor appeared to be more complex than originally proposed. To improve the efficacy of oncolytic viruses, we, therefore, needed to better understand the dynamics between virus replication, release, and spread; immune-mediated viral clearance; and three-dimensional tumor growth.
In the past decade, mathematical models of the intratumoral transport of monoclonal antibodies and other macromolecules have generated valuable insights into cancer treatment (15, 16, 17, 18, 19). In these studies, the macromolecules travel primarily by diffusion and convection (20). However, adenovirus particles are too large to diffuse within a tumor (Chart 8 in Ref. 20). Also, interstitial pressure is nearly constant (at a value roughly equal to the microvascular pressure in the tumor) throughout most of a tumor and is lower at the tumor rim (21), making convective spread of the virus into the tumor difficult. These delivery barriers, which have plagued traditional gene therapy approaches, may be at least partially bypassed by replication-competent viruses, which travel mainly by within-cell and between-cell transport via ongoing cycles of infection, replication, and cell destruction. However, the efficiency of this spread was unknown.
In this article, we formulated and analyzed a mathematical model of a spherical tumor that has been injected with a replication-competent virus. The model focuses on the infection of tumor cells by nearby free virus, the replication of virus within an infected cell, the release of virus during lysis, the repopulation of uninfected cells, and the loss of necrotic debris. For three different injection patterns, simple conditions for virus-induced tumor control are presented. We also determine how these conditions are affected by nutrient-limited central tumor necrosis, an immune response against the virus-infected tumor cells (6, 11, 22), and the infected cell lifetime (i.e., time to virus-mediated killing and release). Recent preclinical and clinical data are used to validate the model and to estimate the key parameter values, so that the tumor control conditions can be discussed within a clinical context. The sensitivity of “tumor eradication or control conditions” to modifications of the key parameter values is explored. These findings should lead to the design of improved oncolytic viruses and treatment approaches.
MATERIALS AND METHODS
The mathematical model describes a spherical tumor and tracks, over space and time, five physical entities within the tumor: (a) uninfected tumor cells; (b) infected tumor cells; (c) necrotic cells (it is mathematically convenient to treat the necrotic debris as cells); (d) free (i.e., extracellular) virus; and (e) an immune response (i.e., viral clearance). The model comprises a system of partial differential equations, which are stated in the Appendix (described in detail in Refs. 23 and 24) and briefly described here. All of the tumor cells are assumed to have radius rc, and their density θ is taken to be constant throughout the tumor. We assume that the volume occupied by free virus particles, which are several orders of magnitude smaller than tumor cells, and immune peptides (e.g., antiviral cytokines) are negligible. The tumor is modeled as an incompressible fluid in which spatiotemporal variations in cell proliferation and removal establish a convective velocity that drives cell motion. In particular, the convective velocity determines the movement (i.e., the growth rate) of the tumor boundary. Because the viruses are designed to be highly selective to cancer cells and have not exhibited significant toxicity in humans to date (2, 11), we neglect the impact of the treatment on normal tissue outside of the tumor mass.
The essential dynamics of the model are as follows. We assume that the uninfected tumor cells proliferate at rate λ. The virus infects tumor cells by binding to receptors on cell surfaces and gaining entry by endocytosis. To model this phenomenon, we assume that the rate at which an uninfected cell becomes infected is proportional to the average virus concentration on the surface of the uninfected spherical cell, the constant of proportionality β describing the ability of the virus to gain entry to the tumor cells (i.e., virus receptor density on tumor cells is not limiting). All of the infected cells are assumed to die (i.e., no lysogenesis occurs) at a constant rate δ, where δ−1 represents the mean infected-cell lifetime. At the time of death, infected cells become necrotic, and this necrotic debris is removed from the tumor at rate μ. Additionally, when an infected cell dies, we assume that N virus particles are released, so that Nδ is the release rate of free virus particles per unit time per infected cell. Because virus particles are present throughout the infected cell at the time of lysis, we assume that the N virus particles are released uniformly throughout a sphere of cell radius rc. The virus is assumed to experience first-order clearance at rate γ.
Finally, recent clinical studies (11, 25) of the adenovirus dl1520 described complete viral clearance over ≤10 days in conjunction with a cytokine-mediated immune response, including major acute increases in TNF.2 Hence, the model assumes that the innate immune response is stimulated at a rate that is proportional (with proportionality constant s) to the product of the infected cell and immune cell concentrations, incur second-order clearance with proportionality constant ω (first-order clearance was inconsistent with the data; Ref. 24), and kill infected tumor cells at a rate proportional (with proportionality constant k) to the product of the infected cell and immune cell concentrations.
Although viruses can be administered intra-arterially, i.v., or intratumorally (i.e., direct injection), the latter method results in superior intratumoral viral titers (26). By varying the initial conditions used to solve the model equations, we consider the impact on efficacy of three types of injection patterns: uniform injection, core injection, and rim injection. In uniform injection, we assume that a fixed fraction p of all tumor cells are initially infected. This is a good approximation to the animal studies in Ref. 26, in which a controlled mixture of pre-infected and uninfected tumor cells are mixed thoroughly and are subsequently injected into a mouse. The second pattern is core injection, in which a fraction p of the cells in the inner core (with radius R0 − w0, where R0 is the initial tumor radius and w0 is the uninfected rim width) of the tumor is infected. A single core injection was used in early mouse experiments with dl1520 (3, 6), early clinical trials with dl1520 (6), and initial trials with oncolytic herpes simplex virus mutants (13, 14) and with hrR3 (27). The third pattern is rim injection, in which a fraction p of cells in the outer rim (with rim width R0 − r0, where r0 is the uninfected core radius) of the tumor is infected. This mode of injection can be viewed as a crude representation of i.v. or intra-arterial administration (28). Fig. 1 depicts core and rim injections.
A central role in our analysis is played by the basic reproductive ratio,
This dimensionless parameter represents the mean number of new virus particles generated by a single virus particle that is inserted into a tumor consisting solely of uninfected tumor cells. In this interpretation, βθ is the number of cells infected per free virus particle per unit time. The mean lifetime of a free virus particle is γ−1, and, therefore, βθ/γ is the mean number of cells infected per particle. This quantity is multiplied by the factor N because each infected cell releases N virus particles at the time of lysis. We rearrange Eq. A and substitute R0γ/Nθ for β in the model, because R0 is easier to estimate from clinical data than β.
RESULTS
Parameter Estimation.
The mathematical results presented below depend on nine key parameters, which are listed in Table 1. The values of the cell radius rc and the tumor cell density θ are standard in the literature, and the value of the tumor proliferation rate λ, which can vary widely across tumor types, was taken to correspond to a tumor doubling time of 3 months, typical for head and neck tumors (6, 7). The infected cell lifetime value in Table 1 is representative of laboratory results (3, 4), and the immune clearance rate ω was determined from the half-life of TNF (25) in Wu et al. (24). The most difficult parameter values to measure are basic reproductive ratio R0, the immune stimulation rate s, and the immune killing rate k. These three values were estimated in a companion article (24) using clinical data from hospitalized patients who received injections of dl1520 for colon carcinoma metastases and were subsequently followed over a 2-week period (25); similar data were obtained from other clinical trials with dl1520 (12). The 21 temporal measurements of virus and TNF levels in the plasma (25), along with the virus and TNF levels predicted by the model, are displayed in Fig. 2. To account for the fact that the model tracks the tumor virus genomes but the data measure the plasma virus genomes, we needed to convert between plasma virus genomes and tumor virus genomes. As explained in our companion article (24), we performed this conversion by assuming that 1% of the virus released from dying tumor cells is released into the blood, in which it is cleared readily (half-life of 15 min) by the reticuloendothelial system (28). We made no attempt to convert TNF plasma levels into TNF tumor levels. To adequately fit the data, our model requires a time delay for the immune system to respond to the virus. Our analysis in Wu et al. (24) implies that a stable periodic solution, in which the oscillations do not vanish over time, arises if the time delay exceeds 27.6 h. Indeed, the beginning of a viral and TNF rebound can be observed in the clinical data in Fig. 2, in which the time delay is ∼42 h.
Tumor Eradication Thresholds.
For each of the three injection patterns, we derive approximate but accurate threshold conditions to predict whether or not the tumor is controlled by the virus in the absence of immune-mediated clearance. We also provide a threshold condition for uniform injection in the presence of immune-mediated clearance. All of these threshold conditions are independent of the percentage of cells initially infected. (See Refs. 23 and 24 for the mathematical derivations of these results, the confirmation of their accuracy, and supporting pictures displaying the spatio-temporal dynamics.) We state these results in Eqs. BCDEFGHI below, provide supporting numerical simulations in Fig. 3, and discuss these conditions in light of existing preclinical and clinical data.
In the uniform injection case, the tumor eventually
The dimensionless quantity R0 − 1 represents the net amount of free virus generated, after one round of infection and lysis, by a single virus particle injected into a pool of uninfected tumor cells. The dimensionless quantity λ(δ−1 + μ−1) is the natural logarithm of the number of uninfected tumor cells generated by a single uninfected cell during the time it takes for a newly infected cell to be removed from the tumor. Hence, these two dimensionless quantities measure the relative growth rates of the infection and the tumor, respectively, and Eq. B implies that the tumor is eradicated if and only if the virus infection can outpace the tumor proliferation, regardless of the size of the initial viral injection.
Using Table 1, we find that R0 = 3.73 and 1 + λ(δ−1 + μ−1) = 1.04; hence, in the absence of an immune response, the virus, if uniformly injected throughout the tumor, is powerful enough to eradicate the tumor (Fig. 3). Indeed, we predict that with no immune response, the virus can eradicate any tumor with λ ≤ ((1/δ) + (1/μ))−1(R0 − 1) = 0.02275 h−1, i.e., any tumor with a doubling time greater than 30.5 h, which covers virtually all solid tumors. This conclusion is consistent with results in a previous study (4), in which repeated injections were able to shrink large head and neck tumors in nude mice (although our model, because it has a continuous state and is deterministic, predicts that the eradication threshold is independent of the fraction of preinfected tumor cells, whereas the true threshold for the fraction of preinfected cells was about 0.03; Ref. 26).
Next, we assume that R0 > 1 + λ(δ−1 + μ−1) (i.e., the tumor is controlled by a uniform injection of the virus, as implied by the values in Table 1), and we investigate whether tumor control can be achieved by injecting only a portion of the tumor. Before describing our results for core and rim injections, we define
to be the wave speed of the infection. In terms of the natural unit of speed in our problem, which is the cell radius divided by the mean infected cell lifetime (rcδ), this wave speed (using Eq. C with R0 = 3.73) is 3.3rcδ = 0.0165 mm/day, or about 1 mm every 2 months, from every infected cell.
In the core injection case (see Fig. 1 a),
In the rim injection case (see Fig. 1 b),
Using the parameter values in Table 1, the uninfected rim width threshold is c0/λ = 2.16 mm in the core injected case in Ref. 4; that is, if a core injection leaves an uninfected rim wider than 2.16 mm, then the wave of infection will never reach the tumor boundary, and exponential growth will result. Similarly, the uninfected core radius threshold is 3c0/λ = 6.45 mm for rim injection in Ref. 5, and, therefore, leaving an uninfected core of at least this size will result in exponential tumor growth. Fig. 3 depicts two cases each, of rim and core injection, in which one case results in tumor control, the other in exponential growth. For a 2-cm-in-diameter tumor, these thresholds correspond to 73.1% (48.2%, respectively) of the tumor volume that needs to be injected in the rim (core, respectively) injection case, to control the tumor. Although this type of result is difficult to validate clinically, investigators on trials with dl1520 observed that a single central core injection was ineffective in controlling tumors in mice and in patients; despite central necrosis, there was tumor escape at the rim (see Fig. 2 of Ref. 6). Consequently, five daily injections, located equidistantly around the tumor periphery (rim) and into the core, were used in Phase II trials (11). Despite a slightly lower total virus dose, this injection technique was able to induce objective tumor responses in a fraction of patients. Similarly, increasing the volume of the viral suspension for intratumoral injection generated better spatial distribution within the tumor and increased efficacy in preclinical models (26), which is also consistent with the predictions of our model.
We next considered the rim injection case in the presence of nutrient-limited necrosis, in which the tumor consists of a necrotic core and a viable rim. In this case, uninfected tumor cells are sandwiched between a necrotic core and a rim of infected tumor cells. Referring to Fig. 1 c, we assume nutrients can readily diffuse through the infected rim (31) and can diffuse a radial distance r0 into the uninfected tumor tissue, where r0 is less than the initial uninfected tumor core radius r0. In this scenario:
and if (λ/3)r0 ≤ c0 ≤cM, then
where the constant cM and the function S1(c0) are too lengthy to include here, but can be found in Ref. 23. But Eqs. F and G together with Table 1 imply that the tumor is not eradicated by rim injection if the viable rim width is greater than 3c0/λ = 6.45 mm, which corresponds to a pretreatment steady-state tumor radius of 8.99 mm. A single diffuse rim injection eradicates the tumor if r0 < 6.06 mm, which corresponds to a pretreatment steady-state tumor radius of 8.44 mm. If 6.06 < r0 < 6.45 mm, then the eradication condition depends on the initial tumor core size r0. In this intermediate region, there are some cases in which repeated injections may eradicate the tumor (see Ref. 23 for details). Nutrient-limited necrosis complements rim injection by sandwiching the uninfected tumor cells, whereas nutrient-limited necrosis and core injection are highly redundant. Hence, rim injection outperforms core injection in the presence of nutrient-limited necrosis.
Finally, we consider uniform injection in the presence of immune-mediated clearance. In this case:
where the threshold T is given by:
Substituting parameter values from Table 1 into Eq. I, we see that efficient viral clearance increases the tumor control condition from R0 > 1.04 in Eq. B to R0 > 13.9 in Eq. I. Hence, R0, which equals 3.73 in Table 1, is less than 13.9, and the virus cannot control the tumor in the presence of this effective immune response (Fig. 3,b). Indeed, in clinical trials with dl1520 (Onyx-015), most of the virus is cleared within 10–14 days in most patients (11); this time frame is insufficient for the virus to eradicate the tumor after a single viral treatment. In other words, the virus is the loser in this three-way race. Because of the relatively low value of R0, we predict that even the slowest-growing tumors could not be eradicated by the virus in the presence of this highly efficient immune-mediated clearance. Moreover, simulations (not shown here) of repeated weekly injections were unable to put a visible dent in the tumor growth curve in Fig. 4 in the presence of the strong immune response.
Immune Suppressors.
One possible strategy to combat the effect of the immune response is to either coadminister an immune suppressor to the host or to engineer/allow the virus to express immune avoidance (“stealth”) genes. This would decrease the stimulation rate s in our model; because the immune response impacts the eradication threshold via the ratio sk:ω, a similar effect could be achieved by coadministering an agent that reduces the immune killing rate k or increases the immune clearance rate ω. For example, because the E3 10.4/14.5 and 14.7 genes that are deleted in dl1520 act to inhibit TNF-mediated clearance, replacement of these genes in a virus could substantially decrease the immune killing rate k. More generally, replacement of immune response-inhibitory genes in combination with immunosuppressive drugs and/or the inherently immunodeficient environment present in many tumors may substantially reduce viral clearance.
As shown elsewhere (24), tumor control requires s ≤ 2.35 × 10−3 mm3/cells·h, which is 20.4 times lower than our estimate for s in Table 1. In other words, the immune suppressor would have to potently shut down immune-mediated clearance of the virus to enable tumor eradication by the virus. Fig. 4 shows the tumor and virus dynamics under a very weak immune response, in which s is reduced to 10−3 mm3/cells·h, and the tumor is kept in check. Moreover, our analysis (see Ref. 24 for details) predicts that the time delay in the immune response is likely to generate periodic oscillations even if an immune suppressor could shut down the immune response. In this case, the tumor size may also oscillate over time, as in Figs. 4 and 5.
Virus Design.
Although to this point, we have discussed the delivery of the virus to the tumor, our results also have implications for the design of replication-competent oncolytic viruses for cancer therapy. Biopharmaceutical companies and academic investigators working in this area possess hundreds of viruses that are potential candidates for cancer therapy. Our results suggest that the most efficacious viruses will be those that generate the greatest wave speeds. Eqs. A and C reveal that the wave speed is proportional to the reciprocal of the infected cell lifetime and depends on the burst size and infectivity rate only through their product. First-generation mutant tumor-selective viruses frequently have a lower burst size and a longer infected cell lifetime than wild-type viruses (2, 4, 8), and these characteristics can be quantified in vitro and in vivo. This model predicts that this viral phenotype should be associated with significantly reduced efficacy versus the parental strain; in fact, experimental results confirm this prediction (2, 4). It is worth noting that murine tumor model data suggest that certain mutant adenoviruses, which generate larger or smaller c0 values in vitro than do wild-type viruses, are also more or less efficacious in vivo, respectively. For example, a recently described E1A mutant adenovirus actually replicates better and spreads faster than wild-type adenovirus. As predicted by our model, this phenotype is associated with significantly greater efficacy in vivo in nude mice (32). Similar examples include adenoviruses engineered to kill faster through the deletion of antiapoptotic genes (e.g., E1B-19kD) or overexpression of cell death genes (e.g., E3-11.6); these viruses also have greater efficacy in vivo (2, 33).
Our analysis, via Eqs. A and C, provides a quantitative basis for the design of replication-competent viruses for cancer treatment. This can be seen in Fig. 5, in which a shorter infected-cell lifetime leads to faster control of the tumor in the absence of immune-mediated clearance and is responsible for pushing the system over the tumor control threshold in the presence of weak immune-mediated clearance (where s = 10−3 mm3/cells·h). However, in the presence of the strong immune response (i.e., the value of s in Table 1), the shorter infected-cell lifetime has virtually no effect. Therefore, second-generation oncolytic viruses should be designed for rapid intratumoral spread and immune avoidance, in addition to tumor selectivity and safety.
DISCUSSION
Replication-selective oncolytic viruses represent a rapidly expanding novel therapeutic platform for cancer. Hundreds of viruses have now been tested preclinically, and approval has been sought and/or testing in humans has been initiated in at least 10 (2). Few therapeutic areas within biotechnology have ever expanded so quickly. However, very little is known about the ability of these relatively large biological particles to spread within a three-dimensional solid tumor. In fact, although these viruses can completely destroy tumor cell monolayers very efficiently, their efficacy against solid tumors has been comparatively modest to date in humans; single intratumoral injections have not led to complete responses to date (2, 6, 13, 14). This poor translation of in vitro efficacy to the in vivo setting was somewhat surprising to many investigators at the time. The mathematical model presented here clearly delineates the factors that make three-dimensional tumor eradication difficult if direct oncolysis is the only mechanism involved. Although thousands of virions can be produced and released from an infected cell, the ability of these released viruses to “seek out” and infect distant tumor cells is severely limited within a solid tumor mass compared with a cell monolayer. However, this model also gives insight into approaches to dramatically improve efficacy in the future. On the basis of this model and confirmatory experimental results to date, it appears highly likely that reproducible tumor eradication will require the following: the pattern of infection within the tumor mass must be diffuse; the rate of virus spread is critical; immune-mediated viral clearance must be suppressed; and the induction of a “bystander effect” (i.e., killing of neighboring uninfected cells) will be greatly advantageous and probably necessary. Of note, bystander effects can be achieved by the base virus itself (e.g., through induction of antitumoral-immunity), through the “arming” of these viruses with therapeutic genes (34) and/or through combinations with standard chemo/radiotherapy (5, 7, 35).
To our knowledge, this is the first attempt to validate a mathematical model of replication-competent viruses for cancer treatment using preclinical and clinical data; the only other mathematical analysis of this problem (36) that we are aware of is discussed elsewhere (24). Our analysis reveals that even in the absence of an antiviral immune response, replication-competent gene therapy pits a linear wave of infection (i.e., a wave moving at constant speed c0 if no necrosis occurs) against exponential tumor proliferation, and to tip the scales in favor of tumor control, the linear infection needs to be administered aggressively and intelligently. The timing of the immune-mediated clearance displayed in the human clinical data in Fig. 2 changes the entire picture: the “strength” of this immune response, as measured by the ratio sk:ω (immune stimulation rate times killing rate, divided by clearance rate), would need to be suppressed by a factor of 20.4 to enable tumor eradication by the virus. In addition, reducing the infected cell lifetime via mutant tumor-selective viruses can have a significant impact if used concomitantly with strategies to mitigate immune-mediated clearance. An important consideration will be, however, whether immunosuppression will result in faster tumor growth and spread.
In addition to these results, we note that the wave speed of the infection increases not only with a shorter infected cell lifetime but also with a higher cell density θ, which varies greatly among tumor types. Hence, our analyses suggest that replication-competent viruses may be more efficacious against dense tumors such as head and neck, and less efficacious against more diffuse tumors, such as those found in the brain (although the degree of fibrosis and stromal cell involvement will also be critical factors). In fact, it is possible that the wave of infection will not survive at the periphery of a highly invasive brain tumor (i.e., glioblastoma multiforme) because the very low tumor cell density at the invasive edges of the tumor may cause R0 to drop below the critical threshold value established by Kirn et al. (2) and Rodriguez et al. (9). Moreover, if the tumor is heterogeneous with respect to its sensitivity to the virus, then R0 could again be reduced to below the critical threshold. In fact, this is a possible explanation for the results of a study stating that, after a single core injection of hrR3 in immunodeficient mice, the histological results suggest that the wave of infection “proceeds from the central aspects of the tumor out towards the periphery, with disappearance at about 50 days” (27). On a related note, although our model is deterministic, it is useful to view the infected-cell lifetime and the distance traveled by a virus particle during one round of infection and lysis as random quantities, possessing their own probability distributions. Models of spatial epidemics (37) more sophisticated than ours have shown that the wave speed is enhanced by increases in the left tail of the lifetime distribution of infected cells and the right tail of the dispersal distribution. Hence, any mechanism that can fatten either of these tails (e.g., co-injecting an agent that breaks down the extracellular matrix, or introducing a mutation that causes premature lysis, such as E1B-19kD deletion or overexpression of adenovirus death protein) should increase the efficacy of the virus; experimental results have confirmed this prediction (Ref. 33 and Kirn, unpublished).3
Although this model appears to be validated by a large amount of preclinical and clinical data to date, as with any mathematical model, its predictions should be viewed with some caution for several reasons. The parameter estimates for R0, s, and k were estimated from 21 data points from only a single patient, although this patient’s virus and TNF levels were representative of other patients’ data that were measured at three different time points. The actual value of R0 may be slightly larger than our estimated value of 3.73 because our model assumed uniform injection, which may have been difficult to achieve in such a large tumor (5.76 cm in diameter). Moreover, a more precise estimate of the timing and magnitude of the immune response would be obtained from TNF levels in the tumor, rather than from the TNF plasma levels that we used. Also, the immune response, tumor susceptibility to virus infection, and tumor growth all differ considerably across patients within a tumor type, and the latter two parameters vary greatly across tumor types. We have also assumed that viral receptors/uptake are not limiting in this model; future models should assess the impact of varying viral uptake efficiency into tumor cells. In addition, no mathematical model can perfectly account for all of the factors involved in a complex biological system such as this. For example, fibrotic tissue planes and other anatomical barriers to viral spread are not accounted for in this model. In addition, tumor vascularity and leakiness may promote spread. Finally, cellular export of adenovirus is not included in this model. Nonetheless, we believe that this analysis, which is supported by comparisons with data from many mouse models and several clinical trials, represents a step forward in elucidating the dynamics of this novel class of cancer treatments. The greatest value of such a model as this may be to provide insights that lead to hypotheses that can be tested. The availability of this model should help to guide the construction and clinical development of these viruses, thus reducing the time consumed in getting regulatory approval and improving success rates.
In summary, our results suggest that replicating virus can be sufficiently potent to eradicate even a fast-growing tumor in the absence of an antiviral clearance. However, our analysis shows that broad spatial distribution of the virus (e.g., achieved by generalized intratumoral distribution) is much more important than high concentrations of the virus, and suggests that administration of the virus needs to be extremely aggressive from a geographical viewpoint; a several-millimeter diameter region, left uninfected in a fast-growing mouse tumor, may be capable of escaping or “outrunning” the virus. Therefore, the lack of objective responses in Phase I trials utilizing a single core injection of target tumors should not be interpreted as evidence for a lack of efficacy of the virus (e.g., the oncolytic herpes simplex virus mutants described by Rampling et al. (13) and Markert et al. (14). Moreover, an efficient innate immune response is easily potent enough to clear the virus prior to direct tumor eradication. Effective immune evasion will be required to achieve tumor control. Taken together, our results suggest that to achieve tumor eradication exclusively via direct oncolysis and spread, a replication-competent virus requires aggressive spatial administration and will probably require immunosuppressive genes, drugs, and/or tumor milieus. Viruses are also being armed with therapeutic genes that can destroy tumors more effectively through direct oncolysis plus a “bystander effect” (34). Alternatively, replication-competent viruses are being used in conjunction with chemotherapy and/or radiation (5, 35); indeed, recent clinical results in this direction appear promising and have led to Phase III approval trials (7). In this way, both infected and neighboring uninfected cells can be destroyed.
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The abbreviation used is: TNF, tumor necrosis factor.
D. H. Kirn, unpublished observations.
Notation for (a) core injection, (b) rim injection, and (c) rim injection with nutrient-limited necrosis.
Notation for (a) core injection, (b) rim injection, and (c) rim injection with nutrient-limited necrosis.
Virus genomes (a) and TNF levels (b) in the plasma over time for one patient, and the corresponding curves predicted by the mathematical model (see Ref. 24 for details). Time 0 is the time of the fourth injection. The virus genomes were not measured on day 5, and were below the level of detection (104 genomes/ml) on days 1 and 6–8. In the virus data: ○, uncensored data; ▵, the censored data (i.e., below the level of detection). The TNF level was not measured on day 9.
Virus genomes (a) and TNF levels (b) in the plasma over time for one patient, and the corresponding curves predicted by the mathematical model (see Ref. 24 for details). Time 0 is the time of the fourth injection. The virus genomes were not measured on day 5, and were below the level of detection (104 genomes/ml) on days 1 and 6–8. In the virus data: ○, uncensored data; ▵, the censored data (i.e., below the level of detection). The TNF level was not measured on day 9.
Tumor radius versus time for three patterns of intratumoral infection. The antitumoral efficacy of uniform injection throughout the tumor on day 1 is compared with injection of the core only, or of the rim only (this figure does not factor in immune-mediated clearance; see Figs. 4 and 5). All of the injections assume baseline infection of 1% of the tumor cells in the appropriate regions of the tumor. This figure demonstrates that uniform injection is associated with the best antitumoral efficacy. For core-only and rim-only injections, examples of tumor growth are given for uninfected tissue sizes on either side of the threshold values predicted by the model. The uninfected tumor rim-width threshold for core injection is 2.16 mm, and the uninfected core-radius threshold for rim injection is 6.45 mm (Eq. 4 and 5, respectively, from Ref. 23). Therefore, an uninfected rim width of 2.86 in the Fig. results in tumor escape, whereas an uninfected rim width of 1.51 results in eradication. Similarly, an uninfected core radius of 8.57 results in tumor escape, whereas an uninfected core radius of 4.61 results in gradual tumor clearance.
Tumor radius versus time for three patterns of intratumoral infection. The antitumoral efficacy of uniform injection throughout the tumor on day 1 is compared with injection of the core only, or of the rim only (this figure does not factor in immune-mediated clearance; see Figs. 4 and 5). All of the injections assume baseline infection of 1% of the tumor cells in the appropriate regions of the tumor. This figure demonstrates that uniform injection is associated with the best antitumoral efficacy. For core-only and rim-only injections, examples of tumor growth are given for uninfected tissue sizes on either side of the threshold values predicted by the model. The uninfected tumor rim-width threshold for core injection is 2.16 mm, and the uninfected core-radius threshold for rim injection is 6.45 mm (Eq. 4 and 5, respectively, from Ref. 23). Therefore, an uninfected rim width of 2.86 in the Fig. results in tumor escape, whereas an uninfected rim width of 1.51 results in eradication. Similarly, an uninfected core radius of 8.57 results in tumor escape, whereas an uninfected core radius of 4.61 results in gradual tumor clearance.
Tumor radius and predicted viral genomes/ml versus time varies significantly depending on the level of immune-mediated clearance of the virus. Uniform injection in the absence and presence of weak (s = 10−3 mm3/cells·h) and strong (s in Table 1) immune-mediated clearance. The dashed curve, the predicted virus genomes in the bloodstream in the presence of the weak immune response. The virus genomes in the presence of the strong immune response is similar to that in Fig. 2 (i.e., peak concentration in the 105 range) and is too low to show up in this figure. Tumor control requires that the effective immune clearance of the virus be severely dampened down or weakened.
Tumor radius and predicted viral genomes/ml versus time varies significantly depending on the level of immune-mediated clearance of the virus. Uniform injection in the absence and presence of weak (s = 10−3 mm3/cells·h) and strong (s in Table 1) immune-mediated clearance. The dashed curve, the predicted virus genomes in the bloodstream in the presence of the weak immune response. The virus genomes in the presence of the strong immune response is similar to that in Fig. 2 (i.e., peak concentration in the 105 range) and is too low to show up in this figure. Tumor control requires that the effective immune clearance of the virus be severely dampened down or weakened.
The effect on efficacy of the rate of virus killing and release from infected tumor cells (i.e., wave speed). A comparison is made between a wild-type virus [δ = (72 h)−1] and a mutant virus that kills and sheds faster from the cell [δ = (24 h)−1]. Results are shown both for a situation in which there is no effective immune-mediated clearance (s = 0) and for one in which there is a weak immune response (s = 10−3 mm3/cells·h). The range of infected cell life-time compared (24 versus 72 hours) was taken from data on wild-type adenovirus (∼72 h in vivo) and mutant adenoviruses that kill tumor cells faster (e.g., 24 h) (Ref. 2 and D. Kirn3). The mutant viruses are clearly more effective if the immune response can be dampened down and/or avoided. If not, neither virus is effective.
The effect on efficacy of the rate of virus killing and release from infected tumor cells (i.e., wave speed). A comparison is made between a wild-type virus [δ = (72 h)−1] and a mutant virus that kills and sheds faster from the cell [δ = (24 h)−1]. Results are shown both for a situation in which there is no effective immune-mediated clearance (s = 0) and for one in which there is a weak immune response (s = 10−3 mm3/cells·h). The range of infected cell life-time compared (24 versus 72 hours) was taken from data on wild-type adenovirus (∼72 h in vivo) and mutant adenoviruses that kill tumor cells faster (e.g., 24 h) (Ref. 2 and D. Kirn3). The mutant viruses are clearly more effective if the immune response can be dampened down and/or avoided. If not, neither virus is effective.
Summary of key model parameters and estimates of their numerical values
Parameter . | Description . | Numerical value . | Reference . |
---|---|---|---|
rc | Cell radius | 0.01 mm | 29 |
θ | Tumor cell density | 106 cells/mm3 | 29 |
λ | Tumor proliferation rate | 3.2× 10 h | 24 |
δ | Infected cell death rate | \(\frac{1}{48}\) h | 3 |
k | Immune killing rate | 15.3 mm3/ng·h | 24 |
μ | Debris removal rate | \(\frac{1}{72}\) h | 30 |
R0 | Basic reproductive ratio | 3.73 | 24 |
s | Immune stimulation rate | 0.048 mm3/cells·h | 24 |
ω | Immune clearance rate | 1.6 ml/ng·h | 24 |
Parameter . | Description . | Numerical value . | Reference . |
---|---|---|---|
rc | Cell radius | 0.01 mm | 29 |
θ | Tumor cell density | 106 cells/mm3 | 29 |
λ | Tumor proliferation rate | 3.2× 10 h | 24 |
δ | Infected cell death rate | \(\frac{1}{48}\) h | 3 |
k | Immune killing rate | 15.3 mm3/ng·h | 24 |
μ | Debris removal rate | \(\frac{1}{72}\) h | 30 |
R0 | Basic reproductive ratio | 3.73 | 24 |
s | Immune stimulation rate | 0.048 mm3/cells·h | 24 |
ω | Immune clearance rate | 1.6 ml/ng·h | 24 |