Gene (2012) 19, 543–549 & 2012 Macmillan Publishers Limited All rights reserved 0969-7128/12 www.nature.com/gt

ORIGINAL ARTICLE Dynamics of tumor therapy with vesicular stomatitis : explaining the variability in outcomes using mathematical modeling

DM Rommelfanger1,2, CP Offord3, J Dev1, Z Bajzer3,4, RG Vile1 and D Dingli1,5

Tumor selective, replication competent are being tested for . This approach introduces a new therapeutic paradigm due to potential replication of the therapeutic agent and induction of a tumor-specific immune response. However, the experimental outcomes are quite variable, even when studies utilize highly inbred strains of mice and the same cell line and virus. Recognizing that virotherapy is an exercise in population dynamics, we utilize mathematical modeling to understand the variable outcomes observed when B16ova malignant melanoma tumors are treated with vesicular stomatitis virus in syngeneic, fully immunocompetent mice. We show how variability in the initial tumor size and the actual amount of virus delivered to the tumor have critical roles on the outcome of therapy. Virotherapy works best when tumors are small, and a robust innate immune response can lead to superior tumor control. Strategies that reduce tumor burden without suppressing the immune response and methods that maximize the amount of virus delivered to the tumor should optimize tumor control in this model system. Gene Therapy (2012) 19, 543–549; doi:10.1038/gt.2011.132; published online 15 September 2011

Keywords: cancer gene therapy; virotherapy; initial conditions; heterogeneity; mathematical modeling; nonlinear least square fitting

INTRODUCTION utilize these exciting novel cancer therapeutics. Various groups have Patients with cancer have variable outcomes—they respond differently developed mathematical and computational models to understand the to therapy and they survive for variable periods of time from dynamics of tumor virotherapy. These models relate to a variety of diagnosis.1 There are many explanations for this variability, including tumor types as well as viruses that kill cells via different mechan- host-specific factors (for example, susceptibility to the specific tumor, isms.15,22–39 In the following analysis, we utilize our modeling pharmacogenomics, general health, physiologic state) as well as approach to try and understand the diverse outcomes observed with tumor-specific features (collectively known as the ‘biology’ of the tumor virotherapy in a seemingly homogenous host and tumor cell tumor and include the specific mutations that lead to tumor devel- population. We utilize data on VSV therapy of B16ova melanoma opment, resistance to therapy).2–4 However, experiments in animal tumor cells that are syngeneic to C57BL/6 mice with a fully competent models using closely inbred strains of mice (for example, C57BL/6) immune response making them a realistic model for the translation of that are implanted with tumor forming cells derived from the same these novel therapeutics into humans. Our results illustrate the critical cell culture and treated identically (for example, with a replicating importance of initial conditions on the outcome of therapy and virus such as vesicular stomatitis virus (VSV)) also result in significant suggest ways in which these conditions can be optimized to improve differences in outcome.5–14 The differences are unlikely to be due to therapeutic outcomes. host genetic factors or the behavior of the implanted cells since they are often from the same culture, as is the batch of virus used for RESULTS attempted control of the tumor. Therefore, the reasons behind this Growth of untreated tumor cells difference must lie elsewhere. In order to determine the growth characteristics of B16ova in Tumor therapy with replicating viruses is unlike any other form of immunocompetent C57BL/6 mice, we fitted serial tumor growth cancer therapy since (i) the therapeutic agent (virus) can be amplified data to the generalized logistic as well as to its special case, the by the cancer cells,15–17 (ii) the virus can be armed to induce a Gompertz model. Data from untreated tumors as well as tumors bystander effect18–21 and (iii) the virus may alert the injected with heat-inactivated (HI) VSV were both considered. Both to induce, or enhance, an attack against the tumor.7,11–14,20 One can models led to good fits (Figure 1); however using the model selection consider tumor virotherapy as an exercise in population dynamics due criteria, we chose the preferred model for each data set. Note that to the complex interactions between the virus, tumor and immune growth of tumors treated with the HI virus was the same as in the system. As a result, dynamic modeling is required to understand these control mice treated with phosphate-buffered saline (PBS), and there- interactions and provide rational predictions about the optimal way to fore, some virus replication is necessary for any anti-tumor effect.12

1Department of Molecular Medicine, Mayo Clinic, Rochester, MN, USA; 2Department of Molecular Pharmacology and Experimental Therapeutics, Mayo Clinic, Rochester, MN, USA; 3Department of Biochemistry and Molecular Biology, Mayo Clinic, Rochester, MN, USA; 4Department of Physiology and Biomedical Engineering, Rochester, MN, USA and 5Division of Hematology, Mayo Clinic, Rochester, MN, USA Correspondence: Dr D Dingli, Department of Molecular Medicine, Division of Hematology, Mayo Clinic, 200 First Street SW, Rochester, MN 55905, USA. E-mail: [email protected] Received 28 March 2011; revised 1 July 2011; accepted 28 July 2011; published online 15 September 2011 Initial conditions and outcome of tumor virotherapy DM Rommelfanger et al 544

1.2 1.2 ) 3 0.9 0.9

0.6 0.6

0.3 0.3 Tumor Volume (cm

0 0 05 10152025 05 10 15 20 25 Time (days) Time (days)

Figure 1 Fitting of tumor growth models to in vivo data. Each curve represents serial data on one tumor with a total of four tumors represented. (a)The tumors were injected with PBS. (b) The tumors were injected with HI virus. The square symbols represent fitting of data by the Gompertz model, which was preferred according to the used model selection criteria (PA¼0.0028; PB¼0.011). The circles correspond to data fitting using the generalized logistic model (PA¼0.0026; PB¼0.024).

Initial tumor size and outcome 0.02 Mass action kinetics would suggest that treating a larger tumor could lead to superior results due to a higher probability of infecting tumor

cells. Therefore, we initially determined whether there were any ) 3 significant differences in tumor size at the time of initiation of therapy. Using the F-test, we determined that tumor volume 7 days after cell 0.015 line injection (before the injection of the virus or control) was not significantly different between the four experimental groups: PBS injected control, HI group and the two intervention groups injected with 8 or 17 doses of virus (P¼0.4070). However, therapy with the virus led to better overall survival: while overall survival was not 0.01 significantly different among the PBS and HI groups (P¼0.4597) nor among the two treated groups (P¼0.9464) by the log-rank test, there was a significant difference between the control groups and the treated groups (P¼0.0019). We next determined whether the individual tumor size at the initiation of therapy impacted the outcome of the 0.005 mice treated with the virus. Using the effect likelihood ratio test, the tumor volume 7 days after tumor cell injection had no impact on overall survival in the two control cases (P¼0.2611), but had sig- Tumor volume 7 days post cell injection (cm nificant influence in the viral-treated groups (P¼0.009) (Figure 2). As the tumor increases in volume, the risk of death increases: an increase 0 of 0.01 cm3 increased the risk of treatment failure and death by a factor Successful Unsuccessful of 3.92. Therefore, treating smaller tumors appears to lead to better Therapy Therapy outcomes and tumor size at the initiation of therapy (y(0)) is a critical Figure 2 The tumor burden at the time of initiation of therapy determines initial condition. outcome. The mice were separated into two groups based on the outcome of therapy. The horizontal dotted line represents the mean volume. Variability in the virus dose available for infection We observed that there were three general patterns of tumor behavior in response to virotherapy with VSV: mice that were cured of their disease, mice where the tumor was controlled while on therapy but We therefore proceeded to consider the impact of v(0) on the grew once therapy was stopped and mice where virotherapy had no outcome, using the estimated average value of y(0) for each group. impact at all on tumor growth (Figure 3). Tumor growth and response Although the virus was administered intratumorally and the total dose data from these three groups were combined and we fitted the model injected was known (kept constant at 5Â108 PFU in 50 mlPBS),we to the averaged tumor growth data for each group. Having established reasoned that it was unlikely that all the virus particles that were the importance of the initial tumor size on the outcome of therapy, we injected remained in the vicinity of the tumor and could successfully attempted to simultaneously fit tumor growth curves for the three infect tumor cells. Therefore, we introduced a correction parameter c groups of mice, while keeping the growth parameters the same. Our that estimated the fraction of each dose of virus that is effectively goal was to consider the initial tumor size as the only variable that present at the tumor site after injection. With this maneuver, the influenced the outcome. This approach did not lead to good fits, simultaneous fitting of cured versus controlled versus uncontrolled implying that y(0) is not the only parameter that influences the tumors improved substantially (Figure 5). Surprisingly, we found that outcome (Figure 4). the fraction of the effective dose size was invariably small and generally

Gene Therapy Initial conditions and outcome of tumor virotherapy DM Rommelfanger et al 545

Successful Virotherapy Delayed Tumor Growth Unimpeded Tumor Growth 0.02 1.2 1.2 ) 3 0.015 0.9 0.9

0.01 0.6 0.6

0.005 0.3 0.3 Tumor Volume (cm 0 0 0 0 25 50 75 100 01020300102030 Time (days) Time (days) Time (days)

Figure 3 Tumors treated with VSV exhibited three behaviors: cure, delayed growth and unimpeded growth. The data were fitted by the model given by Equations (1–5). The arrow on the time scale indicates the end of therapy (Pcure¼0.33; Pdelayed¼0.023; Punimpeded¼0.02).

Successful Virotherapy Successful Virotherapy 0.02 0.02 ) ) 3 0.015 3 0.015

0.01 0.01

0.005 0.005 Tumor Volume (cm Tumor Volume (cm

0 0 0 25 50 75 100 0 255075100

Delayed Tumor Growth Delayed Tumor Growth 1.2 1.2 ) ) 3 0.9 3 0.9

0.6 0.6

0.3 0.3 Tumor Volume (cm Tumor Volume (cm

0 0 0 25 50 75 100 0 25 50 75 100 Time (days) Time (days)

Figure 4 Initial tumor burden and the outcome of virotherapy. Data from a Figure 5 The injected dose of virus is a variable. Treated tumor growth data successfully treated tumor is fitted using the same parameters for a tumor simultaneously fitted as in Figure 4. However, here the fraction c of the total that is transiently slowed. The initial tumor sizes differ: for successful dose injected is introduced as a free parameter fitted for each set of data, 3 3 virotherapy it is 0.0034 cm , while for delayed growth it is 0.0146 cm . yielding c¼0.0958 for successful therapy and c¼0.1201 for delayed growth However, this difference in initial tumor size alone is not enough to explain (Pcured¼0.33; Pfailed¼0.043). the outcome of virotherapy (Pcure¼0.402; Pfailed¼1.79).

o15% of the injected dose. Note that we assumed that this population40 (resulting in a bystander killing of uninfected tumor 7 fraction was constant for a given mouse despite repeated injections cells ). Therefore, we introduced in a model a variable number z0 of and therefore it represents an ‘average’ fraction of injected virus immune cells available at the time of virus administration (see for a given mouse. This parameter was reasonably consistent across Equations (1–5)). Again, we determined that there is variability in all mice. the size of this population of cells; mice with a higher number of innate immune cells that can recognize VSV-infected cells tended to The antiviral immune response have a superior outcome. However, the impact of this variable on the Another potential variable that can influence the outcome is the size of fit was not as strong as that of the initial virus dose (Figure 6). This of the immune cell population that can respond to the infected cell course makes sense, since one does not expect significant variability in

Gene Therapy Initial conditions and outcome of tumor virotherapy DM Rommelfanger et al 546

Successful Virotherapy inbred strains of mice and in vitro cell culture systems, especially when 0.02 experiments are performed using a single cell and virus stocks. Here, we show how mathematical modeling can illustrate and provide

) biologically plausible explanations for the variability in outcomes. 3 0.015 In this work, we have demonstrated that the initial conditions have a determining role on the outcome of therapy, as suggested by other 45–47 0.01 investigators in the field. It appears that tumor cytoreduction before the introduction of the may lead to superior results, in the sense that the initial conditions could be stacked in favor 0.005 of the virus. Perhaps, this is one mechanism by which alkylating agents Tumor Volume (cm such as cyclophosphamide can improve the success of oncolytic virotherapy.6,48 However, such therapy ideally should not interfere 0 with the immune response since it is clear that, at least in this model, 0 25 50 75 100 the bystander killing of uninfected tumor cells by the immune 7 Delayed Tumor Growth response is of some relevance for cure. Some tumors are amenable 1.2 to intralesional injection of the virus. Although this is expected to improve the rate of infection of tumor cells, our analysis suggests that

) only a small fraction of the virus actually infects the tumor. Methods 3 0.9 that can reduce the wastage of virus outside the tumor microenviron- ment should help to improve disease control rates. It has been shown that hydrodynamic forces may have an important role on the 0.6 distribution of the virus in the tumor environment.34,49,50 Therefore, strategies that optimize the delivery of the virus are urgently required 0.3 to also improve on this important initial condition. Tumor Volume (cm The maxim in cancer therapy has been that it is better to treat a smaller tumor rather than a larger one and in general smaller tumors 0 are associated with a superior outcome. This is the basis of the TNM 0 255075100 classification of tumors and its impact on prognosis. However, the Time (days) field of cancer virotherapy has provided the impression that, given Figure 6 Preexisting immunity and the outcome of therapy. Treated tumor the potential amplification of the therapeutic agent due to local new growth data were simultaneously fitted as in Figure 4. However, here z0 virion production from infected tumor cells, that perhaps this (see text) is introduced as a free parameter fitted for each set of data, oncologic paradigm may be broken. Unfortunately, it appears that 8 7 yielding z0¼10 for successful therapy and z0¼5Â10 for delayed growth even with virotherapy, a smaller tumor is a better therapeutic target (Pcured¼0.924; Pfailed¼0.026). and indeed, we have a race between the tumor, the virus and the immune system.27 One criticism of mathematical modeling of virotherapy has been the prediction of oscillatory behavior across the various populations. the innate immune response to the same antigens in these closely This behavior is well known in population dynamics and ecology and inbred mice. it was such an observation that ushered in the era of mathematical biology almost 100 years ago. However, most tumors do not appear to Simulations oscillate in size although this depends on the scales at which tumors In our prior work to understand the dynamics of tumor therapy with are measured (both time scales and geometry). However, in in vivo virus, simulations often led to the appearance of population studies with other oncolytic viruses (MV-Edm-based derivatives), oscillations where the virus and the tumor cells oscillate out of phase clear oscillations have been observed, perhaps due to the lack of an with each other in the absence of an immune response.28,32,33,37 This immune response.33,41 It is possible that such behavior will not be dynamic behavior has been observed in immunodeficient animal observed in the presence of an immune response to the virus. To our models.41 Other models of tumor virotherapy with lytic viruses also knowledge, such oscillations have not been observed with VSV-treated suggest such oscillatory behavior.22,23 However, this behavior has not tumors. Interestingly, the introduction of the correction term c that been observed with VSV oncotherapy. With the inclusion of an accounts for the variable fraction of virus particles that actually infect effective initial population of virus, our model is able to describe tumor cells essentially eliminated this ‘problem’ in our model while nonoscillatory behavior. maintaining biological plausibility. One can postulate that other explanations may exist for the variable DISCUSSION outcomes observed in the experiments and certainly no model can be Tumor virotherapy offers exciting prospects for the field of oncology comprehensive enough to include all possibilities. However, our and many patients have been enrolled in clinical trials that have shown model provides biologically plausible explanations that also make the safety of the approach as well as tantalizing evidence of efficacy in fitting of the data significantly better. The conclusion that the initial various tumor types.42–44 However, the outcomes of therapy are quite tumor size, the extent of innate immune response to the administered variable and the reasons for these differences must be understood in virus and the actual dose of virus available to infect the tumor cell order to optimize the use of such novel . Although the population all influence the outcome is mathematically logical and outcomes may be due to heterogeneity in host and tumor related also biologically.45–47 and clinically relevant. Overall, our analysis genetic factors, these explanations ostensibly are not present in highly suggests that attempts to improve the outcomes of therapy will be

Gene Therapy Initial conditions and outcome of tumor virotherapy DM Rommelfanger et al 547 more successful if these observations are taken into consideration: the Tumor virotherapy is modeled by the following system of differential dice could finally be skewed in favor of therapy rather than the tumor! equations:

0 MATERIALS AND METHODS y ¼ ry lnðK=ðy+xÞÞ À byv À lðz+z0Þy ð1Þ Cell line b 0 e e The murine B16ova melanoma cell line (H2-K ) was derived from B16 cells by y ¼ ryð1 Àðy+xÞ =K ÞÀbyv À lðz+z0Þy ð2Þ transduction with a cDNA coding for the chicken ovalbumin gene.51 The cells –1 were grown in Dulbecco’s modified Eagle’s medium with glucose (4.5 g l )and 0 x ¼ byv À dx À lðz+z0Þx ð3Þ L-glutamine without sodium pyruvate (Mediatech, Herndon, VA, USA) supplemented with 10% (v/v) fetal bovine serum (Life Technologies, Carlsbad, v0 ¼ ax À byv À ov ð4Þ CA, USA) and maintained in a humidified incubator at 37 1Cwith5%CO2.  Virus z z0 ¼ Zxðz+z Þ 1 À À gz ð5Þ VSV Indiana strain, engineered to express the green fluorescent protein (VSV- 0 x 52 GFP) was a gift from G Barber (University of Miami School of Medicine, where y represents the population of noninfected tumor cells, x is the Miami, FL, USA). The virus was amplified by infection of BHK-21 cells at a population of infected tumor cells, v is the population of free virus present multiplicity of infection of 0.01 for 24 h. The culture supernatants were and z is the effective immune response. filtered and purified by sucrose gradient centrifugation. The virus pellet was Introduction of the virus leads to infection of some of the tumor cells at a 1 resuspended in PBS, aliquoted and stored at À80 C. The virus titer was rate byv. Whenever a dose of virus is injected at time t40, the functions are 7 determined by a standard plaque assay on BHK-21 cells. updated to accommodate the fact that the virus population at that time (due to production of new virions from infected tumor cells) is augmented by the Mouse work and monitoring addition of exogenous virus and this larger virus population is now available to All procedures were approved by the Mayo Clinic Institutional Animal Care infect additional tumor cells. The infected cells produce virus at a rate ax and and Use Committee. C57BL/6 mice were purchased from The Jackson die at a rate dx. Not all of the virions produced are infectious and there are Laboratory (Bar Harbor, ME, USA) at 6–8 weeks of age. To establish nonimmune mechanisms of virus inactivation. These mechanisms eliminate 5 subcutaneous tumors, 5Â10 B16ova cells in 100 ml of PBS were injected into free virus at rate ov. With respect to the immune response, we consider that the 8 the right flank of mice. The virus was injected intratumorally (5Â10 PFU in animals already have immune cells that can recognize the virus but these cells 50 ml PBS) every other day starting on day 7 after implantation of the cells in all are amplified in the presence of virus-infected tumor cells. This is compatible mice. Each group was composed of eight mice. HI virus was also used as a with the clonal selection theory of immunity—cells that recognize the epitopes control, apart from a cohort of tumor bearing mice that were injected with PBS are present even before exposure to the antigen and they replicate in response alone. The duration of therapy was variable depending on the protocol to the antigen.40 The immune response is also able to keep tumor growth in (see Results section). Mice were monitored daily for tumor burden and overall check (to some extent)—hence the term l(z+z0)y in Equations (1) and (2). health. Caliper measurements of subcutaneous tumors were taken three times a In the presence of a therapeutic virus, this immune control on tumor growth B 2 week with the animals being killed when tumor size was 1.0Â1.0 cm in two can become more prominent as postulated for VSV previously.7,11–14,20 Initially, perpendicular directions. we introduced different rate parameters (l1 and l2) to describe the immune response against tumor cells and virus-infected tumor cells, respectively, Statistical analysis assuming that l2Xl1. However, data fitting repeatedly showed that these We grouped the mice according to the therapeutic intervention and analyzed two parameters were essentially the same in value. Hence, we simplified the the distribution of tumor sizes at the time of initiation of therapy. The purpose model and consider that the rate of the immune response is the same for was to determine whether there was any correlation between tumor size at the infected versus uninfected tumor cells (l). initiation of therapy and treatment outcome (cure versus failure) as well as We impose a limit on the size of the immune response to both the viral overall survival. We used the F-test, to determine whether there was any infection as well as the tumor, compatible with most other physiological difference in tumor volume at the start of therapy. Overall survival was defined processes. This is achieved by introducing the factor (1Àz/x). Moreover, as the number of days of observed follow-up and not as a function of tumor immune cells die at a rate yz. The initial conditions are y(0)¼y0, x(0)¼0, volume at a particular time. Mice that survived for the full duration of the v(0)¼cvi (where vi is the injected dose and c is a variable from 0 to 1 that experiment have a follow-up value equal to that amount of time (103 days). captures the fact that only a fraction of the injected dose of virus is able to Mice that died before the end of the experiment were uncensored while those infect tumor cells), z(0)¼0. Collectively, these considerations lead to the above that survived were censored accordingly. Differences in overall survival were set of nonlinear differential equations with equilibrium points, which can be tested with the log-rank test. The impact of tumor size at the time of therapy analyzed along the lines similar to those described in Dingli et al.28 and Bajzer on outcome was determined using the Cox proportional hazards model. For all et al.32 A fundamental assumption of the model is that we have mass action statistical analysis, a Po0.05 was considered statistically significant. kinetics and the populations are homogenously mixed. Alternative models based on partial differential equations are more complex and ultimately lead to Mathematical modeling similar results.15,24–27,29–31,34,35 Although stochastic models may be used to The dynamic description has to consider growth of uninfected tumor cells (y), understand these dynamics,35,39 with large populations, the deterministic and the population of infected tumor cells (x), the free virus population (v) and the stochastic models give very similar results (at least on average) and the former cellular immune response (against virus-infected tumor cells as well as unin- are much simpler to analyze. fected tumor cells, z). We start by modeling the growth of untreated tumors that can follow either a Gompertz model or a generalized logistic model (see Model fitting and parameter estimation the first term on the right-hand side of Equations (1) and (2), below). Both The available experimental data were in the form of serial tumor diameters in models have been shown to adequately describe untreated tumor growth under two dimensions. This was used to determine tumor volume (given by a2b/2, a variety of conditions.53 For each untreated tumor, either model was chosen where a is the shorter diameter), and then converted to cell population based on the best fit for that specific tumor33 according to model selection assuming that 1 mm3E1Â106 tumor cells. In order to determine the best criteria (Akaike, Bayesian28,32). Both models impose a maximum size that the tumor growth model for a specific tumor, we fitted the generalized logistic or tumor can reach, known as the carrying capacity (K) with tumor growth its limiting case, the Gompertz model (for which explicit expressions are slowing as this value is approached. This leads to the familiar sigmoidal curves known53,54) to serial growth data for tumors that were injected with PBS alone. for tumor growth.54 We performed nonlinear least squares fitting with the use of custom software

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together with a powerful minimizer based on the simplex-induction hybrid 9 Kottke T, Diaz RM, Kaluza K, Pulido J, Galivo F, Wongthida P et al. Use of algorithm that has been described previously.55 The preferred model was biological therapy to enhance both virotherapy and adoptive T-cell therapy for cancer. determined by using several model selection criteria including the modified Mol Ther 2008; 16: 1910–1918. 10 Lun X, Senger DL, Alain T, Oprea A, Parato K, Stojdl D et al. Effects of intravenously 28,32 Akaike and Bayesian selection criterion as previously described. administered recombinant vesicular stomatitis virus (VSV(deltaM51)) on multifocal and Since the tumor cell population injected across mice is similar and the mice invasive gliomas. J Natl Cancer Inst 2006; 98: 1546–1557. are syngeneic, we expect limited variability in the growth characteristics of the 11 Galivo F, Diaz RM, Thanarajasingam U, Jevremovic D, Wongthida P, Thompson J et al. tumor cells (r, K, e). However, the results suggested that the tumor size at the Interference of CD40L-mediated tumor by oncolytic vesicular stomatitis virus. Hum Gene Ther 2010; 21: 439–450. initiation of therapy was a major determinant of the outcome (see the Results 12 Galivo F, Diaz RM, Wongthida P, Thompson J, Kottke T, Barber G et al. Single-cycle viral section). Moreover, tumor growth was variable across the mice. Presumably, gene expression, rather than progressive replication and oncolysis, is required for VSV this variability could be due to differences in the actual number of cells injected therapy of B16 melanoma. Gene Therapy 2010; 17: 158–170. and the intrinsic variability in tumor forming potential of the injected cells as 13 Wongthida P, Diaz RM, Galivo F, Kottke T, Thompson J, Melcher A et al. VSV oncolytic virotherapy in the B16 model depends upon intact MyD88 signaling. Mol Ther 2010; well as other unquantifiable parameters. In order to account for this variability, 19: 150–158. when fitting the growth function to the untreated tumor cohort, we allowed 14 Wongthida P, Diaz RM, Galivo F, Kottke T, Thompson J, Pulido J et al. Type III y(0) to vary by imposing a maximum value based on the number of cells that IFN interleukin-28 mediates the antitumor efficacy of oncolytic virus VSV in were injected, that is, y(0)A[1,5Â105]. Subsequently, we divided the mouse immune-competent mouse models of cancer. Cancer Res 2010; 70: 4539–4549. 15 Kirn D, Martuza RL, Zwiebel J. Replication-selective virotherapy for cancer: biological data into outcome groups: cured tumors, delayed tumor growth and unim- principles, risk management and future directions. Nat Med 2001; 7: 781–787. peded tumor growth (based on the dynamics of tumor growth). For each 16 Russell SJ. RNA viruses as virotherapy agents. 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Cancer Res 1999; 59: 3861–3865. 19 Dingli D, Peng KW, Harvey ME, Greipp PR, O’Connor MK, Cattaneo R et al. weighted by the respective mean tumor volume as determined experimentally. Image-guided radiovirotherapy for using a recombinant Given that not all of the virus population injected at any time was available to measles virus expressing the thyroidal sodium iodide symporter. Blood 2004; 103: infect tumor cells, we introduced a variable c that can range from 0 to 1. The 1641–1646. value of this parameter was determined by fitting and was different for each 20 Kim JH, Oh JY, Park BH, Lee DE, Kim JS, Park HE et al. Systemic armed oncolytic and immunologic therapy for cancer with JX-594, a targeted poxvirus expressing GM-CSF. group. Initially, we considered that the level of preexisting immunity to the Mol Ther 2006; 14:361–370. virus was the same across all mice and therefore z0 was held constant (estimated 21 Luo C, Mori I, Goshima F, Ushijima Y, Nawa A, Kimura H et al. Replication-competent, to be 104 cells). Four fits were performed assuming four sets of available tumor oncolytic herpes simplex virus type 1 mutants induce a bystander effect following growth parameters (r, K, e) obtained from fitting tumor growth curves for ganciclovir treatment. JGeneMed2007; 9: 875–883. nontreated mice. The best among those fits was retained (Figure 5). Similar fits 22 Wodarz D. Viruses as antitumor weapons: defining conditions for tumor remission. Cancer Res 2001; 61: 3501–3507. 8 were performed assuming c¼1andz0A[0,10 ], which was different for each 23 Wodarz D. Gene therapy for killing p53-negative cancer cells: use of replicating versus group (Figure 6). The same procedure has been adopted when comparing the nonreplicating agents. Hum Gene Ther 2003; 14: 153–159. group of cured mice with the group in which tumor growth was only delayed 24 Wein LM, Wu JT, Ianculescu AG, Puri RK. A mathematical model of the impact of by therapy. Throughout the fitting process, we determined the ratio of model infused targeted cytotoxic agents on brain tumours: implications for detection, design and delivery. Cell Proliferation 2002; 35: 343–361. error variance to data variance and report this value as P for the parameter fits 25 Wein LM, Wu JT, Kirn DH. Validation and analysis of a mathematical model of a (see Supplementary Table S1). replication-competent oncolytic virus for cancer treatment: implications for virus design and delivery. Cancer Res 2003; 63: 1317–1324. 26 Wu JT, Byrne HM, Kirn DH, Wein LM. Modeling and analysis of a virus that replicates CONFLICT OF INTEREST selectively in tumor cells. Bull Math Biol 2001; 63:731–768. 27 Wu JT, Kirn DH, Wein LM. Analysis of a three-way race between tumor growth, a The authors declare no conflict of interest. replication-competent virus and an immune response. Bull Math Biol 2004; 66: 605–625. 28 Dingli D, Cascino MD, Josic K, Russell SJ, Bajzer Z. 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