Intentional bacterial infections can produce efficacious antitumor responses in mice, rats, dogs, and humans. However, low overall success rates and intense side effects prevent such approaches from being employed clinically. In this work, we titered bacteria and/or the proinflammatory cytokine TNFα in a set of established murine models of cancer. To interpret the experiments conducted, we considered and calibrated a tumor–effector cell recruitment model under the influence of functional tumor-associated vasculature. In this model, bacterial infections and TNFα enhanced immune activity and altered vascularization in the tumor bed. Information to predict bacterial therapy outcomes was provided by pretreatment tumor size and the underlying immune recruitment dynamics. Notably, increasing bacterial loads did not necessarily produce better long-term tumor control, suggesting that tumor sizes affected optimal bacterial loads. Short-term treatment responses were favored by high concentrations of effector cells postinjection, such as induced by higher bacterial loads, but in the longer term did not correlate with an effective restoration of immune surveillance. Overall, our findings suggested that a combination of intermediate bacterial loads with low levels TNFα administration could enable more favorable outcomes elicited by bacterial infections in tumor-bearing subjects. Cancer Res; 77(7); 1553–63. ©2017 AACR.

Major Findings

We present the first systematic study that combines in vivo murine experiments and mathematical modeling toward the mechanistic understanding of the therapeutic potential of bacterial infections against solid tumors. In particular, we elucidate the interplay between growing tumors, vascularization, and immune recruitment dynamics in response to bacterial infections. This study suggests that before therapeutic interventions, tumor size and immune recruitment dynamics are critical factors for long-term tumor control. Moreover, an arbitrary increase of bacterial loads can be detrimental for treatment outcomes, which suggests the existence of an optimal bacteria load depending on tumor size. The model predicts that intermediate bacterial loads and administration of low TNFα potentially inducing tumor vascularization can lead to enhanced therapeutic outcomes in silico.

Bacteria of many species invade and colonize solid tumors (1–3). In many cases, this leads to growth retardation of the neoplasia, and in the most favorable situations to complete tumor clearance (1, 2, 4). This phenomenon has been described already 200 years ago (1813) by Arsène-Hippolyte Vautier, a French physician who observed that the tumors of patients shrank when they also suffered from gas gangrene (3, 5). We now know that these symptoms are due to a Clostridium perfringens infection.

Several attempts have been carried out to establish bacteria-mediated antitumor therapy in the clinical practice (3, 4, 6). Most prominent were the attempts of William B. Coley, an American surgeon who successfully treated inoperable skin cancers with heat-inactivated bacteria (3, 7). However, the severe side effects, as well as uncertainties on the underlying mechanisms, prevented the general use of bacteria as a therapeutic agent. Currently, in face of the fact that cancer is the most frequent cause of death in the industrialized world and the second in economically developing countries with a rising incidence (8), novel strategies have to be explored to increase the spectrum and effectiveness of therapies against this devastating disease. In accordance, since two decades, systematic experimentation has been carried out to understand the tumor-invasive properties of bacteria, as well as the mechanisms that lead to bacteria-induced tumor shrinkage or clearance. Recent progress in knowledge of host bacterial pathogen interaction, advances in genome research, as well as the development of new technologies to manipulate bacteria allow now to develop new bacterial strains. They should be metabolically and functionally attenuated to render them safe for application in cancer patients and at the same time therapeutically highly potent (14). In parallel, attempts are made to employ such bacteria and their tumor-targeting features as carriers for therapeutic compounds like toxins, which can be expressed directly in cancerous tissues (15–17).

Quick Guide to Model Equations and Assumptions
Tumor-Effector Cell Recruitment Model

We consider a mathematical model that combines radial tumor growth and immune recruitment dynamics (9) to gather information about specific tumor responses to different therapeutic combinations of bacterial loads and TNFα in mice (Figs. 1A–D and 2A). The proposed model relies on the following main assumptions:

  1. The temporal evolution of the average tumor radius is considered, where invasive and diffusive tumor growth are not taken into account.

  2. The death rate λA reflects the lump effect of apoptotic and necrotic processes.

  3. Innate immunity or base immune surveillance is represented as a minimum presence of active effector cells at any time, even in the absence of tumor cells.

  4. Effector cell recruitment rate is a function of tumor cells following Michaelis–Menten dynamics.

  5. The efficacy of immune killing depends on the ability of effector cells to penetrate the tumor bulk via the tumor-associated vasculature (10, 11). With better vascularization, the effector cells kill tumor cells not only on the surface of the tumor but also further inside.

  6. Effector cells die with a constant rate and get inactivated in dependence on their antitumor activity.

The system variables are the average tumor radius R(t) and effector cell concentration E(t) in the tumor vicinity. The mathematical model is formulated as a system of ordinary differential equations (ODE) given by

formula
formula

where the time coordinate t has been omitted for sake of notation simplicity. LD is an intrinsic length scale resulting from nutrient dynamics, that is, diffusion, supply, and consumption. We assume that B is a dimensionless parameter, where 0 ≤ B ≤ 1 represents the degree of functional tumor-associated vasculature, that is, vessels that allow the flow of blood to the tumor. The functional vascular network is assumed to modulate both the tumor growth and tumor–immune cell interactions as shown in Eqs. A and B. The parameters λM and λA are the mitotic and death rates of tumor cells, respectively. Parameter c represents the killing rate of tumor cells by effectors, r is the immune recruitment rate, and K is the tumor volume at which the recruitment rate is half-maximal. d1 and d0 are the inactivation and death rates of effector cells, respectively, and σ is the background rate of immune effector recruitment. LD, B, λM, λA, c, r, K, d1, d0, and σ are positive constants. Further details about derivation, parametrization, and theoretical analysis of the tumor-effector cell recruitment model can be found in ref. 9.

Therapeutic Potential Definition

Therapeutic potential TP (p): Ωp → [0, 1] is interpreted as the average tumor control, that is, ratio between cases where the tumor radius stays bounded over all the possible cases, with respect to a certain regime of model parameter p ∈ {B, r, R0} given by

formula

where t* = 100 days (about 3 months) is the target time, RS(p) is the tumor radius at the saddle point (Fig. 2B), and H(·) is a Heaviside step function (9). We remark that mice are typically considered in a complete remission stage, that is, permanent absence of disease, if they are tumor-free after 100 days of treatment (12, 13).

Active chemotactic mechanisms for bacteria targeting and colonizing solid tumors have been suggested for Salmonella based on in vitro data (18). However, different species of bacteria that lack these properties were still able to target cancerous sites in immunocompetent mice (19). A passive alternative has been recently proposed (20). Namely, upon systemic application, Gram-negative bacteria elicit a cytokine storm via their endotoxin. The dominating cytokine is most likely the tumor necrosis factor TNFα (20). As a consequence of the cytokine storm, the leaky pathologic blood vessels of the tumor open and a severe hemorrhage is induced. This leaves a large necrotic region behind after the hemorrhage has been cleared. The facultative anaerobe Salmonella might be simply flushed into the tumor during the blood influx. Salmonella then proliferate in the immune-privileged sites provided by the necrotic and hypoxic regions of the tumor, and eventually scavenge on the dead cells. The hemorrhage and induction of necrosis also explains the tumor growth retardation that is observed after application of the bacteria (20). Final tumor clearance is most likely due to the induction of a specific immune response against tumor antigens that is activated by the adjuvant effects of the bacteria (21). However, most of the tumors recover from the original attack by bacteria and eventually continue to grow. Immune escape mechanisms of the neoplasia might be responsible for this phenomenon (22). Therefore, a mechanistic understanding of the biology of bacteria-mediated therapy is of utmost importance for the further development of such a high potential anticancer treatment.

In this work, we consider in vivo murine tumors treated with different bacterial loads and/or TNFα, and record the corresponding tumor volume evolutions before and after treatment. Then, we use a mathematical model that combines vascularized tumor growth and effector cell recruitment dynamics to gain insights into the possible reasons of success or failure of bacterial therapies. Mathematical models have been rather successful in investigating the biology of cancer (23–26) and are becoming an increasingly important resource to address immunologic questions (27–29), as well as useful for optimizing and predicting antitumor therapy outcomes (9, 27, 30–36). Model analysis not only provides a qualitative and quantitative explanation of experimental results, but also allows for novel therapeutic strategies.

Mice, bacteria, and cell lines

BALB/c mice were purchased from Janvier Labs. All recombinant mice were bred at the Helmholtz-Zentrum für Infektionsforschung and all experiments were performed with female, 8- 12-week-old mice if not stated differently and under approval of LAVES (Niedersächsisches Landesamt für Verbraucherschutz und Lebensmittelsicherheit); Permission: 33.9-42502-04-12/0173. S. Typhimurium SL7207 [hisG46, Δ407(aroA544::Tn10)] and E. coli TOP10 were grown on LB agar with 30 μg/mL streptomycin at 37°C. CT26 (ATCC CRL-2638) cells were cultured in IMDM supplemented with 10% FCS, 100 U/mL penicillin, and 100 μg/mL streptomycin, 50 μmol/L 2-mercaptoethanol, 2 μmol/L l-glutamine, and maintained at 37°C and 5% CO2. The F1A11 (H-2d) cell line is a murine fibrosarcoma that expresses β-galactosidase (β-gal) and was obtained by transduction of spontaneously transformed BALB/c fibroblast cell line F1 with the LBSN retroviral vector (37). Cell lines used in the study were obtained in the year 2000.

Tumor growth and bacterial infections

Tumors CT26 and F1A11 were set by injecting 5 × 105 cells in 100-μL PBS subcutaneously. Growth was monitored by caliper. Viable tumor volume was calculated as V = 4/3π(hw2)/8, where h = height and w = width. For intravenous (i.v.) infection, bacteria from glycerol stocks were cultivated on streptomycin LB plates overnight. Single colonies were resuspended in PBS and adjusted to 103 or 5 × 106 bacteria in 100-μL PBS. Bacteria were injected into the tail vein of the animals, as soon as the tumor had reached a volume between 100 and 200 mm3. Similarly, 1 μg of recombinant TNFα in 100-μL PBS was injected intravenously. For analysis, organs were homogenized in 0.1% (v/v) TritonX-100/PBS and homogenates were plated on streptomycin LB plates.

Parameter calibration of the preinjection tumor growth phase (days 0–9)

The proposed mathematical model given by Eqs. A and B was first calibrated for the untreated phase of tumor growth (days 0–9). The preinjection tumor evolutions in Fig. 1 describe similar growth rates than those observed in immunocompromised Rag1−/−, which cannot produce functional T cells and B cells, and wild-type (WT) BALB/c mice experiments (9). As the experimental protocols for untreated tumors are the same, and the tumor volume in these experiments was quantified daily instead of every two days (Fig. 1), we expect more precise parameter estimates. Therefore, we consider the values of model parameters for the untreated (preinjection) phase of tumor growth (days 0–9) as those that we previously estimated from Rag1−/− and (WT) BALB/c murine tumor growth experiments (9).

Figure 1.

Tumor volume evolutions (6–7 mice experiments) without therapeutic interventions until day 10, and treatment responses to different combinations of bacterial loads and TNFα: 103 bacteria (A), 5 × 106 bacteria (B), TNFα (C), and 103 bacteria and TNFα (D).

Figure 1.

Tumor volume evolutions (6–7 mice experiments) without therapeutic interventions until day 10, and treatment responses to different combinations of bacterial loads and TNFα: 103 bacteria (A), 5 × 106 bacteria (B), TNFα (C), and 103 bacteria and TNFα (D).

Close modal

More precisely, we calibrated the intrinsic tumor parameters, that is, the net proliferation rate λp, the degree of functional vascularization B, and the average length of nutrient gradient LD from immunocompromised Rag1−/− mice experiments (9). The control immune-related parameters such as the death rate of tumor cells due to effector cells c, the immune cell recruitment rate r, and the initial concentration of effector cells E0, were estimated from (WT) BALB/c mice experiments (9). The rest of parameter values were taken from earlier studies of tumor-immune dynamics (30, 38–40) and are summarized in Supplementary Table S1.

Model parameters postinjection (days 11–21) were calibrated on the basis of experimental data of tumors treated with different combinations of bacterial loads and TNFα (Fig. 1A–D). As initial conditions, we consider the tumor radius R0 at day 1 postinjection, and estimate the concentration of effector cells E0 at day 1 postinjection, as well as the immune recruitment rate r and degree of functional of tumor-associated vasculature B postinjection. In all cases, we assume that E0 and B vary up to 4.0 × 106 cells and 0.20 at day 1 postinjection, respectively. These ranges of values are based on the potential deviations from the preinjection parameters due to the fast dynamics taking place during the first day postinjection. Immune–tumor dynamics can be also assumed different between individual experiments, implying variations on the immune recruitment rate r, ranging from 0.4 to 0.7 day−1. The rest of parameter values for the model are considered as in Supplementary Table S1.

To reduce the uncertainties due to the finite number of observations, a bootstrapping procedure was implemented to obtain the parameter estimates of average tumor responses (Table 1). The goal was to obtain reliable estimates of the summary statistics by means of a resampling procedure. Different nonparametric sets of bootstrap samples were artificially generated and represent the natural variation of the experimental datasets (41, 42). The resulting means and standard deviations of such samples were considered for parameter calibration. Fitting procedures for each experimental setting start from a large number of different random initial conditions, as well as by randomly perturbing the parameter set to be estimated, where the solutions with lowest residual variances were selected in each case.

Table 1.

Immune recruitment rate r, tumor vascularization B, and initial number of effector cells E0 estimated from the average behavior of the experimental datasets in Fig. 4 

Experimental settingrBE0R0
103 bacteria 0.52 day−1 0.18 1.30 × 106 cells 3.52 mm 
5 × 106 bacteria 0.44 day−1 0.17 2.62 × 106 cells 3.40 mm 
TNFα 0.65 day−1 0.04 1.73 × 106 cells 3.32 mm 
103 bacteria + TNFα 0.56 day−1 0.04 1.71 × 106 cells 3.40 mm 
Experimental settingrBE0R0
103 bacteria 0.52 day−1 0.18 1.30 × 106 cells 3.52 mm 
5 × 106 bacteria 0.44 day−1 0.17 2.62 × 106 cells 3.40 mm 
TNFα 0.65 day−1 0.04 1.73 × 106 cells 3.32 mm 
103 bacteria + TNFα 0.56 day−1 0.04 1.71 × 106 cells 3.40 mm 

Statistical analysis

We have used the nonparametric one-tailed Mann–Whitney U test to statistically test whether two experimental treatments come from the same distribution. A Kruskal–Wallis test was conducted to compare whether more than two experimental samples share the same distribution. Stars represent comparisons with P value less than 0.01.

Treatment with bacteria and/or TNFα has the potential to induce tumor clearance in vivo

Tumors developed in BALB/c mice until day 10, followed by injection of different combinations of bacterial loads and/or TNFα. The subsequent tumor responses to treatment were monitored for the following 11 days, which represents a total experimental time of 21 days. Figure 1 shows the phases of untreated tumor growth and postinjection for four different therapeutic protocols. Injection of a low bacterial load was able to control tumor growth in some but not all cases (Fig. 1A). In two experiments, tumor growth showed only a slight delay in response to the therapeutic intervention. Increasing the bacterial load induces a more homogeneous reaction (Fig. 1B). All monitored tumors were reduced in size, but none of them was eradicated. Injection of TNFα alone or in combination with the low bacterial load induced a consistent reduction of tumor volume in all mice and even leading to tumor clearance in some cases (Fig. 1C and D). These experimental results provide in vivo evidence that the impact of bacteria and proinflammatory factors on tumor growth is diverse and depends on the concentration of injected bacteria and/or TNFα.

While it becomes clear that particular choices of bacterial loads favor tumor burden reduction and in the most favorable situations induce tumor clearance, this observation lacks any mechanistic insight. However, better understanding of the reasons and mechanisms for the diverse treatment outcomes is necessary to make this therapeutic technique useful for clinical practice. Driven by the diversity of outcomes for each experimental setting, we hypothesize that the state of the mice at the time of injection in terms of tumor volume, preexisting number, and subsequent recruitment rate of immune effector cells, as well as the degree of functional vascularization, determines the efficacy of bacterial therapy beyond the experimentation time.

In vivo tumor responses can be reproduced by a mathematical model

The developed mathematical model is particularly designed to allow for long-term mechanistic insights of bacterial and/or TNFα treatment success versus failure (see Quick Guide to Model Equations and Assumptions). For that purpose radial tumor growth, effector cell dynamics, and functional tumor vascularization with impact on both tumor growth and recruitment of effector cells are explicitly considered. Pretreatment model parameters were determined from the phase of untreated tumor growth on the basis of murine experiments (Supplementary Table S1; ref. 9).

The effects of bacterial and TNFα treatments are implicitly modeled in the proposed tumor–immune system interaction, where the tumor radius R0 at day 1 postinjection defines the model initial conditions for the subsequent long-term dynamics. Treatment-related parameters, that is, the initial concentration of effector cells E0, the immune recruitment rate r, and the degree of functional tumor vascularization B are estimated at day 1 postinjection from the experimental data shown in Fig. 1. Experimental evidence suggest that during the first 24 hours postinjection, a dynamic interplay between bacteria, immune cells, and tumor-induced vasculature takes place (3, 17, 19–21). After day 1 postinjection, we observe a relaxation of the aforementioned dynamics and the necrotic core growth, where bacteria settle down and are protected from immune attacks. Accordingly, we assumed in the model that fast processes of tumor reorganization are induced after treatment at day 10 of tumor growth and relax by day 11 (3, 17, 19–21). During this time, related model parameters, that is, functional vascularization B and recruitment of effector cells r are modulated and we assume that they remain constant thereafter. The parameter quadruple (R0, E0, B, r) is then associated with the individual state of the mice at the time of treatment or shortly after, and will be used as such when mice are individually distinguished in the model analysis.

To gain insight on the average tumor responses to each treatment strategy, parameters of interest are first calibrated to the mean tumor temporal trajectories postinjection (Fig. 2A). The model quantitatively recovers the mean tumor responses to bacterial and/or TNFα therapeutical administrations (Fig. 2C–F). The estimated concentrations of effector cells E0, immune recruitment rate r, and degree of functional tumor vascularization B postinjection are summarized in Table 1. Even though mouse-averaged responses to different treatments cannot be considered conclusive for the success or failure of these therapeutic strategies, they can still provide useful information about the overall therapeutic potential of bacterial infections and TNFα protocols.

Figure 2.

A, Average tumor responses to the different treatments in Fig. 1, where bacterial loads or TNFα administration time is indicated by an arrow at day 10. B, Estimates of the critical tumor radius RS for uncontrolled tumor growth depending on the immune recruitment rate r and functional vascularization B. CF, Fitting results of the model for tumor responses in A, where the ranges of experimental observations for each dataset are represented by the shadowed regions.

Figure 2.

A, Average tumor responses to the different treatments in Fig. 1, where bacterial loads or TNFα administration time is indicated by an arrow at day 10. B, Estimates of the critical tumor radius RS for uncontrolled tumor growth depending on the immune recruitment rate r and functional vascularization B. CF, Fitting results of the model for tumor responses in A, where the ranges of experimental observations for each dataset are represented by the shadowed regions.

Close modal

Bacterial infections and TNFα enhance the immune activity and alter vascularization in the tumor bed

The parameter calibration of average tumor evolution postinjection (Fig. 2C–F) reveals that any combination of bacterial infections and TNFα treatment elevates the recruitment rate r (Table 1), when compared with the untreated control cases (Supplementary Table S1). In turn, TNFα treatments imply increased r when compared with exclusive administration of bacteria. The effector cell concentration E0 postinjection in the vicinity of tumors is always higher than without treatment. Interestingly, high bacteria loads (5 × 106) trigger higher E0 but lower r when compared with the corresponding values induced by 103 bacteria. The above observations suggest that bacterial infections with or without TNFα enhance the immune activity in the tumor bed. We also observe the destruction of functional tumor-associated vasculature B when mice are treated with TNFα. In case that bacterial infections are not administered with TNFα, the value of B is predicted to remain almost invariant. All the aforementioned model-driven findings can be deduced from Fig. 3A–D. The obvious question that arises is how these treatment-induced model parameter constellations impact on short-, that is, within the experimentation time, and long-term tumor dynamics. Figs. 4A–D and 5A–D show the model-fitting of the individual short-term evolution of tumor volume and effector cell amounts after administration of different combinations of bacterial loads or TNFα (Fig. 1A–D).

Figure 3.

Parameter estimates for the preinjection tumor growth phase and treatment responses to different combinations of bacterial loads or TNFα. Immune recruitment rate r (A), number of effector cells E0 (B), functional vascularization B (C), and tumor radius (D; crossing lines column). Preinjection parameter values in Supplementary Table S1 and mean tumor radius at time of injection (Fig. 2A). Solid color columns: r, E0, B, and mean tumor radius R0 at day 1 postinjection estimated from murine experiments (Fig. 1). *, significant differences and straight lines imply no difference.

Figure 3.

Parameter estimates for the preinjection tumor growth phase and treatment responses to different combinations of bacterial loads or TNFα. Immune recruitment rate r (A), number of effector cells E0 (B), functional vascularization B (C), and tumor radius (D; crossing lines column). Preinjection parameter values in Supplementary Table S1 and mean tumor radius at time of injection (Fig. 2A). Solid color columns: r, E0, B, and mean tumor radius R0 at day 1 postinjection estimated from murine experiments (Fig. 1). *, significant differences and straight lines imply no difference.

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Figure 4.

Model fitting of the individual short-term tumor responses to treatment strategies with 103 bacteria (A), 5 × 106 bacteria (B), TNFα (C), and 103 bacteria + TNFα; data as in Fig. 1A–D (D).

Figure 4.

Model fitting of the individual short-term tumor responses to treatment strategies with 103 bacteria (A), 5 × 106 bacteria (B), TNFα (C), and 103 bacteria + TNFα; data as in Fig. 1A–D (D).

Close modal
Figure 5.

Short-term evolution of effector cell amounts estimated from tumor responses to treatments with 103 bacteria (A), 5 × 106 bacteria (B), TNFα (C), and 103 bacteria + TNFα; data as given in Fig. 4A–D (D).

Figure 5.

Short-term evolution of effector cell amounts estimated from tumor responses to treatments with 103 bacteria (A), 5 × 106 bacteria (B), TNFα (C), and 103 bacteria + TNFα; data as given in Fig. 4A–D (D).

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Tumor volume is critical for long-term bacterial therapy outcomes

In addition to mean tumor responses to different therapeutic strategies, we investigate individual mice and whether differences at the time of injection allow to predict treatment success or failure. Initially, we focus on tumor responses to infections induced by 103 bacteria. This is a particularly interesting experimental setting. It can be considered on average as a failure scenario (Fig. 2A). However, a variety of treatment responses was found (Fig. 1A). To calibrate r, E0, and B with respect to this treatment strategy, we started from the tumor radius R0 at day 1 postinjection and the preinjection estimates given in Supplementary Table S1 for the other model parameters. Figure 3 shows the resulting parameter values that accurately fit the different tumor responses (Fig. 4A; Supplementary Table S2). Functional levels of tumor-associated vasculature postinjection are similar to those in the untreated tumor growth phase (Fig. 3C), but higher concentration of effector cells and immune recruitment rates postinjection are achieved when compared with the preinjection estimates (Fig. 3A and B). However, treatment induces higher effector cell recruitment rates insufficient to reduce, in most cases, the tumor volume by the end of the experimentation time (Fig. 4A).

The diverse treatment responses can be grouped into three cases (Fig. 4A): (i) unfavorable tumor responses, experiments 2 and 6, (ii) almost invariant tumor volumes before and after administration of bacteria, experiments 1, 3, and 5, and (iii) tumor reduction as in experiment 4. Model simulations support that the treatment responsiveness of the tumor depends on its size at the time of injection. Only observing the experimental curves, we deduced that for too big tumors at day 1 postinjection this treatment strategy provides no benefit. However, for sufficiently small tumor sizes, the bacterial treatment–induced immune recruitment dynamics are able to keep tumors under control or reduce tumor volumes during the experimentation time.

Indeed, model analysis demonstrates the existence of a critical tumor radius RS, above which tumors evade immune surveillance (9, 34). Considering the mean values of the estimated parameters (Supplementary Table S2), we can calculate a critical tumor radius, RS = 4.1 mm that corresponds to a critical volume VS = 288.82 mm3, which correctly predicts tumor escape in experiments 2 and 6 as R0 > RS at the time of injection (Fig. 4A). A decreasing number of effector cells postinjection is predicted in both cases (Fig. 5A). Figure 2B shows numerical estimates of RS depending on the recruitment rate r and functional tumor vasculature B. There exists a critical immune recruitment rate rS depending on B, below which tumor growth control is not achieved (9). But, for increasing r and decreasing B, the chance of long-term tumor control increases as RS acquires higher values. The mean value of r predicted in response to this bacterial load is higher than the corresponding critical immune recruitment rate rS ≃ 0.49 day−1 (Figs. 2B and 3A). Accordingly, model simulations indicate long-term control for tumors with radii R0 < RS, that is, murine experiments 1, 3, 4, and 5 (Fig. 4A), whereas for the remaining cases tumors evade from immune surveillance.

High bacterial loads result in short-term positive therapeutic responses but not in long-term tumor control

We now consider murine experiments treated with 5 × 106 bacteria (Fig. 1B). Irrespective of tumor sizes, we observe positive treatment responses during the experimentation time. Tumor sizes are considerably reduced in all experiments, but complete neoplasia clearance is not achieved at day 11 postinjection. In some cases, the treatment response curves even suggest tumor regrowth. Although all treatments induce large necrotic regions, experimental measurements include always viable cancer cells at the tumor border allowing for relapse. Parameter estimates for this therapeutic choice are represented in Fig. 3 (Supplementary Table S3). Fitting results predict a similar tumor-associated vasculature B to the one estimated from the untreated tumor growth phase (Fig. 3C).

At a first glance, comparing with tumor evolutions upon injection of 103 bacteria, we could naïvely think that higher bacterial loads lead to improved therapeutic outcomes. However, the model predicts that long-term tumor control is not achieved with this specific treatment choice. The most striking difference to other scenarios is that the number of effector cells at day 1 postinjection is inversely related to the immune recruitment rate (Fig. 3A and B). Higher concentrations of effectors at day 1 postinjection are associated with rapid tumor reduction, but lower immune recruitment rates allow tumor regrowth toward the end of the experimentation time (Fig. 4B). In fact, long-term tumor evasion is predicted as the mean recruitment rate estimated is below the critical immune recruitment threshold rS ≃ 0.49 day−1 (Figs. 2B and 3A). This results in a reduction of the number of effector cells postinjection (Fig. 5B). Therefore, the model predicts that tumor control cannot be obtained by the administration of this bacterial load. These findings evidence that an arbitrary increase of bacterial loads does not necessarily imply better long-term therapeutic outcomes. Combined with the previous tumor responses to infections induced by 103 bacteria, we conjecture the existence of an optimal bacterial load for therapeutic benefits.

Intermediate bacterial loads induce optimal in silico outcomes

Model results suggest that bacteria directly affect the immune system dynamics without major damage to the preexisting tumor-associated vascular functionality. Increasing bacterial loads are associated with higher numbers of effector cells E0 and lower immune recruitment rates r at day 1 postinjection. While tumor evasion is always predicted after administration of 5 × 106 bacteria, positive long-term therapeutic responses are obtained in the majority of cases (4/6) treated with 103 bacteria. Assuming a monotonically decreasing and increasing dependency of recruitment rates r and initial concentrations E0 of effector cells on bacterial loads as shown in Fig. 6A and B, respectively, the model can provide insights into the therapeutic potential (TP) of intermediate bacterial loads as defined in Eq. C (see Quick Guide to Model Equations and Assumptions).

Figure 6.

Model predictions with respect to increasing bacterial loads. A and B, Linear interpolations of the immune recruitment rate r and number of effector cells E0 at day 1 postinjection for bacterial loads ranging between 103 and 5 × 106 bacteria. C, Predicted long-term tumor control depending on tumor size and bacterial load. D, Therapeutic potential (TP) of bacterial therapies for tumors with sizes between 3.5 to 4.5 mm. The y-axis represents the fraction of tumors controlled 100 days after the administration of different bacterial loads. C and D, Simulations of tumor control in 100 days (light gray) and uncontrolled tumor growth (dark gray).

Figure 6.

Model predictions with respect to increasing bacterial loads. A and B, Linear interpolations of the immune recruitment rate r and number of effector cells E0 at day 1 postinjection for bacterial loads ranging between 103 and 5 × 106 bacteria. C, Predicted long-term tumor control depending on tumor size and bacterial load. D, Therapeutic potential (TP) of bacterial therapies for tumors with sizes between 3.5 to 4.5 mm. The y-axis represents the fraction of tumors controlled 100 days after the administration of different bacterial loads. C and D, Simulations of tumor control in 100 days (light gray) and uncontrolled tumor growth (dark gray).

Close modal

Figure 6C shows that there exists a critical bacterial load, around 105 bacteria, above which therapeutic benefits cannot be achieved irrespective of tumor size. In this case, the degree of functional tumor vascularization B was fixed to 0.17, which is in the range estimated after bacterial treatments (Supplementary Table S2; Supplementary Table S3). The therapeutic potential of bacterial infections is predicted to increase below this critical threshold (Fig. 6D). However, for sufficiently large tumors, that is, R0 > 4.4 mm, bacterial treatments alone do not provide positive outcomes and additional therapeutical strategies are required. Interestingly, these findings indicate that for the failure cases treated with 103 bacteria (Fig. 4A), that is, murine experiments 2 and 6 with R0 = 4.28 and 4.23 mm, respectively, an educated bacterial load increase could lead to long-term tumor control. Thus, model predictions support that administration of an optimal amount of bacteria could significantly improve long-term treatment outcomes in silico.

TNFα augments effector recruitment dynamics and reduces tumor-associated vascular functionality

Treatment responses in Fig. 1C suggest that an exclusive administration of TNFα induces tumor control in mice. The tumor-associated vasculature is found to be damaged under these conditions (Fig. 3C; Supplementary Table S4). This model prediction is supported by experimental evidence in mice demonstrating that TNFα is a potent antivascular cytokine able to initiate apoptosis in cancer cells (43, 44). TNFα targets tumor vasculature by inducing hyperpermeability and destruction of vascular lining, which results in the accumulation of cytostatic drugs inside tumors and damage to the vascular structure (45).

The critical tumor radius in this therapeutic choice, RS = 5.6 mm, is higher than R0 for all experiments, which indeed suggests long-term tumor control (Supplementary Table S4). The number of effector cells at day 1 postinjection is on average higher than those estimated from infections induced by 103 bacteria, but lower than those with 5 × 106 bacteria (Fig. 3B). Fitting of tumor responses in these cases predicts that TNFα induces higher effector recruitment rates compared with all other therapeutic strategies (Fig. 3A). This prediction is supported by recent experimental data in mice, where an increased effector cell migration toward tumor bulk is observed after TNFα administration (46). After complete tumor clearance, the effector cell concentration is decreased (Fig. 5C).

Therapeutic combinations of 103 bacteria and TNFα result in positive short-term treatment responses, and even in some cases complete tumor clearance (Fig. 1D). The functional levels of tumor-associated vasculature and number of effector cells at day 1 postinjection are estimated similar to those by exclusive administration of TNFα (Fig. 3B and C). However, time-evolution curves postinjection indicate that some tumors might escape from immune surveillance and relapse (Fig. 4D). The analysis of the individual experiments reveals an association between successful therapy outcomes and high immune recruitment rates. For experiments 2 and 3, the estimated values of r are lower than the corresponding critical immune recruitment rate, rS = 0.48 day−1 (Supplementary Table S5). This suggests tumor evasion accompanied by a reduction in the concentration of effector cells postinjection (Fig. 5D). In the remaining cases, r > rS and R0 < RS = 5.6 mm suggest long-term tumor control. Thus, the combination of bacterial infections with administration of TNFα does not always improve the therapeutic success rate when compared with the exclusive administration of TNFα.

Low TNFα dosages could improve the efficacy of intermediate bacterial loads

Recent experiments in mice evidenced that systematic administration of TNFα has dual effects by remodeling tumor stroma and enhancing adaptive immunity (47). Low doses of intratumoral TNFα stabilize tumor blood vessels and promote antitumor immune responses mediated by increased effector cell infiltration. These findings indicate that a low dose of TNFα can be a potent adjuvant for active immunotherapy (47). On the other hand, larger dosages act as an anti-vascular agent as considered in the murine experiments (Fig. 1).

TNFα vasomodulatory effects on tumor growth dynamics are investigated in the framework of the proposed mathematical model. In our previous works (9, 34), we demonstrated the existence of an optimal combination of immuno- and vasomodulatory interventions for tumor shrinkage. These ideas could be realized by treatment strategies with combinations of bacterial loads and TNFα. We quantify the long-term tumor control dependency on functional tumor vascularization B and immune recruitment rate r, as well as on the initial conditions R0 and E0. Supplementary Figure S1 shows that for both high and low functional levels of tumor-associated vasculature B, a therapeutic “window of opportunity” arises for the long-term tumor control in mice. Tumor vascular destruction, that is, low B values, is associated with high TNFα dosage, which are rather prohibitive for humans (48). On the other hand, for low doses of TNFα, which potentially lead to high values of B via induced vascular normalization, the model simulations suggest that the tumor control probability is inversely related to the tumor size but favored by larger immune recruitment rates. This is supported by preclinical studies demonstrating that vascular normalization increases infiltration of effector cells into tumors (49–51). In addition, Supplementary Fig. S2 evidences the existence of an optimal amount of effector cells that maximizes the regime of favorable outcomes (total green area) in mice at high functional vascularization. This optimal E0 concentration coincides with the intermediate bacterial loads around 105 bacteria (Fig. 6). Therefore, an educated combination of low TNFα and intermediate bacteria loads is predicted to optimize treatment outcomes.

In the current study, we investigated the therapeutic potential of bacterial infections against tumor growth. We focused on the interplay between growing tumors, vascularization, and immune recruitment dynamics after bacterial-based therapeutic interventions. We conducted in vivo tumor experiments in mice treated with different combinations of bacteria loads and/or TNFα. To assess the experimental observations, we used a mathematical model of tumor–effector cell interactions, in which the functional tumor-associated vasculature and induced immune responses were predicted to play a crucial role in long-term bacterial therapy outcomes. Pretreatment model parameters were calibrated on the basis of untreated tumor growth murine experiments (9), and compared with estimates postinjection in treated mice with four different bacterial and/or TNFα treatment protocols.

Mice experiments show that tumor growth retardation, and even eradication in the most favorable situations, can be obtained with bacterial infections. This study suggests that tumor sizes at the first day postinjection and the underlying immune recruitment dynamics contain predictive information for short- and long-term bacterial treatment outcomes. Model results suggest that an arbitrary increase of bacterial loads does not necessarily result in better long-term tumor control. This implies the existence of an optimal bacterial load for tumor remission depending on tumor size at the time of injection. The administration of adequate bacterial loads combined with immunostimulatory agents is predicted to enhance the tumor control probability. These results are valid under the assumption that the immune recruitment rate is time-invariant during the experimentation time and even for longer periods. Although this assumption is plausible for short times, for example, during the experimentation time, for long-term one may expect the recovery of recruitment rates close to the lower pretreatment control values. This typically implies an underestimation of tumor growth for the fitted model parameter values when compared with control estimates. However, this fact does not disqualify the tumor–effector cell recruitment model predictions. For instance, in the case of 5 × 106 bacteria, the estimated large values of the effector recruitment rate induce a lower bound of the tumor growth rates, in comparison with the corresponding control values. Even for these underestimated tumor growth rates, the model predicts evasion, meaning that for the control recruitment rates, the tumor is surely uncontrollable. Finally, concerning the long-term controllable cases, we can be more confident as they are evident from the short-term experimental observations.

In the mice experiments of tumor growth we conducted, the TNFα dose administered was enough to destroy functional tumor-associated vasculature. Although the exclusive administration of TNFα leads always to tumor control, the combination with bacteria seems not always to result in positive treatment outcomes. This suggests the existence of an antagonistic mechanism between TNFα and bacterial treatments, raising the question whether bacteria and TNFα also coregulate each other. At this point, the exact biological mechanism of this antagonistic behavior remains to be elucidated. Even though we cannot ensure the statistical significance of these findings due to the small sample size, the model analysis suggests that the resulting tumor control failure is associated with high effector cell concentrations E0 postinjection and an undercritical immune recruitment rate r leading to tumor evasion. Parameters r and E0 result from the fast processes of tumor reorganization during the first day postinjection (3, 17, 19–21), which are out of model's resolution and this particular experimental setting. In addition, the model analysis revealed that E0 at day 1 postinjection is decisive for the short-term tumor dynamics, but the long-term evolution is exclusively dependent on r (9, 34). Increasing r was predicted to always improve the therapeutic potential, but E0 nonmonotonically impacts onto the tumor control probability (9). To shed further light on these short timescale mechanisms, new experiments need to be designed that monitor the immune and vascularization dynamics during the first hours postinjection.

TNFα is immunogenic at almost any dose. On the other hand, TNFα damages the tumor blood vessels at high doses, whereas it improves the functionality of tumor vasculature at low doses (21, 47). However, as the administration of high TNFα concentrations to humans may endanger the patient's life (48), low dosage is the only plausible treatment choice. On the basis of the model results, we hypothesized that the administration of an intermediate bacterial load triggering accurate immune responses followed by a low TNFα dosage enhancing vascularization could provide beneficial therapeutic outcomes depending on tumor size at the time of injection. As this treatment strategy potentially improves tumor vascularization, it could be further optimized when combined with nonimmunosuppressive adjuvant therapeutic modules such as chemo- and radiotherapy (52–54).

Although the biological processes involved are more complicated than the current model description, this study represents an important step toward the understanding of the complex link between bacteria-induced immune responses and tumor growth dynamics. In particular, the proposed model provides a minimal framework, with respect to immunologic details and components, to understand our experimental data. Adding complexity can be justified upon further data acquisition. We think that the proposed tumor-effector cell recruitment model may serve as a predictive tool, in combination with in vivo experimentation, for the design and assessment of novel treatment strategies combining bacterial infections and other therapeutic modules. Further, we firmly believe that the development of this approach together with parametrization on human data can reveal the antitumoral potential of bacterial treatments.

No potential conflicts of interest were disclosed.

Conception and design: H. Hatzikirou

Development of methodology: H. Hatzikirou, J.C. López Alfonso

Acquisition of data (provided animals, acquired and managed patients, provided facilities, etc.): S. Leschner, S. Weiss

Analysis and interpretation of data (e.g., statistical analysis, biostatistics, computational analysis): H. Hatzikirou, J.C. López Alfonso, M. Meyer-Hermann

Writing, review, and/or revision of the manuscript: H. Hatzikirou, J.C. López Alfonso, S. Weiss, M. Meyer-Hermann

Administrative, technical, or material support (i.e., reporting or organizing data, constructing databases): J.C. López Alfonso, M. Meyer-Hermann

Study supervision: S. Weiss, M. Meyer-Hermann

Other (model implementation and simulations): J.C. López Alfonso

S. Weiss gratefully acknowledges the support of the Deutsche Krebshilfe.

J.C. López Alfonso, M. Meyer-Hermann, and H. Hatzikirou were supported by the German Federal Ministry of Education and Research (BMBF) for the eMED project SYSIMIT (01ZX1308D). M. Meyer-Hermann was also supported by the German Federal Ministry of Education and Research (BMBF) for the eMED project Sys-Stomach (01ZX1310C), and the Helmholtz Initiative for Personalized Medicine.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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