Model of Host-Pathogen Interaction Dynamics Links In Vivo Optical Imaging and Immune Responses

ABSTRACT Tracking disease progression in vivo is essential for the development of treatments against bacterial infection. Optical imaging has become a central tool for in vivo tracking of bacterial population development and therapeutic response. For a precise understanding of in vivo imaging results in terms of disease mechanisms derived from detailed postmortem observations, however, a link between the two is needed. Here, we develop a model that provides that link for the investigation of Citrobacter rodentium infection, a mouse model for enteropathogenic Escherichia coli (EPEC). We connect in vivo disease progression of C57BL/6 mice infected with bioluminescent bacteria, imaged using optical tomography and X-ray computed tomography, to postmortem measurements of colonic immune cell infiltration. We use the model to explore changes to both the host immune response and the bacteria and to evaluate the response to antibiotic treatment. The developed model serves as a novel tool for the identification and development of new therapeutic interventions.

level but only at distinct time points and locations. Using a system for combined bioluminescence and X-ray computed tomography (CT) imaging, bacterial burden can be quantified and localized with precision.
However, in order to enable the interpretation of in vivo images in terms of postmortem-derived disease parameters, a model that links whole-body in vivo imaging results to cellular and molecular data obtained postmortem, not all of which can be visualized, is crucial.
A model of the underlying biological processes that give rise to the signals that are measured with whole-body imaging can place processes occurring at the cellular or molecular scale in the context of processes taking place over a much larger scale, in different parts of the animal, or at different time points during the course of disease. In other contexts, such as in the investigation of inflammatory bowel disease, detailed models of host-pathogen interaction in the gut have been described (18,19). In this study, we develop a model of C. rodentium infection to create a direct link between state-of-the-art in vivo whole-body imaging results and detailed biological knowledge at the cellular scale. We show how the model can be used to analyze changes in host immune response, to study mutations in the pathogen, and to analyze and simulate the response to antibiotic treatment.

RESULTS
Dynamics of C. rodentium colonization, clearance, and recruitment of immune cells. We first establish the general dynamics of host-pathogen interaction during C. rodentium infection. These results are used to determine the model, to link experimental postmortem data to results from in vivo imaging, and to simulate the contribution of host immunity to bacterial clearance.
Following oral administration, C. rodentium colonizes the cecum and colon of mice (Fig. 1A). Bacteria are continually shed into the feces, and the level of bacteria in the stool correlates with the burden of attached bacteria in the colon (6). We measured the bacterial levels in the stool for wild-type C57BL/6 mice over the course of infection (Fig.  1C); the peak of infection is at day 8 postinoculation, and the infection is cleared at around day 21. Clearance of C. rodentium is achieved through a combination of innate and adaptive defense mechanisms and competition by the microbiota. Neutrophils, B cells, and T cells play a major role in the elimination of the pathogen (6). We measured recruitment of immune cells to colonic samples obtained postmortem (Fig. 1B). Neutrophils start accumulating in the colon at the early stage infection, followed by B cells at a later stage; CD4 ϩ T cells are detected from day 14 postinfection.
We modeled how a change in bacterial shedding depends on the host immune response. We included neutrophils and antibody-producing B cells as elimination factors, and we accounted for their dependence on the presence of CD4 ϩ T cells. We relied on the following assumptions: elimination of bacteria by neutrophils is proportional to the measured levels of neutrophils, and B cells first need to be activated before they are effective; B cells are activated by CD4 ϩ T cells; and elimination through the antibody response mediated by B cells is proportional to the levels of active B cells. We further assume that the T cell population only effectively activates B cells after day 8.
Under these assumptions, we model the change in bacterial levels, C, in the colon over time (dC/dt) by the simple equation dC/dt ϭ ␥ · C Ϫ · N · C Ϫ ␤ · B · CD4 ϩ T · C Ϫ · C. The positive growth term (␥ · C) results in an increase in the bacterial load over time, and the negative elimination terms (neutrophils [ · N · C] and B cells [␤ · B · CD4 ϩ T · C]) and the negative treatment term ( · C) result in decreases of the bacterial load over time. The growth parameter, ␥, represents the bacterial replication rate per day, and the parameters and ␤ represent bacterial elimination rates per day per active immune cell population. N is the concentration of neutrophils, B is the concentration of B cells, and CD4 ϩ T is the concentration of CD4 ϩ T cells. The last term in the model corresponds to elimination as a result of treatment. Initially we set the treatment parameter, , to zero, but we will use this term later. We modeled the initial value of the level of bacteria in the stool by C 0 ϭ fC O , where C O is the concentration of bacteria during oral gavage and the parameter f represents the fraction that is present in the colon at day 2 of the infection.
The in vivo dynamics of this system have not previously been modeled, and therefore we lack reliable estimates of model parameters (such as rates of killing of bacteria due to different parts of the immune system). We therefore used the data  Dynamic Modeling of Host-Pathogen Interaction Infection and Immunity generated here to obtain estimates of the parameters. We used a method that results in not just one point estimate of the parameters, which would only correspond to one good fit of the model to the data, but instead generates multiple probable parameter estimates that all result in good model fits. This provides an estimate of the uncertainty in the parameters, which may be due to biological variability or noise in the data (i.e., we infer parameters in a Bayesian setting) (20). We obtain estimates for the parameters by fitting the model to measurements of the bacterial load in the stool, obtained from wild-type C57/BL6 mice infected with wild-type C. rodentium, assuming a direct correlation between the levels of attached bacteria in the colon and bacteria in the stool ( Fig.  1C and D). We used stool data to calibrate the model instead of imaging data, as stool data are more abundantly available. In the next section we will link the model to in vivo imaging data of the bacterial load in the abdomen. We used the measured immune cell infiltration in the model as the input (N, B, and CD4 ϩ T cells). We estimated the immune cell infiltration for time points between measured time points using linear interpolation. To estimate parameters, we used an algorithm (the Bayesian inference algorithm Metropolis-Hastings) which allowed us to obtain multiple probable estimates for parameter values (we generated samples from the posterior parameter distribution). This is a sequential method that starts with some initial parameter estimates (sampled from prior parameter distributions that are defined according to our current knowledge about the parameter values) and iteratively updates these estimates based on how well model simulations using the sampled parameter values fit the data (21,22). Whether the estimate of the bacterial load is close to the measured CFU per gram was determined by comparing it to the measured variance in the bacterial levels at each time point (i.e., a Gaussian likelihood). We used this set of parameter estimates to simulate trajectories of bacterial concentrations. Figure 1C shows the mean and the interval that includes 90% of the modeled bacterial loads (i.e., the 90% confidence interval) calculated from these simulated trajectories, which closely matched the measured trajectory of bacterial load.
The model allows us to measure the relative impact of the immune cell populations. In particular, the negative correlation between and ␤ shows that a decrease in the rate of elimination by neutrophils is compensated for by an increase in the rate of elimination by B cells. Furthermore, we find that the rate of elimination by neutrophils increases when the bacterial growth rate is higher (positive correlation between ␥ and ) ( Fig. 1D).
In vivo bioluminescence imaging data. We linked the model results to longitudinal whole-body C. rodentium imaging in vivo. We measured disease progression in four mice infected with bioluminescent C. rodentium using the IVIS system for optical tomography and X-ray CT from day 1 to the peak of infection at day 8. Reconstruction of the three-dimensional (3D) bioluminescence distribution for a representative mouse clearly shows a local increase in bioluminescence signal in the abdomen (Fig. 2B). We quantify the bacterial load by summing the total bioluminescence signal in the abdomen for each mouse at each time point ( Fig. 2A). For the purpose of comparison with model results, we normalized these signals to the one obtained at day 8. We determined the trend in the means and standard deviations of the quantified imaging data using a method that assumes a certain smoothness over time and for which no additional parameters are necessary (we use Gaussian process [GP] regression, a nonparametric Bayesian method for nonlinear regression [23,24]). The trends apparent in the quantified in vivo imaging data correspond nicely to the confidence interval calculated with the dynamic model ( Fig. 2A).
Using the model, we predicted the outcome of imaging experiments. In Fig. 2C we show the estimated mean bacterial loads on three different days. In this way, the model can serve as a tool for experimental design (25,26) and to monitor imaging outcomes against predictions during the course of an ongoing experiment.
For the same four mice, we also acquired the 2D bioluminescence distribution at day 8 and the ex vivo 2D bioluminescence distribution of the mouse intestines (Fig. 2D). We quantified the signal in these images in the same way as that described above and compared quantification across the three acquisition methods ( Fig. 2D and E). The quantified signals correspond to the predicted confidence interval.
Perturbations in the host immune response. Changes in the host immune cell response, as observed, for example, when using knockout mice, can result in a change in bacterial death rate due to altered immune system activity. We investigated the effects different cells of the immune system have by the model parameters corresponding to the activity of neutrophils () and B cells (␤), dissecting the contributions of these different immune cell populations (Fig. 3). We either increased parameters by 50%, decreased them by 50%, or set them to zero, and we compared the resulting model simulations.
A change in the neutrophil term ( Fig. 3, top row) has an impact on the height of the peak of infection: even a small decrease in neutrophil effectiveness may lead to inability to clear the infection. A change in the B cell term (Fig. 3, bottom row), being active during the later stage of disease progression, has a less pronounced effect. Setting activity of either cell type to zero would lead to inability to clear the infection.
The perturbations considered here can be related to knockout mice described in the literature. A decrease in the accumulation of neutrophils and macrophages to the site   of infection has been observed in MyD88 Ϫ/Ϫ mutant mice. These mice lack MyD88, which is part of the signaling pathway of Toll-like receptors that activates inflammatory cytokine production, and as a result the recruitment of neutrophils and macrophages to the site of infection is impaired (27). Bacterial levels quickly expanded followed by death from day 7 when the level of colonizing bacteria reached high levels. A similar quick expansion toward high levels of bacterial burden is also observed in the corresponding simulation (Fig. 3, top right image). Setting both B cells and the T cell population to zero (Fig. 3, bottom right image) is related to rag1-deficient mutant mice. Infection in this mouse model experimentally becomes a chronic infection (28). The corresponding simulation shows a similar pattern of chronic infection, i.e., the immune response does not clear the infection over the time scales considered here (or in the experiments).
Immune response to infection with C. rodentium espO deletion mutant. We next used the model to investigate changes in the immune response when wild-type C57BL/6 mice are infected with C. rodentium carrying a deletion of the T3SS effector EspO, a homologue of OspE from Shigella, shown to stabilize focal adhesion and block cell detachment (29,30) (Fig. 4A).
We measured the levels of neutrophils, B cells, and CD4 ϩ T cells in colonic tissue samples at four time points, as well as the level of bacterial shedding in the stool at 14 time points, spanning the course of infection. For the espO mutant, we observed a significantly higher bacterial load at days 14, 15, and 16 postinfection compared to infection with wild-type C. rodentium. The recruitment of immune cells to the colon following infection with the mutant was generally similar to what we observed during wild-type infection (Fig. 4B), with the main difference being that T cell infiltration was reduced on day 14.
We then further analyzed the dynamics by fitting our model against C. rodentium ΔespO stool measurements to characterize the response dynamics in this mutant (estimating parameter values in the same way as was described above), using the immune cell populations measured for the mutant as the starting point for the analysis (Fig. 4B). We found that the estimated parameter correlations are similar to the parameters we obtained when fitting the wild-type infection dynamics, and that the order of magnitude of the estimated parameters was the same (Fig. 4C and D). Elimination due to the B cell term in the model depends on B cell abundance, their activation by CD4 ϩ T cells, and the estimated rate parameter. At day 14, the total elimination rate corresponding to the active B cell term in the model was smaller during infection with the ΔespO mutant than with the wild type, mainly due to the difference in T cell abundance. This provides a possible explanation for the prolonged high bacterial load during infection with the ΔespO mutant.
In order to validate this finding, we proceeded to measure the level of antibodies against EspA experimentally following wild-type and ΔespO mutant infections (Fig. 4E), and we found a significantly lower level of C. rodentium-specific antibodies at day 14 during infection with the ΔespO mutant, confirming the model prediction.
Therapeutic response. Finally, probing a pivotal aspect of bacterial infection biology, we used the calibrated model to investigate the therapeutic response to two antibiotic treatments. We evaluated antibiotic treatment of C. rodentium infection with the bactericidal antibiotic ciprofloxacin and with bacteriostatic chloramphenicol. We measured the C. rodentium bacterial burden in vivo during antibiotic chemotherapy using 2D bioluminescence imaging between days 6 and 14 postinfection (Fig. 5A). Treatment was started on day 6 postinfection and continued until the end of the experiment. We quantified the imaging data by summing the signal (total flux) from the abdomen for each mouse (Fig. 5B), corrected for a constant background signal level of 1e5 photons/s, determined by quantifying the uninfected mouse images. The measured bacterial loads showed a sharp decrease when treatment with either antibiotic was started, while the standard deviation was larger during treatment with chloramphenicol, suggesting more variable therapeutic effectiveness. The quantified 2D imag-  ing data of wild-type infected mice show excellent correspondence to the calibrated model (Fig. 5C, first image). We simulated treatment with ciprofloxacin by including the treatment term in the model (), resulting in a steep decrease of the bacterial load, with a narrow confidence interval (Fig. 5C, second image). We modeled treatment with chloramphenicol by changing the growth term in the model (␥). The bacterial growth rate is a net rate of bacterial multiplication and death, and stopping multiplication would make this term negative. This results in a decrease in the bacterial load, but the progression is now more variable than it would be if the treatment resulted in the direct killing of bacteria (Fig. 5C, third image). Finally, we can use the model for experimental design (25,26) to determine the effect of treatment during a specific time window. We can evaluate any combination of starting and stopping times for the treatment by simulating with the model in order to identify which would be the most relevant experiments to carry out. We simulated treatment with a single dose of chloramphenicol and ciprofloxacin (at day 6) and found that compared to treatment with a single dose of chloramphenicol, treatment with a single dose of ciprofloxacin would have a larger beneficial effect (Fig. 5D).

DISCUSSION
In this study, we developed a mechanistic modeling approach that links bacterial colonization levels measured in vivo to the immune response, including possible treatments. Whole-body bacterial colonization was measured in the stool and observed in vivo, and immune cell populations (neutrophils, CD4 ϩ T cells, and B cells) in the gastrointestinal tract were obtained postmortem. The modeled colonization levels correspond very well to the colonization levels observed with both 2D-and 3Dquantified in vivo imaging data. We used the mechanistic model to unravel the effect of each cell type, as well as different antibiotic treatments, on the course of disease.
By simulating the outcomes corresponding to observed changes in host immune response due to mutations described in the literature using our dynamic model, we analyzed a range of separate literature studies within a single modeling framework. The model allows us to draw conclusions regarding the role of each of the aspects of the immune response in infection clearance in wild-type (C57BL/6) mice. Moreover, the results obtained from wild-type mice suggest that the sometimes disparate behavior found in different experimental studies can be combined into a clear picture of infection dynamics with the help of the general model presented.
Second, we have analyzed the immune response to the C. rodentium ΔespO mutant and found prolonged colonization compared to the wild type. Using the model, we then predicted that a lower total elimination rate of B cells at day 14 could explain the prolonged infection, which we validated experimentally by measurement of IgG antibodies against EspA. This shows how to use the model to identify gaps in our knowledge that require further investigation. Such successful predictions also serve to validate the model. The success of the current study opens the prospect of using a similar approach to analyze the influence of other effectors on the immune response, such as Tir, which can affect the recruitment of neutrophils (31).
Finally, we have used the dynamic model to evaluate the response to antibiotics, showing that simulations with the model accurately describe two different effects: the effect of antibiotics that directly kill bacteria and the effect of antibiotics that stop growth. Comparison with experiments shows that the model captures the variation of colonization levels in time both qualitatively and quantitatively. Therefore, these types of simulations can be used as a tool for experimental design and to compare imaging results with expectations during imaging experiments. The simulations also allow feasibility studies prior to experiments to indicate whether the experiments would result in a detectable change in the bacterial load in vivo, for example, in the context of probiotic experiments (32), and to explore different treatment regimens in terms of timing and duration.
What is already becoming clear from the current study is that the use of bacteriostatic antibiotics risks that there remains for a considerable time, probably longer than a typical or practical therapeutic time course, a pool of bacteria that are ready to lead up to recurrent disease, persistence, and the evolution of antibiotic resistance. Here, modeling of the type developed in the current study offers a rational way to prioritize therapeutic approaches.
The modeling approach that we provide in this study can be applied to a wide range of experimental studies. In this work we have used quantified bioluminescence data, but the same type of model can be used in combination with other state-of-the-art optical imaging modalities, such as fluorescence optical tomography and X-ray CT (33). The emerging range of markers that is being developed for visualizing many different disease parameters in vivo strongly increases the number of links that can be made between in vivo and postmortem observations. Of particular interest are markers that visualize the species included in the model in vivo, such as markers for neutrophils and T cells.
Previous studies have also combined in vivo imaging and postmortem experiments to analyze C. rodentium infection. For example, Hall et al. (15) have studied the role of natural killer cells during C. rodentium infection using in vivo optical imaging, and Cronin et al. (34) have studied the use of C. rodentium as a vehicle for cancer treatment. The modeling approach that we describe here is ideally suitable to relate the reported in vivo colonization levels to the postmortem-derived parameters in these studies to provide a direct connection. Compared to available simulation platforms (19), the limited number of host-pathogen interactions included in our model makes it specific to the experimental setup, enabling us to directly validate theoretical results against experiments. More detailed spatial information can be included using a model with multiple compartments, and different scale levels can be included by combining this with agent-based modeling, as has been done for investigating tuberculosis (35). However, the spatial distribution of bacteria would be required, for example, by visualizing this in vivo using X-ray CT contrast agents.
The relatively simple modeling approach that we employ, using an ordinary differential equation (ODE) model in combination with Bayesian parameter inference, facilitates the interaction between theory and experiment and enables combined research toward the discovery of major biological breakthroughs. In this way, the developed modeling framework serves as an interface between theory and experiment and can play an important role in the future for specifically identifying gaps in our current knowledge, to define key points that require further investigation, both in terms of modeling and experiment, as well as for the development of suitable treatments.

MATERIALS AND METHODS
Bacterial strains and growth conditions. The bacterial strains and primers used in this study are listed in Table 1. Bacteria were grown in Luria-Bertani (LB) medium supplemented with kanamycin (50 mg ml Ϫ1 ) and nalidixic acid (50 mg ml Ϫ1 ) as required.
C. rodentium espO deletion mutant strain ICC1333 was constructed using the PCR one-step Red recombinase method (37). Briefly, the mutation was obtained using a PCR product containing the Animals. All animal experiments were performed in accordance with the Animals Scientific Procedures Act of 1986 and were approved by the local Ethical Review Committee. Pathogen-free female 18to 20-g C57BL/6 mice were housed in HEPA-filtered cages with sterile bedding, food, and water.
Mice inoculated with the wild-type strain and/or with phosphate-buffered saline (PBS) (mock infection) were included in parallel with the mutant strain. A minimum of 5 and a maximum of 8 mice per group was used for each experiment. Each experiment was repeated a minimum of two times.
Oral gavage of mice. Mice were inoculated by oral gavage with 200 l of overnight LB-grown C. rodentium suspension in PBS (ϳ5 ϫ 10 9 CFU) or mock infected with PBS. The number of viable bacteria used as the inoculum was determined by retrospective plating onto LB agar containing antibiotics. Stool samples were recovered aseptically at regular intervals after inoculation, and the number of viable bacteria per gram of stool was determined by plating onto LB agar. At appropriate time points, mice were culled and the colonic tissues collected for flow cytometry analysis. Cardiac punctures were performed for serum analysis.
For studies where mice were treated with antibiotics, at day 6 postinfection (p.i.) mice were gavaged without anesthesia and with 200 l ciprofloxacin (100 mg/kg of body weight/day), chloramphenicol 100 mg/kg/day), or sterile distilled water (vehicle control) daily for 7 days.
Sample collection for flow cytometry. A 4-cm segment of the terminal colon was cut, opened longitudinally, and rinsed in sterile PBS. The tissue samples were placed in 4 ml of RPMI 1640 supplemented with 10% fetal bovine serum (FBS), penicillin-streptomycin (P/S), GlutaMAX, DNase (10104159001; Roche), and liberase (540112001; Roche) in a C-Mac tube (Miltenyi Biotec), followed by tissue dissociation using a gentleMACS dissociator (Miltenyi Biotec). First, the tissue was homogenized using the intestine setting followed by incubation at 37°C and 5% CO 2 for 30 min in a shaking incubator, and then a final dissociation step was performed using the lung 2 setting on the gentleMACS dissociator. The digested preparation was disrupted to a single-cell suspension by passage through a 70-m sieve (no. 352350; BD Labware/Falcon, USA) and resuspended in RPMI 1640 supplemented with 10% FBS and P/S at 0.5 ϫ 10 6 to 1 ϫ 10 6 cells/ml. Extracellular antigen analysis. Cells were stained for surface markers (such as CD4 ϩ CD8 Ϫ T cells, B220 ϩ CD3 Ϫ B cells, and CD11b ϩ Ly6G ϩ neutrophils) in PBS containing 1% bovine serum albumin with 0.5% sodium azide (PBA) for 30 min at 4°C and fixed with IC fixation buffer (eBioscience) as described previously (31). Nonviable cells were excluded using a fixable near-infrared dead cell staining kit with excitation at 633 or 635 nm (L10119; Invitrogen). Antibodies were purchased from BD Pharmingen or eBioscience.
Data acquired on a BD Fortessa III with 20,000 live lymphocytes or myeloid events were analyzed with the FlowJo (TreeStar) analysis program. Data are shown as a percentage of live myeloid or lymphocyte gates. Myeloid and lymphocyte gates are determined by their position on the live/dead versus forward scatter plots generated by the cytometer. A fluorescence-minus-one (FMO) control was included for each fluorescent marker, and the expression of a particular marker was calculated by subtracting FMO fluorescence values from fluorescent antibody levels.
Purification of recombinant C. rodentium EspA. A pET28a-CR-EspA construct (pICC929) ( Table 1), expressing C. rodentium EspA with a carboxyl-terminal 6-His affinity tag, was generated by PCR amplification from C. rodentium genomic DNA using primers EspA-CR-NcoI-Fw and EspA-CR-EcoRI-Rv ( Table  1) and ligation of the NcoI-and EcoRI-digested amplicon into pET28a digested with the same enzymes. The resulting pICC929 plasmid was transformed into E. coli K-12 strain BL21(DE3) Star cells, which lack RNase E, to stabilize mRNA (Novagen), and the protein was expressed under isopropyl-␤-D-thiogalactopyranoside (IPTG) induction and affinity purified with nickel resin using standard protocols. The yield, size, stability, and purity of the expressed protein was assessed by SDS-PAGE. Protein concentration was determined by the bicinchoninic acid method (Pierce Biotechnology, Inc., Rockford, IL, USA) (38).
Detection of C. rodentium anti-EspA IgG in serum. An enzyme-linked immunosorbent assay (ELISA) was used to detect serum IgG and antibodies to C. rodentium EspA. Sample dilutions were added to the antigen-coated wells (0.5 to 1 mg of C. rodentium EspA per well) of a microtiter 96-well plate. After incubation for 1 h at room temperature and subsequent washing, wells were incubated with horseradish peroxidase-conjugated donkey anti-mouse IgG. After further washing, tetramethylbenzidine substrate (TMB; Sigma) was added. Color formation was stopped after 10 min with 2.5 N sulfuric acid. The optical density was measured at 450 nm (OD 450 ) with background correction at 620 nm. Fold change was calculated as the ratio to EspA IgG level detected in PBS mock-infected mice.
In vivo optical imaging of C. rodentium-infected mice. Whole-animal bioluminescence imaging (BLI) was performed daily following infection using an IVIS SpectrumCT (PerkinElmer) (39). In some studies, representative mice from each group were imaged using diffuse light imaging tomography with integrated CT imaging (DLIT-CT) as previously described (6,39). At necropsy on either day 8 or 14 p.i., whole organs or washed excised gastrointestinal tissues with the mucosa exteriorized were imaged. Regions of interest were identified, and the total photon flux was quantified (photons/s) using Living Image software 4.3.1, service package 1 (PerkinElmer).