The first step toward ABM model development was describing the BJI system and its behaviors and characterizing its components and their roles. A systematic literature search was conducted to retrieve the agents’ behaviors, rules, and interactions, and to identify the relevant parameters and their ranges of applicable values. By giving some parameters a range of values, we enriched the ABM model with the capacity of simulating variable behaviors of the agents and testing several hypotheses. We used the information from the basic biology to characterize the cellular and molecular attitudes of different components of BJI biological system during the first stage of infection; namely those cover the innate immune response, the bone remodeling cells and signals, and the bacteria development. These biological characteristics will be simplified and formulated as agents’ rules. The different parameters were retrieved by first looking for human studies, but regarding the lack of several information, we extended our search to include general characteristics of the components, animal models of BJI, and mathematical models of the bone remodeling process.

Search was conducted in PubMed and ScienceDirect databases exploring the relevant journal papers from their inception to date, July 2017. Literature search strategy, i.e., search queries and inclusion/exclusion criteria, are described in detail in Supplementary Material and Supplementary Table S1.

The goals of the simulation were to illustrate the relationships between variation in the bone cell population and the evolution of infectious processes. The goal was also to investigate the efficiency of the innate immune system in defending the bacteria during the first stage of the BJI for several scenarios without treatment intervention. Further, the simulations were conducted to detect sensitive parameters that affected the dynamics of the system.

To verify the model stability and performance, and to investigate the effect of bacterial inoculum size on the system dynamics, the initial number of bacteria were adjusted to three different values (5, 5 × 10, 5 × 10² CFU/mm²). These values were assumed to represent three different infected inoculum states – low (5 CFU/mm²), medium (5 × 10 CFU/mm²) and high (5 × 10² CFU/mm²). We ran the simulations for n = 100 iterations for each of the infected inoculum state, while all other changeable parameters were set to the median of their range (Table 3). For each iteration, the dynamics of the population for bacteria, osteocytes, and neutrophils agents for a time duration of 300 h (t = 300 ticks) was tracked. The generated data at each time click (1 h) were saved to an output data (.csv) file for analysis. Subsequently, the mean and standard deviation for the population at each time step for each type of these agents was quantified. Further, the relation between two agents over time under the same initial conditions was analyzed. We benefitted from the 3D surface graphs and the generated data from the previous step to analyze the relationships between each of bacteria and neutrophils populations over time, bacteria and osteocytes populations over time, and neutrophils and osteocytes populations over time.

The systematic search process identified 42 articles for bone cell characterizations, 48 articles for immune cell characterization, 12 articles for bacteria characterization, and 29 articles for interactions (Supplementary Figures S1–S4). Following the systematic literature search, the agent interactions and the parameter ranges were identified from the retrieved articles. Flowcharts of the review steps along with the list of all articles used for the study are reported in Supplementary Figures S1–S4 and Supplementary Table S2.

A representative example of the model space was taken during one simulation at three different times (Figure 1). The initial state of the model was represented at the beginning of the simulation (0 h), for initial inoculum infection state of (5 × 10² CFU/mm²), where this initial number of bacteria were randomly distributed in the adjacent surface. It also demonstrated the low presence of immune cells macrophages and neutrophils, and the initial state of bone tissue before being damaged (Figure 1A). The progress of the infection during the simulation (60 h) later showed that the bacteria propagated toward entering the bone tissue and destroying it. The monocytes also had been activated in this step besides the other immune cells (Figure 1B). The simulation state at (150 h) showed the system after 6 days, where the damage happened to the bone tissue, and the bacteria count decreased because of engulfing by the immune cells (Figure 1C). This model is 2D what makes it suitable to represent trabeculae bone, but it could be also considered as a cross-section of cortical bone.

The mean and the standard deviation of the dynamics of the population for 100 iterations under different infection inoculum states (low, medium, high) revealed interesting behavior for each of the bacterial, neutrophil (PMN), and osteocyte population (Figure 3). The bacterial population inclined toward the same steady non-null counts, regardless of the inoculum state (Figures 3A1–A3). In addition, the population intensity of the first few days was proportional to inoculum size. It was also observed that the behavior of the bacteria varied compared to the mean behavior with small variance magnitude even if the output illustrated (by the effect of time characteristic) a high-frequency oscillation in the bacteria population. At the same time, it showed low fluctuation in the context of the general trend of the bacterial population (Figures 3A1–A3). It was also observed that the bacterial population faced quasi-extinction, followed by re-growth for the inoculum in the medium and high infection state for all 100 iterations.

On the other hand, the PMN dynamics followed the evolution of the bacteria with a slight delay, reaching a non-zero stable level on the 12th day. In addition, it illustrated that the PMN population was not subject to rapid variations represented by the low-frequency oscillations, which are comparatively smoother than those introduced by the bacteria (Figures 3B1–B3). Asymptotic behavior of PMN population for the three inoculum states proposed a non-null mean value with fixed frequency oscillations and decreasing magnitude with time (Figures 3B1–B3).

Concerning the bone tissue loss in each state, the osteocyte population dynamics was similar in mean population intensity, while it differed from the mean behavior with important variance magnitudes whatever the inoculum. The bone cell population predicted a trend of unexpected high-frequency oscillations after t = 200 h (Figures 3C1–C3).

Comparisons of system response for different inoculum states and over time for the same type of agent population suggested that the high inoculum infection state was associated with close to elimination of bacteria population and full elimination of PMN population. It was also associated with the highest population counts for both agent types in the first few days of the infection (Figure 3). The results presented that bacteria and PMN tended toward a steady asymptotic non-null state, regardless of the inoculum infection state. With regards to bone tissue damage, the first response stage had a similar degradation phase of the bone infected site represented by decreasing count of osteocytes for each inoculum state, where it was noted that osteocytes passed by a common minimum level by the 7th-8th day for all three inoculum states, representing 2% of the loss in the infected site mass (Figure 3). During the recovery phase, it was observed that the smallest inoculum infection state caused the most severe and relatively stable loss on bone cells by 12th day, the medium inoculum infection state showed better progressive recovery, and the high inoculum infection state displayed an intermediate recovery, with sub-optimum recovery at 300 h. The displayed loss of bone tissue compared to baseline for each inoculum infection state was 1.4, 1.2, and 1%, respectively (Figure 3).

The 3D representation of the relation between two types of agents over time suggested minimum levels of bacteria and PMN population count around the fourth day of infection, while it suggested a delay in minimum levels of population in osteocytes count with regard to the minimum levels of bacteria and PMN (Figure 4).

For the proposed inoculum infection states, on the12th day, the observed bacterial population for the inoculum of the order of 100 was stable and not extinct, which could also be considered as an indication for latent infection. Since the PMN populations followed, in the same manner, the bacteria populations but in a slight delay to reach the stable non-null levels on the 12th day, it could also be used as an indication of the latent infection. If so, it should be validated in terms of PMN count in the biological laboratory test. In addition, the relationship between the osteocytes and the inoculum infection state raises the question of whether it states a restitutio ad integrum for the medium infection state that introduced the best progressive recovery.

Further, the observed stable non-extinct level of bacteria population 10 days after the infection showed the inability of the innate immune cells to completely eliminate the bacteria. This points out the imperative role of further defense methods: the adaptive immune response, which initiated by the 4th–7th day post the infection, and the need for therapeutic intervention.

In terms of population fluctuations (intra-simulation), the observed small variance within the bacterial population dynamics highlighted their stable behavior even with the rapid, strong oscillation which could be an artifact of the internal clock management in NetLogo algorithm. The fluctuations observed in the PMN population had a less important effect compared to the bacterial state, which reflects a spatial impact. In fact, greater the probability that a PMN encountered a bacterium, more likely it would have reacted, which ultimately reduced the fluctuations.

On the other hand, the predicted high-frequency oscillating trend of the osteocytes population after t = 200 h was unexpected. It was expected to display a rapid decreasing phase, and then a slower increasing phase. This considerable fluctuations in the osteocytes population appeared interesting as it could partially explain the inter-individual variability. Further, these fluctuations increased in intensity over time, raising the question if it could be explained as a pre-chaotic behavior “positive feedback.” It was also noted that the PMN and the bacteria populations reach their minimum levels simultaneously before rising again. This weakness in the innate immune response could provide a better possible window to start the therapeutic treatment.

The BJI system model developed in this study is the first to incorporate experimental data with the ABM modeling approach and extract information in the form of a dynamic system and it lays the foundations for an in-depth and detailed BJI simulation system. The interactions between the agents and signals provide a comprehensive ability to analyze the system considering the spatial characteristics and the inter-agent variability. Additionally, the observed results corroborate with clinically and microbiologically available pathophysiological pathways and they are generally in line with expectations, as that the investigated populations move toward equilibrium (Wagner et al., 2003). This model has also shown its predictive ability for the evolution of bone mass with respect to bacterial inoculum states and time. The proposed model framework introduces a flexible and interactive virtual laboratory to test and explain several existing hypotheses or even to design new ones.

One of the major limitations of the ABM approach is that it generally cannot be released simply, unless the entire model is given. Evidence is also difficult to obtain, and implementation details can cause many problems. For that, we had to start with a less complex model (at the same time integral one) and to build later upon it. The developed model presents certain limitations, and all the results should be evaluated within the framework of these limitations. The first limitation is model validation due to the lack of experimental data, which might be acquired at a later stage (e.g., the bone mass). The 2D representation represents a limitation with regards to the ability of agents to interact in a 3D space. However, we adopted the simplicity in building this first model aiming at developing a feasible modeling framework and understanding the complex integration of available physiological data with the ABM modeling framework. Future work is aimed at enhancing the model by a more realistic architecture, tissue specification, and patient-based data. The current model lacks the second stage of the immune response, the adaptive immune response, which is necessary to investigate the further progression of the infection. The bacteria biofilms, which are lacking in the current model, also play an important role in identifying the behavior of the bacteria itself and the system response due to their resistance to immune defense and antibiotic agents. Since biofilms are common during this infection, it is worthwhile to model them in the future. In this model we were limited to S. aureus, other pathogens will be investigated in the future. In this first model, the BJI was modeled without prostheses, which will be added in the future. Other agents including bacterial survival factors have a significant contribution in introducing an additional approximation to mimic the real system. The adaptability of the present model depends on the type of modifications that are foreseen. All modifications needed would be investigated through the same process we described in this study: first, identify if (i) new agents (e.g., cells) should be integrated and (ii) already modeled interactions are sufficient or should be modified or completed by new ones; second, list all relevant parameters and parameters values related to these modifications through literature search; finally, model agents, messengers and implement parameters values in the new model. This process can be applied if we want to introduce variants or new pathogens that may interact with each other and with the existing agents or if we want to model cancellous bone rather than cortical bone or if children and adults BJIs differ from critical aspects.

This work has drawn attention toward the incorporation between modeling power and current literature using the ABM modeling framework of infection dynamics. It also highlights the progression of the simulation environment that generates data that would be used in the extraction and synthesis of the model in the form of differential equation systems. This model could in fine be integrated within a global predictive approach to propose more personalized therapeutic treatments. The future global approach would combine microbiological and imaging data of the patient to extract infection parameters (bacteria, immune system) and patient-specific bone morphology. As further data are made available, the model would be refined to better estimate an in-depth comprehension of BJI’s pathophysiology. Furthermore, studying the dynamics of the BJI including critical parameters such as the agent interactions and cross-talk signals could make it possible to test new or alternative therapeutic pathways for their efficacy.