A mathematical model of neuroinflammation in severe clinical traumatic brain injury

This study was approved by the Institutional Review Board at the University of Pittsburgh. We utilized clinical, demographic, and CSF inflammatory biomarker data from n = 89 individuals with severe TBI who were between 16 and 75 years of age and had admission Glasgow Coma Scale (GCS) scores ≤ 8 with positive findings on head computed tomography. Participants were excluded under the following circumstances: a penetrating TBI, documented prolonged cardiac/respiratory arrest at the time of injury, evidence of brain death in the first 3 days of injury, or an Abbreviated Injury Scale (AIS) score ≥ 5 in a non-head region. Based on International Classification of Disease (ICD)-9 codes reported at the time of acute care discharge, no individuals in this study had a history of or concurrent malignant neoplasms (implicating cancer) at the time of injury. Individuals with TBI received care consistent with The Guidelines for the Management of Severe Head Injury [18].

Neuroinflammatory markers were measured using Luminex™ bead array assays (Millipore, Billerica, MA). These markers included interleukin (IL)-1β, IL-4, IL-5, IL-6, IL-7, IL-8, IL-10, IL-12, tumor necrosis factor alpha (TNF-α), soluble vascular adhesion molecule-1 (sVCAM-1), soluble intracellular adhesion molecule-1 (sICAM-1), and soluble Fas (sFAS). All inflammatory markers were considered in principal component and cluster analyses, as reported previously [19]. However, only a subset of these markers (IL-1β, IL-4, IL-10, and IL-12) was utilized in the mathematical model to represent hallmark pro- and anti-inflammatory mediators of microglia activity and brain tissue integrity [2, 20].

Demographic and clinical variables were collected via in-person interview and electronic medical record abstraction. The variables reported include age, sex, body mass index (BMI), injury severity scale (ISS) score, best GCS score in 24 h, and length of stay in acute care. The functional capacity of patients was assessed using the GOS at 6 months to measure long-term global recovery. Individuals received the following GOS scores accordingly: (1) dead, (2) vegetative state, (3) severe disability, (5) moderate disability, and (5) good recovery [21].

Principal component analysis (PCA) previously reported by R.G. Kumar et al. identified sets of inflammatory markers that contribute the greatest variance to acute CSF inflammatory profiles (days 0–5 post-TBI) [19]. Individuals were assigned a score for each significant principal component (PC) based on their levels of the inflammatory markers that contribute to that particular PC. A non-hierarchical k-means cluster analysis was then conducted on the scores for significant PCs for all individuals, characterizing patient subpopulations with similar acute neuroinflammatory profiles post-TBI. This analysis yielded two major cluster groups that were distinguished by distinct CSF inflammatory profiles for days 0–3 post-TBI. Within clusters 1 and 2 identified by R.G. Kumar et al. [19], we grouped individuals in each cluster based on GOS scores. Unfavorable and favorable outcome groups were defined as individuals exhibiting a 6-month GOS = 1,2,3 (dead/vegetative state/severe disability) or a 6-month GOS = 4,5 (moderate disability/good recovery), respectively. These clustering techniques produced the following patient groups: an unfavorable outcome group with a relatively high inflammatory load (cluster 1), a favorable outcome group with a lower inflammatory load (cluster 2A), and an unfavorable outcome group with a similar lower load (cluster 2B).

We built upon the initial statistical work described above by R.G. Kumar et al. [19], which identified patient subgroups with distinct acute neuroinflammatory profiles, by using mathematical modeling techniques to investigate potential mechanisms in post-TBI neuroinflammation that may underlie the observed patient heterogeneity. A system of ODEs was derived to represent the temporal dynamics of four cytokines (IL-1β, IL-4, IL-10, and IL-12), which were determined from the literature to constitute the minimal set needed to represent the range of roles played by cytokines in regulating both microglial responses and subsequent secondary CNS tissue damage time courses [2, 20]. The equations consist of terms that represent biological processes—such as production, inhibition, saturation, or decay of particular inflammatory mediators—and capture changes in those inflammatory mediator levels over time. ODE models provide a framework for representing multiple interactions and dependencies between mediators post-injury.

To guide the derivation of the ODE model, a simplified conceptualization of microglial contributions to acute neuroinflammation post-TBI was formed (Fig. 1). This theoretical model was limited to a core set of mediators previously identified as having distinct roles in microglial behavior. The corresponding reduced mathematical model, much in the spirit of previous reduced ODE models that have been proven to be useful in the analysis of acute inflammatory responses [13‐17], aimed to characterize acute neuroinflammatory and microglial dynamics through seven differential equations for the following biological variables: M1-like microglia (M1), M2-like microglia (M2), IL-1β (IL1), IL-12 (IL12), IL-10 (IL10), IL-4 (IL4), and secondary tissue damage (D). The differential equations include 52 parameters with direct biological interpretations, as shown in Table 1. The following sections detail the acute neuroinflammatory processes post-TBI that are encompassed within our ODE model and a priori information that guided its derivation.

Our ODE model (Eqs. (3)–(9)) includes a large number (52) of parameter values, mostly representing rate constants, half-activation, or saturation levels, and exponents that affect sensitivity to changes in levels of evolving quantities. These parameter values were constrained by previous literature and biological requirements (e.g., positivity), but many have not been measured experimentally. Our approach was to tune parameter values through parameter optimization methods to produce outputs consistent with observed clinical inflammatory biomarker trajectories while remaining within the constraints that we imposed. Optimization was performed over the 45 parameters listed in Table 1, as well as over initial condition values for M1, M2, IL1, IL12, IL10, IL4, and D. Model integration and parameter optimization were performed using Matlab (MathWorks, Natick, MA).

For each patient group, the model was fit to computed CSF cytokine averages over the first 5 days following TBI in 6-h increments. A moving-average smoothing procedure was applied to the cytokine data to diminish the effect of outliers and short-term fluctuations in the data. Specifically, we defined overlapping 12-h bins, each shifted by 6 h relative to the previous bin, consisting of data from 0 to 12, 6–18, 12–24 h post-TBI, and so on. For each cytokine, all values that occurred during the time range encompassed by a bin were averaged together.

Initial parameter estimates were guided by available values reported in the literature [14, 16]. The Nelder-Mead simplex method was then employed as a nonlinear optimization algorithm to determine sets of initial conditions and parameter values that best fit model outputs, obtained by numerical integration of Eqs. (3)–(9) from a given set of initial conditions, to the averaged clinical data [36]. This method was implemented by using the fminsearch function in Matlab to attain optimal model fits to cytokine data from each subcluster. The Matlab function ode15s was used to solve the system of ODEs. Goodness of fit to patient data was determined by evaluating an error function that compared the model-generated cytokine values to mean patient cytokine values at every 6-h mark, summing the squared differences and dividing by the squared standard deviation to normalize. Additionally, the following microglia heuristics were set in place: M1 > M2 prior to day 2, M2 > M1 after day 3, and M1, M2 > 2 for the entire time course. A penalty value of 100 was added to the error calculation for each of these conditions that was violated. Empirically, smaller penalty values did not yield enforcement of the desired conditions.

Parameter estimation techniques led to a representative parameter set that fit cytokine data for each patient group. This baseline parameter set was then randomly perturbed to provide new parameter sets that would serve as starting estimates for an additional 100 iterations of subsequent parameter estimation. Perturbations were randomly selected from a uniform distribution from 0.5 to 1.5 times the baseline parameter set values. The parameter optimization procedure was repeated for each of the 100 distinct parameter sets for each patient group, yielding an ensemble of model fits. The use of this method was intended to allow for the possibility that diverse parameter sets could provide similar model fits and to capture the variability in biological characteristics that would naturally occur across a patient population.

The ensembles of 100 model trajectories and parameter sets from which they were generated were compared across patient groups to investigate relative differences in cytokine, microglia, and tissue damage temporal dynamics. Statistically comparing the distributions of each parameter value between clusters was used to elucidate physiological mechanistic differences that potentially explain divergent clinical outcomes post-TBI.

In previous work, R.G. Kumar et al. identified groups of patients with similar neuroinflammatory profiles in the acute phase with no a priori knowledge of long-term outcome [18]. However, further analysis revealed significant associations between cluster group assignment and long-term outcome based on GOS score. The vast majority (93.1%) of individuals in cluster 1 experienced unfavorable outcomes (GOS = 1,2,3), whereas members of cluster 2 varied greatly in their recovery status 6 months post-TBI. We thus grouped cluster 2 into cluster 2A, consisting of the 45.5% of individuals in cluster 2 reporting favorable outcomes (GOS = 4,5), and cluster 2B, composed of the remaining 54.5% with unfavorable outcomes (GOS = 1,2,3). Grouping by 6-month GOS score was not implemented in cluster 1 due to the small number of favorable outcome patients belonging to this group (n = 2). GOS scores at 6 months post-TBI (Table 2) and clinical characteristics (Table 3) are reported for each of these patient clusters. Those in cluster 1 were substantially older compared to individuals in cluster 2, and hospital length of stay was shorter, a finding likely due (in part) to higher acute mortality rates in this patient cluster. These groupings show that (1) different acute neuroinflammatory response profiles may lead to similar outcomes, and (2) similar acute neuroinflammatory responses may lead to disparate outcomes.

An ODE model (Eqs. (3)–(9)) for the combined temporal evolution of M1- and M2-like microglia, levels of cytokines IL-1β, IL-12, IL-10, IL-4, and a secondary tissue damage variable D was derived based on well-supported biology underlying the acute inflammatory response to TBI (“Ordinary differential equation model development” section). Parameter optimization methods were used to find collections of the 52 model parameters for which model trajectories best fit the averaged and smoothed data for each patient group (“Parameter optimization” section). Ensembles of 100 model fits, which were produced by a range of initial parameter sets along with averaged cytokine data values, are shown for each patient cluster in Figs. 2, 3, and 4.

Divergent responses are observable among the model trajectories for microglia, cytokines, and tissue damage across the three patient clusters. Cluster 1, which had the highest inflammatory load associated with unfavorable outcomes, demonstrated the most prolonged and elevated microglial expression of the three clusters. Cluster 1 trajectories were also characterized by relatively constant levels of pro-inflammatory cytokines IL-1β and IL-12, high initial values of IL-10, and rapid decay of IL-4. Interestingly, two types of damage responses were observed: a more common response in which damage rose only slowly, and a minority response with a rapid rise in damage following by a slow increase, a finding that likely captures heterogeneity expected in any patient population.

Optimal model fits to patient data were achieved by tuning biological parameters that appear in our set of differential equations (Eqs. (3)–(9)) describing the time course of acute neuroinflammation post-TBI. The resulting distributions of values for each parameter were compared to identify those parameters with the most significant variation across patient clusters (“Parameter distribution analyses” section). Each model parameter has a corresponding biological interpretation (Table 1), and the neuroinflammatory role of each identified parameter was considered in order to understand its contribution to the relative temporal dynamics of cytokines, microglia, and tissue damage that differentiate long-term outcome. Table 4 shows a summary of the most dissimilar parameter distributions between clusters, based on Bhattacharyya metrics of overlap (BC) and spread (BD). Parameter distribution differences were ranked by ascending overlap (BC) and descending spread (BD). We report and discuss the most disparate parameter distributions that were selected under the following criterion: BC ≤ 0.38, and BD ≥ 0.97.

In conjunction with Bhattacharyya tests applied to compare parameter distributions between clusters, a parameter sensitivity analysis was implemented through Matlab to ensure that the significantly different parameters under consideration also significantly influence model behavior (“Sensitivity analysis” section). Model sensitivities to values of all parameters reported in Table 4 exceeded our sensitivity threshold of 2. An additional file provides the ranges and averages of each dissimilar parameter distribution by pairwise cluster comparisons (see Additional file 1).

R.G. Kumar et al. found considerable differences between clusters 1 and 2 in terms of steroid hormones [18]. Motivated by previous work showing the existence of three patient groups with significantly different cortisol trajectories over the first 6 days post-injury [38], we investigated CSF cortisol level differences between patient groups (Table 5) to supplement our inflammatory marker comparisons. In post-hoc analyses, we found that cluster 1 had higher cortisol levels than both cluster 2A and 2B over the full time course (p < 0.001 for all comparisons). For days 0–3, cluster 2A had significantly lower levels of cortisol compared to cluster 2B (p = 0.002). However, transitioning into day 4–6, differences in average cortisol levels were no longer significant between cluster 2A and 2B (p = 0.340).

Initial statistical work by our group applied principal component and cluster analysis to identify variability in acute neuroinflammatory biomarkers among individuals following TBI [18]. These analyses suggested sets of cytokines that demonstrate similar acute expression patterns and may contribute to similar courses of inflammation and tissue recovery, helping to justify the inclusion of a small number of specific cytokines in our current modeling work. In this study, we sought to utilize cytokine dynamics to infer the neuroinflammatory state and subsequent microglial activation profile of each patient cluster.

This methodology is a progressive direction in the TBI field in that relative concentrations of cytokines can be tracked and used to inform hypotheses regarding early injury severity patterns and patient prognosis. While studies have shown that absolute levels of acute anti-inflammatory mediators are predictive measures of initial brain damage and complications, such as intracranial pressure [43], our model illustrates the relative relationships and time courses of mediators to provide a more comprehensive view of neuroinflammation post-TBI. The model was designed on the premise that cytokines are expressed simultaneously following injury, collectively contribute to microglial polarization profiles, and may serve as useful biomarkers patterns to gauge the level of neuroprotection or neurotoxicity in the local microenvironment [3, 41, 44].

In extension to cytokine measurement studies that characterize neuroinflammation post-TBI, this modeling framework is a novel method of investigating (1) temporal dynamics of inflammatory mediators, (2) interdependent cytokine pathways and feedback interactions, and (3) cytokine influences on differential microglia expression and tissue damage responses. Differential equations are well suited for representing post-TBI neuroinflammation because the relative concentration changes of inflammatory mediators can be modeled with respect to time. This initial report serves as a proof-of-concept that implementing mechanistic modeling can further our understanding of inflammatory network dynamics, kinetics, and phenotypic polarization.

While early delivery of anti-inflammatory agents has been an intuitive strategy for containing post-TBI neuroinflammation, the lack of consistent success suggests there is more to consider for this approach. Suppressing a single neuroinflammatory mechanism does not facilitate healthy tissue recovery due to complex interdependencies of neuroinflammation [3, 39]. There has been a shift of perspective in the TBI field that acknowledges not only the dual role of neuroinflammation post-TBI, but also the dichotomy of roles such as perpetuating damage and maintaining homeostasis by individual mediators, including cytokines and microglia phenotypes [9, 45]. The utility of ODE modeling in this context is to elucidate the benefit or detriment of particular mediators relative to time-post-injury and expression of other local mediators.

Model simulations performed on patient-specific data from different outcome groups produced quantitative projections of cytokine dynamics and qualitative predictions of microglia and tissue damage dynamics. The ensembles of parameters, which were tuned to generate optimal fits for each patient cluster, were statistically compared to generate hypotheses regarding differences in the neuroinflammatory regimes of each cluster. These analyses are a contribution to ongoing attempts to characterize the functional roles and heterogeneous effects of microglia and related cytokines in acute TBI neuroinflammation [9, 46]. This report presents the first computational model in the field that aims to model the temporal evolution and M1/M2 phenotypic balance of microglia. In the following sections, we discuss the inflammatory trends, unique parameter differences, and additional clinical considerations for each respective patient cluster and long-term outcome.

R.G. Kumar et al. had identified a group of individuals (cluster 1) with relatively high day 0–3 CSF inflammatory loads, almost all of which experienced poor long-term outcomes [18]. After subgrouping cluster 2 individuals by 6-month GOS score, we identified another poor outcome group yet with a relatively lower acute inflammatory load (cluster 2B). We hypothesized that the disparity in acute neuroinflammatory profiles could drive differential, yet both detrimental, courses of inflammation post-TBI that hinder recovery.

Cluster 2A model ensembles best demonstrate a beneficial physiological response to TBI. Microglial activity was present for a length of time that was neither permissive nor indiscriminate in pro-inflammatory mediated damage (as in cluster 1), or sub-physiological in minimizing tissue damage associated with other forms of secondary injury (as in cluster 2B). This observation is reflected in significant parameter differences regarding the decay of microglia and IL-10, as well as release rates of the pro-inflammatory cytokines. Although we observe a gradual increase in the damage expression, it is lower in magnitude than both unfavorable outcome clusters and appears to be well contained as it plateaus near day 5. Particularly of note, the relative ratio of M2:M1 microglia in cluster 2A was approximately 1:1 after the initial injury, becoming larger than 1 as the microglial response progressed. Conversely, M2:M1 ratios in clusters 1 and 2B were nearly 1:4 initially and became even smaller over time. These findings may support the physiological importance of microglia, even perhaps of the stereotypically neurotoxic M1-like phenotype, in the acute injury phase post-TBI.

There is considerable need in the TBI field to characterize neuroinflammation, particularly with respect to the contributions of microglial functionality, in order to assess acute injury progression and tailor intervention strategies to enhance neuroprotection for particular patient subgroups. In this study, we implemented CSF cytokine time courses as proxies to indicate the state of neuroinflammation post-TBI and inform microglia activation and polarization dynamics. However, the consistent collection of informative neuroinflammatory data is not always clinically feasible. The refinement of cerebral microdialysis (CMD) and CSF cytokine measurement methodology will contribute to ongoing efforts to centrally monitor TBI-induced neuroinflammation [50].

While cytokine data was available from a large patient cohort, more limited samples were available to contribute to each 6-h smoothed mean. Serial sampling and cytokine data at a greater temporal resolution would improve model trajectories to provide more accurate predictions of the progression of neuroinflammation post-TBI. The inflammatory data remained oscillatory in nature despite a smoothing procedure for averaging. Marked changes in inflammatory mediator dynamics were largely absent, with most levels present at consistently low levels. As a result, the fits to cytokine data did not produce noticeable fluctuations between microglia phenotypes.

There is inherent abstraction in mathematically modeling biological processes. In our model, microglia subtypes and tissue damage are qualitatively projected with arbitrary units rather than quantified by cell count or tissue volume. This limits the direct interpretation of these time courses to an estimate of the M1- or M2-like “state” of the brain following injury. Moreover, parameter values cannot be taken as literal rates with established units. Although model parameters were initially guided by existing experimental literature, scaling procedures on cytokine data and confounding factors in experimental situations complicate the direct rate interpretations. Our determination of differences between parameter value distributions should be viewed as a relative test of neuroinflammatory distinctions between clusters, which integrated additional statistical metrics to inform the degree to which the distributions differed. Due to limitations of the statistical methodology, we in fact de-emphasized the value of traditional hypothesis testing alone and considered two Bhattacharyya metrics to inform our comparative parameter analyses. Parameter value differences were ranked by lowest overlap (BC) and highest spread (BD) to highlight the most dissimilar parameter distributions between clusters.

Additionally, as a reduced model, the set of ODEs derived were limited to prototypical markers of microglia activation and acute inflammation to encompass basic regulatory components in the inflammatory network: initiation, propagation, phenotype switching, and inhibition. The model was designed to represent recent findings that IL-4 and IL-10 act as switch-like factors in microglia polarization. However, the self-regulating and immunosuppressive role of these anti-inflammatory mediators may be overemphasized in this model, based on the relatively rapid return of microglia to baseline levels predicted in all clusters. Inclusion of additional mediators may create a more comprehensive and nuanced illustration of the inflammatory network that drives the microglial response to TBI.

In general, the understanding of microglia classification and roles is still evolving in the neurotrauma field. At first, a spectrum of activation states was proposed, ranging from M1 to M2 extremes, with multi-functional subtypes of M2 in between [46, 51]. Not only are more nuanced views now being considered as alternatives to these rigid phenotype classifications, a layer of complexity has been added as simultaneous expression of M1 and M2 phenotypic markers on the same cell has now been observed in animal models [9]. This developing research area calls for further classification of molecular profiles and associated functional roles of microglia, particularly advancements that successfully translate in vitro findings to in vivo scenarios. Macrophage research and dynamics have informed our ODEs and have paved the way for much of our understanding of phenotype polarization and functionality; however, it is necessary to investigate these parallels in the brain with microglia in order to extend M1 and M2 characterizations explicitly. A key strength of our work is that the modeling framework that we utilize need not be interpreted in terms of a strict M1/M2 dichotomy. Rather, the framework allows for a flexible interpretation of the M1 and M2 variables as cell counts, states, or even associated microglia functions. The focus is on the relative contribution and effects of the pre-specified cytokines on the model behavior of those variables as well as the physiological processes of pro- and -inflammatory cytokine production, tissue damage, and healing that M1 and M2 represent in our model; as long as these elements are present and interrelated in the biological response to damage, our predictions related to these quantities are not dependent on any specific M1/M2 dichotomy.

More generally, our model encodes physiological interactions among biological quantities thought to contribute to inflammatory response dynamics as well as to tissue damage and healing in the acute phase post-TBI. This framework requires making assumptions about which physiological processes contribute and in what ways. We acknowledge that alternative sets of assumptions could lead to different conclusions, but we have attempted to tailor our modeling choices to reflect current understanding derived from previous experimental, clinical, and computational work, albeit with some simplifications to retain tractability. Nonetheless, any experimental or clinical work aimed at predicting outcome, such as suggesting early-warning signs for patient risk groups and targets for therapeutic interventions, would necessarily proceed based on some theoretical framework, typically reflecting the prevailing scientific viewpoint, which would exert a strong impact on the study performed. Among the advantages of the modeling approach used here in this report, we note that the underlying assumptions are clearly stated (see “Ordinary differential equation model development” section) and the parameter fitting process does not impose any additional biases; rather, this modeling approach evaluates all model parameters’ contributions to cluster differences, giving all an equal chance to emerge as significant.

Lastly, microglia responses in the acute phase may be a transient phenomenon overshadowed by subsequent chronic elevations [46, 52], which have been observed as late as 11 months to 17 years post-TBI in humans [40], and accompanying pathologies; nonetheless, these transients may contribute to long-term influences of acute neuroinflammation on patient outcome [18].

PCA and cluster analysis by R.G. Kumar et al. identified a cluster of individuals (cluster 1) with elevated inflammation with respect PC1 markers. However, the outcomes of cluster 2 individuals in this study were variable, despite negative PC1 scores for the majority [18]. This work indicates that elevations of PC1 markers (IL-5, IL-6, IL-8, IL-10, sVCAM, sICAM, and sFAS) were predictive of outcome for a subset of the patients in the study; however, other predictive measures were yet to be unveiled to distinguish patient prognosis in cluster 2. In the current study, we grouped cluster 2 patients by 6-month GOS score to investigate differences in their inflammatory and microglial dynamics post-TBI beyond classification of particular acute inflammatory markers. Despite having greater cortisol levels in the first 3 days following TBI, cluster 2B individuals were also projected to be in an immunosuppressed state due to early microglia decay and greater sensitivity to IL-10 in negative feedback mechanisms that control cytokine production.

Our findings on cluster 2B emphasize the important point that microglial responses underlying poor patient outcomes post-TBI are likely heterogeneous; in particular, poor outcomes in some patients might relate to a sub-physiological microglial response. This interpretation of results exemplifies the utility of mathematical modeling in exploring how early patient stratification based on inflammatory marker expression may lead to informed understanding of acute recovery trajectories and guide decision-making for specific immunomodulatory therapy types that may benefit from additional pre-clinical evaluation.