Panorama general de los procedimientos de afiliación y re afiliación al

The pro- and anti-inflammatory responses were modeled after a mass-spring-damper response to a bump (see Figure3.2). This physical system was simplified using transfer functions and their step response. Transfer functions provide a mathematical formalism where model pa- rameters can be directly linked to dynamics characteristics of the system response (such

Figure 3.2: Schematic of a mass-spring-damper system serving as a physical analog to the inflam- matory response. The cytokine response was likened to a mass-spring-damper physical system. Cytokine trajectories were modeled as the unit step response of a second-order transfer function to a pathogen presence (analogous to a car suspension responding to a “road bump”). The steady state gain of the response was set to unity.

as oscillation, overshoot, etc.), thereby facilitating model identification from data. Further- more, transfer functions represent entire classes of mathematical systems exhibiting a desired behavior. The inflammatory response (output) may be characterized as the response to a rectangular wave of infection for a specified duration (input). However, the short temporal length of the cytokine data from the ProCESS trial (up to 72 hours after trial enrollment), in combination with an assumption that subjects still had ongoing infection by 72 hours, led to the simplification of this input to a step change (as shown in Figure3.2, bottom right). A similar cytokine modeling approach (authors used a state-space realization of second-order transfer functions) was taken in a journal article by Yiu, et al [73].

Analysis of the cohort cytokine data revealed three types of step response dynamics, which are represented by the second-order transfer functions in Figure3.3). The first transfer

Figure 3.3: Sample unit step responses of the three transfer functions used. Each response reaches a steady state output value of 1.0 but the trajectory to reaching steady state is characteristically different for each model. The first category, oscillatory response, is a three-parameter system (τ , ζ, τ3) characterized by a damped oscillating approach to 1.0. The second category, overshoot response, is a three-parameter system (τ1, τ2, τ3) characterized by a fast rise followed by a decline to 1.0. The third category, rising only response, is a two-parameter system (τ1, τ2) characterized by a rising trajectory to 1.0 with no overshoot.

function enforced an oscillatory response to capture the primary rise and fall motif and a secondary lower-magnitude peak. Such peaks have previously been observed in the clinic [74]. The damping parameter of this transfer function, ζ, was of particular interest. ζ controls the oscillatory nature of the system and bounds were set to prevent it from taking low values near zero, thereby preventing the system from oscillating for many periods at a relatively high magnitude. The second transfer function captured the primary rise and fall motif only. Finally, the third transfer function enforced a rising only behavior to capture subjects whose inflammatory responses keep rising.

The data from the convenience cohort was normalized to account for extensive magnitude differences between subjects.The normalization criterion consisted of dividing each patient’s IL-6 and IL-10 trajectory by their respective 72-hour measurements. The normalization procedure was assumed to normalize both the magnitude of inflammatory response and the magnitude of infection for each patient. A benefit of this normalization was that it removed the need to estimate a gain for the transfer function model. The dynamics of the resulting normalized pro- and anti-inflammatory responses were clustered.

The first step was fitting the 390 subjects’ IL-6 and IL-10 responses (after taking the log10 of cytokine concentrations) to each type of transfer function response. The 0, 6, and 24 cytokine measurements were normalized by dividing by their respective IL-6 or IL-10 72-hour measurement. The normalized IL-6 and IL-10 trajectories were fit against each of the three transfer function categories in Figure 3.3. Parameter fitting was performed using the Levenburg-Marquardt algorithm provided by the Python lmfit package.

This approach fully addressed the problem previously illustrated by Figure 3.1. Step responses for second order transfer functions always begin at 0. This enabled the objective function to also fit a discrete hourly time shift parameter (bounded to [0,78] hours) to estimate the pre-hospital time for each patient. Each patient was then classified into one of nine categories (three possibilities each for IL-6 and IL-10, yielding 9 combinations) based on the lowest sum of squared errors.

The next step further split each of the 9 categories by identifying similar clusters of subjects in parameter space. Splitting within each category allowed for the possibility of different behaviors of the oscillatory, overshoot, and rising-only responses. Parameters for each subject were assumed to be distributed about a Gaussian mixture. Gaussian mixtures were identified via the expectation maximization algorithm and subjects were segregated in accordance to the highest probability of membership to a certain Gaussian. Expecta- tion maximization is a well known statistical algorithm that (i) calculates a probability of membership for each subject (expectation-step), (ii) calculates all Gaussian component pa- rameters (µi, σi) via the maximum likelihood approach in accordance to each component’s

membership (maximization-step), and then (iii) repeat steps (i)-(ii) iteratively. The learning step and mixture modeling were performed via the sklearn package in Python. The number

of components in the Gaussian mixture model for each of the 9 categories was a user input and was varied over multiple simulations until a minimum Bayesian Information Criteria (BIC) was obtained. Low membership clusters (n < 10) were removed and those subjects were reassigned to the next most similar cluster.

The third step generated a master inflammatory response trajectory for each remaining cluster. The parameters from each cluster were averaged and used to generate a representa- tive IL-6 and IL-10 trajectory for the cluster. Subjects within each cluster were then shifted in time (original measurement intervals were preserved) to best align their IL-6 and IL-10 trajectories to the respective master response trajectory.

Clinical outcomes were defined by 14-day all-cause mortality and 14-day multiple organ failure rates. This was done for the same reasons outlined in the previous chapter. No distinction was made between baseline MOF and 14-day MOF because this method iden- tifies patient time zeros, which overrides the clinical definition of baseline (time of clinical presentation).

Statistical analysis of the resulting clusters was conducted on patient demographics, outcomes, and clinical biomarkers taken within 6 hours of trial enrollment. Statistically significant differences (p < 0.05) were identified using the non-parametric Dunn’s test for continuous variables and the Chi-squared test for categorical variables. Multiple pair-wise testing, to specifically identify which clusters were different from each other, was performed via Dunn’s test and Chi-squared tests. Bonferroni corrections were applied to counteract the error effects of multiple comparisons.