2.2. Mathematical Model of Acute Inflammation
The starting point for the present model comes from [
26], a 4-state phenomenological description of the inflammatory response process.
Pathogen, P, induces inflammatory response leading to activated phagocytic cells, . This cell activation leads to pathogen death and also tissue damage, D. Moderation of the inflammatory response is provided by anti-inflammatory mediators, represented by . This abstract model of the inflammatory response to pathogen provides multiple steady states and the ability to achieve clinically-relevant states of recovery (low P and D at long time), septic sepsis (high P and D at long time), and aseptic sepsis (high and D, but , at long time).
The acute inflammation model developed herein consists of eight ordinary differential equations (ODEs). The dependent variables used in the model include: endotoxin concentration (); total number of activated phagocytic cells (), which includes all the activated immune response cells (such as neutrophils, monocytes, etc.); a non-accessible tissue damage marker (); concentrations of pro-inflammatory cytokines IL-6 () and TNF (); concentration of the anti-inflammatory cytokine IL-10 (); a tissue damage driven non-accessible IL-10 promoter ((t)); and a non-accessible state representing slow acting anti-inflammatory mediators ((t)).
A schematic diagram of the model capturing all the major interactions between the eight states is shown in
Figure 1. Introduction of
P into the system activates
N. Once activated,
N up-regulates production/release of all the inflammatory mediators (TNF, IL-6, IL-10, and
) [
40]. The pro-inflammatory cytokines have a positive feedback on the system; thereby, they further activate
N, and up-regulate other cytokines [
40,
41]. On the other hand, the anti-inflammatory cytokine and mediators have a negative feedback on the system. Hence, IL-10 and
inhibit the activation of
N and up-regulation of other cytokines [
42,
43]. Mathematically,
also serves to provide overall stability to the model (i.e., a low standing level of
, serves to return inflammation to its baseline value after small perturbations). The model also incorporates tissue damage due to activated phagocytic cells, represented by a damage marker,
D; this corresponds biologically to neutrophil-induced damage to lung tissue after a systemic inflammatory response. Tissue damage further up-regulates activation of
N [
44] and also contributes to up-regulation of IL-10 [
45,
46].
The endotoxin insult is injected intraperitoneally in the rats as a bolus administration, which initiates the inflammatory cascade. The ODE describing the dynamics of
can be written as:
decays exponentially with a rate equal to
. The decay rate was fixed at 3
, which is consistent with the values obtained from published literature [
47,
48,
49]. The initial conditions (
t = 0) for Equation (
5) are either 3, 6, or 12 mg/kg depending on the endotoxin dose level.
Resting phagocytic cells are activated by the presence of endotoxin in the system. The equations representing activation of
can be mathematically written as:
Here, Equation (
6) represents the total number of activated phagocytic cells (
). Parameters
and
represent the rate of activation and elimination of
, respectively. Activation of
is driven by
and
, through
, as shown in Equation (
7). Throughout this work, functions with nomenclature
and
represent up-regulating (
) and down-regulating (
) effects of inflammatory mediator
j on mediator
i. These up- or down-regulating functions are dimensionless and bounded, having values between 0 and 1. Functions,
and
indicate the up-regulating effects of
and
on
, respectively. Both these functions are Michaelis-Menten type equations, as shown in Equations (8) and (9). As the concentrations increase, the values of the up-regulating functions also increase, approaching 1 asymptotically. Gain parameters
and
scale the up-regulating functions
and
to capture the appropriate effects on
, respectively. The inhibitory effects of
and
are captured by the down-regulating functions
and
, respectively. Here, as the variables increase, the values of the functions decrease, approaching 0 asymptotically (see Equations (10) and (11)). Parameters
,
,
,
, and
are the half-saturation constants determining the concentration level of the variables at which the corresponding up-regulating or down-regulating function will reach half of its saturation point. The initial condition (
t = 0) for Equation (
6) is
= 0.
The tissue damage caused by the inflammatory response to endotoxin challenge is modeled as follows:
Parameters
and
represent the rate of generation and the rate of elimination of the unobserved tissue damage marker,
. Elevated
further contributes to the activation of
(
7) [
44] and production of IL-10 [
45,
46]. Parameter
is the half-saturation constant. A 6th-order Hill function was utilized in order to accurately capture the data. Further explanation regarding the choice of the Hill function coefficient is provided in the
Appendix A. The initial condition (
t = 0) for Equation (
12) is
= 0.
The anti-inflammatory mediator,
, represents a combination of various inflammation inhibitory mediators, including the cytokine Transforming Growth Factor-
β1 (TGF-
β1) and cortisol. The
equation is written as:
Parameters
and
represent the rate of
production/secretion and clearance, respectively. At basal conditions, the system is assumed to be slightly anti-inflammatory. This was achieved by introducing a constant,
, into Equation (
13). Hence, at
t = 0 and
= 0,
=
.
The dynamics of IL-6, which is predominantly considered a pro-inflammatory mediator, can be mathematically written as:
The up-regulation of IL-6 production is governed by activated
. Production of IL-6 is further up-regulated by the presence of elevated
and
itself, and this is captured by the up-regulating functions
and
, respectively. The inhibitory effects of the anti-inflammatory cytokines were captured by the down-regulating function
. The clearance rate of IL-6 is represented by the parameter
. Once again, Section A further explains the selection of the Hill function coefficient. Parameters
,
,
, and
are the half-saturation constants. The initial condition (
t = 0) for Equation (
15) is
= 0.
Pro-inflammatory TNF concentration can be represented by the following equations:
The rate of production of TNF due to activation of
is governed by the parameter
, and the rate of clearance of TNF is represented by the parameter
. A power of 1.5 was assigned to
instead of a Michaelis-Menten or Hill type expression in order to capture the rapid production and elimination of TNF. Further justification for the proposed
-state (Equation (
18)) is provided in
Appendix A. The function
represents the up-regulating effect of
on its own production. The functions
and
represent the inhibitory effect of anti-inflammatory cytokine
and pro-inflammatory cytokine IL-6 (which in some instances, such as this, acts as an anti-inflammatory mediator [
50]), respectively. Parameters
,
, and
are the half-saturation constants. A 6th-order Hill function for
modeled the rapid suppression of
on the
dynamics. The initial condition (
t = 0) for Equation (
18) is
= 0.
The dynamics of
, which is a strong anti-inflammatory cytokine, can be represented by the following equations:
Here, Equation (
22) captures the circulating
levels. Unlike the other measured cytokines, the
dynamics demonstrate two distinct peaks separated by approximately 4 to 6 h when perturbed by endotoxin challenge. The first surge of IL-10 production is predominantly attributed to
, which is captured by a 3rd-order Hill equation multiplied by the parameter
(first RHS term of Equation (
22)). Production of IL-10 is further up-regulated by the presence of elevated pro-inflammatory cytokines like IL-6, which is represented by the up-regulation function
. The production of IL-10 in the basal state is represented by the constant
(as observed in experimental data). Hence, at
t = 0 and
= 0,
=
. It has been shown that the rate of elimination of IL-10 is inversely proportional to circulating
[
51]. This phenomenon is captured by a down-regulating function,
, coupled with the parameter
, as shown in Equation (
22). Further discussion of the Equation (
22) structure is presented in
Appendix A.
The second surge in IL-10 production is attributed to tissue damage
[
45,
46], and the dynamics of this
D-induced effect are captured by the variable
in Equation (
22). The dynamics of
are represented by the ODE (22); here the rate of production of
is represented by the parameter
coupled with a 4th-order Hill equation (first term in RHS of Equation (22)), which is driven by
. Once again, this is data-motivated (see the further discussion in
Appendix A). The rate of elimination of
is given by parameter
. Parameters
,
,
, and
are the half-saturation constants.
The model developed above is a simplification of our earlier model [
27,
52] that eliminates nonlinear up- and down-regulatory functions that did not display significant dynamics over the endotoxin challenge range employed here. The parameter values used here are those published in Table 1 of [
27]. Alternative modeling options for the various interactions are discussed in
Appendix A.
2.10. Model Predictive Control
A nonlinear MPC (NMPC) algorithm was implemented using HA device flow rate(s) as the manipulated variable(s); the controlled variables were activated phagocytes (
N); damage (
D); and circulating cytokine levels IL-6, TNF and IL-10. The reference trajectory was defined by the system response to a 3 mg/kg endotoxin dose, which is uniformly survivable without treatment. More precisely, let
be the system state at time
t following an endotoxin dose of
mg/kg and null treatment. The reference for the controlled variable
i at step
k is defined:
The NMPC objective function was:
These terms penalize predicted error in the controlled variable (
) from the reference (
), changes in the flow rates (
), and the use of large HA flow rates (
) when the effect is negligible. Standard statistical notation is employed throughout (prediction at time
given information up to time
k). Weights for each controlled variable
i are given by:
Weights for manipulated variables are given by: and , where c is the number of independent adsorption columns in the HA device. The size of was , with the HA configuration establishing the number of adjustable flow rates. Flow rates were constrained to be non-negative, while the sum of flows was constrained to be less than mL/min.
The MPC time step was h, the prediction horizon was steps, and the move horizon was . Control simulations were also performed with and , but no substantial difference in performance was observed. Predicted trajectories were based on the deterministic endotoxemia model coupled to an HA device model with n = 20 or n = 5 discretizations (five discretizations used in conjunction with PF). Constrained minimization was performed using fmincon in MATLAB with multiple initializations at each time step.
NMPC was performed with and without state estimation. In the former case, the full system state was passed to the controller at each time step (
Figure 2B). In the later, state estimates based on serial cytokine measurements were generated by a PF algorithm (
Figure 2C).