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Modeling and Hemofiltration Treatment of Acute Inflammation^{ †}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Inflammation

#### 1.2. Sepsis

#### 1.3. Mathematical Models of Inflammation

#### 1.4. Endotoxemia: A Model of Sepsis

#### 1.5. Hemoadsorption: A Potential Treatment for Sepsis

#### 1.6. Model-Based Treatment

#### 1.6.1. Model Predictive Control

#### 1.6.2. State Estimation: Mapping Diagnostics into Model Representations

#### 1.7. Manuscript Overview

## 2. Materials and Methods

#### 2.1. Experimental Data

#### 2.2. Mathematical Model of Acute Inflammation

#### 2.3. Parametric Sensitivity by Finite Difference Method

#### 2.4. In Silico Treatment

#### 2.5. Stochastic Endotoxemia Model

#### 2.6. Observation Model

#### 2.7. Hemoadsorption Model

#### 2.8. WBC Capture Model

#### 2.9. HA Device Configurations

#### 2.10. Model Predictive Control

#### 2.11. HA Performance Metric

#### 2.12. Particle Filter State Estimation

#### 2.13. State Estimation Performance Metric

## 3. Results

#### 3.1. Parameter Sensitivity Analysis

#### 3.2. Controlling the Inflammatory Response

#### 3.2.1. MPC Using HA

#### 3.2.2. HA Efficacy: Cytokine Versus WBC Capture

#### 3.2.3. Differential Capture of Inflammatory Mediators

#### 3.2.4. Particle Filter State Estimation

#### 3.2.5. Hemoadsorption Control with State Estimation

## 4. Discussion

#### 4.1. Trading Off Biological Fidelity and Model Structure

#### 4.2. Biological Fidelity Challenges Parameter Estimation

#### 4.3. Cell Capture Is Predicted Key to HA Efficacy

#### 4.4. HA Devices with Differential Cytokine and WBC Capture May Have Little Benefit

#### 4.5. MPC Reference Trajectory

#### 4.6. State Estimation

#### 4.7. Real-Time Control of HA

#### 4.8. Limitations

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Model Structure Justification

#### Appendix A.1. Model Selection Technique

#### Appendix A.2. Model Selection Comparisons

**Figure A1.**Model simulation comparison of IL-6 between the proposed model Equation (15) (solid line) and its alternate versions, AV-1, Equation (A4) (dashed line) and, AV-2, Equation (A5) (dotted line), against experimental data (circle) (mean ± SD) in response to endotoxin challenge of 3 mg/kg (

**top**) and 12 mg/kg (

**bottom**).

**Figure A4.**Model simulation comparison of damage-mediated IL-10 between the proposed model with a 6th-order Hill function in Equation (22) (solid line), AV-7 with a 4th-order Hill function in Equation (22) (dashed line), and AV-8 with a 2nd-order Hill function in Equation (22) (dotted line) against experimental data (circle) (mean ± SD) in response to endotoxin challenge of 3 mg/kg (

**top**) and 12 mg/kg (

**bottom**).

**Table A1.**Calculated AIC and BIC values (based on 3 mg/kg and 12 mg/kg data) of the proposed model and its alternate versions. Note: all base models have the same AIC and BIC values, they are reported separately for convenient comparison of related submodels.

Model (Equation) | AIC | BIC |
---|---|---|

IL-6 (Equation (14)) | 274.1 | 299.7 |

AV-1 (Equation (A4)) | 303.3 | 328.2 |

AV-2 (Equation (A5)) | 280.0 | 305.6 |

TNF (Equation (18)) | 274.1 | 299.7 |

AV-3 (Equation (A6)) | 282.2 | 307.8 |

AV-4 (Equation (A7)) | 277.0 | 303.2 |

IL-10 (Equation (22)) | 274.1 | 299.7 |

AV-5 (Equation (A8)) | 290.2 | 315.1 |

AV-6 (Equation (A9)) | 287.4 | 312.4 |

Effect of D on IL-10 (Equation (12)) | 274.1 | 299.7 |

AV-7 | 279.4 | 305.0 |

AV-8 | 281.1 | 306.7 |

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**Figure 2.**(

**A**) Diagram of endotoxemia model coupled to a hemoadorption (HA) device model. N, interleukin (IL)-6, tumor necrosis factor (TNF) and IL-10 are assumed to circulate through the ex vivo HA circuit. Cytokine capture is modeled in a two step process: reversible diffusive entry into bead pores, followed by irreversible binding to the internal bead surface. White blood cell (WBC) capture is treated as reversible binding at the bead surface. (

**B**) Schematic of in silico HA control experiments without state estimation. Subject is modeled by the deterministic endotoxin model. Complete state information is passed to the controller once per hour. (

**C**) Schematic of in silico HA control experiments with PF state estimation. Subject is modeled by the stochastic endotoxin model. State estimates are based on noisy cytokine measurements obtained once per hour.

**Figure 3.**Parametric relative sensitivity analysis of the inflammation model for the eight model states.

**Figure 4.**HA performance is measured by relative absolute error, $RA{E}^{HA}$, calculated from time of intervention through 24 h. $RA{E}^{HA}=0$ implies perfect control, while $RA{E}^{HA}=1$ indicates the controller performed no better than null treatment. The x-axis corresponds to the time of HA intervention, where $t=0$ is the time of endotoxin administration. Control simulations were based on the deterministic endotoxemia model (solid lines) or stochastic model with PF state estimation (dashed lines). Panels correspond to HA device configurations in Table 2. (

**A**) Configuration A, which captures cytokines only, performs poorly. Configurations that capture white blood cells, (

**B**–

**D**), perform well. Hypothetical multi-channel HA devices with differential specificity (

**C**,

**D**) do not substantially improve performance over (

**B**). In all cases, HA performance declines when treatment is delayed.

**Figure 5.**Simulated endotoxemia with model predictive control (MPC) of HA device A (blue) or null treatment (red). Device A captures cytokines but not WBC. MPC guides IL-6 levels close to the target trajectory, but other cytokines, activated phagocytes (N) and tissue damage (D) are not substantially impacted. Endotoxin dose was 12 mg/kg and HA intervention began immediately at $t=0$.

**Figure 6.**Simulated endotoxemia with MPC of HA device B (blue) or null treatment (red). MPC with device B, which captures both cytokines and WBC, is able to guide the inflammatory state to the reference trajectory. Endotoxin dose was 12 mg/kg and HA intervention began immediately at $t=0$.

**Figure 7.**Trajectories of manipulated variables for the device configurations in Table 2; panel labels correspond to device configurations. Response is to a 12 mg/kg endotoxin simulation with immediate HA intervention. Higher HA flow rates are required to achieve control without

**A**vs. with

**B**WBC capture. Differential column configurations

**C**and

**D**demonstrate time-dependent HA column flow for separate columns.

**Figure 8.**An example of particle filter state estimation on simulated data. circles mark observed cytokine measurements, error bars indicate the standard deviation of the state estimate, lines show the true model state.

**Figure 9.**Sample simulation of the stochastic endotoxemia model with MPC of HA (blue) or null treatment (red). State estimates based on noisy cytokine measurements were generated by PF. Endotoxin dose was 12 mg/kg with intervention at 2 h. HA was applied using configuration D.

IL-6 | TNF | IL-10 | N | |
---|---|---|---|---|

${k}_{ad}$ | 0.62 | 0.188 | 0.682 | 0.177 |

${k}_{des}$ | 0.013 | 0.015 | 0.072 | 0.010 |

${k}_{bnd}$ | 0.006 | 0.007 | 0.022 | 0 |

HA | Specificity | ||
---|---|---|---|

Configuration | Column 1 | Column 2 | Column 3 |

A | TNF, IL-6, IL-10 | - | - |

B | N, TNF, IL-6, IL-10 | - | - |

C | N | TNF, IL-6, IL-10 | - |

D | N | TNF, IL-6 | IL-10 |

No. | Parameter | No. | Parameter | No. | Parameter | No. | Parameter |
---|---|---|---|---|---|---|---|

1 | ${d}_{p}$ | 11 | ${k}_{NTNF}$ | 21 | ${k}_{IL6}$ | 31 | ${k}_{IL10IL6}$ |

2 | ${k}_{N}$ | 12 | ${k}_{NIL6}$ | 22 | ${d}_{IL6}$ | 32 | ${x}_{IL10IL6}$ |

3 | ${x}_{N}$ | 13 | ${k}_{D}$ | 23 | ${x}_{IL6}$ | 33 | ${k}_{IL10}$ |

4 | ${d}_{N}$ | 14 | ${d}_{D}$ | 24 | ${x}_{IL6IL10}$ | 34 | ${d}_{IL10}$ |

5 | ${k}_{NP}$ | 15 | ${x}_{D}$ | 25 | ${k}_{IL6IL6}$ | 35 | ${x}_{IL10}$ |

6 | ${k}_{ND}$ | 16 | ${k}_{CA}$ | 26 | ${x}_{IL6IL6}$ | 36 | ${s}_{IL10}$ |

7 | ${x}_{NTNF}$ | 17 | ${d}_{CA}$ | 27 | ${k}_{TNF}$ | 37 | ${x}_{IL10IL10}$ |

8 | ${x}_{NIL6}$ | 18 | ${s}_{CA}$ | 28 | ${d}_{TNF}$ | 38 | ${k}_{IL10Y}$ |

9 | ${x}_{NCA}$ | 19 | ${k}_{IL6TNF}$ | 29 | ${x}_{TNFCA}$ | 39 | ${d}_{IL10Y}$ |

10 | ${x}_{NIL10}$ | 20 | ${x}_{IL6TNF}$ | 30 | ${x}_{TNFIL6}$ | 40 | ${x}_{IL10Y}$ |

Parameter Groups | Parameters |
---|---|

${\theta}_{P}$ | ${d}_{p}$ |

${\theta}_{N}$ | ${k}_{N}$, ${x}_{N}$, ${d}_{N}$, ${k}_{NP}$, ${k}_{ND}$, ${x}_{NTNF}$, ${x}_{NIL6}$, ${x}_{NCA}$, ${x}_{NIL10}$, ${k}_{NTNF}$, ${k}_{NIL6}$ |

${\theta}_{D}$ | ${k}_{D}$, ${d}_{D}$, ${x}_{D}$ |

${\theta}_{{C}_{A}}$ | ${k}_{CA}$, ${d}_{CA}$, ${s}_{CA}$ |

${\theta}_{IL6}$ | ${k}_{IL6TNF}$, ${x}_{IL6TNF}$, ${k}_{IL6}$, ${d}_{IL6}$, ${x}_{IL6}$, ${x}_{IL6IL10}$, ${k}_{IL6IL6}$, ${x}_{IL6IL6}$ |

${\theta}_{TNF}$ | ${k}_{TNF}$, ${d}_{TNF}$, ${x}_{TNFCA}$, ${x}_{TNFIL6}$ |

${\theta}_{IL10}$ | ${k}_{IL10IL6}$, ${x}_{IL10IL6}$, ${k}_{IL10}$, ${d}_{IL10}$, ${x}_{IL10}$, ${s}_{IL10}$, ${x}_{IL10IL10}$ |

${\theta}_{{Y}_{IL10}}$ | ${k}_{IL10Y}$, ${d}_{IL10Y}$, ${x}_{IL10Y}$ |

States | Relative Sensitivity (%) | |||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{\theta}}_{\mathit{P}}$ | ${\mathbf{\theta}}_{\mathit{N}}$ | ${\mathbf{\theta}}_{\mathit{D}}$ | ${\mathbf{\theta}}_{{\mathit{C}}_{\mathit{A}}}$ | ${\mathbf{\theta}}_{\mathit{I}\mathit{L}\mathbf{6}}$ | ${\mathbf{\theta}}_{\mathit{T}\mathit{N}\mathit{F}}$ | ${\mathbf{\theta}}_{\mathit{I}\mathit{L}\mathbf{10}}$ | ${\mathbf{\theta}}_{{\mathit{Y}}_{\mathit{I}\mathit{L}\mathbf{10}}}$ | |

$P\left(t\right)$ | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

$N\left(t\right)$ | 13.8 | 64 | 0.2 | 1.9 | 2.8 | 5.7 | 11 | 0.5 |

$D\left(t\right)$ | 11.8 | 41.0 | 28.7 | 1.6 | 2.4 | 4.9 | 9.4 | 0.3 |

${C}_{A}\left(t\right)$ | 13.0 | 45.8 | 0.1 | 23.3 | 2.3 | 5.1 | 10.0 | 0.2 |

$\left[IL6\right]\left(t\right)$ | 5.4 | 17.0 | 7.0 | 1.4 | 36.3 | 5.1 | 23.3 | 4.6 |

$\left[TNF\right]\left(t\right)$ | 9.8 | 23.9 | 0.3 | 21.2 | 3.9 | 33.3 | 7.3 | 0.2 |

$\left[IL10\right]\left(t\right)$ | 8.5 | 31.0 | 30.8 | 1.1 | 3.5 | 3.9 | 8.0 | 12.9 |

${Y}_{IL10}\left(t\right)$ | 8.7 | 29.6 | 33.2 | 1.1 | 1.6 | 3.6 | 6.9 | 15.3 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Parker, R.S.; Hogg, J.S.; Roy, A.; Kellum, J.A.; Rimmelé, T.; Daun-Gruhn, S.; Fedorchak, M.V.; Valenti, I.E.; Federspiel, W.J.; Rubin, J.;
et al. Modeling and Hemofiltration Treatment of Acute Inflammation. *Processes* **2016**, *4*, 38.
https://doi.org/10.3390/pr4040038

**AMA Style**

Parker RS, Hogg JS, Roy A, Kellum JA, Rimmelé T, Daun-Gruhn S, Fedorchak MV, Valenti IE, Federspiel WJ, Rubin J,
et al. Modeling and Hemofiltration Treatment of Acute Inflammation. *Processes*. 2016; 4(4):38.
https://doi.org/10.3390/pr4040038

**Chicago/Turabian Style**

Parker, Robert S., Justin S. Hogg, Anirban Roy, John A. Kellum, Thomas Rimmelé, Silvia Daun-Gruhn, Morgan V. Fedorchak, Isabella E. Valenti, William J. Federspiel, Jonathan Rubin,
and et al. 2016. "Modeling and Hemofiltration Treatment of Acute Inflammation" *Processes* 4, no. 4: 38.
https://doi.org/10.3390/pr4040038