# Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics

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## Abstract

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## 1. Introduction

## 2. The Models Description

#### 2.1. The Biphasic Model

#### 2.2. The Multiscale Model

## 3. Sensitivity Analysis

#### 3.1. Sensitivity Analysis in the Biphasic Model

#### 3.2. Sensitivity Analysis in the Multiscale Model

## 4. Machine Learning

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Influence of Multiscale Model Parameters on Time-to-Cure

## References

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**Figure 3.**For visualization purposes, the dynamics with example trajectories in which the parameter ${\u03f5}_{a}$ was simulated with 3 parameter values 0.97, 0.985, and 0.999 is displayed using our in-house multiscale model HCV simulator [29]. The other parameters were fixed as follows: $\beta $ (0.00000005 mL/day/virion), $\delta $ (0.14/day), $\rho $ (8.18/day), c (22.3/day), d (0.01/day), $\alpha $ (40.0 vRNA/day), $\mu $ (1.0/day), $\kappa $ (6.36), $\gamma $ (0.24/day), ${\u03f5}_{s}$ (0.65), and s (130,000 (cells/mL)).

**Figure 4.**One-way sensitivity analysis of time-to-cure using the multiscale model. Depicted are minimum and maximum time-to-cure values for each parameter that were varied in a defined range (i.e., $\kappa $ (3.5–6.7), $\gamma $ (0.2–0.96/day), ${\u03f5}_{s}$ (0.4–0.8), ${\u03f5}_{a}$ (0.9–0.999), and s (4.62–6.48 log(cells/mL)), while the other parameters were fixed as follows: $\beta $ (0.00000005 mL/day/virion), $\delta $ (0.14/day), $\rho $ (8.18/day), c (22.3/day), d (0.01/day), $\alpha $ (40.0 vRNA/day), and $\mu $ (1.0/day).

**Figure 5.**Distribution of relative errors obtained for predicting time-to-cure using machine learning with the biphasic and multiscale models. Measurements of all time points (0, 2, 7, 14 and 28 days) were used for the machine learning.

**Figure 6.**Distribution of relative errors obtained for predicting time-to-cure using machine learning with the biphasic and multiscale models. Measurements of reduced time points (0, 14 and 28 days) were used for the machine learning.

**Table 1.**Central values of the biphasic model for the PRCC. The (min, max) of the parameter space are shown.

c | $\mathit{\epsilon}$ | $\mathit{\delta}$ |
---|---|---|

$6.0\phantom{\rule{3.33333pt}{0ex}}(3,12)$ | $0.5\phantom{\rule{3.33333pt}{0ex}}(0.25,1)$ | $0.5\phantom{\rule{3.33333pt}{0ex}}(0.25,1)$ |

**Table 2.**Central values of the multiscale model for the PRCC. The (min, max) of the parameter space are shown.

s | d | $\mathit{\beta}$ | $\mathit{\delta}$ | $\mathit{\rho}$ | c |
---|---|---|---|---|---|

130,000 (65,000, 260,000) | $0.01\phantom{\rule{3.33333pt}{0ex}}(0.05,0.02)$ | $5\times {10}^{-8}\phantom{\rule{3.33333pt}{0ex}}(0.55\times {10}^{-8},25\times {10}^{-8})$ | $0.14\phantom{\rule{3.33333pt}{0ex}}(0.07,0.28)$ | $7.95\phantom{\rule{3.33333pt}{0ex}}(3.975,15.9)$ | $22.5\phantom{\rule{3.33333pt}{0ex}}(11.25,25)$ |

${\mathbf{\epsilon}}_{\mathbf{s}}$ | $\mathbf{\alpha}$ | ${\mathbf{\epsilon}}_{\mathbf{\alpha}}$ | $\mathbf{\kappa}$ | $\mathbf{\mu}$ | $\mathbf{\gamma}$ |

$0.5\phantom{\rule{3.33333pt}{0ex}}(0.25,1)$ | $40\phantom{\rule{3.33333pt}{0ex}}(20,80)$ | $0.5\phantom{\rule{3.33333pt}{0ex}}(0.25,1)$ | $6.36\phantom{\rule{3.33333pt}{0ex}}(3.18,12.72)$ | $1\phantom{\rule{3.33333pt}{0ex}}(0.5,2)$ | $0.24\phantom{\rule{3.33333pt}{0ex}}(0.12,48)$ |

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**MDPI and ACS Style**

Reinharz, V.; Churkin, A.; Dahari, H.; Barash, D. Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics. *Mathematics* **2022**, *10*, 2136.
https://doi.org/10.3390/math10122136

**AMA Style**

Reinharz V, Churkin A, Dahari H, Barash D. Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics. *Mathematics*. 2022; 10(12):2136.
https://doi.org/10.3390/math10122136

**Chicago/Turabian Style**

Reinharz, Vladimir, Alexander Churkin, Harel Dahari, and Danny Barash. 2022. "Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics" *Mathematics* 10, no. 12: 2136.
https://doi.org/10.3390/math10122136