# Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Development of Mathematical Models

#### 2.1.1. The Standard Biphasic Model

#### 2.1.2. The Multiscale HCV Model

#### 2.2. Data Description

#### 2.3. Solving the Model Equations

#### 2.4. Parameter Estimation

#### 2.4.1. Preliminaries

#### 2.4.2. Optimization by a Constrained Version of Nonlinear Least Squares (Gauss–Newton Method)

- Start with an initial guess ${x}_{0}$ and iterate for $k=0,1,2,\dots $
- Solve $mi{n}_{\mathsf{\Delta}{x}_{k}}{\parallel J\left({x}_{k}\right)\mathsf{\Delta}{x}_{k}+r\left({x}_{k}\right)\parallel}_{2}$ to compute the correction $\mathsf{\Delta}{x}_{k}$.
- Choose a step length ${\alpha}_{k}$ so that there is enough descent.
- Calculate the new iterate ${x}_{k+1}={x}_{k}+{\alpha}_{k}\mathsf{\Delta}{x}_{k}$.
- Check for convergence.

#### 2.4.3. Optimization by Derivative-Free Methods (COBYLA Method)

Algorithm 1: COBYLA method. |

#### 2.5. Method Scope and Other Approaches

#### 2.5.1. Parameters Change When Transforming a PDE Multiscale Model to a System of ODEs

#### 2.5.2. Problematic Issues in Strategies Relying on Canned Methods

## 3. Results

`scipy.optimize.curve_fit`method, which is a Python implementation of a simple Levenberg–Marquardt scheme as a canned method. The next column to the right are the values obtained previously by the use of Levenberg–Marquardt along with the numerical method to solve the model equations as outlined in [57]. In the left columns are the values obtained by our new efficient methods. The small differences assure us that the significant efficiency achieved, thereby making our simulator a practical and useful tool, did not result in less accuracy.

## 4. Discussion

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Details of the COBYLA Method

**Figure A1.**Illustration of the original simplex method. The points ${x}_{0},{x}_{1}$, and ${x}_{2}$ form the initial simplex. (

**A**) The point $\overline{x}$ is the midpoint of the line joining ${x}_{0}$ and ${x}_{1}$, and $\widehat{x}$ is the reflection of ${x}_{2}$ through this line. If $f\left(\widehat{x}\right)<f\left({x}_{1}\right)$, ${x}_{2}$ is replaced by $\widehat{x}$, shifting the location of the simplex. (

**B**) If $f\left(\widehat{x}\right)\ge f\left({x}_{1}\right)$, ${x}_{2}$ is replaced with $(1/2)({x}_{2}+{x}_{0})$ and ${x}_{1}$ is replaced with $(1/2)({x}_{1}+{x}_{0})$, reducing the volume of the simplex.

**Figure A2.**Illustration of the Nelder–Mead method. The points ${x}_{0},{x}_{1}$ and ${x}_{2}$ form the initial simplex. The vertex ${x}_{2}$ is replaced with a vertex of the form ${x}_{\mathrm{new}}=\overline{x}+\theta (\overline{x}-{x}_{2})$. If $f\left(\widehat{x}\right)<f\left({x}_{0}\right)$, $\theta =2$, if $f\left({x}_{0}\right)\le f\left(\widehat{x}\right)<f\left({x}_{n-1}\right)$, $\theta =1/2$, and if $f\left({x}_{n-1}\right)\le f\left(\widehat{x}\right)$, $\theta =-1/2$.

**Figure A3.**Minimization of $\widehat{f}$ The candidate vertex ${x}^{*}$ is computed by minimizing $\widehat{f}\left(x\right)$ subject to the constraints ${\widehat{c}}_{1},{\widehat{c}}_{2}\ge 0$ within the trust region $\left|\right|x-{x}_{0}{\left|\right|}_{2}\le \rho $. (top) The region of optimization (green) is the intersection of the trust region $\left|\right|x-{x}_{0}{\left|\right|}_{2}\le \rho $ with the half planes defined by the affine constraints ${\widehat{c}}_{1}\ge 0$ and ${\widehat{c}}_{2}\ge 0$. (bottom left) The function, $\widehat{f}$, to be minimized is represented graphically by the plane (blue) passing through points $({x}_{0},f\left({x}_{0}\right)),({x}_{1},f\left({x}_{1}\right)),({x}_{2},f\left({x}_{2}\right))$. (bottom right) The vertex ${x}^{*}$ is defined to be the point within the region of optimization (green) at which $\widehat{f}$ (blue) is minimized.

**Figure A4.**Minimization of $\widehat{M}$ Should the constraints ${\widehat{c}}_{i}\left(x\right)\ge 0$ be inconsistent with one another within the trust region $\left|\right|x-{x}_{0}{\left|\right|}_{2}\le \rho $, the candidate vertex ${x}^{*}$ is chosen to minimize $\widehat{M}:=max\{-{\widehat{c}}_{i}\left(x\right):i=1,\dots ,m\}$. (top) The constraints ${\widehat{c}}_{1}\left(x\right)\ge 0$ and ${\widehat{c}}_{2}\left(x\right)\ge 0$ are inconsistent within the region $\left|\right|x-{x}_{0}{\left|\right|}_{2}\le \rho $. (bottom left) Graphs of the affine functions $-{\widehat{c}}_{1}\left(x\right)$ (blue) and $-{\widehat{c}}_{2}\left(x\right)$ (green). (bottom right) The vertex ${x}^{*}$ is defined to be the point within the trust region (black circle) at which $\widehat{M}$ is minimized.

**Figure A5.**Illustration of the new vector ${x}^{\mathsf{\Delta}}$, generated to improve the shape of the simplex. The vertex ${x}_{1}$ is replaced with either ${x}^{\mathsf{\Delta}}={x}_{0}+\gamma \rho {v}_{\ell}$ or ${x}^{\mathsf{\Delta}}={x}_{0}-\gamma \rho {v}_{\ell}$, whichever point results in a smaller value of $\widehat{\Phi}$.

## Appendix B. Parameter Estimation in the Biphasic Model

**Figure A6.**Biphasic model fitting example with data taken from [36] of a patient who was treated with mavyret. The LSF method (default) is recommended for use.

**Figure A7.**Biphasic model fitting example with data taken from [36] of a patient who was treated with epclusa. In this particular case, COBYLA was selected instead of LSF and succeeded to yield a fit.

## Appendix C. Parameter Estimation in the Multiscale Model

**Figure A8.**Fitting the parameters c and $\rho $ of the multiscale model to generated data points using the LSF method.

**Figure A9.**Fitting the parameters c and $\rho $ of the multiscale model to generated data points using the COBYLA method.

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**Figure 2.**Start fit that emanates from data of a patient reported in [48]. The fitting curve corresponds to default parameters before fitting with our methods. The multiscale model is used.

**Figure 3.**End fit using Gauss–Newton (LSF) that emanates from data of a patient reported in [48].

**Figure 4.**End fit using COBLYA that emanates from data of a patient reported in [48].

**Figure 5.**Comparison between the line fits of different methods inside the simulator window for the retrieved data points of patient HD that was reported in [48].

**Figure 6.**Comparison between the line fits of different methods for the retrieved data points of patient HD that was reported in [48].

s ($\mathrm{cells}/m{L}^{-1}$) | Influx rate of new hepatocytes |

d (${\mathrm{d}}^{-1}$) | Target cell loss/death rate constant |

$\beta $ ($\mathrm{m}\mathrm{L}{\mathrm{d}}^{-1}{\mathrm{virion}}^{-1}$) | Infection rate constant |

$\delta $ (${\mathrm{d}}^{-1}$) | HCV-infected cell loss/death rate constant |

$\rho $ (${\mathrm{d}}^{-1}$) | Virion assembly/secretion rate constant |

c (${\mathrm{d}}^{-1}$) | Virion clearance rate constant |

$\alpha $ (vRNA${\mathrm{d}}^{-1}$) | vRNA synthesis rate |

$\mu $ (${\mathrm{d}}^{-1}$) | vRNA degradation |

$\kappa $ | Enhancement of intracellular viral RNA degradation |

$\gamma $ (${\mathrm{d}}^{-1}$) | Loss rate of vRNA replication complexes |

${\epsilon}_{s}$ | Treatment vs. secretion/assembly effectiveness |

${\epsilon}_{\alpha}$ | Treatment vs. production effectiveness |

**Table 2.**Default parameters that are used herein. Parameter s comes from Equation (9), taking $\overline{V}$ as the max Virions value.

$\alpha $ | 40 d^{−1} | $\beta $ | $5\times {10}^{-8}$ d^{−1} |

c | $22.3$ d^{−1} | $\delta $ | $0.14$ d^{−1} |

$\mu $ | 1 d^{−1} | d | $0.01$ d^{−1} |

$\rho $ | $8.18$ d^{−1} | s | $\left(\overline{V}\beta c+dc\right)/\left(\beta N\right)$$\mathrm{cells}$/$\mathrm{m}$$\mathrm{L}$ |

**Table 3.**Values of the parameters when fitted to the patient digitized data. The rightmost column has the values when the retrieved data points are fitted to the long-term approximation as in [57]. The left columns contain the fitted parameter values by our efficient methods. Except for the rightmost column, all methods are combined with the Rosenbrock numerical scheme. The fixed parameters have the values shown in Table 2. Run-time comparison is reported in seconds in the last row.

Gauss–Newton (LSF) | COBYLA | Levenberg–Marquardt | Long-Term | |
---|---|---|---|---|

${\epsilon}_{s}$ | 0.609 | 0.598 | 0.602 | 0.6000 |

${\epsilon}_{\alpha}$ | 0.995 | 0.994 | 0.995 | 0.994 |

$\kappa $ | 6.210 | 6.375 | 6.219 | 6.160 |

$\gamma $ (d^{−1}) | 0.137 | 0.177 | 0.139 | 0.140 |

accuracy (sum error${}^{2}$) | 0.538 | 0.582 | 0.538 | 0.587 |

run-time (s) | 194 | 3698 | 70118 | <1 |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Churkin, A.; Lewkiewicz, S.; Reinharz, V.; Dahari, H.; Barash, D. Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics. *Mathematics* **2020**, *8*, 1483.
https://doi.org/10.3390/math8091483

**AMA Style**

Churkin A, Lewkiewicz S, Reinharz V, Dahari H, Barash D. Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics. *Mathematics*. 2020; 8(9):1483.
https://doi.org/10.3390/math8091483

**Chicago/Turabian Style**

Churkin, Alexander, Stephanie Lewkiewicz, Vladimir Reinharz, Harel Dahari, and Danny Barash. 2020. "Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics" *Mathematics* 8, no. 9: 1483.
https://doi.org/10.3390/math8091483