# Understanding the Complex Patterns Observed during Hepatitis B Virus Therapy

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model

#### 2.2. Monte Carlo Sampling

- MC.1
- The initial condition (2) exists and is stable, i.e., ${r}_{I}>\delta $ and $1<{R}_{0}<\frac{{r}_{T}}{{r}_{I}-\delta}$.
- MC.2
- The components of initial condition (2) are bound by $0<\overline{T}<K$ cells per ml, $0<\overline{I}<K$ cells per mL, and $0<\overline{V}<1.9\times {10}^{10}$ virus per ml.
- MC.3
- The drug-dependent virus asymptotic removal condition (6) holds, i.e., $0<\mathcal{R}<1$. This condition guaranties that we are only examining drug therapy that is efficient in removing the hepatitis B virus.
- MC.4
- The total hepatocyte population is bound as follows $0.2K\le T\left(t\right)+I\left(t\right)\le K$ cells per mL, for all t. This condition guarantees that the total liver cell population never decays below $20\%$ of the healthy hepatocyte concentration K [23].

## 3. Results

#### 3.1. Parameter Regions for Bi-Phasic and Tri-Phasic Virus Dynamics

#### 3.2. Relationship between $V\left(t\right)$ Patterns and the Duration of Treatment

#### 3.3. Liver Toxicity and the Role of Initial Conditions

## 4. Discussion

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**A**) Samples for bi-phasic (dark blue dots) and tri-phasic (red to light blue dots) $V\left(t\right)$ patterns in the three-dimensional ${r}_{T}/{r}_{I}$-$\delta $-$\u03f5$ space; (

**B**) Projection of the samples in (

**A**) onto the $\delta $-$\u03f5$ space; (

**C**) Projection of the samples in (

**A**) onto the ${r}_{T}/{r}_{I}$-$\delta $ space; (

**D**) Projection of the samples in (

**A**) onto the ${r}_{T}/{r}_{I}$-$\u03f5$ space.

**Figure 3.**Distribution of the tri-phasic $V\left(t\right)$ samples versus the length in days of: (

**A**) flat phase; (

**B**) first slope; (

**C**) last slope.

**Figure 4.**Samples for bi-phasic (dark blue dots) and tri-phasic (red to light blue dots) $V\left(t\right)$ dynamics for: (

**A**) fixed ${r}_{T}/{r}_{I}=5$; (

**B**) fixed $\delta =0.08$; (

**C**) fixed $\u03f5=0.7$. The other parameters are as in Table 2.

**Figure 6.**Density of bi-phasic (blue) and tri-phasic (pink) $V\left(t\right)$ samples versus: (

**A**) Liver turnover; (

**B**) Net liver gain, for $\u03f5=0.7$, ${r}_{T}/{r}_{I}=5$, $0.01\le \delta \le 0.1$ ${\mathrm{d}}^{-1}$, and $\tau =100$ days.

**Figure 7.**Density of bi-phasic (blue) and tri-phasic (pink) $V\left(t\right)$ samples versus liver turnover for $V\left(t\right)$ curves that achieved clearance (of one virion) in (

**A**) one year; (

**B**) two years. The parameters are fixed $\u03f5=0.99$ ad ${r}_{T}/{r}_{I}=2.5$, and variable $0.01\le \delta \le 0.1$ ${\mathrm{d}}^{-1}$.

**Figure 8.**Density of bi-phasic (blue) and tri-phasic (pink) $V\left(t\right)$ samples versus $\delta $, for fixed $\u03f5=0.7$ and ${r}_{T}/{r}_{I}=5$.

**Figure 9.**Samples for bi-phasic (dark blue dots) and tri-phasic (red to light blue dots) $V\left(t\right)$ dynamics for $\u03f5=0.7$, ${r}_{T}/{r}_{I}=5$, $0.01\le \delta \le 0.1$ ${\mathrm{d}}^{-1}$, $\tau =100$ days in respect to: (

**A**) liver turnover and initial uninfected liver population $\overline{T}$, (

**B**) liver turnover and initial uninfected liver population $\overline{T}$. (

**C**) Length of the third phase as a function of the initial conditions $\overline{T}/\overline{I}$.

Figure 1 | $\mathit{\beta}$ | ${\mathit{r}}_{\mathit{T}}$ | ${\mathit{r}}_{\mathit{T}}/{\mathit{r}}_{\mathit{I}}$ | K | $\mathit{\delta}$ | p | c | $\mathit{\eta}$ | $\mathit{\u03f5}$ |
---|---|---|---|---|---|---|---|---|---|

(A) | $1\xb7{10}^{-10}$ | 3 | 5 | $1.9\xb7{10}^{7}$ | 0.07 | 100 | 0.67 | 0.5 | 0.99934 |

(B) | $2.5\xb7{10}^{-7}$ | 0.9 | 3.3 | $1.9\xb7{10}^{7}$ | 0.22 | 5 | 0.7 | 0.5 | 0.991 |

(C) | $1.9\xb7{10}^{-6}$ | 0.8 | 2.9 | $1.9\xb7{10}^{7}$ | 0.25 | 1.4 | 0.18 | 0.2 | 0.9988 |

(D) | $6.6\xb7{10}^{-8}$ | 0.5 | 7.2 | $1.9\xb7{10}^{7}$ | 0.06 | 164 | 1 | 0.5 | 0.997 |

**Table 2.**Fixed and boundaries for the parameters in model (1).

Parameter | Description | Value | Reference |
---|---|---|---|

$\beta $ | Infectivity rate | ${10}^{-10}$ mL/(virions × d) | [22] |

K | Hepatocyte-carrying capacity | $1.9\times {10}^{7}$ cells/mL | [12] |

p | Viral production | 100 virions/(cell × d) | [22] |

c | Viral clearance | 0.67 ${\mathrm{d}}^{-1}$ | [21] |

$\eta $ | Efficacy of therapy in blocking infection | 0.5 | [10] |

Parameter | Description | Range | References |

${r}_{T}$ | Uninfected cell division rate | (0,4] ${\mathrm{d}}^{-1}$ | [12,22] |

${r}_{I}$ | Infected cell division rate | (0,4] ${\mathrm{d}}^{-1}$ | [12] |

$\delta $ | Infected cells killing rate | [0.01,0.1] ${\mathrm{d}}^{-1}$ | [10,11,12,22] |

$\u03f5$ | Efficacy of therapy in blocking viral production | [0.2,1] | [10,12] |

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**MDPI and ACS Style**

Carracedo Rodriguez, A.; Chung, M.; Ciupe, S.M. Understanding the Complex Patterns Observed during Hepatitis B Virus Therapy. *Viruses* **2017**, *9*, 117.
https://doi.org/10.3390/v9050117

**AMA Style**

Carracedo Rodriguez A, Chung M, Ciupe SM. Understanding the Complex Patterns Observed during Hepatitis B Virus Therapy. *Viruses*. 2017; 9(5):117.
https://doi.org/10.3390/v9050117

**Chicago/Turabian Style**

Carracedo Rodriguez, Andrea, Matthias Chung, and Stanca M. Ciupe. 2017. "Understanding the Complex Patterns Observed during Hepatitis B Virus Therapy" *Viruses* 9, no. 5: 117.
https://doi.org/10.3390/v9050117