# Modeling the Interplay between HDV and HBV in Chronic HDV/HBV Patients

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background

_{1}and p

_{2}represent the production rate constant of HDV and HBV from infected cell number, I

_{0}. We assumed that the clearance rate constant, c, is the same for HDV and HBV and that it is within the range of the HDV clearance rate estimated in [14,15]. Parameter ε represents the efficacy of the treatment for HDV, with 0 < ε < 1, and g is the assumed additional treatment inhibitory effect in blocking HDV production as previously conducted under antiviral treatment for HBV [16,17]. Parameter n governs the HBV production rate increase under anti-HDV treatment.

_{0}and B

_{0}are the HDV and HBV levels at the onset of treatment, respectively. The number of the infected cells, I

_{0}, was kept constant under treatment and prior to treatment onset, where ε = g = 0, and n = 1.

## 3. Modified Model

_{0}~0, as D(t) decreased under treatment, because D

_{0}> κ, the production of HBV increased, e.g., at time t

_{1}after the initiation of treatment. When D(t

_{1}) = κ, the pre-treatment HBV production p

_{2}doubled (Figure 3). The time t

_{1}at which p

_{2}double depends on the pre-treatment ratio κ/D

_{0}as shown in Figure 3.

_{1}in Equation (3), and p

_{2}for HBV in Equation (4) is replaced by

_{0,}was kept constant under treatment and prior to treatment onset, where ε = g = 0 as assumed in the original model. Furthermore, we assumed that HDV decay begins at time τ (days), corresponding to the delay observed in the data and possibly reflecting Lonafarnib pharmacokinetics.

## 4. Results

#### 4.1. The Modified Model Simulates the Measured Data

#### 4.2. Analytic Solutions

#### 4.3. Analytic Solutions for HDV

#### 4.4. Analytical Solution for HBV and Plots

_{2}F

_{1}(α, β; ɣ; z), where α, β, and ɣ are the function parameters and z is the variable of the Gaussian hypergeometric function.

## 5. Discussion and Conclusions

^{−4}(D

_{0}/D(t))

^{n}in Equation (2) with a new term (1+(k/D)

^{n}) as shown in Equation (5). The term (1+(κ/D)

^{n}) allowed starting with a pre-treatment HBV production rate, which could increase as a result of HDV decline by the several suggested mechanisms that were reviewed in [18]. The value of κ, which was unique for each patient (Figure 4), could represent, in part, the interplay between HDV and HBV.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**One-way sensitivity analysis on the impact of parameter c on predicted HBV/HDV kinetics under anti-HDV treatment. Model parameters were set to D

_{0}= 1,202,260, B

_{0}= 282, κ = 26,137, n = 1.25, g = 0.094, τ = 0.1, and ε = 0.97, with c being set at c = (0.423, 0.4465, 0.47, 0.4935, 0.517). The arrows depict the direction of increase of the parameter c.

**Figure A2.**One-way sensitivity analysis on the impact of parameter n on predicted HBV/HDV kinetics under anti-HDV treatment. Model parameters were set to D

_{0}= 1,202,260, B

_{0}= 282, κ = 26,137, c = 0.47, g = 0.094, τ = 0.1, and ε = 0.97, with n being set at n = (1.125, 1.1875, 1.25, 1.3125, 1.375). HDV predicted kinetics were not affected by different n values.

**Figure A3.**One-way sensitivity analysis on the impact of parameter κ on predicted HBV/HDV kinetics under anti-HDV treatment. Model parameters were set to D

_{0}= 1,202,260, B

_{0}= 282, n = 1.25, c = 0.47, g = 0.094, τ = 0.1, and ε = 0.97, with κ being set at κ = (23,523, 24,830, 26,137, 27,444, 28,751). The arrows depict the direction of increase of the parameter κ. HDV predicted kinetics were not affected by different n values.

**Figure A4.**One-way sensitivity analysis on the impact of parameter g on predicted HBV/HDV kinetics under anti-HDV treatment. Model parameters were set to D

_{0}= 1,202,260, B

_{0}= 282, κ = 26,137, c = 0.47, n = 1.25, τ = 0.1, and ε = 0.97, with g being set at g = (0.0846, 0.0893, 0.094, 0.0987, 0.1034).

**Figure A5.**One-way sensitivity analysis on the impact of parameter ε on predicted HBV/HDV kinetics under anti-HDV treatment. Model parameters were set to D

_{0}= 1,202,260, B

_{0}= 282, κ = 26,137, c = 0.47, n = 1.25, τ = 0.1, and g = 0.094, with ε being set at ε = (0.95, 0.96, 0.97, 0.98, 0.99).

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**Figure 1.**The model Equations (1) and (2) failed to reproduce HBV kinetics in additional patients reported in Yurdaydin et al. [5]. Model parameter values were D

_{0}= 1.2 × 10

^{6}, B

_{0}= 282, c = 0.51, g = 0.089, ε = 0.97.

**Figure 3.**One-way sensitivity analysis on the impact of parameter κ on predicted HBV kinetics, B, (red curves) under anti-HDV treatment. Model parameters were set to D

_{0}= 1,202,260, B

_{0}= 282, c = 0.47, n = 1.25, g = 0.094, τ = 0.1, and ε = 0.97. HDV predicted kinetics, D (blue curve) was not affected by different κ values.

**Figure 4.**Model agreement with measured HDV RNA (circles) and HBV DNA (triangles) kinetics in four representative patients. Model parameters were (

**a**) B

_{0}= 30,200, D

_{0}= 1.14815 × 10

^{6}, ε = 0.53, g = 0.077, n = 1.8, and κ = 9799, (

**b**) B

_{0}= 151, D

_{0}= 6.30957 × 10

^{5}, ε = 0.97, g = 0.054, n = 2, and κ = 25,503, (

**c**) B

_{0}= 5.88844 × 10

^{5}, D

_{0}= 5.01187 × 10

^{7}, ε = 0.97, g = 0.020, n = 1.1, and κ = 1.23198 × 10

^{6}, and (

**d**) B

_{0}= 282, D

_{0}= 1.20226 × 10

^{6}, ε = 0.97, g = 0.094, n = 1.25, and κ = 26,137. The other model parameters c and τ were fixed to 0.47 and 0.1, respectively, except for Patient 2 with τ = 0.6, as depicted in Figure 4a.

**Figure 5.**(

**a**) shows the HBV Mathematica analytical solution (i.e., Equation (13) herein), HBV Berkeley Madonna fit for Patient 10, which was a result of fitting the modified model Equations (1) and (5) with measured data using Berkeley Madonna. (

**b**) follows a similar approach as conducted in 5a but for HDV using D.

**Figure 6.**(

**a**) shows the HBV Mathematica analytical solution (i.e., Equation (14) herein), HBV Berkeley Madonna fit for Patient 2, which was a result of fitting the modified model with measured data using Berkeley Madonna as reported in [4] and the HBV raw data. (

**b**) follows a similar approach as conducted in 6a but for HDV using D.

**Figure 7.**(

**a**) shows the HBV Mathematica analytical solution (i.e., Equation (15) herein), HBV Berkeley Madonna fit for Patient 4, which is a result of fitting the modified model with measured data using Berkeley Madonna 7. (

**b**) follows a similar approach as conducted in 7a but for HDV using D.

**Figure 8.**(

**a**) shows the HBV Mathematica analytical solution (i.e., Equation (16) herein), HBV Berkeley Madonna fit for patient 5, which was a result of fitting the modified model with measured data using Berkeley Madonna 8. (

**b**) follows a similar approach as conducted in 8a but for HDV using D.

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**MDPI and ACS Style**

Mhlanga, A.; Zakh, R.; Churkin, A.; Reinharz, V.; Glenn, J.S.; Etzion, O.; Cotler, S.J.; Yurdaydin, C.; Barash, D.; Dahari, H. Modeling the Interplay between HDV and HBV in Chronic HDV/HBV Patients. *Mathematics* **2022**, *10*, 3917.
https://doi.org/10.3390/math10203917

**AMA Style**

Mhlanga A, Zakh R, Churkin A, Reinharz V, Glenn JS, Etzion O, Cotler SJ, Yurdaydin C, Barash D, Dahari H. Modeling the Interplay between HDV and HBV in Chronic HDV/HBV Patients. *Mathematics*. 2022; 10(20):3917.
https://doi.org/10.3390/math10203917

**Chicago/Turabian Style**

Mhlanga, Adequate, Rami Zakh, Alexander Churkin, Vladimir Reinharz, Jeffrey S. Glenn, Ohad Etzion, Scott J. Cotler, Cihan Yurdaydin, Danny Barash, and Harel Dahari. 2022. "Modeling the Interplay between HDV and HBV in Chronic HDV/HBV Patients" *Mathematics* 10, no. 20: 3917.
https://doi.org/10.3390/math10203917