A Hepatitis C Virus Infection Model with Time-Varying Drug Effectiveness: Solution and Analysis

<div><p>Simple models of therapy for viral diseases such as hepatitis C virus (HCV) or human immunodeficiency virus assume that, once therapy is started, the drug has a constant effectiveness. More realistic models have assumed either that the drug effectiveness depends on the drug concentration or that the effectiveness varies over time. Here a previously introduced varying-effectiveness (VE) model is studied mathematically in the context of HCV infection. We show that while the model is linear, it has no closed-form solution due to the time-varying nature of the effectiveness. We then show that the model can be transformed into a Bessel equation and derive an analytic solution in terms of modified Bessel functions, which are defined as infinite series, with time-varying arguments. Fitting the solution to data from HCV infected patients under therapy has yielded values for the parameters in the model. We show that for biologically realistic parameters, the predicted viral decay on therapy is generally biphasic and resembles that predicted by constant-effectiveness (CE) models. We introduce a general method for determining the time at which the transition between decay phases occurs based on calculating the point of maximum curvature of the viral decay curve. For the parameter regimes of interest, we also find approximate solutions for the VE model and establish the asymptotic behavior of the system. We show that the rate of second phase decay is determined by the death rate of infected cells multiplied by the maximum effectiveness of therapy, whereas the rate of first phase decline depends on multiple parameters including the rate of increase of drug effectiveness with time.</p></div

HCV viral load undergoes biphasic decay upon initiation of silibinin treatment at time .

<p>The transition time between the first and second phases, , is calculated by maximizing the curvature in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769.e173" target="_blank">equation (14</a>), and is marked by a vertical dashed line. VE model fit of Canini et al. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Canini1" target="_blank">[19]</a> (solid line) and HCV viral load data (dots) for (a) Patient 46, with transition time days, and for (b) Patient 48, with transition time days.</p

Truncated series solutions for the VE model compared with the exact solution (11) under silibinin treatment (; see Table 1 for parameters).

<p>Legend: (i) Series terms with exponents , , , and terms, included in the approximation (16), from the series solution (19); (ii) Series terms with exponents from (i) and also the and terms missing from the approximation.</p

Probability distribution of number of viral replications following activation of a latently infected cell.

<p>(a) Probability of the number of rounds of viral replication following the activation of a single latently infected cell, assuming drug efficacy <i>ε</i> = 0.999, 0.99, and 0.9, which are associated with reproductive ratios <i>R</i> = 0.0023, 0.023, and 0.23, and initial viral loads <i>V</i>₀ = 3, 3.1, and 4 copies/mL, respectively. (b) The probability of the maximum number of rounds of viral replication achieved as a function of the number of latent cell activations, using <i>ε</i> = 0.99, which is associated with <i>R</i> = 0.023 and initial viral load <i>V</i>₀ = 3.1 copies/mL. Note that these are discrete distribution functions, with the dots in (a) indicating probability of cells achieving <i>k</i> generations; the lines are included for clarity and have no meaning.</p

Example of a biphasic decline of HCV, following a short delay, after initiation of interferon- therapy at .

<p>Fit of Neumann et al. model (solid line) to data for Patient 1E (dots) from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Neumann1" target="_blank">[3]</a>.</p

Different exponential terms in approximate solution (16) compared with the exact solution and for silibinin treatment parameters, for which (see Table 1).

<p>(a) Exponential terms from (16) plotted separately. (b) Exponential terms from (16) plotted in combined form.</p

Approximation to viral dynamics compared to exact dynamics under mericitabine treatment, 750 mg qd, .

<p>(a) For patient 92102 from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Guedj3" target="_blank">[16]</a>, characterized as “flat”. (b) For patient 92103 from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Guedj3" target="_blank">[16]</a>, characterized as “non-flat”. (c) Different exponential terms in approximate solution (16) compared with the exact solution for patient 92103, characterized as “non-flat”. Parameter estimates from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Guedj3" target="_blank">[16]</a>: For patient 92102, , , , , , , and ; for patient 92103, , , , , , , and .</p

Residual Viremia in Treated HIV⁺ Individuals

<p>Total latent reservoir size (black, solid line) with pre-existing portion (blue, dashed lines) and new latent cell infections (red, dash-dotted line) for latent cell fraction <i>f</i> = 10⁻⁴ and drug efficacy <i>ε</i> = 0.6 (reproductive ratio <i>R</i> = 0.92), assuming <i>δ</i> = 1 day⁻¹ and <i>t</i>_1/2 = 44 months.</p

Approximate and analytic solution of the VE model under danoprevir () or telaprevir () treatment with patient data.

<p>(a,c) Approximate solution (16) compared to the analytic solution (11) for (a) danoprevir or (c) telaprevir treatment. (b,d) Different exponential terms in approximate solution compared with the exact solution, with decay phases indicated, for (b) danoprevir or (d) telaprevir treatment. Danoprevir treatment: data from patient 04-94XD (dosing 200 mg tid) in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Rong2" target="_blank">[25]</a> with associated parameter estimates for VE model , , , , , , and [unpublished]. Telaprevir treatment: data from patient 6 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Guedj2" target="_blank">[6]</a> with associated parameter estimates , , , , , , and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Guedj2" target="_blank">[6]</a>.</p

Model parameter estimates obtained for different drug treatments of chronic HCV.

<p>Model parameter gives the viral clearance rate, the infected hepatocyte death rate, gives the exponential scale at which the drug reaches its maximum value from its minimum value , gives the delay in the drug activity, and gives the efficacy of treatment in blocking new cell infection.</p><p>*Clearance rate fixed at days⁻¹ from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Neumann1" target="_blank">[3]</a>, not estimated.</p><p>**Efficacy of treatment in blocking new cell infection fixed at from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003769#pcbi.1003769-Guedj3" target="_blank">[16]</a>, not estimated.</p><p>***Unpublished.</p><p><b>Notes</b>: (<b>i</b>) In fitting viral load data, authors investigating telaprevir and mericitabine used the more general VE model (12), while those investigating silibinin, danoprevir, and sofosbuvir used the simple VE model (3) with Efficacy of treatment in blocking new cell infection was only used in the silibinin model (effectively 0 in other models). (<b>ii</b>) qd  =  daily dosing, bid =  bi-daily dosing. (<b>iii</b>) Parameter estimates derive from HCV treatment studies on patients who were treatment naïve, except in the case of mericitabine, where all patients were interferon non-responders.</p