The model and simulations

An overview of the model is given below. It includes the description of the algorithm and main parameters, as well as the discussion of the source of parameter values. The model is adapted from Sanche et al.[24], the main difference being the consideration of two compartments with different exposures to the drugs.

The model is implemented using an algorithm which iteratively computes the number of events involving active CD4 cells during small time intervals: i) the number of virion-producing cells that die out, ii) the number of new cell infections, and iii) the fraction of these infections that involve a newly mutated virus. Step i) is computed using a constant cell death rate (d_y). Step ii) is computed from reproduction numbers Rij, where i and j indicate the strain of the virus and the compartment, respectively. Details on reproduction numbers are given below. The fraction of de novo mutation is based on probabilities of SNP mutations estimated from empirical data and reported elsewhere.[43] Total infection activity is translated in plasma virions, by assuming one virion-producing cell supplies one plasma virion, a relationship that is consistent with independent data.[5, 44]

The model assumes compartments are isolated from each other in terms of viral infection, i.e. a virus produced by one cell can only infect cells within the same compartment. This has been confirmed with experimental data: the infection of new cells is considered an essentially local phenomenon and the genetic makeup of the viral populations suggests a high degree of compartmentalization.[21, 45] The modeled viral dynamics is very similar in each compartment, the main difference being the level of drug exposure influencing the reproduction numbers.

Reproduction numbers Rij are expressed as R₀ (1-s_i) fu,ji, where R₀ is the mean number of CD4 cells becoming infected by viruses produced by a single infected cell when susceptible cells are abundant and when no drug is present, s_i is a fitness cost for strain i, and fu,ji is the fraction of CD4 infection events unaffected by the drugs for strain i in compartment j. In this study, fu,ji was either estimated for wild-type virus from viral load data (index i is omitted in this case, since only wild-type virus was considered), or computed from drug concentrations. In the latter case, plasma drug concentrations are first simulated from reported pharmacokinetic models. Since the concentrations vary over time t, so do the fu,ji. In the particular case where only one drug is used, fu,ji takes a relatively simple form (Eq 1):

where C_p(t) is the plasma drug concentration at time t, k_p is the coefficient adjusting for plasma protein binding, k_l,j is the coefficient adjusting for the degree of drug penetration in the compartment j, respectively, IC500 is the concentration inhibiting 50% of infection events by the wild-type strain in vitro in a medium devoid of plasma proteins, m is the Hill coefficient, and finally ρ_i and σ_i are two factors adjusting the values of IC500 and m, respectively, for the resistant viral strain i. For the wild-type strain, ρ_i = 1 andσ_i = 0. It should be noted that to estimate the impact of concomitant drug use, fa,ji(t) no longer takes a simple analytical form (see Jilek et al. and Sanche et al.).[24, 35]

Parameters φ_j are used in all simulations to limit the number of cells each compartment may contain. The overall number of cells is itself dictated by parameter λ, which is an entry rate of uninfected activated CD4 cells.

Many parameter values are patient-specific. The death rate d_y is randomly selected for each patient from a distribution of values based on empirical measures of the first phase decay.[46] The distribution has a median value corresponding to a half-life of about 0.7 day. The same applies for λ (directly linked to the distribution of viral set points)[28] and R₀ (based on the growth rate of viral loads during rebounds).[36, 37] Inter-individual variability for pharmacokinetic parameters is based on the reported values of population pharmacokinetics models.[31, 33, 34, 47] All other model parameters have a priori assigned values reported in Sanche et al.[24], with the exception of φ_j and f_u,j.