Anti-proliferative therapy for HIV cure: a compound interest approach

In the era of antiretroviral therapy (ART), HIV-1 infection is no longer tantamount to early death. Yet the benefits of treatment are available only to those who can access, afford, and tolerate taking daily pills. True cure is challenged by HIV latency, the ability of chromosomally integrated virus to persist within memory CD4+ T cells in a non-replicative state and activate when ART is discontinued. Using a mathematical model of HIV dynamics, we demonstrate that treatment strategies offering modest but continual enhancement of reservoir clearance rates result in faster cure than abrupt, one-time reductions in reservoir size. We frame this concept in terms of compounding interest: small changes in interest rate drastically improve returns over time. On ART, latent cell proliferation rates are orders of magnitude larger than activation and new infection rates. Contingent on subtypes of cells that may make up the reservoir and their respective proliferation rates, our model predicts that coupling clinically available, anti-proliferative therapies with ART could result in functional cure within 2–10 years rather than several decades on ART alone.


.1 Equilibrium solutions leading to decoupled equations for latently infected cells.
Our model for HIV including latently infected cells and antiretroviral therapy (ART) is described completely in the main body with all rates specified in Table 1. Here we present the ordinary differential equations (ODEs) without further clarification:Ṡ Equilibrium solutions (denoted by the asterisk) can be calculated by settingṠ,L,Ȧ,V = 0. The system has two equilibrium solutions: 1. "Viral-free equilibrium", the values at which the system rests prior to infection 2. "Setpoint equilibrium", the values at which the system has durable viral infection: commonly referred to as "viral setpoint" in clinical HIV care.
where = 1 − (1 + ξ/θ L )τ . We use the notation to simplify the appearance of the equations and choose the lower-case , as this factor encapsulates all the latent dynamics. For τ 1, we have ∼ 1, and τ is far less than one in the literature (see Table S4).

Stability analysis of equilibrium solutions: Calculating the basic reproductive number
confirms that decoupling latent cell equations is valid when ART efficacy is above the critical value.
By linearizing our system of equations, we address the local stability of the equilibrium points. We begin by defining our state variables as a vector x = [S, L, A, V ] T such that we can express the system of ODEs as F(x) = ∂ t x. Then, we Taylor expand this function around the equilibrium point x * where F(x * ) = 0: By construction the first term in the expansion is zero. Calling a small ∆x 1 = x − x * , we can neglect terms of O(∆x 2 ) and higher. Using x = x * + ∆x we can rewrite and because x * is not a function of time we are left with the linear equation we desire: The matrix J = ∂ x F is referred to as the Jacobian matrix for our model, and is in complete form: We can evaluate the Jacobian at both equilibria derived above (Eq. S2 and Eq. S3). The eigenvalues of the Jacobian then govern how perturbations near equilibrium behave. If all eigenvalues λ j of the Jacobian have negative real components Re(λ j ) < 0, perturbations decay back to equilibrium and the equilibrium is deemed stable. 1 Before infection, we input the equilibrium solutions (Eq. S2) and the Jacobian of the viral free equilibrium (subscript vfe) is We can calculate the eigenvalues of this rather sparse matrix. Taking the determinant of J − λ1 4×4 and setting this equal to zero we find We immediately identify the eigenvalue λ 1 ≈ −δ S . Then, noting that the trailing term τ β πξα S /δ S is many orders of magnitude smaller than the other quantities (see parameter values in Table 1 of main body), we drop this term and identify a second eigenvalue λ 2 ≈ θ L . We remind the reader that θ L = α L − δ L − ξ is a negative number and defined as the net clearance rate of the latent reservoir per Ref. 14 Thus, we have identified two negative eigenvalues, meaning that perturbations to the viral free equilibrium return to equilibrium along those respective eigendirections. Solving for the remaining eigenvalues using the expression within the square brackets leads to We define the quantity calling it the basic reproductive number that depends on the ART drug efficacy . By following through calculations (noting bounds on the factor 4 δ A γ (δ A +γ) 2 ∈ [0, 1]) it can be shown that if R ART 0 < 1 both λ 3/4 < 0 such that the viral free equilibrium is stable. Letting R ART 0 = 1 makes λ 3 = 0 and λ 4 < 0, while R ART 0 > 1 results in λ 3 ≥ 0 while λ 4 < 0. The basic reproductive number controls the stability of the viral free equilibrium, only when R ART 0 > 1 will an infection take off. Note that the same result can be verified using the next-generation matrix method 2 To calculate the "critical drug efficacy" c , we solve for the drug efficacy that makes R ART 0 ≤ 1. This yields or using the typical definition of the basic HIV dynamics model R 0 = πα S β δ S δ A γ , we see that For our parameter values, R 0 ∼ 8 as expected for HIV, guaranteeing instability of the viral free equilibrium for any introduction of virus.
Observations from patient data show that primary infections in humans lead to a durable viral setpoint. The Jacobian of our model evaluated at the endemic equilibrium can be written by inserting the equations Eq. S2 into Eq. S7. By calculating the eigenvalues of this Jacobian, we assess the stability of the setpoint equilibrium. Numerical computation of the eigenvalues of this matrix (using eig() in Matlab) for varying values of the drug efficacy are presented in Fig. S1.
Here we find that above the critical drug efficacy c (for our parameters c ∼ 85%), the real part of the third eigenvalue becomes positive. Therefore, while the setpoint equilibrium is stable for low drug levels, < c , above the critical efficacy the setpoint is no longer stable. At precisely this drug level the viral free equilibrium becomes stable again, which can be calculated numerically, or seen from the analytical derivation above.
Furthermore, calculating the eigenvalues of the viral free equilibrium shows that above the critical efficacy, the viral free equilibrium becomes stable (all eigenvalues having negative real parts). For our parameter range, the eigenvalues (-0.2, -0.0006, -0.3, -24) approximate the death or clearance rates of susceptible cells, latent cells, active cells, and virus respectively. This result agrees with our analytical approximation of the first 2 eigenvalues above. Only the eigenvalue related to the active cell rate is noticeably increased relative to its natural rate, presumably due to coupling between two eigendirections. Most importantly, the disparate timescales allow adiabatic decoupling of the processes. 3 That is, because the solution for the return to equilibrium corresponds to the solution of or using our eigenvalue decomposition, the complete solution to the dynamics of a perturbation from the viral free equilibrium can be expressed as where v j are the 4 eigendirections, not explicitly written here. The timescale for decay of each eigendirection is proportional to 1/λ j . For example, the fastest eigendirection λ 4 becomes negligible in the timescale of a day, corresponding to viral clearance. Two of the other eigenvalues have timescales of around 1 week, corresponding to clearance of active cells. However, the time required to completely return to viral free equilibrium (t * ) depends on the slowest rate or the smallest magnitude eigenvalue as t * ≈ 1/ min |λ j |. 1 In our case, the smallest eigenvalue is approximately θ L so that complete return to viral free equilibrium ultimately depends only on the latent cell clearance rate. This justifies our approximation that virus is negligible soon after ART initiation and that the model for long term ART can be simplified toL which has the solution (S17)  Figure S1: Setpoint stability depends on the efficacy of antiretroviral therapy (ART). Plots of the equilibrium concentrations and the eigenvalues of J (Eq. S7) evaluated at those viral setpoint equilibrium (Eq. S3) values at varying ART efficacy ART . The critical therapy thresholds are illustrated by the vertical dashed line (with our parameters and Eq. S12 we can calculate c ∼ 85%). At this value the real part of the third eigenvalue becomes positive making the viral equilibrium unstable.
In summary, our model system rests at the viral free equilibrium prior to infection. Any virus results in HIV infection which progresses and settles at a stable viral setpoint equilibrium. However, ART disrupts this equilibrium, making it unstable and driving the system to return to viral free equilibrium. This movement toward viral-free equilibrium is slow and is limited by the rate of latent cell decay. But, this limiting decay is much slower than the other decays, making it possible to ignore those dynamics, and focus solely on the exponential clearance of the latent cells.

Deriving the basic reproductive number R 0 with the next-generation matrix method
Following Diekmann, Heesterbeek, and Roberts, 2 we calculate the basic reproduction number of our complete model we begin by breaking apart our model into 2 matrices in the 'next-generation' (NG) fashion. We assume that the model is at the uninfected steady state so thatṠ = 0 and define S = S 0 = α S /δ S . Then, in matrix notation, calling where F is the matrix that describes new infections in each compartment and V is the matrix that describes the rates for leaving each compartment and for moving between them Then we want write the relationship so that the largest eigenvalue of M NG = FV −1 will be our reproductive number, and we can see that at least 1 eigenvalue must be greater than 1 for an infection to take off. We have and thus The next generation matrix admits two eigenvalues of zero and one that is positive. The positive eigenvalue is the largest eigenvalue of M N G and therefore we have the basic reproductive number of our model: Comparing this value to the approximate derivation above (Eq. S11), we see that because ξ/θ L < 1, the approximation is fairly accurate. Additionally, we could have carried through the calculation using infectivity as altered by ART efficacy without loss of generality.

Allowing direct active to latent transitions has a neglible impact on reservoir size.
The contribution of actively infected cells to latency can be ignored such that we can consider the latent cell equation alone. This follows from a heuristic argument made by Conway and Perelson. 4 On ART, virus can be created by latent cell activation, which occurs rarely. However, the infectivity of HIV is reduced by c on ART such that most susceptible cells are protected from infection. Further, the likelihood of any new infected cell becoming latent is quite small (τ ). Thus, the product of the rare events-activation, new infection, and latency-is effectively zero.
Still, in some modeling works a transition from active to latent cells has been included. 5,6 If we add a transition term to the model as a rate φ, the equations for the latent and active pool (decoupled from the virus and susceptible as before) areL We verified numerically that only for unnaturally large values of φ > α L (or likewise φ > δ L ) does including this transition impact the reservoir size. In that case, there is a transient increase in the initial number of latent cells before the typical latent clearance dynamics take hold. This result is a consequence of the rapid death rate of active cells relative to the other rates.

Impact of duration and rates on clearance time
In the main article, we proposed that optimal latent clearance is achieved by increasing the duration of therapy rather than increasing the potency of the therapy equivalently (Fig. 2). If we examine the remaining fraction of latent cells we can study the effect of multiplying any rate by a factor r, denoted L (r) or multiplying the duration of time by factor d, denoted L (d) . The ratio of the percent remaining after doing either of these multiplications tells us which procedure is more valuable. For example, if we multiply the activation rate only, Then for L (r) to be smaller than L (d) , we must have Note if we set d = r, we find that d < 1 because (α L − δ L ) is a negative number. Therefore, the decrease in the latent pool due to multiplying the duration of therapy will always be larger than an equivalent multiplication of the rate (unless the r < 1, which is a nonsensical proposition equivalent to making therapy less effective).

Model parameters
2.1 Latent cell parameters: θ L , α L , δ L , ξ Our most critical parameter values are those that describe the proliferation, death, and activation rates of latently infected cells-the sum of which is their net clearance rate θ L . Measuring these parameters in vivo in humans presents experimental challenges, and results depend on the models chosen to interpret the experimental data. 7 We acknowledge that there is uncertainty in these parameter estimates and thus provide an uncertainty and sensitivity analysis to demonstrate theoretical ranges for plausible outcomes (Fig. 4). Note that despite the variability of proliferation estimates, for even the slowest proliferation estimates we have encountered in the literature, proliferation rates are still two orders of magnitude larger than the estimated activation rate.
We use the proliferation rate α L from Macallan et al. who measured in vivo turnover with deuterated glucose labeling in memory T cells in healthy adults and found that on average, 1.5% of CD45R0 + CCR7 + T cells (central memory) proliferate daily, whereas 4.7% of CD45R0 + CCR7 − T cells (effector memory) and 0.2% of CD45R0 − CCR7 + T cells (naïve) proliferate per day, corresponding with exponential growth rates of α i = 0.015, 0.047, and 0.002, respectively. 8 We use Macallan et al.'s estimates from "healthy" adults (i.e. HIV-negative) because this study includes proliferation estimates for the CD4 + T cell subsets of interest.
In fact, several experimental works have demonstrated similar rates of memory CD4 + T cell proliferation among HIVnegative and HIV-infected on long-term ART. 9,10 Whereas these studies examine the entire pool of CD4 T cells, rather than the smaller pool of latently infected cells, there is no empirical evidence to suggest that proliferation rates of latently infected cells would be lower than their uninfected counterparts. Indeed, Chomont et al. demonstrated that systemic proliferation events are tightly linked to the size and composition of the latent reservoir. 11 These studies show that CD4 + T cell proliferation rates are similar among HIV-uninfected and HIV-infected persons on long-term ART. However, net proliferation or proliferation of naïve versus memory cell rates are measured rather than the effector and central memory subsets that are now recognized as being important reservoir subsets, and thus we adopted the Macallan et al. rates.
Note that in our model, as in all T cell proliferation references we cite, proliferation rates reflect a combined impact of clonal expansion and homeostatic proliferation, as these are not yet distinguishable experimentally.  12,13 Assuming that the reservoir contains 1 million cells on average, the rate of activation of a single cell per day ξ = 5.7 × 10 −5 . 13 Note that the activation rate ξ is several orders of magnitude smaller than the proliferation rate α L .
The net clearance rate of the latent reservoir is estimated from Siliciano et al.'s quantitative viral outgrowth assay as θ L = -5.2 ×10 −4 per day. 14 This result was corroborated by Crooks et al. 15

Susceptible and active cell parameters:
In the complete model, the production of CD4 + T cells from the bone marrow and thymus is described by α S ; and δ S is the rate of susceptible T cell death. In several early HIV modeling papers, a value of 10 per µL-day was estimated for the production rate 16,17 with δ S estimated at 0.02. Luo et al. used Bayesian statistical modeling to estimate HIV model parameters using data from 10 patients who underwent a series of 3-5 ART treatment interruptions with viral loads taken three times weekly following interruptions and then weekly following initiation of treatment. 12 They estimated α S = 295, δ S = 0.18, and δ A = 1. Huang et al. also use Bayesian methods (Markov Chain Monte Carlo) and fit their model to data from Ref., 18 an AIDS clinical trial comparing dosing regimens for indinavir and ritonavir. 19 Huang et al.'s model also incorporated adherence, drug concentrations, and drug susceptibilities. They find α S = 98.1, δ S = 0.08, and δ A = 0.37. Note in Table S1 that the ratio of α S to δ S varies between 500-1639 with the more recent experiments in relative agreement near 1500. Thus, we chose α S = 300, δ S = 0.2 to reflect this ratio.
The death rate of productively infected cells δ A was initially thought to be 0.24. 16 Later, Perelson et al. find δ A = 0.5 based on frequent sampling of 5 patients after giving ritonavir monotherapy. 20 Using a fixed viral clearance rate γ = 23 from, 21 Markowitz et al. determined δ A = 1 based on potent antiretroviral therapy with lopinavir-ritonavir, tenofovir, lamivudine, and efavirenz in 5 chronically-infected patients. 22 We chose δ A = 1.0 to reflect more recent experimental findings. We assign α A = 0 because the relative rate of proliferation of actively infected cells likely occurs at negligible rates compared to the death rate of these cells.
Parameter Perelson 16 Huang 19 Luo 12 Table S1: Proliferation and death parameters for susceptible and actively infected cells.

Estimating the infectivity β
Perelson et al. calculate the infectivity (using physical diffusion) of a virus finding β = 2.4 × 10 −5 µL per day. 16 This value is ubiquitous but Luo et al. and Huang et al. also estimate parameter values for β as 3.9×10 −3 and 1.7×10 −5 , respectively. 12,19 Given that the range of these values spans two orders of magnitude, we chose a value between these extremes: 10 −4 .

Viral parameters: π, γ
The viral 'burst rate' π is the amount of virus a single actively infected cell emits in a day (roughly its lifetime). Rong and Perelson note that π is problematic, as its experimental value is under question and affects the values of many of the other parameters. 5 Haase et al. used radioactively-labeled RNA probes and quantitative image analysis to determine the number of viral particles per mononuclear cell in biopsies from fixed lymph tissue finding a mean value of 74. 23 Using quantitative, competitive, real-time PCR to measure the mean viral RNA copy number per infected cell from freshfrozen cervical lymph nodes from 9 HIV patients with varied viral loads, Hockett et al. found π = 10 3.6 = 4 × 10 3 . 24 Both of these estimates do not necessarily reflect the number of copies produced in a day per cell or in a cell's lifetime, rather the amount that was being produced at the instant the experiments were performed. We adopt π = 10 3 . For the viral clearance rate γ, we use Ramratnam et al.'s estimate γ = 23, obtained from viral load measurements taken over 5 days before, during, and after apheresis in 4 patients assuming a constant rate of viral production. 21 Parameter Perelson 16 Table S3: Viral burst and clearance rates. Note: 12 used the data from 21 but used the geometric mean rather than the arithmetic mean, i.e. 18.8 rather than 23.

The latency fraction: τ
The 'latent cell fraction' τ is the rate at which newly infected cells join the latent cell pool, whereas 1 − τ is the rate at which they join the actively infected pool. The few estimates of this parameter fall throughout a wide range. Despite this, the choice of τ within the given range does not affect our cure estimates or our estimate of the critical epsilon ( c ). Because Conway and Perelson's estimates are based on modern measurements of the reservoir, we chose τ = 10 −4 , the upper bound on their estimates. 4 Parameter Callaway 17 Jones 25 Conway 4 Units τ 10 −6 10 −3 10 −4 unitless Table S4: Fraction of infected cells that enter a latent state.

Parameter estimates, ranges, and confidence intervals
See  Table S5: A summary of all parameters used in our simulations. * δ L is back-calculated from known α L , θ L , and ξ. 95% confidence intervals are given in parentheses () where applicable from experimental data. Otherwise, the range is taken from our literature search or from the ranges given in the cited works not assumed to be normally distributed; these values are given in square brackets [].

Determining the Hill coefficient and IC50s
In dose-response relationships, the concentration of drug that inhibits the response by 50% is called the IC50. The slope at the steepest point along the dose-response curve is called the Hill slope (or coefficient). 27 The Hill coefficient and the IC50s for MMF were calcuated using the drc package in the R statistical computing language. Specifically, the drm fitting command was used to fit the experimental in vitro proliferation data to the 'LL.4' (four parameter log-logistic function) function: in which f (x) represents the fraction of proliferating cells at mycophenolic acid (MPA) concentration x. The upper limit of cellular proliferation in the experiment is c (no drug) and d is the lower limit (maximal drug effect). 28 The active metabolite of MMF is mycophenolic acid, which was used in the titration experiments.

Previous studies in HIV-infected patients treated with ART and MMF
MMF has been given to HIV-infected patients in various settings, either experimentally as an antiviral drug or as part of standard regimens after kidney transplantation. Below we review the studies in which MMF was given to HIVinfected patients and either viral load reduction, reservoir reduction as measured by viral co-culture (QVOA), or time to viral rebound after treatment cessation were measured. Two trials assessed reservoir reduction using viral co-culture. Chapuis et al. revealed a reduction of the viral reservoir in the ART and MMF combination group but no reduction in the group on ART alone. 10 Sankatsing et al. also studied MMF and ART combination treatment and found a mean daily decay rate of latently infected cells (using viral co-culture) of 0.017 infected cells/10 6 cells in patients on MMF and 0.004 infected cells/10 6 cells in patients on ART alone. Despite the fact that this trend was not statistically significant(only eight patients in the MMF + ART group and nine in the ART-only group), the reservoir decay rate in the MMF-treated patients was almost five times as high as in the ART-only patients.
Both García et al. and Millán et al. demonstrated a delay in viral rebound after cessation of ART in patients who received MMF in addition to ART of 2-3 weeks. 29,30 Margolis et al. gave MMF to five HIV-infected patients failing antiretroviral therapy with an average viral load of 10 4.78 copies/mL. Four of the five patients experienced a greater than 0.5 log decrease in viral load compared to entry whereas three of five sustained the 0.5 log decrease after one year on ART combined with MMF.
Jurriaans et al. published a case of an HIV-infected patient who received a five drug ART regimen plus MMF and sero-reverted. 31