# Influence of follicular dendritic cells on decay of HIV during antiretroviral therapy

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Edited by Stirling A. Colgate, Los Alamos National Laboratory, Los Alamos, NM, and approved July 17, 2000 (received for review February 15, 2000)

## Abstract

Drug treatment of HIV type 1 (HIV-1) infection leads to a rapid
initial decay of plasma virus followed by a slower second phase of
decay. To investigate the role of HIV-1 retained on follicular
dendritic cells (FDCs) in this process, we have developed and analyzed
a mathematical model for HIV-1 dynamics in lymphoid tissue (LT) that
includes FDCs. Analysis of clinical data using this model indicates
that decay of HIV-1 during therapy may be influenced by release of
FDC-associated virus. The biphasic character of viral decay can be
explained by reversible multivalent binding of HIV-1 to receptors on
FDCs, indicating that the second phase of decay is not necessarily
caused by long-lived or latently infected cells. Furthermore, viral
clearance and death of short-lived productively infected cells may be
faster than previously estimated. The model, with reasonable parameter
values, is consistent with kinetic measurements of viral RNA in plasma,
viral RNA on FDCs, productively infected cells in LT, and
CD4^{+} T cells in LT during therapy.

Treatment of HIV type 1
(HIV-1) infection with reverse transcriptase (RT) and protease
inhibitors leads to decay of plasma virus (1–4), decay of virus
associated with follicular dendritic cells (FDCs) (5–8), partial
recovery of CD4^{+} T cells in blood (1, 9–11) and
lymphoid tissue (LT) (12), and partial restoration of the FDC network
(13). A feature of HIV-1 dynamics is biphasic decay of virus during
therapy. At the start of therapy, plasma virus decays quickly, but by 2
wk, a second phase is reached and the rate of viral decay slows
considerably (4, 14). Second-phase dynamics may be caused by one or
more processes, including viral production by long-lived infected
cells, activation of latently infected cells, and release of HIV-1 from
viral reservoirs (3, 4).

The pool of HIV-1 on FDCs is a significant viral reservoir (15, 16).
During the asymptomatic untreated stage of infection, the FDC network
harbors ≈10^{11} copies of HIV-1 RNA (5, 17). This
pool of virus, which composes a large fraction of the viral burden in
an infected patient (17), may influence HIV-1 dynamics, given its large
size and the observed loss of virus from FDCs during antiretroviral
therapy (5–8). Here, to assess the influence of FDCs on HIV-1
dynamics, we use a mathematical model, which includes FDC-associated
virus, to analyze quantitative kinetic measurements of viral and
cellular dynamics in blood and LT (18). We consider measurements of
plasma viral load (14), FDC-associated HIV-1 RNA (5), infected
mononuclear cells in LT (5), and CD4^{+} T cells in
LT (12) during treatment for eight patients. The model combines earlier
models for HIV-1 dynamics (19) with a recent model for the reversible
binding of HIV-1 to FDCs (20).

## Model

Fig. 1 illustrates our model of
HIV-1 dynamics during therapy with RT and protease inhibitors. We
consider uninfected cells, short-lived productively infected cells, and
long-lived chronically infected cells. We consider two types of free
and bound viral particles: particles unaffected by therapy, which are
potentially infectious, and therapy-modified particles, which are
noninfectious because they lack functional *gag* and
*pol* gene products caused by inhibition of HIV-1 protease. We
assume the concentration of free virus in blood is the same as that in
extracellular fluid throughout the body.

### Cells.

Dynamics of cells are characterized by (4, 19, 21)
1
2
3
where *T*, *T**, and *C** are the
numbers of uninfected, productively infected, and chronically infected
cells, respectively, and *V* is the number of free potentially
infectious viral particles (Fig. 1). Uninfected cells die with rate
constant μ, are generated at constant rate λ, and proliferate
according to a logistic law, in which *p* is the rate constant
and *T _{c}* is the carrying capacity. [Only

*T*appears in the logistic growth law, because

*T*≈

*T*+

*T** +

*C** (Table 1). The carrying capacity

*T*is the target cell population at which proliferation is assumed to shut off because of limiting factors or homeostatic mechanisms.] Productively and chronically infected cells die at different rates, characterized by δ and μ

_{c}_{C}. The overall rate at which cells are infected is given by (1 −

*e*)(

_{r}*k*+

*k*)

_{C}*VT*, where

*k*and

*k*characterize the rates of productive and chronic infection, respectively. The quantity

_{C}*e*represents the efficacy of treatment with RT inhibitors. Before treatment,

_{r}*e*= 0. For therapy with RT inhibitors that are 100% effective, analytical expressions can be derived from Eqs. 1–3 for

_{r}*T*,

*T**, and

*C** as a function of treatment time

*t*. These equations are available as supplemental material on the PNAS web site (www.pnas.org).

### Free Virus.

Dynamics of free viral particles are characterized by (4, 19)
4
5
where *V̂* is the number of free viral particles
that are noninfectious because of therapy, *R* is the number
of free receptors on FDC, and *B*_{1} and
*B̂*_{1} are the numbers of
potentially infectious and noninfectious viral particles on FDC that
are bound to one receptor (Fig. 1). Productively infected cells produce
viral particles at rate *N*δ per cell, where *N* is
the viral burst size, whereas chronically infected cells produce viral
particles at rate π per cell. The quantity
*e _{p}* is the efficacy of protease inhibitors.
Before treatment,

*e*= 0. The rate constant

_{p}*c*characterizes clearance of free viral particles. The parameter α is an apparent rate constant for association of free viral particles with receptors on FDCs, and

*k*is the rate constant for dissociation of singly bound viral particles.

_{r}### FDC-Associated Virus.

Dynamics of viral particles on FDCs are characterized by (20)
6
7
where *B _{i}* and

*B̂*

_{i}represent the number of unmodified and therapy-modified viral particles, respectively, bound to

*i*receptors (Fig. 1). A viral particle can bind up to

*n*receptors. Eq. 6 governs the dynamics of potentially infectious viral particles; the governing equations for therapy-modified noninfectious viral particles are obtained from this equation by replacing

*V*with

*V̂*and each

*B*

_{i}with

*B̂*

_{i}. Eq. 7 is a conservation equation, in which

*R*represents the total number of FDC receptors. The forward and reverse crosslinking rate constants,

_{T}*k*and

_{x}*k*

_{−}

_{x}, characterize reactions on the surface of FDCs. The total amount of virus bound to FDCs is given by the sum Σ

_{i=1}

^{n}(

*B*+

_{i}*B̂*).

_{i}These equations are based only on considerations of ligand-receptor binding (20). The model omits other processes that might influence the decay of FDC-associated virus, such as turnover of FDCs, internalization of HIV-1, shedding of HIV-1 on iccosomes (22), stripping of HIV-1 by cells that interact with FDCs, structural breakdown of virions, and recovery of the FDC network during therapy (13).

## Parameter Estimates

Parameters were estimated for eight patients (Tables 1 and
2) who participated in the study of
Notermans *et al.* (18) and for whom the level of plasma HIV-1
RNA (14), the number of infected mononuclear cells in LT (5), the
amount of FDC-associated HIV-1 RNA (5), and the number
CD4^{+} T cells in LT (12) were monitored during
treatment. Four patients received triple therapy with a protease
inhibitor (ritonavir) and two RT inhibitors (lamivudine and
zidovudine), and the others received ritonavir monotherapy, followed by
the triple combination after 3 wk. We assume drugs are 100% effective.
Thus, for triple therapy patients, *e _{r}* =

*e*= 1 for

_{p}*t*≥ 0, and for ritonavir monotherapy patients,

*e*= 0 and

_{r}*e*= 1 for

_{p}*t*≥ 0 (viral dynamics are unaffected by the shift to triple therapy at 3 wk; unpublished results). We also assume dynamics are in a steady state before treatment.

### Decay of Long-Lived Infected Cells.

To estimate μ_{C}, we use Eq. 3 and counts of
infected mononuclear cells in LT at wks 3 and 24 of treatment (5). A
count of infected cells at wk 3 was unavailable for patients 20490 and
20496. For these patients, we assume μ_{C}
= 0.0038 d^{−1}, which is approximately the slowest
observed rate of decay (Table 2). Subsequent estimates of key
parameters (e.g., δ) are essentially unchanged if we assume more
rapid decay (unpublished results).

### Baseline Total Body Numbers.

We determine the baseline total body numbers of free viral particles
(*V*_{0}), FDC-associated viral particles
(*F*_{0}), short-lived infected cells
(*T*^{*}_{0}), and uninfected cells
(*T*_{0}) from baseline measurements (5, 12, 14). We
use Eq. 3 to determine the baseline number of long-lived
infected cells (*C*^{*}_{0}) from our
estimate of μ_{C} (Table 2) and the count of
infected mononuclear cells at wk 3 or 24 (5). Unit conversions to and
from total body numbers are based on 700 g of LT (5, 12, 17), 15
liters of extracellular fluid (23, 24), and two copies of HIV-1 RNA per
viral particle.

### Recovery of Cells.

We analyze CD4^{+} cell recovery only for triple therapy
patients; for these patients, viral and cellular dynamics are
uncoupled. We assume a 6-mo half-life for uninfected cells: μ =
0.0038 d^{−1}. This half-life is probably a lower limit (25,
26), but subsequent estimates of parameters (e.g., δ) change little
if the half-life is longer (unpublished results). The relative
contributions of cell generation in the thymus, characterized by λ,
and proliferation, characterized by *p* and
*T _{c}*, are uncertain. We consider the extremes:
cell expansion is caused only by

*de novo*generation, in which case

*p*= 0, or cell expansion is caused only by proliferation, in which case λ = 0. For the case

*p*= 0, we determine the value of λ that best simultaneously fits the counts of CD4

^{+}T cells (12) and infected mononuclear cells (5) in LT. For the case λ = 0, we determine the values of

*p*and

*T*that best simultaneously fit these data. In the fitting procedure, we use Eqs. 1–3 to calculate the sums

_{c}*T*+

*T** +

*C** and

*T** +

*C**, i.e., the total body numbers of (CD4

^{+}T) cells and infected cells. The value of δ, which appears in Eq. 2, is determined as described below.

### Decay of Short-Lived Infected Cells.

For triple therapy patients, we find δ from the steady-state forms of
Eqs. 1–3: δ = [λ +
*pT*_{0}(1 −
*T*_{0}/*T*_{c}) −
μ*T*_{0} −
μ_{C}*C*^{*}_{0}]/*T*^{*}_{0}.
Thus, estimates of δ are derived from cellular data. Similar
estimates are obtained for the cases *p* = 0 and λ
= 0 (unpublished results). For the monotherapy patients,
*e _{r}* = 0 during therapy, and cellular and viral
dynamics are coupled (Eqs. 1–3). For this
reason, we derive δ from viral data (see below). Estimates also can
be obtained from viral data for triple therapy patients; these
estimates are similar to those obtained when δ is derived from
cellular data (unpublished results).

### Production of Virus by Infected Cells.

To focus on the potential of FDCs to influence HIV-1 dynamics, we set
π = 0, which ensures that long-lived infected cells do not
contribute to viral dynamics. We find *N*, the burst size of
productively infected cells, from the steady-state forms of Eqs.
4 and 6: *N* =
*cV*_{0}/(δ*T*^{*}_{0}).

### Decay of Free and FDC-Associated Virus.

The rate constants *k _{r}* and

*k*

_{−}

_{x}had been estimated earlier (20):

*k*

_{−}

_{x}=

*k*≈ 0.1 s

_{r}^{−1}. A reasonable value for α

*V*

_{0}is 0.3 d

^{−1}if

*n*= 20 (20). Typically, if

*n*≠ 20, the dimensionless crosslinking constant

*K*, where

_{x}R_{T}*K*≡

_{x}*k*/

_{x}*k*

_{−}

_{x}, can be adjusted to yield kinetic behavior similar to that with

*n*= 20 (20). The average

*V*

_{0}for patients that we consider is 2 × 10

^{9}(Table 1) (14). Thus, we specify

*n*= 20 and α = 1.5 × 10

^{−10}d

^{−1}for all patients. The same qualitative results are obtained with a range of values for α (unpublished results). For treatment times of interest, we assume viral particles on FDCs are characterized by a single (mean) valence, although variation in

*n*can influence the long-term dynamics of FDC-associated virus (20). For triple therapy patients, we determine the values of

*c*and

*K*that best simultaneously fit measurements of plasma (14) and FDC-associated (5) virus. In this procedure, we numerically integrate Eqs. 4–7 and the set of equations for therapy-modified virus on FDC derived from Eq. 6. As part of the fitting procedure, the values of

_{x}R_{T}*R*,

*R*, and

_{T}*B*for

_{i}*i*= 1, … ,

*n*at

*t*= 0 are determined by using the steady-state forms of Eqs.

**4, 6**, and 7, the baseline number of FDC-associated viral particles

*F*

_{0}(5), and the identity

*F*

_{0}= Σ

_{i=1}

^{n}

*B*. These calculations are described in the supplemental material. For monotherapy patients, the fitting procedure is used to determine δ as well as

_{i}*c*and

*K*. Here, the equations for viral dynamics are coupled with Eqs. 1–3. To simplify our analysis, we assume that

_{x}R_{T}*T*(

*t*)≈

*T*

_{0}in Eqs. 2 and 3. To find

*k*and

*k*, which appear in Eqs. 2 and 3, we use the steady-state forms of these equations:

_{C}*k*= δ

*T*

^{*}

_{0}/(

*V*

_{0}

*T*

_{0}) and

*k*= μ

_{C}_{C}

*C*

^{*}

_{0}/(

*V*

_{0}

*T*

_{0}).

## Results

We have developed a model for HIV-1 dynamics that includes FDC
(Fig. 1; Eqs. 1–7) and used this model to
analyze decay of plasma virus (14), decay of FDC-associated virus (5),
decay of infected cells in LT (5), and recovery of CD4^{+} T
cells in LT (12).

### Release of Virus from FDCs Can Explain Biphasic Plasma Viral Decay.

Viral loads and best-fit theoretical time courses of viral decay are shown in Fig. 2 for two triple therapy patients. As is typical, the model is consistent with the data for these patients. Plots for all patients are available as supplemental material (Figs. 5 and 6). In previous analyses, the first and second phases of plasma viral decay have been attributed, respectively, to death of short- and long-lived infected cells (3, 4, 14). Here, time courses have been calculated on the basis that long-lived infected cells make no contribution to viral load (i.e., π = 0), but the model is still capable of matching the observed viral decay. Thus, production of virus by long-lived infected cells is not required to explain second-phase dynamics if release of virus from FDCs contributes to the plasma viral load.

### Influence of FDCs on First- and Second-Phase Dynamics.

FDC may influence the first phase of plasma viral decay. We find that
death of short-lived infected cells is faster than first-phase decay,
as can be seen by comparing our estimates of δ with those determined
earlier for the same patients by Notermans *et al*. (14) who
used a model in which first-phase decay matches the rate of cell death
(Table 2). Our higher estimates of δ are more consistent with direct
counts of infected mononuclear cells in LT between d 0 and 2 of
treatment (Table 2) (5). FDCs also may influence the second phase of
plasma viral decay. Because the FDC reservoir is the only source of
virus that we consider at treatment times much larger than the
half-life of short-lived infected cells, the rate of second-phase decay
corresponds to the net rate at which virus is lost from FDCs.

### Rate of Viral Clearance.

Based on analysis of plasma viral decay, Perelson *et al.* (3)
determined that 3 d^{−1} is a lower bound on the value of
*c*, the rate constant for viral clearance. We typically
estimate much larger values (Table 2), which are more consistent with
recent direct measurements of *c* (27–29). Higher values for
*c* are needed to explain the data because viral clearance is
offset by release of virus from FDCs.

### Persistence of Potentially Infectious Virus.

As illustrated in Fig. 3, potentially
infectious virus may persist during combination therapy, even if
therapy is 100% effective. The source of persistent potentially
infectious virus is the FDC reservoir. This virus, which is present at
the start of therapy, is not directly affected by antiviral drugs.
Although therapy-modified virus competes with pretherapy virus for
receptors on FDCs and displaces this potentially infectious virus to
some extent, the amount of therapy-modified virus produced in triple
therapy patients, *NT*^{*}_{0}, is
insufficient to replace the virus on FDCs at the start of treatment,
*F*_{0}, i.e.,
*NT*^{*}_{0} < *F*_{0}
(Tables 1 and 2). Elsewhere (30), we show that when RT inhibitors are
absent, more therapy-modified virus is produced, because of new
infections generated by infectious virus from FDCs, and noninfectious
virus displaces pretherapy virus to a greater extent than that shown in
Fig. 3. Consistent with this finding, we find greater displacement of
pretherapy virus on FDCs by therapy-modified virus for ritonavir
monotherapy patients, as shown in the supplemental material (Fig. 7).

### Cellular Dynamics.

The model is consistent not only with measurements of viral dynamics but also measurements of cellular dynamics in LT. In Fig. 4, cell counts and two best-fit theoretical time courses are shown for each of two patients. Plots for all triple therapy patients are given in Fig. 8 of the supplemental material. One theoretical time course is based on target cell recovery caused only by proliferation and the other is based on recovery caused only by generation of new cells. The two time courses are similar, indicating that the relative contributions of proliferation and generation are indeterminate on the basis of this analysis alone.

## Discussion

Decay of HIV-1 during antiretroviral therapy is biphasic, with a slow second phase. Second-phase decay has been attributed to long-lived infected cells (4). By considering release of virus from FDCs (5), we have been able to reproduce biphasic viral dynamics without including production of virus by long-lived cells (Fig. 2). However, our analysis does not rule out a role for long-lived infected cells in viral dynamics: such cells and the FDC pool of virus both may contribute to the second-phase viral load. We find that a long-lived infected cell population, under the assumption of 100% effective drugs, is still required to explain the observed persistence of infected cells (Fig. 4). An alternative explanation for the persistence of infected cells is ongoing infection allowed by drugs that are less than 100% effective (31, 32).

Release of virus from FDCs can lead to biphasic plasma viral decay for
two reasons. First, the amount of virus on FDCs is substantial (17) and
its release (5, 20) can be expected to influence plasma measurements.
For triple therapy patients, consistent with measurements of virus in
cells and on FDCs (17), we predict that the initial viral load on FDCs,
*F*_{0}, is greater than the amount of
virus produced by infected cells during treatment,
*NT*^{*}_{0} (Tables 1 and 2). Second,
dissociation of virus from FDCs during treatment is biphasic because of
reversible, multivalent binding of HIV-1 to FDCs (20). As viral
particles dissociate from receptors on FDCs during first-phase decay,
receptors are freed to interact with the particles that remain. These
particles then attach, on average, to a greater number of receptors,
causing the net rate of release to slow. A steady rate of dissociation,
the second phase of decay, is reached when most receptors are free and
the average number of bonds holding a viral particle on the cell
surface changes little with time.

If the FDC network is a major source of second-phase virus, which is consistent with the available data (Fig. 2), FDCs may affect the long-term outcome of therapy. We predict that virus on FDCs is released continuously during treatment and that a significant fraction of the virus released at all times is virus present before therapy (Fig. 3). The infectivity of this virus depends on the length of time FDCs retain HIV-1 in an infectious state, which is unknown. However, protein antigens, which are quickly degraded in blood, remain intact on FDCs for months to years (33). Also, it has been shown in short-term experiments that HIV-1 on FDCs is infectious (34) and that target cells become infected when cocultured with FDC isolated from mice that were inoculated with HIV-1 42 d earlier (35) and even 9 mo earlier (G. F. Burton, personal communication).

In our calculations, plasma and FDC-associated virus decay in parallel
(Fig. 3), and release of virus from FDCs limits the overall rate of
viral decay. The influence of FDCs on first-phase decay is indicated by
our parameter estimates (Table 2). We find that first-phase decay is
slower than clearance of free virus, which is characterized by the rate
constant *c*, and death of cells in the productive state of
infection (models in which the eclipse phase of the viral life cycle is
explicitly considered (36) show that δ is the death rate of cells in
the posteclipse phase), which is characterized by the rate constant
δ. In contrast, in earlier models (4, 14), the rate of first-phase
decay matches δ. We obtain estimates for *c* and δ that
are higher than expected on the basis of models without FDCs (3, 4,
14), because viral clearance and cell death are buffered by release of
virus from FDCs. Direct measurements also indicate that *c*
(27–29) and δ (5) (Table 2) are higher than earlier model-based
estimates.

We have found that plasma viral decay is influenced by release of virus from FDCs and shown that this release may be the cause of slow second-phase decay (Fig. 2). This finding is significant because the virus seen in plasma, which is monitored to make clinical decisions, is of unknown origin. We now suggest that part of this virus is derived from the pool of virus on FDCs at the start of therapy and that plasma viral measurements do not necessarily reflect the current population of replicating virus, which may be masked by the release of stored virus. Furthermore, we suggest that blocking HIV-FDC interactions, which should significantly speed dissociation of FDC-associated virus (37), may increase the rate of viral decay during treatment and reduce the treatment time required to achieve a given level of virological control. Speeding viral decay in the face of less than perfect therapy will reduce the potential for development of drug resistance. The ability to speed viral decay also may affect the design of interrupted treatment strategies aimed at inducing immunological control of HIV infection. With faster decay, treatment periods can be shortened and compliance and drug side effects are less of an issue. Finally, speeding decay may aid in the identification of other persistent viral reservoirs, all of which must be considered in eradication strategies.

Our analysis leads to several predictions. Earlier studies (4, 14), in
which second-phase decay is attributed to long-lived infected cells,
suggest a sharp break between the two phases of viral decay. In
contrast, we predict a gradual slowing between the first and second
phases (Fig. 2), consistent with data from some studies (38). We also
predict that a significant fraction of virus present during the second
phase of therapy-induced decay will carry processed *gag* and
*pol* gene products (Fig. 3). This finding is contrary to the
predictions of earlier models (3), according to which virus present
during second-phase decay is therapy modified if protease inhibitors
are 100% effective. Finally, because we find that second-phase virus
may include virus present before treatment (Fig. 3), we predict that
virus arising after cessation of treatment will reflect some of the
viral diversity present before treatment. A recent report indicates
that viral reservoirs other than latently infected cells play a role in
re-emergence of plasma viremia after discontinuation of antiretroviral
therapy (39).

## Acknowledgments

We thank A. T. Haase, J. E. Mittler, and J. M. Murray for helpful discussions. This work was performed in part under the auspices of the U. S. Department of Energy and was supported by National Institutes of Health Grants RR06555 and AI28433 (A.S.P.) and the German Cancer Research Center (N.I.S.).

## Footnotes

↵‡ W.S.H. and N.I.S. contributed equally to this work.

↵¶ Present address: Department of Medical Informatics, Biometry and Epidemiology, Friedrich-Alexander University of Erlangen-Nürnberg, 91054 Erlangen, Germany.

↵‡‡ To whom reprint requests should be addressed at: T-10, MS K710, Los Alamos National Laboratory, Los Alamos, NM 87545. E-mail: asp{at}lanl.gov.

This paper was submitted directly (Track II) to the PNAS office.

Article published online before print:

*Proc. Natl. Acad. Sci. USA*, 10.1073/pnas.190065897.Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.190065897

## Abbreviations

- HIV-1,
- HIV type 1;
- FDC,
- follicular dendritic cell;
- LT,
- lymphoid tissue;
- RT,
- reverse transcriptase

- Received February 15, 2000.

- Copyright © 2000, The National Academy of Sciences

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