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Journal of Virology, March 2001, p. 2597-2603, Vol. 75, No. 6
0022-538X/01/$04.00+0 DOI: 10.1128/JVI.75.6.2597-2603.2001
Copyright © 2001, American Society for Microbiology. All rights reserved.
Release of Virus from Lymphoid Tissue Affects Human
Immunodeficiency Virus Type 1 and Hepatitis C Virus Kinetics in
the Blood
Viktor
Müller,1,*
Athanasius
F. M.
Marée,2 and
Rob J.
De Boer2
Collegium Budapest, Institute for Advanced
Study, 1014 Budapest, Hungary,1 and
Theoretical Biology, Utrecht University, 3584 CH Utrecht, The
Netherlands2
Received 19 July 2000/Accepted 20 December 2000
 |
ABSTRACT |
Kinetic parameters of human immunodeficiency virus type 1 (HIV-1)
and hepatitis C virus (HCV) infections have been estimated from plasma
virus levels following perturbation of the chronically infected
(quasi-) steady state. We extend previous models by also considering
the large pool of virus localized in the lymphoid tissue (LT)
compartment. The results indicate that the fastest time scale of HIV-1
plasma load decay during therapy probably reflects the clearance rate
of LT virus and not, as previously supposed, the clearance rate of
virus in plasma. This resolves the discrepancy between the clearance
rate estimates during therapy and those based on plasma apheresis
experiments. In the extended models plasma apheresis measurements are
indeed expected to reflect the plasma decay rate. We can reconcile all
current HIV-1 estimates with this model when, on average, the clearance
rate of virus in plasma is approximately 20 day
1, that of
LT virus is approximately 3 day1, and the death rate of
virus-producing cells is approximately 0.5 day1. The fast
clearance in the LT compartment increases current estimates for total
daily virus production. Because HCV is produced in the liver, we let
virus be produced into the blood compartment of our model. The results
suggest that extending current HCV models with an LT compartment is not
likely to affect current estimates for kinetic parameters and virus
production. Estimates for treatment efficacy might be affected, however.
| |
INTRODUCTION |
Our understanding of the dynamics of
human immunodeficiency virus type 1 (HIV-1) infection relies largely on
the analysis of changes in the viral load in plasma after initiation of
treatment with potent antiretroviral drugs. Quantitative mathematical
models have been fitted to clinical data to provide estimates for the virion clearance rate, the average life span of productively infected cells, the viral generation time, and the total daily production of
virus (14, 15, 20, 21). These studies have identified two
time scales during the first 1 to 2 weeks of potent antiviral therapy.
The slower of the two was supposed to reflect the turnover of
virus-producing cells, with an average death rate (
) of about 0.5 day1. For the faster decay of virus particles a
"clearance" rate (c) of about 3 day1 was
found. The estimation of c from the total virus decline was later suggested to be unreliable (7), but an independent
measurement of the decline of plasma infectivity (21)
supported the estimate of 
3 day1.
A recent plasma apheresis study yielded significantly faster estimates
for the HIV-1 virus clearance (22). Large quantities of
plasma were removed from infected patients to create an additional clearance term for the virus. During the 1 to 2 h of apheresis, the virus level in plasma dropped significantly and rapidly returned to
baseline levels after the apheresis was stopped. A mathematical model
was fitted to the data to obtain estimates for the natural clarance
rate c. The estimates for four patients ranged from 9 to 36 day1, with an average of approximately 23 day1. Although the early estimates of approximately 3 day1 were established as lower bounds, we show below that
the large discrepancy in the estimates for c can be due to
the compartmentalization of the virus pool. The estimates for the
turnover rate, , of virus-producing cells have also been published
as lower bounds (21) and have also been questioned
(5). In HIV-1-infected patients, no correlation between
and the magnitude of the cytotoxic response was found
(17). This resulted in a debate on the importance of the
cytotoxic immune response (4) and in more sophisticated immune control models (13).
It has long been known that the major site of HIV-1 infection is the
lymphoid tissue (LT) (3, 19). The trafficking of lymphocytes between the plasma and the LT compartment has been shown to
be of potential importance in HIV-1 infection (12, 23).
Here we extend this approach to the distribution of virus. A series of
studies have confirmed that most of the virus, i.e., on the order of
1010 virions in a typical patient, resides in the LT
compartment (1, 6, 18, 24). Most of the virus is bound to
the surfaces of follicular dendritic cells (FDCs). Surprisingly, virus
in the LT was found to decline at about the same rate as free virus in plasma during therapy (1): during the first two days of
therapy, virus in LT declined at a rate of about 0.4 day1. Later the decay slows down, which may be due to
multivalent binding (9). The early parallel declines of
virus in LT and plasma suggest that on this time scale of a few days
there is a (quasi-) steady state between the two virus compartments
that is maintained by an exchange of viral particles. The decay of virus in LT introduces a third time scale, which is intermediate to
those for c and and which hence influences the
established estimates.
Mathematical analysis of plasma apheresis (22) and
antiviral treatment data (14) has also been carried out
for hepatitis C virus (HCV). To explain the rapid drop of the viral
load in plasma in about 1 day, which is followed by a slower
second-phase decay, Neumann et al. suggested that alpha interferon
(IFN-
) blocks the production of new HCV virus particles
(14). The first-phase decay was dominated by a
c of approximately 6 days1, and the slow
second-phase decay, with 0.01 day1 < < 0.4 day1 was conjectured to reflect the death rate of
virus-producing cells.
The HCV estimates from the plasma apheresis study are in much better
agreement with the treatment estimates: in two patients virion
clearance rates were estimated to be 5.5 and 9.9 day1.
Below we suggest that the reason for this difference between HIV-1 and
HCV is the different localization of virus production. Whereas HIV-1 is
mostly produced in the LT, HCV is produced in the liver, which connects
to the blood plasma compartment. By implementing this difference in the
models we are able to show that the impact of an LT virus reservoir on
turnover parameter estimates is likely to be much smaller for HCV than
for HIV-1. Total virus production estimates are also more reliable.
| |
MATERIALS AND METHODS |
The models.
The models consider virus-producing infected
cells, I, free plasma virus, VP, and
virus in the lymphoid tissue, VL. To set the
pretreatment steady state, we write a simplified term allowing for
limited target cell availability (see Appendix). Thus, in the pretreatment situation new infections occur at a rate (
) given by
= maxV/(h + V), where max is the maximum value of and h is a saturation parameter. Because we set
max = 0 during highly active antiretroviral therapy
(HAART), most results do not depend on the form of this equation.
Altenatively, we could have allowed for a density-dependent to
model an immune response.
Virus is produced at rate p, with units of per infected cell
per day, and is cleared from the blood at rate c and from
the LT at rate cL. Assuming a constant immune
response on the time scale of treatment, the death rate of infected
cells, , is a fixed parameter. Between the two compartments there is
an exchange of virus at rates i, for influx into the blood,
and e, for efflux from the blood into the LT. Most of the
virus in the LT is in fact bound onto FDCs. Because the kinetics of
virus association and dissociation seems to be fast (9),
we treat LT virus as a single population. Our LT clearance rate
cL should therefore be interpreted as an average
over the bound and the free virus in the LT. In the Appendix we briefly
discuss how this simplification can be derived.
In the paper we consider "typical" patients with most of the virus
in the LT, i.e.,
VL 
VP, and "late-stage" patients with
equal
amounts of virus in the LT and in the plasma. This difference
is
probably caused by the destruction of the FDC network during
disease
progression, which reduces the amount of virus that can
be bound to
FDCs (
11). The LT virus population in our model
is a
mixture of FDC-bound virus and free virus in the LT. Virus
bound to
FDCs with multiple bonds is probably protected from fast
clearance and
has a longer residence time in the LT than free
virus (
9).
Destruction of the FDC network during disease progression
is therefore
expected to increase
cL and the rate of
transport
(
i) of our mixed-virus population in the LT to the
plasma. In
this paper we vary both parameters to (i) model late-stage
patients
with high
i and (ii) study the effect of the
unknown
cL in the
LT. Since the steady-state
plasma viral load is proportional to
the amount of virus released from
the LT into the plasma (see
below), increasing
i decreased
the
VL/
VP ratio.
In the HIV model virus-producing CD4
+ T cells reside in the
LT. Conversely, in the HCV model, where the infected cells are
mostly
hepatocytes, virus is released into the blood plasma compartment.
Thus
for the HIV-1 model we write
|
(1)
|
|
(2)
|
|
(3)
|
whereas in the HCV model produced virus enters the blood
so we write
|
(4)
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|
(5)
|
|
(6)
|
Due to the density-dependent infection term, the
nontrivial equilibrium of both models is always
stable.
We make a clear distinction between "true" virus clearance,
cL and
c, and the "transport"
terms,
i and
e: the "total clearance"
rates
are defined by
e +
c
and
i + cL. As there is
no known
biological mechanism that would transport large amounts
of virus from
the blood to the LT,
e should be small. The flux
of virus
from the LT to the blood probably follows the flow of
lymph to the
blood. Later we will show that the plasma apheresis
experiments suggest
that this process is also likely to be slow,
hence
i should
also be small. Because this flux is not passive
diffusion, we write the
explicit flux parameters
e and
i and formulate
our system in terms of total body counts of productively infected
cells, plasma virions, and LT virions. By appropriate scaling
one can,
however, rewrite the model in terms of concentrations
and conventional
diffusion-like transport
terms.
During a treatment blocking new infections, we set
= 0. As
this makes the model linear, one may solve the eigenvalues of
the full
system (equations 1 to 3) to find that
1 =
and
|
(7)
|
We have argued above, and suggest below, that transport
parameters
e and
i are probably small. If the
product
ei is sufficiently
small, i.e., if 4
ei
is
(
)
2, one obtains the simple eigenvalue structure
1 =
,
2 =
,
and 3 =
, such that the solution
becomes
|
(8)
|
The kinetics of the viral load in the plasma,
VP, thus has three time scales, set by
,
, and
Plasma apheresis.
The clearance rate of plasma virus has
been studied with plasma apheresis experiments (22).
During plasma apheresis the removal of plasma increases the clearance
of free plasma virus. The changes in plasma virus concentration were
modeled by the equation
|
(9)
|
where
P is the total input of virus into
the blood and
c' is the sum of the natural clarance rate
c and the rate of removal
by apheresis,
. The input
P represents virus production and was
considered to be
constant during the 1 to 2 h of apheresis. Estimates
for
c and
P were obtained by a nonlinear fitting
procedure of
the plasma virus concentrations before, during, and after
apheresis.
In total less than 10
8 virus particles were
removed during an
experiment.
Although 10
8 particles correspond to a significant fraction
of the total plasma virus pool,
VP, it is less
than 1% of the total
virus load,
VP +
VL. Thus, the mere fact that one finds an
observable
decline in the plasma virus load by removing less than 1%
of the
total body virus suggests that the exchange between the LT
compartment
and the blood cannot be too rapid, i.e., transport rates
i and
e have to be sufficiently small. In our
two-compartment model
the input term in equation 9 becomes
P =
iVL and the clearance
term becomes
c' =
c +
e +
. Because typically
VL is
VP, it is
reasonable to assume that
P (=
iVL) hardly varies during the
apheresis.
Thus equation 9 applies equally well to our two-compartment
model
for HIV-1, with the only change being that the estimated
clearance
rate of 23 day
1 (
22) should
reflect the total clearance,
=
c +
e.
For HCV, the input of free plasma virus is production plus influx from
the LT (see equation 6), i.e.,
P =
pI +
iVL, which
can again be treated as constant during
plasma apheresis. The
clearance rate in equation 9 remains
c' =
c +
e +
. Thus, the
plasma apheresis estimates for
the total clearance,
=
c +
e,
are 5.5 and 9.9 day
1 (
22). These estimates are close to
those obtained from treatment
of HCV-infected patients, which suggested
a
of 6 day
1 (
14).
Because the transport rate of virus from the plasma to the LT is
probably small and because the plasma apheresis experiments
suggest a
limited exchange on a time scale of hours, we will assume
that the
efflux rate,
e, is small, i.e., we set
e = 1 day
1 for both viruses. This seems a conservative choice
because the
rapid total clearance rates measured during the plasma
apheresis
experiments would largely reflect the true clearance,
c (
22).
Accordingly, we set
c at 24 day
1 for HIV-1 and 7 day
1 for HCV. For
robustness all simulations were checked for a
c of 5 day
1 and an
e of 20 day
1 for
HIV-1 and for a
c of 2 day
1 and an
e of 6 day
1 for
HCV.
| |
RESULTS |
Antiviral treatment: HIV-1. (i) The 1- to 2-week time scale of
total virus decline.
Combination therapy of HIV-1 infection with
protease and reverse transcriptase inhibitors blocks new infections and
renders newly produced virus noninfectious. As a consequence, the
population of virus-producing cells declines exponentially at rate (10). After a fast initial transient dominated by the fast
clearance of plasma virus, the decline of the plasma virus load was
believed to approach the same slope, (21). In models
allowing for additional virus compartments, such as LT, this need not
remain true. Assuming for simplicity a 100% drug efficacy and
considering the total virus load (i.e., infectious plus noninfectious
virus), we model similar treatment by setting the maximum infection
rate max = 0.
To study how the different compartments affect the estimate of
, we
first ignore the fast transient by a quasi-steady-state
(QSS)
assumption for the plasma virus,
VP = iVL/. Solving the
QSS model for
max = 0, one obtains for the LT virus
|
(10)
|
where
|
(11)
|
Because of the QSS assumption the plasma virus load
remains proportional to equation 10. The two different true clearance
rates,
cL and
c, of virus turnover
are combined into a composite
decay rate,
. Note that for the
supposedly small value of
e,
After a fast initial transient,
the behavior of the system will
be dominated by the slowest exponent in equation 10,
i.
e.,
by or
.
Thus,
the observed slope at which virus declines in the blood will
not reflect the death rate,
,
of productively infected cells
whenever is <
.
This we call the "
masking"
of infected-
cell
turnover rates.
In equations 11
can only be smaller than under the following condition:
cL +
ci/(
e +
c)
cL +
i =
<
. To remain consistent with
the data (
10,
21), the
slower of the two exponents should
be about 0.5 day
1. We therefore set
= 1 day
1 and we restrict ourselves to values of
0.5 day
1. When the FDC network is degraded during
disease progression,
one may expect the flow rate,
i, of
virus from the LT to the blood
to increase because a smaller fraction
of the LT virus is bound
to the FDCs (
11). In terms of
equations 11, disease progression
is thus expected to increase
and
the
VP/VL ratio.
We studied the full model numerically to find parameter values that are
consistent with the treatment and the plasma apheresis
estimates and to
investigate when we find masking (i.e.,
<
).
In all
numerical examples we found similar behavior for the QSS
model and the
full system, confirming the validity of equation
10. Typically, in
asymptomatic patients one finds that most of
the total body virus is
confined to the LT (
6,
18). A patient
with a QSS
I of 10
8 virus-producing cells and
VP of 10
8 plasma virus particles
might harbor as many as 5 × 10
10 virus particles in
the LT (
6). The first 10 day of the treatment
of such a
patient is depicted in Fig.
1a, where we first set 0.5
day
1 for the decay rate of LT virus. All other parameter
values are
set such as to obtain the required initial steady state
(Fig.
1). Because
VL is
VP, we find that
is <
in this example, and
the observed decline of the plasma
virus load is dominated by
the decay rate,
,
of LT virus and not by the death rate,
,
of virus-producing cells.
This provides an example of the masking
of
.
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FIG. 1.
Viral load during HAART as obtained by equations 1 to 3. (a and c) Typical situation with a large LT virus compartment
(VL VP) for slow and rapid LT
clearance rates, respectively. (b) Late-stage patient, with equal
amounts of virus in the LT and blood compartment and a slow LT
clearance rate. In panel a there is masking of , whereas in panels b
and c the plasma viral load is coming down at rate . Parameters:
c = 24 day1, e = 1 day1, = 1 day1, and h = 109. We set max = 1.02 × 108 to attain the stable pretreatment equilibrium and set
max = 0 at day 0. Other parameters are as follows:
i, 0.05 (a and c) and 25 day1 (b);
cL, 0.5 (a and b) and 5 day1 (c);
p, 274 (a and b) and 2,524 particles cell1
day1 (c). Initial steady state, VL = 5 × 1010 particles, Vp = 108 particles, I = 108
cells (a and c) and VL = 1.4 × 108 particles, VP = 1.4 × 108 particles, I = 1.3 × 107 cells (b).
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Because patients in a more-advanced stage of the disease can have a
much smaller compartment of LT virus (
11), our next
example is a case with equal amounts of virus in the blood and
in the
LT. To achieve
VL =
VP,
we increased the influx rate,
i,
of virions from LT to blood
500-fold. All other parameter values
were unchanged. The time course of
treatment is depicted in Fig.
1b. The decline of virus compartments
approaches that for the
productively infected cells (
) (the graphs
of
VL and
VP coincide
with the graph depicting
). In this case
is >
, and indeed
there is no masking. Because
c is >
, such a situation is
expected
when a sufficiently large fraction of the total-body virus is
located in the blood (see equations
11).
The turnover rate of LT virus is unknown and need not be a simple
exponential decline (
9). Equation
10 only shows that the
slower of the
and
timescales is reflected in the parallel
declines of plasma and LT virus. Figure
1c depicts the same system
also
for a 10-fold-higher clearance of LT virus. To allow for
the same
initial steady state, we compensated by changing the
production rate
accordingly (Fig.
1c). Because
is >
in Fig.
1c,
masks
cL, and a realistic
of 1 day
1
is observed. The latter scenario implies a rapid clearance of
virus
from the LT, i.e., implies that
is >
.
Summarizing, for cases representing typical asymptomatic patients where
VL is
VP, there are
two parameter settings that remain
consistent with the previous
observations (
10,
21,
22).
First, the turnover of LT virus
may be slow, i.e.,
0.5
day1,
and would be expected to be responsible for
the observed decline of virus in the blood.
Second,
the turnover of LT
virus may be fast,
i.
e.,
>
, such that the observed decline of virus in
the blood
reflects the turnover,
, of productively infected cells.
In the
first scenario LT virus turnover masks
, whereas in the
second
masks LT virus turnover. Finally, because the masking
in the
slow-turnover scenario depends on the
VL/
VP ratio, it may
depend on disease stage. Such a dependence is not expected for
the
rapid-turnover
scenario.
(ii) The rapid time scale of infectious virus.
During complete
protease inhibitor treatment the infectious virus titer was expected to
fall with a slope of c (21). Data from a single
late-stage patient revealed that this slope was approximately 3 day1 (21). For the same patient was
estimated to be 0.53 day1. Considering the infectious
virus titer only, we modeled complete protease inhibitor treatment by
setting p = 0. Assuming, as above, that the product of
the transport parameters, ei, is sufficiently small, we
obtain
|
(12)
|
with
and
time scales. Since, the turnover rate,
, of the LT virus has the slower time scale, it is expected to
dominate
the dynamics. In Fig.
2 we consider the
full model with
p = 0 for same three cases as in Fig.
1. Indeed we see that the
decay rate of LT virus dominates, and that it
masks clearance
rate
in the blood, in Fig.
2a and c,
representing typical patients
with
VL
VP (i.e., small
i). The observed
slope of 3 day
1 (
21) could therefore
correspond to the total clearance,
, of LT virus. To model the late-stage patient of Fig.
2b, with
VL =
VP, we increased the
rate of transport,
i, of virus from
the LT to the plasma.
Increasing
i increases
, such that
may come to represent the slower time
scale. Indeed in Fig.
2b
we observe a rapid slope of approximately 24 day
1.
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FIG. 2.
Blocking viral production in HIV-1 infection. The plasma
virus titer follows the decay rate of LT virus if the LT reservoir is
large (a and c) and follows the decay rate of plasma virus if it is
small (b). The parameters and initial steady states are the same as in
the corresponding panels of Fig. 1. Treatment is modeled by setting
p = 0.
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(iii) Interpretation.
The dynamics of plasma virus is
characterized by three time scales, each of which can be estimated from
existing data sets. Plasma apheresis (22) provides an
estimate of approximately 23 day1 for the total clearance
of plasma virus. For patients with a large LT virus pool, the observed
decline of infectious virus at rates of at least 3 day1
(21) should reflect the slower of clearance rates
, and
, which is then necessarily the latter. Total plasma virus and LT virus
decline at a rate of 0.5 day1 (10), which in
typical patients with VL
VP should be dominated by the slower of
and , which would then be . Hence the model is consistent
with current data when is 23 per day,
is 3 per day, and is 0.5 per day.
HCV.
The biphasic decline of plasma virus titers during
IFN- treatment of HCV infection suggests that the inhibition of
virus production is incomplete and that the second-phase slope reflects the decline of residual production by the death rate, , of
virus-producing cells (14). This was modeled by decreasing
virus production by factor . Because we cannot set = max = 0 during IFN- treatment, we fixed the
infection rate, , at its pretreatment value to keep it constant on
the short time scale of interest. For the plasma viral load we obtain
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(13)
|
For HCV the typical total blood pool of virus is on the
order of 10
10 particles and the size of the LT virus
compartment is unknown.
We investigated two scenarios, one with a large
LT virus compartment
and one with an LT virus compartment equal to the
blood compartment.
Obviously, when the LT virus compartment is
sufficiently small,
it will fail to affect the behavior of the new
model. In "Plasma
apheresis" we suggested setting
= 8 day
1; hence we set
e = 1
day
1 and
c = 7 day
1. The
fast initial drop of viremia does not allow for a much smaller
true
clearance rate,
c (not shown), which further confirms the
assumption of a small efflux,
e. We set
= 0.5 day
1 and
VP(0) = 10
10 particles, while the remaining free parameters, i.e.,
p,
, and
i, were set so as to obtain the
required pretreatment steady state.
Figure
3 shows different drug efficacies (i.e.,
= 0.99 and 0.80
[
14]), depicting the slopes of the
three decay rates and the
total body count of plasma and LT virus. For
comparison with the
model of Neumann et al. (
14), Fig.
3
also depicts the solution
of their one-compartment model.
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FIG. 3.
HCV treatment dynamics where treatment blocks 99 (a and
c) or 80% (b and d) of virus production. The initial decline reflects
, the total viral clearance in the blood. In panels a and b the
LT virus compartment is large and the second-phase decline of plasma
virus (heavy solid line) follows the total plasma LT clearance rate,
.
In panels c and d the plasma and the LT virus compartments are of equal
size, and the rate of the second-phase decline is between
and . Symbols are as in Fig. 1. Heavy dash-dotted line,
behavior of the original one-compartment model (14).
Parameters: c, 7 day1, e, 1 day1; , 0.5 day1;
cL, 0.05 day1; i, 0.05 (a and b) and 0.95 day1 (c and d). The pretreatment
plasma viral load is identical in all panels (VP = 1010 particles), which was achieved by setting and
p. The initial total body counts of LT virus are
1011 particles (a and b) and 1010 particles (c
and d).
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|
For Fig.
3a and b where
VL(0) = 10
VP(0), we have to allow for a small decay rate
of LT virus, e.g.,
cL = 0.05 day
1.
This constraint stems from the pretreatment ratio of the two
virus
pools, i.e.,
VL/
VP =
e/
.
As
in the original model (
14),
the decline of plasma virus is
biphasic (
Fig.
3).
The two models agree on the rapid first phase,
which
fits well to a slope of . Thus, no masking is
expected for the decay rate of plasma
virus. The models differ with
respect to the second-phase decay,
however. When the LT virus
compartment is large, the kinetics
of the two-compartment model is
significantly slower than that
of the earlier model. In this case a
slow decay of LT virus may
mask
. The masked death rate could be
even higher than the 0.5
day
1 in our example. Thus, if
the two-compartment model reflects reality
better, the death rate of
virus-producing cells could be much
larger than the previous estimates
(
14). Additionally, the depth
of the drop in the virus
load during the rapid first phase is
significantly smaller in the
two-compartment model than it is
in the previous model. Since this
initial drop of viremia was
used to assess the efficacy of the IFN-
treatment, the drug efficacy,
, could also be an underestimate. Note
that this effect is strongest
at high efficacy (cf. Fig.
3a and
b).
Figure
3c and d depict the situation with equally large virus
compartments. Although for both phases the slopes are in a fairly
good
agreement with those of the original model, the second-phase
decline of
plasma viremia reflects an intermediate rate between
and
.
In the two-compartment model, the initial drop of plasma
viremia
remains small, however. To obtain equal pool sizes, we
increased the
influx,
i, of virus from the LT compartment into
the blood.
This situation also allows us to study higher LT virus
decay rates.
Increasing the decay rate to 0.5 day
1 had little effect
on the difference in the initial drop in the
viremia (not shown). For
even smaller LT compartment sizes, the
masking effect of
cL disappears
altogether.
Summarizing, the two models always agree on the estimates for
during the initial phase. They differ in the
estimation of
treatment efficacy. Whenever the LT compartment is
sufficiently
large, they also differ in the estimate for
. This
interpretation
is consistent with the observation that the HCV
estimates for
from the IFN-
treatment and the
plasma apheresis experiments
were in good agreement (
14,
22).
| |
DISCUSSION |
During asymptomatic HIV-1 infection the LT virus compartment is
known to be large. We have investigated how this LT compartment may
affect current procedures for estimating the virus clearance rate,
c, and the average lifetime of productively infected cells, 1/. We have found good agreement between all previous studies when
the total clearance rate in the blood, , is
23 day1, the total clearance rate in the LT,
(=cL + i) is 3 day1, and
is 0.5 day1. As virus infusion experiments suggest
a relatively slow efflux from the blood, the true clearance rate might
be close to the above estimate, i.e., c 20 day1. The dominance of the LT virus compartment in turn
requires a small influx rate, which implies that
cL is 3 day1. Our results
suggest that the LT total clearance rate,
, and the total plasma clearance rate, , can be
estimated from the deline of infectious virus (21) and by
plasma apheresis (22), respectively. The death rate of
virus-producing cells, , can indeed (10, 21) be
obtained from the decline of total plasma or LT virus.
Importantly, when the LT virus compartment is large, the influx rate,
i, has to be small and the decline of infectious virus follows the LT clearance rate, cL (which was
found to be 3 per day in the single studied case [21]).
This implies high estimates for the total daily production of HIV-1
particles. An infection involving a total body burden of 5 × 1010 particles would require a production of at least
1.5 × 1011 particles per day. Late-stage patients,
having a smaller LT virus compartment (11), would require
a lower daily production.
We let virus production in HCV infection take place in the blood
compartment because the liver releases virus into the blood. As yet,
there are no good estimates available for the LT virus compartment in
HCV infection. Irrespective of its size, however, allowing for a LT
virus compartment hardly affects current estimates for the total viral
clearance rate, . Indeed, the HCV estimates for
obtained from plasma apheresis (22) are
in good agreement with those obtained from IFN- treatment
(14). Because HCV production takes place in the blood
compartment, we can only have a large LT virus compartment in our model
by allowing for a very slow clearance rate, cL,
in the LT and a slow influx into blood. If our HIV-1 estimate of
3 day1 also holds for HCV, we would expect much
a smaller pool of LT virus in hepatitis infections. This implies that
current estimates of (14) are not likely to be
confounded by the LT virus compartment and that the high plasma viral
loads associated with hepatitis B virus and HCV (14, 16)
need not reflect a much higher total body viral burden or production
than that estimated for HIV-1. The LT virus compartment may
nevertheless remain responsible for an underestimation of the efficacy
of IFN- treatment.
The current model allowing for three time scales is still lacking the
possibly important slow time scale of virus attached to the FDC network
by multiple bonds (8, 9). Our QSS assumption for virus
bound to the FDC network is expected to be valid during the first
few day of treatment only. To reliably study the later stages, one
would have to merge the current model with that of Hlavacek
et al. (8, 9). Their work incorporates multiply bound
virus particles but neglects the breakdown of FDC-associated virus and
does not account for the observed rapid initial decline of plasma infectivity.
Another simplification of our model is the omission of virus production
by infected cells in the blood compartment. In our typical patient,
about 2.5 × 109 virions must enter the plasma per day
to balance the rate of loss observed in the plasma apheresis
experiments. With the most likely parameter setting of Fig. 1c, this
could be achieved by production from about 106 productively
infected cells, i.e., 1% of the number present in LT. This is close to
the ratio of total CD4 T cells in blood to CD4 T cells in LT. Even
though infection is likely to induce cells to home to lymph nodes
(2), the contribution of blood resident cells cannot be
excluded completely. However, the results of the present work are not
affected by this possibility. The behavior of the large LT virus pool
during the first day of HAART is hardly affected by the plasma pool.
Hence our first result remains valid: the 0.5-per-day decay rate of LT
virus is the slower of and .
Nor is the decline of plasma infectivity affected, as the production of
infectious virions is blocked both in the plasma and in LT. This
decline of 3 per day therefore also remains the slower of
and L. As the
estimates obtained by plasma apheresis reflect ,
irrespective of the source term of plasma virus, our interpretation
remains valid for all three time scales. Virus production in the blood
might only affect the short initial transient of total plasma virus,
which is hard to analyze due to the confounding effect of
pharmaceutical delay. In all, production of virus in the blood cannot
be excluded by our analysis but does not affect our results.
To summarize, the presence of an LT virus compartment in current models
of viral infection may confound parameter estimation. Current data
(1, 21) suggest that the average total clearance rate
, of virus in the LT compartment is relatively fast. We can reconcile
most data when the following relation exists: > 3 > . Such a fast LT clearance implies that
current estimates of from plasma apheresis and of
from antiviral treatment are hardly confounded by the LT virus
compartment and that virus production during HIV-1 infection is even
higher than was estimated previously. Finally, an LT clearance rate of
approximately 3 day1 would reconcile the difference
between the estimates for the plasma virus clearance as obtained by
plasma apheresis and HAART.
| |
APPENDIX |
A standard equation for simple target cell dynamics is
dT/dt = 
TT *TV, where susceptible target cells,
T, arise at rate , die at rate T, and are
infected at rate *. For the pretreatment steady state one
sets dT/dt = 0 to obtain T(0)
= 
. Substituting this into infection term TV, one
obtains maxV/(h + V), where
max = and h = T/*.
Virus in the LT is either free or bound to FDCs. As exchange with the
blood involves free virus only and binding to FDCs is reversible, we
write for the HIV-1 model VL = V*L + B and
|
(A1)
|
|
(A2)
|
where
V*L
is free virus,
B is bound LT virus,
i*
is the influx rate of free LT virus into the blood, and
k1 and
k2 are association
and dissociation rates, respectively.
k2 for
monovalent binding
was estimated as 0.1 per s (
9),
which is orders of magnitude
faster than the other processes in
the system. As the association
and dissociation of virus mostly involve
virions attached to FDCs
by one or a few ligands, one expects this
fraction of bound virus
to remain at QSS with free virus. Setting
dB/dt = 0 one obtains
B =
k1V*L/(
k2 +
cL). Adding equation A2 to equation A1 one obtains
equation
2 with
i =
i*
.
| |
ACKNOWLEDGMENTS |
V.M. was supported by the Hungarian Soros Fund and the
Hungarian Scientific Research Fund (OTKA).
We thank Alan Perelson and Bill Hlavacek (Los Alamos) for extensive
discussions and Béla Novák, Béla Gy
rffy, and
Péter Simon (Budapest) for helpful comments.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: Collegium
Budapest, Institute for Advanced Study, Szentháromság u. 2, 1014 Budapest, Hungary. Phone: 36 1 224 8306. Fax: 36 1 224 8310. E-mail: muller_v{at}ludens.elte.hu.
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0022-538X/01/$04.00+0 DOI: 10.1128/JVI.75.6.2597-2603.2001
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Top
Abstract
Introduction
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