Molina-Par?s C., Lythe G. (editors) Mathematical Models and Immune Cell Biology

Подождите немного. Документ загружается.

Chapter 18

Using Mathematical Models to Explore the Role

of Cytotoxic T Lymphocytes in HIV Infection

Helen Fryer and Angela McLean

Abstract The combination of mathematical modelling and data analysis to under-

stand the within-host dynamics of human immunodeﬁciency virus (HIV) infections

has been one of the most informative uses of mathematical biology in the last

decade. Simple models of viral dynamics together with viral load measurements

provided an early estimate that the turnover of HIV infected cells is very rapid:

most do not survive beyond 1 day. Although this estimate was initially a surprise

to the ﬁeld, further corroborating evidence has made it widely accepted. More

recently, within-host models have been used to investigate the efﬁcacy of cyto-

toxic T-lymphocytes (CTLs) in controlling HIV infection. Though there is clear

experimental evidence that they play some role, the magnitude of this role remains

contentious. Models have offered three insights on this topic. Firstly, in chronically

infected humans fewer than 20% of HIV-infected cell death is attributed to killing by

CTLs. Secondly, CTLs are more efﬁcient in acute infection than chronic infection,

but not dramatically so, and thirdly, CTLs are markedly more efﬁcient in simian than

human immunodeﬁciency viral infection. Although based on simple models and re-

peatable data, the main prediction of this work that CTL vaccines might work in

macaques but not in humans is yet to gain recognition. This is despite the fact that

this prediction was borne out by the failure of STEP vaccine. We contend that in

time this assertion too will become more widely accepted.

With an estimated 33 million people affected by HIV/AIDS across the world [1],

a huge research effort is now focussed on understanding the immune response to

Human Immunodeﬁciency Virus (HIV) in a bid to creating an effective vaccine.

Early HIV vaccines aimed to elicit antibodies, but quickly proved to be completely

ineffective [2]. The reason for this is now clear: even during natural infection, HIV

is able to evade antibody neutralisation, primarily by obstructing antibody binding

sites. Over the last 15 years, several pieces of evidence have demonstrated that cy-

totoxic T lymphocytes contribute to the control of HIV infection in vivo. These

H. Fryer (



)

The Institute for Emerging Infections, The Oxford Martin School,

Department of Zoology, University of Oxford, Oxford OX13PS, UK

e-mail: helen.fryer@zoo.ox.ac.uk

C. Molina-Par´ıs and G. Lythe (eds.), Mathematical Models and Immune Cell Biology,

DOI 10.1007/978-1-4419-7725-0

18,

 Springer Science+Business Media, LLC 2011

363

364 H. Fryer and A. McLean

include a temporal correlation between the reduction of viremia in acute infection

and the appearance of HIV-speciﬁc CTLs [3, 4]; the selection of viral mutants that

evade killing by CTLs [5, 6]; and associations between particular HLA class I alle-

les and life expectancy [7–9]. Though the magnitude of CTLs’ contribution to viral

regulation is contentious, these ﬁndings have encouraged the vaccine ﬁeld to direct

research towards developing a vaccine that elicits CTLs. The aim of such a vaccine

would be to dampen viral replication in hosts who become infected.

In 2004 the ﬁrst large-scale human efﬁcacy trial (the STEP Study) of a CTL-

based vaccine candidate began. Considerable hope was pinned on this vaccine, but

in 2007 the trial was halted when interim results showed the vaccine to be en-

tirely ineffective both at preventing infection and reducing viral load in those who

become infected [10, 11](Table18.1 and Fig. 18.1). This result is regarded as a

Table 18.1 Results of the STEP study. The number of HIV infections ac-

cording to baseline Ad5 antibody titre. The Merck vaccine trial was halted

early when it became apparent that the infection rate in individuals with

high immunity to the Ad5 vector was higher in those who were vaccinated

than in those who received the placebo. Even in individuals with low Ad5

immunity, the vaccine provided no protection [15]

Number of HIV infections

Ad5 antibody titre prior to innoculation Vaccine Placebo

<18 20/382 20/394

18 < Ad5200 8/140 4/142

200 < Ad5 1000 14/229 7/229

Ad5 > 1000 7/163 2/157

Total 49/914 33/922

Fig. 18.1 Boxplots of the viral setpoints measured in individuals who became infected in the

STEP study. There was no difference in the viral setpoints measured in the vaccine (N D 46)

and placebo recipients (N D 30). When this analysis was restricted to patients who had low Ad5

immunity, the same result was obtained (data not shown) [15]

18 CTL in HIV Infection 365

major setback for the HIV vaccine ﬁeld, not least because the candidate (the Merck

HIV DNA/adenovirus complex) had shown the most promise of several proposed

CTL-based vaccines that were tested in monkeys challenged with simian-human

immunodeﬁciency virus (SHIV)[12]. The failure of this vaccine has reignited doubt

that CTLs play a pivotal role in controlling HIV replication and has thus led the ﬁeld

to question whether CTLs will ever prove a useful tool for an HIV vaccine. In addi-

tion, the failure has highlighted the inconsistency between CTL efﬁcacy in monkeys

and humans. In light of these issues, this chapter reviews how mathematical mod-

els have been used together with experimental results to address the question: how

effective are CTLs are in controlling HIV-1 infection?

The chapter begins with a discussion of experimental and observational results

that provide insight into the role of CTLs in controlling HIV infection. This is fol-

lowed by a review of models of viral dynamics that have been used to interpret

patient viral load data in relation to this question. Finally, a model of the infection of

CD4 T cells with either the wildtype viral strain or a CTL escape mutant strain [13]

is discussed. We show how this model has been used to analyse CTL escape mutant

outgrowth data to quantify the magnitude of the CTL response. Further, we show

how it has been used to compare the strength of the HIV-speciﬁc CTL response in

humans to that of the SIV/SHIV-speciﬁc response in macaques [14].

Experimental and Observational Results that Provide Insight

into the Role of CTLs in Controlling HIV Infection

Today there is considerable evidence to suggest that CTLs contribute to the control

of HIV infection in vivo. What is not clear is exactly how important their contribu-

tion is at each stage of infection. Measuring their contribution directly is not possible

and there is some evidence against a signiﬁcant role for CTLs; notably certain corre-

lations predicted on the basis of a strong CTL response have not been observed. Data

indicating an important role for CTLs in controlling HIV in infected individuals are

summarised in Fig. 18.2.

One of the early observations in favour of a role for CTLs came in 1994 when

a temporal correlation was established between the reduction in viremia in acute

infection and the appearance of HIV-speciﬁc CTLs [3, 4]. The appearance of the

CTLs preceded the appearance of antibodies, indicating that antibodies were not

responsible for the observed effect (Fig. 18.2(a)). Other evidence is provided by

the selection of CTL escape mutations in vivo [5, 6]; by in vitro experiments that

have revealed CTLs capable of exerting potent antiviral effects towards HIV [16–19]

(Fig. 18.2(b)); and by the high levels of anti-HIV-1 CTLs seen in established HIV-1

infection. These can often be detected directly from the peripheral blood without

in vitro stimulation. This conﬁrms that the CTLs are ‘at work’, rather than simply

waiting to be stimulated into action.

Probably the best evidence in favour of a signiﬁcant contribution by CTLs, how-

ever, comes from clear associations between progression to AIDS and different

366 H. Fryer and A. McLean

Fig. 18.2 (a) The time course of natural HIV infection in a typical individual. A temporal cor-

relation exists between the containment of viremia during acute infection and the appearance of

HIV-speciﬁc CTLs. The measurements shown are the CD4

T cell count (cells/l), the plasma

viral load (RNA copies/ml), the CTL response (% effector CTL in PBMC) and the neutralising

antibody titre [31]. (b) Evidence that CTLs target HIV-infected cells in vitro. The mean percent

speciﬁc CTL cytotoxicity and corresponding standard error is presented for four seropositive and

ﬁve seronegative individuals. The percent speciﬁc cytotoxicity is shown for the effector:target ra-

tio 25:1. Cytotoxicity is measured in vitro against two vaccinia recombinants: one containing the

env region of the HIV genome and one containing the bacterial lacZ gene used as a control. The

HIV-speciﬁc response to the env/Vac complex was signiﬁcantly elevated in seropositive individu-

als compared to the response to the lacZ/Vac (control) complex in the same hosts and compared to

the env/Vac complex in seronegative individuals. Note that some background cytotoxicity directed

against the Vaccinia virus is expected in all hosts [16]. (c) The association between the HLA B35Px

allele and progression to AIDS. Infected individuals who carry at least one copy of the HLA B35Px

allele progress to AIDS more rapidly than those who contain no copies of HLA B35px [20, 22].

(d) The effect of depletion of CTLs during primary SIV infection in macaques. The early control

of virus fails when CTLs are depleted [20,22]

18 CTL in HIV Infection 367

HLA alleles. These alleles encode for the HLA molecules involved in the process

that results in CTL targeting. Notably, HLA types B57, B51 and B27 are associ-

ated with slow progression and HLA B35Px is associated with rapid progression

(Fig. 18.2(c)). Patients who are homozygous at any of the three HLA class I loci

also progress substantially more rapidly to disease [7–9]. More recently, Kiepiela

and colleagues have also shown that the part of the genome targeted by CTLs

is an important determinant of disease progression. They found that an increased

breadth of gag-speciﬁc responses is associated with lower levels of viremia, whereas

an increased breadth of env-speciﬁc and accessory/regulatory-speciﬁc responses is

associated with increased levels of viremia. The reason for these protein-speciﬁc

differences is unclear.

HIV research that cannot be conducted in humans is instead tested in non-human

primates infected either with simian immunodeﬁciency virus (SIV) or the hybrid

virus simian-human immunodeﬁciency virus (SHIV). The primates used for these

tests are typically rhesus macaques or pigtail macaques, chosen because upon in-

fection they reproduce many of the key elements of HIV infection in humans. Not

only do they experience CD4

T cell depletion leading eventually to AIDS-like

symptoms, but they also have similar immune systems to humans. Like humans,

macaques have been shown to elicit CTL responses against immunodeﬁciency virus

infection. Furthermore, when antibodies speciﬁc for CTLs were used to experimen-

tally deplete CTLs during acute SIV/SHIV infection, the early control of virus failed

[20–22](Fig.18.2(d)). When the antibodies were instead given in the chronic phase,

the virus level rose until the effects of the antibody wore off. Depletion of B-cells

had no effect. These results favour a role for CTLs in controlling the set point in

natural HIV infection, though should be viewed in light of the unknown extent to

which results from macaques can be extrapolated to humans.

Evidence of CTL involvement in controlling HIV infection is therefore clear.

Exactly how important they are in this process, however, is put into question by

the absence of certain results. If CTLs were crucial in containing HIV, one might

expect that counts of HIV-speciﬁc CTLs would correlate with viral load. Despite

early reports to this effect [23–25], it is now accepted that no such simple correla-

tion exists [26,27]. In an examination of CTL responses directed against the entire

HIV genome in 57 patients, no correlation was found between either the breadth or

the magnitude of the CTL responses and viral load [27]. A second study into the

effects of drug therapy interruptions also showed no correlation between CTL count

and either viral rebound rates at the start of the interruption, or viral clearance rates

at the restart of therapy [26]. Furthermore, neither measures of absolute numbers

nor changes in numbers of CTLs were able to predict viral load in this experiment.

At the very least, these results indicate that different CTLs have different efﬁcacies,

a ﬁnding supported by the study by Kiepiela et al. [28] and by two recent stud-

ies that suggest that a correlation may exist between the number of polyfunctional

CTL (as measured by ﬁve different immunological markers) and plasma viral load

[29,30].

368 H. Fryer and A. McLean

Modelling HIV-1 Viral Dynamics to Understand What Patient

Viral Load Data Tells Us About the Importance of CTLs

in Controlling HIV

One way in which the importance of CTLs in controlling HIV infection can begin to

be addressed is by investigating how well observed viral load data can be explained

by mathematical models of viral dynamics. Viral dynamics during HIV infection

have been widely analysed and it is now accepted that there are three main phases

to the dynamics in natural infection [31](Fig.18.2(a)). During the ﬁrst few weeks

(acute infection), viral load rises to a peak before declining and settling to an equi-

librium called the viral set point. The start of this decline is typically associated with

the appearance of CTLs speciﬁc for HIV. Once at the set point, viral load remains at

that level for a period of several years (the chronic phase), during which there is

both rapid viral replication and rapid viral loss. The set point differs between hosts

and is predictive of life expectancy – higher set points correlate with shorter life ex-

pectancies. Eventually this balance is disrupted and viral load gradually increases

in concordance with a decline in CD4 count that eventually results in AIDS. Infor-

mation about viral dynamics can also be attained by disrupting the natural course of

infection. This can be achieved when antiretroviral therapy is started. These drugs

disrupt the viral life cycle and when given during chronic infection, lead to viral

load decaying in an exponential manner.

A Simple Model of Virus Infection

A simple model that has been proposed to explain some of the observed HIV viral

dynamics is shown in Fig. 18.3 [32]. The model consists of three variables: the

Fig. 18.3 Asimple

within-host model of viral

infection. T, I and V represent

the number of target cells,

infected cells and free virions,

respectively. Free virions are

produced at a rate, ˛,per

infected cell and infect target

cells at a rate proportional to

the abundance of free virions,

ˇV. In addition, target cells

are produced at rate  and die

at a rate d per cell; free

virions are cleared at rate 

per virion; and infected cells

die at rate ı per cell

18 CTL in HIV Infection 369

population size of target cells (i.e. uninfected CD4

T cells), T, infected cells, I,

and free virions, V. Free virions are produced at a rate, ˛, per infected cell and in-

fect target cells at a rate proportional to the abundance of free virus, ˇV. In addition,

target cells are produced at rate  and die at a rate d per cell; free virions are cleared

at rate  per virion and infected cells die at rate ı per cell. The model is prescribed

by the following set of coupled ordinary differential equations.

T D   dT  ˇV T

I D ˇV T  ıI

V D ˛I  V: (18.1)

Fitting the Simple Target-Cell-Limited Model to Viral Load Data

from Hosts with Natural HIV Infection

In the simplest form of this model, the death rate of infected cells, ı, can be as-

sumed to be constant through time. As such, the model includes no explicit CTL

response. Despite this, the model is nevertheless able to reproduce many of the viral

dynamics observed in HIV-infected individuals; namely it predicts that virus will

grow rapidly up to a peak before declining and settling at a stable equilibrium at

which there is continuous production and loss of virus. It is perhaps surprising that

a model that includes no explicit introduction of a CTL response during acute infec-

tion is able to reproduce the early containment of virus so often attributed to CTLs.

In fact, in this model it is saturation of the target cell population that is the cause of

viral suppression. The simple model of infection of target cells therefore puts into

question the hypothesis that acute viral containment is due to CTLs and suggests

that the observed temporal correlation may be coincidental. In support of this asser-

tion, Phillips [33] has cited three studies in which infected individuals suppressed

their primary viremia without mounting any detectable speciﬁc anti-viral immune

responses.

Although saturation can explain the drop in viral load qualitatively, how well it

can explain the drop quantitatively has been questioned. Stafford et al. [34] ﬁtted

viral load data from ten patients in acute infection. They showed that, by assuming

only small variations in different model parameters across patients, the model could

accurately explain both the increase in viremia up to the peak and the set point that

was eventually reached, but could not explain the seemingly overly-rapid rates of

viral decay seen in some individuals. These fast decay rates could be explained by

the appearance of CTLs during acute infection. The fact that viral set points can

be explained without inclusion of a CTL response, however, implies that even if

the CTL response is important during acute infection, it is not explicitly important

during chronic infection.

370 H. Fryer and A. McLean

Does Progression to AIDS Happen Because of a Failing

CTL Response?

It has often been suggested that it is the failure of the CTL response that eventu-

ally disrupts the equilibrium observed during chronic phase and leads to increasing

viremia and eventual progression to AIDS. If this were true, it would favour the the-

ory that CTLs are crucial in controlling HIV during chronic infection, as has been

shown to be true in SIV/SHIV infected in macaques. In these monkeys, depletion of

CTLs during chronic infection leads to an increase in viremia.

It has been proposed that, if progression to AIDS results from a failing CTL

response, then individuals with more advanced disease would kill infected cells

more slowly. Conversely if AIDS results from the virus evolving to infect cells more

quickly, infected cells would be cleared more quickly in individuals with more ad-

vanced disease. To investigate this issue, curves showing the decay of viremia

following treatment with a protease inhibitor (PI) – a type of antiretroviral – in

chronically infected humans have been compared to decay curves predicted by a

version of the simple target-cell limited model that is adapted to include PI ther-

apy during chronic infection. Protease inhibitors do not affect the survival or rate of

virion production of infected cells, but do prevent the production of infectious viri-

ons. To model the impact of PI therapy, infectious virions V

and non-infectious

virions V

are regarded separately and the production rate of infectious virions

is set to zero. Furthermore, for the short period over which viral decay has been

measured (approximately 1 week) the number of target cells is assumed to remain

constant at the steady state value attained prior to therapy, T

. Equations represent-

ing the adapted model are given by:

I D ˇV

 ıI

DV

D ˛I V

: (18.2)

During the steady state achieved prior to therapy, the abundance of target cells, pro-

ductively infected cells and free virus can be regarded as constant. Thus by setting

T D I D V D 0 in the original model (18.1), expressions can be found for ˛ and

(˛ D ı=ˇT

and I

D ˇV

=ı) which enable (18.2)tobesolvedtogivean

expression for the decay of virus (V D V

C V

) upon commencement of therapy:

V.t/ D V

t

V

  ı



  ı

ıt

 e

t

/ ıte

t

(18.3)

This function ﬁts patient data in which plasma viral load after PI treatment is seen

to decay in an exponential fashion following an initial lag [35](Fig.18.4(a)). Im-

portantly, this equation shows that the exponential viral decay rate will be faster

if clearance of infected cells (rate ı) is faster. It is noteworthy that this correlation

would hold true irrespective of whether clearance is CTL mediated. The hypothesis

18 CTL in HIV Infection 371

Fig. 18.4 The decline of viremia upon commencement of treatment with a protease inhibitor in

individuals with chronic HIV infection. (a) An example of the predicted decline in viral load fol-

lowing therapy in patients late in chronic infection and in patients early in chronic infection under

the assumption that patients with further progressed disease clear infected cells more slowly (see

(18.3)). The rate of viral decline would be faster in patients early on during chronic infection. (b)A

comparison of pre-therapy CD4 counts and viral clearance rates in 20 individuals. No relationship

exists between these two variables, implying that progression to AIDS does not result from a failing

CTL response [36]

that individuals with more advanced disease clear infected cells more slowly can

therefore be tested by comparing therapy-induced viral decay rates to pre-therapy

CD4 T cell counts. Interestingly, a comparison of such data shows that not only is

there no relationship between pre-therapy CD4 T cell counts and infected cell clear-

ance rates, but that infected cell clearance rates are remarkably consistent across

different individuals [36]. These results imply that people do not progress to AIDS

because clearance of infected cells diminishes. They therefore also provide no evi-

dence that CTLs are important in controlling HIV during chronic infection.

In summary, a simple within-host mathematical model of viral dynamics can

be used to understand what observed viral dynamics reveal about the importance

of the CTL response in HIV infection. This analysis shows that much of the ob-

served viral dynamics seen in natural infection can be explained qualitatively by a

model that includes no explicit CTL response. This includes containment of virus

during acute infection through saturation of the target cells. When the model and

the data are compared quantitatively, there remains no explicit evidence that CTLs

are important during chronic infection; however, in some individuals, rates of viral

containment during acute infection are faster than predicted by the simple model,

suggesting that CTLs may indeed be functionally important during acute infection,

as has been shown to be to be the case for macaques using CTL depletion experi-

ments. To investigate further whether CTLs are important during chronic infection,

viral clearance rates have been compared to pre-therapy CD4 T cell counts. No rela-

tionship has been found between these two variables, suggesting that progression to

AIDS does not occur because of a failing CTL response. This comparison therefore

372 H. Fryer and A. McLean

also provides no evidence in support of CTLs being functionally important dur-

ing chronic HIV infection. This is in disagreement with results from macaques in

which depletion of CTLs during chronic SIV/SHIV infection leads to signiﬁcantly

increased levels of viremia.

Modelling Within-Host Dynamics of CTL Escape and Reversion

to Estimate the Proportion of Cell Lysis Attributable to CTLs

There is good evidence to show that the average life expectancy of productively

infected CD4

T cells is considerably less than that of CD4

T cells present in

uninfected individuals [35–40]. What is not known is what proportion of this cell

death is attributable to CTLs and what proportion is attributable to other factors–

notably viral cytotopathicity and by-stander activation.

Longitudinal data relating to the outgrowth of CTL escape mutants within indi-

viduals is able to provide us with insight into this issue (Fig. 18.5(a)). For a start,

it is clear from the fact that selection of CTL escape mutants occurs, that CTLs are

responsible for some proportion of infected cell death. Secondly, it is intuitive that

the rate of outgrowth of a particular CTL escape mutant should reﬂect three main

quantities: the strength of the CTL response that is being evaded, the ﬁtness con-

straints associated with the mutant and the rate of cell turnover. Fitness costs cannot

be measured directly. However, ﬁtness costs are believed to drive the reversion of

Fig. 18.5 Escape and reversion data and theoretical ﬁts. (a) Three examples of the selection of

an escape mutant in an individual who makes an immune response to the epitope of interest. The

function 1=.ge

kt

C1/ is ﬁtted to each of the escape curves to estimate the value of the escape rate,

k. The estimated escape rates for these three events are, 0.012, 0.119 and 0.41 day

1

.(b)Three

examples of the reversion of an escape mutant in an individual who does not make an immune

response to the epitope of interest. The function 1=.ge

t

C1/ is ﬁtted to each the reversion curves

to estimate the value of the ﬁtness cost (or reversion rate), . The estimated ﬁtness costs for these

three events are, 0.016, 0.005 and 0.005 day

1