Proceedings of the 2005 IEEE
Engineering in Medicine and Biology 27th Annual Conference
Shanghai, China, September 1-4, 2005
Mathematical Analysis of the HIV-1 Infection : Parameter Estimation,
Therapies Effectiveness and Therapeutical Failures
Djomangan Adama OUATTARA
IRCCyN : UMR CNRS 6597, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes
1, rue de la Noë, Bp 92101, 44321 Cedex 3, Nantes, France
Email: [email protected]
Abstract— This paper presents a method to estimate accurately the parameters involved in the HIV-1 infection. An
application to the evaluation of the drugs effectiveness and
therapeutical failures is presented.
I. I NTRODUCTION AND M ODELLING
Research on HIV/AIDS infection is performed in various
domains including mathematical modelling. The mathematical modelling is about modelling the evolution of the disease using mathematical tools, mainly differential equations.
There exists many contributions in this domain and the major
achievements were focused on measuring the kinetics of the
virus: [1], [2], [3], [4], [5], [6]. These contributions were
obtained from data over a short period of time, under the
assumption that parameters are not time varying. In this
paper, we consider clinical data available over a large period
of time (over 6 years), for 2 representative patients. The
parameters of the model are computed for each therapy and
their evolution are used to evaluate the efficiency of drugs
as the RTI, the Reverse transcriptase Inhibitor and the PI,
the Protease Inhibitor. The goal of this paper is threefold:
half-saturation1 constant of the proliferation process. System
(1) models the infection process, but some phenomena are
neglected. The cytotoxicity of CD8 T-cells (CTL) or the
latently infected T-cells ([7], [8]) are not modelled here. So,
we do the first assumption that all these phenomena can be
neglected in the above 3D model. Then, their effects will
be concealed in the explicit parameters of the model. The
second assumption is that drugs mainly affect parameters β
and k when therapy is initiated. Parameters s, δ, µ, c, r, K
are supposed non-sensitive to drugs. In fact, RTIs block
the production of new viral particles since the reverse
transcription of the viral RNA into DNA is blocked. This
has a direct impact on k. Similarly, the action of PIs have
a direct impact on β. These drugs disturb the cleavage
of viral proteins. As a result, new produced virus will
be defective and thus, non-infectious. The mathematical
decoupling of these two drugs is important and even
if the effects of the one can influence the other during
multi-therapies, we do the assumption that this phenomenon
is negligible compared to their straightforward effects.
1) to present how one can have an accurate estimation of
the parameters involved in the HIV-1 infection,
2) to present how one can evaluate the effectiveness of
various drugs entering in multi-therapies,
3) to derive a tool to help in the diagnosis of therapeutical failures : immunological failures and virological
failures.
Let us present
time model.
⎧
⎨ Ṫ
Ṫ ∗
⎩
v̇
the basic 3-dimensional (3D) continuous= s − δT − βT v + p(v, T ),
= βT v − µT ∗ ,
= kT ∗ − cv.
(1)
This model includes the dynamics of the infected CD4+
T-cells, T , the uninfected CD4+ T-cells, T ∗ , and the virions,
v. The free virus particles infect the uninfected cells at a
rate proportional to the product of their abundance (βT v)
and are removed from the system at the rate cv. We suppose
in this model that the healthy CD4+ T-cells are produced
at a constant rate s. µT ∗ represents the rate at which the
infected cells are removed from the system. The term
Tv
p(T, v) is equal to r K+v
. It models the proliferation of the
CD4+ T-cells. r is the maximal proliferation rate. K is the
0-7803-8740-6/05/$20.00 ©2005 IEEE.
Fig.1: The HIV/AIDS life cycle.
II. E STIMATION P ROCEDURE
The identifiability of continuous-time nonlinear systems
was studied in [5],[9]. The case of discrete-time systems is
presented in [10]. The outputs of the system are T + T ∗ ,
821
1 When
v = K, r(v) =
r
.
2
the overall measured quantity of CD4 T-cells and v, the
measured viral load. With these two outputs, we can show
that the 3D model is identifiable. At least, 11 measurements
are necessary to estimate all the parameters of the model (e.g.
6 measurements of the CD4 load and 5 of the viral load). The
method used to estimate the parameters of the model is the
Nelder-Mead Simplex method [11]. This method was used in
[6] for identification of the parameters of the 3D continuoustime model. However, the main drawback of continuous time
models is that the necessary time to solve the differential
equations is very long compared to discrete time-models.
So, the model used here to estimate the parameters is the
3D discrete-time model. This model is described as follows:
⎧
⎨ Tt+1 = s + (1 − δ)Tt − βTt vt + p(Tt , vt ),
T∗
= βTt vt + (1 − µ)Tt∗ ,
(2)
⎩ t+1
vt+1 = kTt∗ + (1 − c)vt .
where Tt = T (t), Tt∗ = T ∗ (t), vt = v(t) for each t =
t0 , t1 , · · · , tN . This model is the first order approximation
of the Taylor expansion (along the time t) of the continuoustime 3D model. Since the necessary time to solve this system
is very short, we set up a new method to improve the
parameter estimation. This method is described as follows:
A. Estimation Procedure
1) Principle: Let consider an optimization algorithm (e.g.
steepest descent, simplex, etc.). Running this algorithm to
minimize a quadratic function on a large number of initial
conditions (i.c.), one can see that the solutions are distributed
in the neighborhood of the real optimum.
Example 2.1: Let consider f (x) = 0.01x2 + 0.05x +
0.075, and compute
x̂ = min{f (x)}. Choosing a
x
set of 5000 i.c. using an uniform distribution on the
interval [−30, 30], we perform 5000 estimations of the
minimum of f . The solutions distribution is as follows:
The distribution of the estimated solutions
0.015
0.01
0.005
0
−5
−3
x 10
0
x
Zoom on the optimun x* = −2.5
5
10
5
0
−2.6
−2.5
−2.4
−2.3
x
Fig.2: Distribution of the results with tolerances on x and f (x)
equal to 0.25. The estimated optimum x̂ = median(x) = −2.4908
with IQR = 0.1. The real optimum is x∗ = −2.5.
The IQR (interquartile range) measures the dispersion of the
results. It will depend on the chosen precision (tolerance) and
the convexity of f. The IQR gives an important information
on the confidence on the results. In the HIV/AIDS case, for
ethical and financial reasons, it is not possible to collect
a large amount of data. Therefore, it is very difficult to
provide a robust estimation of the parameters with traditional
methods. However, with the method presented here, we are
now able to compute accurately all the 8 parameters of the
above 3D model, and this with the strict minimum number
of data (11 measurements). For the 3D model case, 2000
i.c. are used according to a uniform law on the intervals
Vθi , where Vθi is an admissible interval of the parameter θi .
For example, Vs can be equal to [1e − 20, 20] because the
production of CD4 T-cells is always positive and smaller than
20 CD4/mm3 . Note that, this method provides information
on all local optimum (in the interval Vθ ). The characteristic
of each optimum is a normal like distribution. Figure 3 below
illustrates the results obtained for one patient.
III. M AIN R ESULTS
In this section, we present results obtained with 2 representative patients from the CHU of Nantes, France (Nantes
University Hospital). The patients we study here are the
same who have been presented in ([12], [13]). However,
results presented here have been improved using the method
described above.
A. Patient 97 (43 years old)
The estimates of parameters are displayed in Table 1.
• In a first period 2409 ≤ t ≤ 2681, the patient is
treated with two RTIs: zidovudine (AZT) and lamivudine (3TC), and one PI: saquinavir soft gel (SQV).
During this period, the infection decreases (the viral
load drops from 65000 copies/ml to 5000 copies in 113
days) before increasing again to 21000 copies/ml.
• In the second period 2681 ≤ t ≤ 4788, the patient’s
therapy consists of two RTIs : 3TC and stavudine (d4T)
and one PI ritonavir (ABT-538). The viral load drops
under the threshold of 50 copies/ml.
Evolution of k and β: The production rate of virions, k,
ranges from 2694 to 928 copies/ml and per infected T-cells,
depending on the treatment which is applied. We conclude
that the RTIs effectiveness are better in the second period
than in the 1st since the parameter k is 3 times lower in the
2nd period. This conclusion on RTIs effects is in accordance
with the results presented in [13]. Values of β suggest that
PIs effectiveness are approximately the same in periods 1
and 2.
Proliferation: In [13], proliferation were not taken in to
account in the model and β were sometimes negative. We
explained how the proliferation can disturb the estimation
of β (and/or δ). Here, taking into account the proliferation,
we can insure that PIs effects are approximately the same in
the 2 periods. The estimate of r confirms our intuition on
proliferation. Proliferation rate, r, is higher (4 times higher)
in the 2nd period. The high value of K, in the 2nd period,
shows why the proliferation effect were on β and not on δ. In
r
fact, if v K, p(v, T ) K
T v. In this case, if proliferation
r
is neglected, the estimate of β will be β̂ = β − K
< 0, since
r
>
β
(proliferation
is
greater
than
virus
infectivity).
On
K
the other hand, if v K, p(v, T ) rT. The effect of the
CD4 proliferation will be on δ.
822
Param.
The estimates of the parameters are displayed in Table 2.
•
•
•
s (mm−3 d−1 )
The first line of antiretroviral therapy is a bi-associated
combination of RTI (AZT+3TC): from t = 2032 to t =
2156 (for 124 days). Then a tritherapy: AZT+3TC+IDV
is started for the next 769 days. IDV (indinavir) is a PI.
During the second period, the patient receives
several associations : AZT+3TC+ABT-538+IDV
(treatment stopped by the patient’s decision);
AZT+3TC+NFV(nelfinavir) (treatment stopped due
to a virological failure); AZT+3TC+IDV (treatment
stopped for toxicity). All these therapies have not been
shown to be efficient.
In the last period, the patient is treated with
AZT+3TC+EFV(efavirenz). No PI is involved.
Evolution of k and β: In the last period of treatment,
parameter β increases (1450 times higher), and k drops to a
level 9 times lower than in the 1st period. We conclude that
the 2nd period’s RTIs: AZT+3TC+EFV, are more efficient
than the 1st period’s RTIs AZT+3TC for patient 556. The
lack of PI clearly yields the high increasing of parameter β.
The higher value on r suggests an higher rate of proliferation
in the 2nd period. These results confirm the 1st results in
[13].
Recommendations for patient 556: For this patient, the
evolution of parameters enable us to deduce some hints
for a better treatment. The RTI of the second period
(AZT+3TC+EFV) combined with the PI of the first period
(IDV) could lower both parameters β and k.
Estimates θ̂
Param.
IQR
0.31
0.55
[0.22, 0.76]
0.016
[0.0092, 0.025]
µ (d−1 )
0.11
0.07
[0.082, 0.15]
β (ml−1 d−1 )
0.23
0.19
[3.53e−7, 6.51e−7]
k (d−1 )
2694.3
3735.2
[1297, 5032.3]
r (d−1 )
9.54e−3
0.022
[0.232e−4, 0.022]
14750
[0.027, 14750]
251.53
Estimates θ̂
IQR
CI50%
0.60
0.16
[0.52, 0.68]
δ (d−1 )
1.44e−3
5.0e−4
[1.17e−3, 1.67e−3]
µ (d−1 )
0.24
0.17
[0.19, 0.37]
c (d−1 )
0.57
0.59
[0.31, 0.90]
4.19e−7
9.65e−7
[2.4e−8, 9.89e−7]
k (d−1 )
928
1624.9
[593.91, 2218.8]
r (d−1 )
0.04
0.12
[4.2e−4, 0.12]
4031.4
39743
[37.14, 39780]
β (ml−1 d−1 )
K (ml−1
0.037
[0.044, 0.074]
0.092
0.068
[0.061, 0.13]
1.02e−9
1.79e−8
[5.42e−14, 1.79e−8]
k (d−1 )
2604
5186.5
[152.04, 5338.5]
r (d−1 )
1.08e−4
8.04e−3
[2.79e−9, 0.008]
1009
[3.59e−5, 1009]
9.53
b.
No parameter available
c.
3312 d ≤ t ≤ 4420 d
AZT+3TC+EFV
Estimates θ̂
IQR
CI50%
0.13
0.24
[5.55e−4, 0.24]
δ (d−1 )
2.79e−3
6.78e−3
[4.30e−4, 7.20e−3]
µ (d−1 )
0.12
0.10
[0.079, 0.18]
c (d−1 )
0.64
0.97
[0.30, 1.26]
1.50e−6
2.13e−6
[6.19e−7, 2.75e − 006]
s (mm−3 d−1 )
β (ml−1 d−1 )
k (d−1 )
296.24
604.73
[102.21, 706.94]
r (d−1 )
3.23e−3
7.75e−3
[1.04e−5, 7.76e−3]
17.36
146.5
[0.087, 146.59]
K (ml−1 )
Table 2: Parameters estimates of patient 556. In the first period (a.),
he was treated with 3TC+AZT and after, with 3TC+AZT+IDV. In
the second period (b.), estimation was impossible because of poor
data. In the third period (c.), he was treated with AZT+3TC+EFV.
−3
x 10
1st local optimum
0.02
2nd local optimum
0.01
0.04
15
0.03
10
0.02
5
0.01
0
0
0
0
1
2
0
3
0.05
0
0.1
1
2
3
β
δ
−6
x 10
0.015
0.01
0.02
0.01
0.005
0.01
0.005
0
0
0
0.2
0.4
0.6
0
0.8
2
µ
4
k
0
6
4
x 10
0
0.5
1
1.5
2
c
0.6
0.2
0.4
0.1
0.2
0
0
0.02
0.04
r
0.06
0
5
K
10
5
x 10
Fig.3: Parameters distributions for patient 97 on the 1st period. The
asymmetric distribution of k is due to positive constraints. All the
parameters are constrained to be positive during the optimization
process. For parameter s, we can distinguish the signature of two
local optimums (the first is at ∼ 0.25 and the second at ∼ 0.8).
Compared to the others parameters, the distributions of r and K
suggest a lower confidence in their estimate. From this method,
one has information on the locations of all the optimums. So, it is
possible to improve the results obtained here by choosing smallers
intervals Vθ in the neighborhood of the estimated optimums.
2681 d ≤ t ≤ 4788 d
3TC+d4T+ABT-538
b.
s (mm−3 d−1 )
0.060
c (d−1 )
0
2.99e−7
Param.
µ (d−1 )
Param.
[0.16, 0.34]
4.71e−7
K (ml−1
[8.44e−5, 3.77e−3]
K (ml−1 )
CI50%
0.017
)
[0.13, 0.70]
3.68e−3
s
δ (d−1 )
c (d
0.57
2.09e−3
0.015
s (mm−3 d−1 )
−1
0.46
δ (d−1 )
β (ml−1 d−1 )
2409 d ≤ t ≤ 2681d
AZT+3TC+SQV
a.
2032d ≤ t ≤ 2925d
3TC+AZT then 3TC+AZT+IDV
−−→
Estimates θ̂
IQR
CI50%
a.
B. Patient 556 (42 years old)
IV. A NALYSIS OF T HERAPEUTICAL FAILURES
Table 1: Parameters estimates of patient 97. In the first period
(a.), he was treated with AZT+3TC+SQV and in the second (b.)
with 3TC+d4T+ABT-538. CI50% = 50% confidence interval, IQR
= interquartile range. Refer to the Fig. 3 to see the parameters
distributions on the 1st period for this patient.
To begin with, recall four notions of therapeutical
failures[14], [15]:
Immunological failure is defined when the amount of
CD4 T-cells remains below the level of 200/mm3 after
6 months of treatment. Two indicators can be used to
characterize this failure : the ratio s/δ̃ and tλ . They are
823
defined as follows:
• The ratio s/δ̃ represents the theoretical highest set point
of the CD4 T-cells count, i.e. when βT v = 0 (if the
treatment is 100% efficient). Parameter δ̃ is computed
as follows:
– δ̃ = δ if the proliferation term p(v, T ) is not,
taken into account in the model when estimating
parameters, and
rv̄
– δ̃ ≈ (δ − K+v̄
) if proliferation term is considered
in the model.
v̄ is the mean of the viral load.
• tλ is the theoretical necessary time for the CD4 T-cells
to reach a given level λ when starting from the initial
level T0 . tλ can be computed as follows:
δ̃
δ̃
1
tλ = × − loge 1 − λ + loge 1 − T0
s
s
δ̃
(3)
with λ < s/δ̃. So, to characterize immunological
failure, the criterion t200 can be deduced from (3).
Virological failure is due to a persistent replication of
the viral load under treatment. It usually results from a bad
inhibition of HIV replication or viral resistance. So, it can be
considered mainly as the consequence of PIs failures (β) and
RTIs failures (k). Virological failure is a re-interpretation of
the results in the previous section and it will not be discussed
further here.
Biological failure: is defined when viral load is higher
than 30,000 copies/ml and CD4 T-cells account is lower than
200/mm3 . It can be considered as the superposition of both
virological and immunological failure.
Clinical failure: This situation is characterized by the
clinical manifestation of opportunistic diseases. This failure
is generally related to a biological failure with a low account
of CD4 T-cell and a high level of viral load. Clinical failure
will not be considered here. The effect of opportunistic
diseases is not modelled in the 3D model.
A. Patient 97
In this period, mean(v) = 28346 mm−1 copies/ml, so
δ̃ = 0.00754 d−1 . Remark that, this is exactly the value
of δ we estimated in [13] when proliferation were not
taken into account. So, the ratio s/δ̃ = 41.11 CD4/mm3
is very low. The first treatment will never be able to increase
CD4 T-cells load over the 41.11 CD4/mm3 . In the second
period, s/δ̃ increases to an acceptable level (416 CD4/mm3 ).
However, t200 = 13 months > 6 months. This informs
about the delay of 13 months which is needed to reach the
200CD4/mm3 level with an optimal combination therapy.
Clinical data show that it tooks 10.5 months to reach this
level.
B. Patient 556
In the 1st period δ̃ = 0.0019. This is in accordance with
the results without proliferation in [13] (a value of 0.0021
were calculated for δ). s/δ̃ equals 230.08 CD4mm3 in the
first period. t200 = 33 months yields the diagnosis of an
immunological failure. Clinical data show that approximately
25 months were necessary to reach the 200 CD4/mm3 level.
V. C ONCLUSION
Mathematical analysis of the HIV model allows to evaluate
the effectiveness of the drugs involved in the therapies.
This mathematical methodology gives new hints to clinical
doctors for the diagnosis of patients. At this stage, we
conclude that the mathematical analysis is able to evaluate
the effectiveness of the immune system and to predict the
type of failure, if any. The two criteria, s/δ̃ and tλ enable
us to predict the kinetics of the immune system. This is
a new contribution in research on HIV/AIDS disease. It is
possible to compute the theoretical "maximum" level of CD4
T-cells and the delay needed to reach any level below this
maximal one. It requires only 11 measurements. Open issues
offered now include a real time estimation of parameters for
patients starting a treatment and to feed back this information
in the decision for an optimal therapy which minimizes the
secondary effects for a maximal impact on the viral load.
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