Personalized life expectancy and treatment benefit index of antiretroviral therapy
 Yanni Xiao^{1},
 Xiaodan Sun^{1}Email author,
 Sanyi Tang^{2},
 Yicang Zhou^{1},
 Zhihang Peng^{3},
 Jianhong Wu^{4} and
 Ning Wang^{5}
DOI: 10.1186/s1297601600470
© The Author(s) 2017
Received: 24 September 2016
Accepted: 29 December 2016
Published: 18 January 2017
Abstract
Background
The progression of Human Immunodeficiency Virus (HIV) within host includes typical stages and the Antiretroviral Therapy (ART) is shown to be effective in slowing down this progression. There are great challenges in describing the entire HIV disease progression and evaluating comprehensive effects of ART on life expectancy for HIV infected individuals on ART.
Methods
We develop a novel summative treatment benefit index (TBI), based on an HIV viral dynamics model and linking the infection and viral production rates to the Weibull function. This index summarizes the integrated effect of ART on the life expectancy (LE) of a patient, and more importantly, can be reconstructed from the individual clinic data.
Results
The proposed model, faithfully mimicking the entire HIV disease progression, enables us to predict life expectancy and trace back the timing of infection. We fit the model to the longitudinal data in a cohort study in China to reconstruct the treatment benefit index, and we describe the dependence of individual life expectancy on key ART treatment specifics including the timing of ART initiation, timing of emergence of drug resistant virus variants and ART adherence.
Conclusions
We show that combining model predictions with monitored CD4 counts and viral loads can provide critical information about the disease progression, to assist the design of ART regimen for maximizing the treatment benefits.
Keywords
Viral dynamic model HIV Antiretroviral therapy Life expectancy Treatment benefit indexBackground
Human immunodeficiency virus (HIV), the pathogen causing acquired immune deficiency syndrome (AIDS), exhibits highly complex interaction with human immune system [1, 2]. HIV infection typically results in a vast virus replication during the acute infection phase that is followed by a chronic phase where the viral load approaches a much lower quasisteady state, and then followed by a sharp and sudden rise of viral loads when the immune system collapses [3–7]. Typical stages of HIV infection are well documented [6, 8], and the antiretroviral therapy (ART) is shown to be effective in slowing down the progression to AIDS and improving the life quality of HIV patients [9–11]. Most existing models however failed to describe the entire HIV disease progression trajectory partly, especially they could not model the significant increase of viral loads after the development of AIDS.
There has been substantial progress in modelling antiretroviral intervention, with particular success in predicting longterm viral dynamics [12–17]. A challenge in describing the entire HIV disease progression trajectory arises from the temporal variability of the infection rate and the viral reproduction rate [13, 16, 18]. One purpose of this study is to propose a novel viral dynamic model which can describe a typical disease progression including acute infection, chronic latency and AIDS stage on the basis of the classic viral dynamic model frame [8, 19–21]. We then show that parametrizing the infection rate and viral reproduction rate through three key parameters in the Weibull function [22, 23] permits us to extend the classical viral dynamics model in such a way that accurate description of the viral dynamics during the entire HIV disease progression within a host is possible. We also demonstrate, using a longitudinal cohort study in China, how parameters of the relevant Weibul functions can be estimated by fitting the viral dynamics model prediction to patient data, and how these parameterized Weibul functions in combination with the viral dynamics model yields important information about the comprehensive effects of ART on the life expectancy (LE).
Estimating the LE is important to inform the patients of their prognosis at the individual level, and to predict the future demographic and socioeconomic impact of HIV/AIDS at the population level. Several studies have investigated the prolonged LE of patients due to ART in highincome countries or resourceconstrained settings at the population level [24–28], using observed mortality rates in various cohort studies. There are many challenges in determining the timing of infection, predicting the LE of HIV infected individuals and quantifying the comprehensive effects of ART on life expectancy. In the study here, based on parametrized temporal variability of infection rate and viral reproduction rate through the Weibul function which are incorporated in the classical viral dynamics model, our another purpose is to establish a predictive formula at the individual level for the LE of patients receiving ART, and to simulate how this individual LE is related to ART treatment specifics such as drug efficacy, sensitivity, adherence, treatment starting time and the time of emergence of drug resistant virus variants.
Methods
Definitions of the parameters used in the model
Variables  Definitions  Initial  Reference 

T  Uninfected CD 4^{+} cell population size  1200 μ l ^{−1}  
T ^{∗}  Infected CD 4^{+} helper cell population size  0  
V _{ I }  HIV population size  100 μ l ^{−1}  
τ  Prolonged LE  interim variable  
Parameters  baseline values [ranges]  
s  Rate of supply of CD 4^{+} T cell from precursors  15 μ l ^{−1} day ^{−1}  [21] 
d  Death rate of uninfected CD 4^{+} T cells  0.02 day ^{−1}  [21] 
\(k(\bar {k})\)  Infection rate per virion  2.1818×10^{−7} μ l ^{−1} day ^{−1}(lk)  [21] 
δ  Death rate of infected CD 4^{+} T cells  0.35 day ^{−1}[0.2 0.6]  [21] 
\(\lambda (\bar {\lambda })\)  Number of free virus produced by lysing a CD 4^{+} T cell  3928.6 μ l ^{−1} day ^{−1}(l λ)  [21] 
c  Death or clearance rate of free virus  2.4 day ^{−1} [1.5 3.5]  [21] 
\(\beta _{k}(\bar {\beta }_{k})\)  Scale parameter of Weibull function  1500(225) [502000]  see text 
\(\beta _{\lambda }(\bar {\beta }_{\lambda })\)  Scale parameter of Weibull function  200(200) [10500]  see text 
\(\alpha _{k}(\bar {\alpha }_{k})\)  Shape parameter of Weibull function  1.1(1.1) [0.22]  see text 
\(\alpha _{\lambda }(\bar {\alpha }_{\lambda })\)  Shape parameter of Weibull function  0.04(0.04) [0.0050.2]  see text 
T _{ m }  LE since infection without therapy  11 [5,16] ×365 days  [34] 
\(\eta (\bar {\eta })\)  Drug efficacy of combination therapy  0.95 [0.5 1] (q η)  [21] 
\(\tau _{50}(\bar {\tau }_{50})\)  Drug sensitivity of combination therapy  30 [20, 100] ×365 days (p τ _{50})  see text 
τ _{ m }  Maximum LE after infection  (T _{ d }−T _{ s })days  – 
T _{ d }  Time of natural death  74 ×365 days  [37] 
T _{ s }  Time to initiate treatment after infection  Determined by B _{ CD4}  – 
T _{ r }  Time of emergence of drug resistant virus  Random variable in [T _{ s },T _{ e }]  – 
T _{ e }  LE after infection with treatment  In [T _{ m },T _{ d }]  – 
T _{1000}  Time to virological failure  Determined by viral loads  – 
B _{ CD4}  Baseline CD4 counts to initiate the treatment  350 [100450]cells/ μ l  – 
d _{ a }  Drug adherence rate  90% [50100%]  – 
where a parameter overbar indicates the same parameter but now associated with ART, and T _{ e } is the time when the patient under ART will die and this will be further explained below. In what follows, we will write \(\bar {k}=lk\) and \(\bar {\lambda }=l\lambda \) for a positive constant l.
Recall that \(\bar {\eta }\) and \(\bar {\tau }_{50}\) denote the reduced drug efficacy and drug sensitivity due to emergence of drug resistant variants, we have \(\tau _{50}<\bar {\tau }_{50}\) and \(\eta >\bar {\eta }\). Therefore, if we write \(\bar {\tau }_{50}=p\tau _{50}\) and \(\bar {\eta }=q\eta \), then p≥1 and q≤1. We will address the issue of DA and its impact on TBI in the following.
The smallest one of real roots t _{12} and \(\bar {t}_{12}\), lying in the interval [T _{ m },T _{ d }], is what we want to find, and denoted by T _{ e }. Then T _{ e }−T _{ m } gives the prolonged LE due to ART, indicates the integrated treatment benefits. Based on the feasibility of four roots we can provide the formula of T _{ e } and one of the possible cases is discussed in the following. Note that here T _{ d } is set to be equivalent to or greater than the T _{ e }. Due to the fact the extension of the life by ART has been increasing, we then assume that the LE of patients with ART can be as long as the average LE of individuals without infection.
This formula shows how the drug efficacy (η or \(\bar {\eta }\)), sensitivity (τ _{50} or \(\bar {\tau }_{50}\)), time of emergence of drug resistant virus (T _{ r }) affect the prolonged LE, and consequently the disease progression. In particular, \(T_{e}= (B_{1}+\sqrt {\triangle })/(2\eta)\) indicates that the drug efficacy is quite poor and the patient dies before the emergence of drugresistant virus; while \(T_{e}= (\bar {B}_{1}+ \sqrt {\bar {\triangle }})/(2\bar {\eta })\) indicates that drugresistant variants emerge during ART when the patient is alive.
Based on above definitions and analyses, we can see that the natural HIV disease progression is defined in the interval [T _{ i },T _{ m }], and ART prolongs the LE till T _{ e }. From the mathematical point of view, we can simulate the model (1) with (3–4) in the interval [T _{ i },T _{ m }) or model (1) with (5) in the interval [T _{ i },T _{ e }) to produce the whole disease progressions without or with ART.
where \(h_{i}^{1}\) and \(h_{i}^{2}\) represent the accumulative number of days before i+1 days, when drug doses are missed during treatment intervals [T _{ s },T _{ r }) and [T _{ r },T _{ e }], respectively. Let \(\tau _{i+1}^{c_{1}}=\tau _{i}^{[T_{s}, T_{r})}(i+1)\) or \(\tau _{i+1}^{c_{2}}=\tau _{i}^{[T_{r}, T_{e}]}(i+1)\), so if the dose is missed at the first treatment day (i.e. T _{ s }), then we have \(\tau _{T_{s}}^{c_{1}}=0\).
The data We consider a longitudinal cohort study that recruited 464 HIV infected individuals from Aihui, Hubei and Yunnan provinces from November 2003 and the cohort has been followed until now. The ART information and clinical/lab biomarker data were collected, including viral loads every 6 months and CD4 T cell counts every 3 months. Due to the cost, viral loads for majority of patients were not tested for each followup, and hence data on viral loads are missing. We analyzed the data anonymously.
The simulation method By using the least square method and fitting the proposed model to the data together with information on first data point and/or the date of death for the death cases we estimate some model parameters which are associated with Weibull function (shape, scale and location parameters), drug sensitivity and efficacy, ART initiation time, time of emergence of drugresistant variants and LE. Other model parameters such as the rate of supply of CD4+ T cell from precursors s, death rate of uninfected CD4+ T cells d, the baseline infection rate per virion k and etc are chosen from literature and listed in Table 1. Numerical simulations for the proposed model are carried out using Matlab 8, for the duration from infection (T _{ i }) to death (T _{ m } or T _{ e } with ART).
Results
Mimicking the entire HIV disease progression
Treatment benefit and LE
We apply our model to examine the integrated effect of ART on the LE of a patient. The maximum/optimal prolonged LE is given by the difference of the LE (T _{ e }) when the patient is on ART and the LE (T _{ m }) when the patient is without ART, that is, τ _{ m }=T _{ e }−T _{ m } (see critical time points in Fig. 1). The actual prolonged LE defined as the Treatment Benefit Index (TBI) at any given time t since the initiation of ART (T _{ s }) is given by a saturated function τ(t) (6). This index summarizes the integrated effect of ART on the LE of a patient, and more importantly as will be shown in next section, this summative index can be reconstructed from the individual clinic data. The actual LE of the patient is then determined when the viral reproduction function reaches infinite, and this can be analytically calculated by finding the smallest root of a simple algebraic equation T _{ m }+τ(t)−t=0. As the TBI tracks the prolonged LE during the ART, it is natural to observe (simulations not reported here) that early initiation of ART delays disease progression and results in long LE, and further simulations show that late emergence of drug resistant variants or strong sensitivity leads to an increase in LE.
Effect of adherence on drug efficacy

Case 1: Random pattern Suppose does missing is a random event due to uncertainty, we randomly take the days from the interval [T _{ s },T _{ e }] with a proportion of 1−d _{ a }. We run the simulations with various DA rates and again obtain that the greater DA rate the longer mean LE.

Case 2: Regular pattern Let w _{1} (w _{2}) be the numbers of days of drugon (drugoff), then DA rate yields \(d_{a}=\frac {w_{1}}{w_{2}+w_{1}}\). Therefore, we call the regular pattern as nw _{1}:nw _{2} pattern with n=1,2,3,⋯, where n depicts the frequency of onoff pattern switching with fixed adherence rate d _{ a }.
with r _{1}>1 and close to 1, r _{2}<1 and close to 1. Since no reliable information of how DA influences on the time of emergence of resistant virus variants, we simply do not consider the effects of DA on time of emergence of resistant strain here.
In the case of randomly missed dose pattern, our simulations demonstrate that TBI decreases and disease progresses faster with less DA rate. The simulated distributions of LE again illustrate a comparative advantage of increasing adherence (not shown here). For a given adherence 60% (or 90%), the mean LE is 16.6944 (or 23.2532) years with range of 271 (177) days. Similarly, lower drug efficacy and decreased sensitivity in such a scenario give a shorter mean LE.
A case study: reconstruction of TBI from the clinical data
Estimated parameter values for patients 1–7
Par.  P1  P2  P3  P4  P5  P6  P7 

β _{ λ }  117.95  107.65  152.83  136.51  109.63  109.92  116.75 
β _{ k }  1542.68  1614.66  1873.36  1270.91  1476.14  1471.95  1606.20 
α _{ λ }  0.32  0.13  0.44  0.31  0.08  0.08  0.04 
α _{ k }  0.56  0.99  0.84  1.08  1.11  1.11  1.07 
T _{ m }  4014  3946.7  4571.1  3739.2  3995  4002.1  4019.7 
η  0.78  0.71  0.60  0.78  0.89  0.78  0.96 
τ _{50}  117.89  39.34  383.85  30.23  52.64  55.92  31.76 
τ _{ m }  15915  18055  4496  7574  17382  17277  20596 
q  0.84  0.77  0.89  0.75  0.78  0.64  0.65 
p  3.72  8.13  1.23  13.96  1.89  2.19  3.14 
\(\bar {\beta }_{k}\)  73.35  1614.7  1869.7  12.70  1294.6  931.08  217.12 
\(\bar {\alpha }_{k}\)  0.56  0.99  0.84  0.80  1.10  1.11  1.07 
l  0.80  0.99  0.99  0.86  0.80  0.99  0.90 
T _{ s }  2983.6  3681  3290.4  3358.7  3719.5  3521  3839.7 
T _{ r }  3796.47  5070.56  3812.76  3929.77  5544.5  4424.67  4232.49 
T _{ e }  4279.6  5116  4592.1  4038.7  4909  4615.4  6706.3 
T _{ d }  18899.09  21736.01  7786.16  10932.50  21101.71  20798.12  24436.16 
B _{ CD4}  200  100  200  50  –  –  – 
Estimated parameter values for patients 815
Par.  P8  P9  P10  P11  P12  P13  P14  P15 

β _{ λ }  108.12  109.99  127.17  107.15  174.51  112.38  139.62  113.85 
β _{ k }  1619.24  1444.27  1170.06  1626.31  1586.35  1490.67  1660.41  1576.88 
α _{ λ }  0.03  0.13  0.22  0.04  0.13  0.06  0.05  0.08 
α _{ k }  1.18  1.10  0.93  1.09  1.23  1.09  1.17  1.18 
T _{ m }  3128.4  4026.7  3701.9  3118.1  3254.5  3952.6  4300.6  3220.4 
η  0.99  0.87  0.95  0.94  0.96  0.93  0.93  0.95 
τ _{50}  30  30.09  30.71  53.27  30.22  55.79  39.33  65.58 
τ _{ m }  21897  18753  21340  20003  21530  17541  21454  20267 
q  0.55  0.65  0.88  0.58  0.90  0.76  0.53  0.57 
p  4.65  4.98  3.25  2.61  3.30  2.11  3.04  2.18 
\(\bar {\beta }_{k}\)  746.11  619.94  961.22  189.82  1164.1  1321.1  814  772.8 
\(\bar {\alpha }_{k}\)  1.18  1.09  0.87  1.09  1.13  0.93  0.66  1.11 
l  0.92  0.86  0.84  0.92  0.83  0.93  0.81  0.93 
T _{ s }  2441.4  3498.6  3445.7  2472.7  2599.4  3002  3215.7  2569 
T _{ r }  4526.31  5048.56  4863.17  2872.21  3906.73  4790.24  4418.25  4498.8 
T _{ e }  7268.8  6339.7  6938.3  5054.7  6722.5  5473  6294.4  4708.4 
T _{ d }  24338.38  22251.46  24785.25  22476.19  24129.89  20542.58  24669.66  22835.56 
B _{ CD4}  200  200  150  –  200  300  300  200 
Discussions
It is known that HIV infection typically results in a vast replication of virus during the acute phase. The viral load then becomes much lower and approaches a quasisteady state, and finally increases significantly after the development of AIDS [6, 35]. The viral loads change overtime and behave as the ‘bathtub curve’, which shows three stages over the life time and hence is very well depicted by the proposed mixed Weibull function [23] for the temporal variability of infection rate and viral reporduction rate. Despite intensive and promising progress in HIV/AIDS viral dynamics modeling, it remains a challenge to provide approximation of the entire HIV disease progression dynamics. Here, by linking the viral reproduction rate and infection rate to the Weibull function with biologically interpretable shape, location and scale parameters, we showed that the viral dynamics model can describe a typical disease progression including acute infection, chronic latency and AIDS stage. In particular, when life expectancy is assumed to be infinity, the threeparameter Weibull function becomes unity and our proposed model reduces to the classic model of HIV dynamics [8, 19, 21].
We have also shown that our model can be used to predict the LE of an HIV infected individual and the time of virological failure. The accurate description of the entire HIV disease progression makes it possible to use this model to predict the transmission probability at different stages based on viral loads, and this is important when we consider new infections generated by a particular infected individual. Our model can also be used to determine the timing of infection for an infected individual based on individual parameters, monitored data on CD4 cell counts and viral loads, which is difficult to get. This estimation of the infection time for each infected individuals provides vital information on estimating new infections at the population level. In addition, the knowledge about the timing of infection for HIVinfected individuals in various communities enables effective contact tracing and facilitates treatment resource allocation.
Simulating the proposed model shows early initiation of ART can result in long LE (great T _{ e }) and prolonged LE (T _{ e }−T _{ m }), in agreement with those in previous studies [26]. Since the waiting time for the emergence of resistant genomes is substantial [36] and is incorporated in our introduced TBI, we developed a continuous (rather than an impulsive model) model of HIV dynamics with switching to describe differences of drug efficacy and sensitivity after emergence of drug resistant virus variants. Our results show that later emergence of drug resistant virus variants leads to longer (prolonged) LE and more persistent viral suppression. The estimated piecewise TBIs are increasing functions with treatment duration, with a great/low slop before/after the emergence of drug resistant virus variants. Therefore, we could estimate the time of emergence of drugresistant variants for an infected individual, which may provide information on the time for switching to the secondline regimen without resistance testing. It is known that individualized therapy is hampered by limited availability of viral load and resistance testing, making it difficult to determine whether the remaining antiviral potency of previously used drugs outweighs their toxicity [27]. Hence, our estimation makes individualized therapy more feasible and costeffective.
Conclusions
The proposed novel modeling approach led us naturally to the introduction of the treatment benefit index (TBI) to summarize the integrated effect of ART in terms of prolonged LE. Moreover, this TBI can be reconstructed from clinical data with predicting the time of virological failure. Our model can be used to determine the timing of infection for an infected individual based on individual parameters, monitored data on CD4 cell counts and viral loads. Main results show that combining model predictions with monitored CD4 counts and viral loads can provide critical information about the disease progression, to assist the design of ART regimen for maximizing the treatment benefits.
Abbreviations
 AIDS:

Acquired immune deficiency syndrome
 ART:

Antiretroviral therapy
 DA:

Drug adherence LE: Life expectancy
 HIV:

Human immunodeficiency virus
 TBI:

Treatment benefit index
Declarations
Acknowledgements
Not applicable.
Funding
The authors are supported supported by the National Megaproject of Science Research No. 2012ZX10001001, by the National Natural Science Foundation of China (NSFC,11571273 (YX), 11471201(ST)), and by the Fundamental Research Funds for the Central Universities (08143042 (YX)), by the Jiangsu Province Science & Technology Project of Clinical Medicine (BL2014081 (ZP)), by Jiangsu Province Science & Technology Demonstration Project for Emerging Infectious Diseases Control and Prevention (BE2015714 (ZP)), by the International Development Research Center, Ottawa, Canada(104519010), and by the Natural Sciences and Engineering Research Council as well as the Canada Research Chairs Program (JW).
Availability of data and materials
Please contact author for data requests.
Authors’ contributions
NW supervised the project. YX, XS, ST, YZ, ZP, JW and NW conceived and designed the study. ZP collected the experiment data. ST and XS conducted all data analyses and simulations. YX, ST, JW and NW wrote the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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