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A stochastic Markov branching process framework to evaluate HIV eradication
Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems Available online at www.sciencedirect.com Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Proceedings, 2nd IFAC Conference on Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018
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IFAC PapersOnLine 51-13 (2018) 85–90
A stochastic Markov branching process A stochastic Markov branching process A stochastic Markov branching process framework to evaluate HIV eradication A stochastic Markov branching process HIV eradication framework to evaluate framework to evaluate HIV eradication ∗ ∗∗ framework to evaluate HIV eradication Alessandro Boianelli Esteban A. Hernandez–Vargas ∗ ∗∗ ∗ ∗∗
Alessandro Boianelli ∗ Esteban A. Hernandez–Vargas ∗∗ Alessandro Boianelli Esteban A. Hernandez–Vargas ∗Alessandro Boianelli ∗ Esteban A. Hernandez–Vargas ∗∗ Department of Drug Discovery Sciences, Boehringer Ingelheim ∗ ∗ of Drug Discovery Sciences, Boehringer Ingelheim ∗ Department Pharma GmbH Co KG, 88397, Biberach, Germany. Department of Drug& Sciences, Boehringer Ingelheim Pharma GmbH & Discovery Co KG, 88397, Biberach, Germany. ∗∗ ∗ Frankurt Institute for Advanced Studies (FIAS), Department of Drug Sciences, Boehringer Ingelheim Pharma GmbH & Discovery Co KG, 88397, Biberach, Germany. ∗∗ ∗∗ for Advanced Studies (FIAS), ∗∗ Frankurt Ruth-Moufang-Strasse Frankfurt am Main, Germany. Pharma GmbHInstitute & 1, Co60438 KG, 88397, Biberach, Germany. Frankurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. ∗∗
Frankurt Institute for Advanced (FIAS), 1, 60438 FrankfurtStudies am Main, Germany. Ruth-Moufang-Strasse Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, HIV can persist within a small pool of long-lived resting memory CD4+ T cells. In concert Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, HIV can persist within a small pool of long-lived resting memory CD4+ T cells. In concert with ART, different latency reversal agents (LRAs) are under development tocells. activate latent Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, HIV can persist within a small pool of long-lived resting memory CD4+ T In concert with ART, different latency reversal agents (LRAs) are under development to activate latent reservoirs thus reducing viral persistence. Based on aresting stochastic Markov branching process with HIV can persist within a small pool agents of long-lived memory CD4+ Ttocells. In concert with ART, different latency reversal (LRAs) are under development activate reservoirs thus reducing viral persistence. Based on a stochastic Markov branching processlatent with HIV reservoirs dynamics in isolation, simulations suggest folds increase the activation with ART,thus different latency reversal agents (LRAs) are that under10Markov development toinactivate latent reservoirs reducing viral persistence. Based on a stochastic branching process with HIV reservoirs dynamics in isolation, simulations suggest that 10 folds increase in the activation rate from latently to actively infectedsimulations could reduce extinction time for all in reservoirs to 50 reservoirs thus reducing viral persistence. Based on athe stochastic Markov branching process with HIV reservoirs dynamics in isolation, suggest that 10 folds increase the activation rate from latently to actively infected could reduce the extinction time for all reservoirs to 50 months. However, when viral dynamics and reduce newsuggest infection are incorporated a to more HIV reservoirs dynamics in isolation, simulations thatcycles 10 folds increase in the in activation rate from latently to actively infected could the extinction time for all reservoirs 50 months. However, when viral dynamics and new infection cycles are incorporated in a more realistic simulations point outinfection that LRAs would least 30to fold rate fromscenario, latently stochastic to actively infected could reduce the extinction timerequire for all at reservoirs 50 months. However, when viral dynamics and new cycles are incorporated in a more realistic scenario, stochastic simulations point out that LRAs would require at least 30 fold increase in the activation ratesimulations to induce concomitant eradication of are the require differentatsub-reservoirs. months. However, when viral dynamics and new infection cycles incorporated in a30more realistic scenario, stochastic point out that LRAs would least fold increase in the activation rate to induce concomitant eradication of the different sub-reservoirs. realistic scenario, stochastic point out that LRAs would least 30 fold increase in the activation ratesimulations to induce concomitant eradication of the require differentatsub-reservoirs. © 2018, IFAC Federation of Automatic Control) Hosting byofcure, Elsevier Ltd. Allsub-reservoirs. rights reserved. increase in Stochastic the(International activation rate to Markov induce concomitant eradication the different Keywords: Modeling, Branching Process, HIV Reservoirs Keywords: Stochastic Modeling, Markov Branching Process, HIV cure, Reservoirs Keywords: Stochastic Modeling, Markov Branching Process, HIV cure, Reservoirs Keywords: Stochastic Modeling, Markov Branching Process, HIV cure, Reservoirs 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Although antiviral1.therapy potently suppresses the virus, Although antiviral therapy potently suppresses the virus, a functional or eventherapy sterile HIV cure suppresses remains a challenge. Although antiviral potently the virus, a functional or sterile HIV cure remains a HIVantiviral infection, a subset of latently cells Although potently suppresses the virus, aUpon functional or even eventherapy sterile HIV cure remainsinfected a challenge. challenge. Upon HIV infection, a of latently infected cells carrying inactive integrated proviral DNA (HIV a functional or even sterile HIV cure remains a reservoirs) challenge. Upon HIV infection, a subset subset of latently infected cells carrying inactive integrated proviral DNA reservoirs) is rapidly established and (2012)]. DifUpon HIV infection, a[Eisele subset of Siliciano latently infected cells carrying integrated proviral DNA (HIV (HIV reservoirs) is rapidlyinactive established [Eisele and Siliciano (2012)]. Different studies have suggested thatSiliciano macrophages [Groot carrying inactive integrated proviral DNA (HIV reservoirs) is rapidly established [Eisele and (2012)]. Different studies have suggested that macrophages [Groot et al. (2008)] and cells [Perreau etDifal. is rapidly established [Eiselehelper and Siliciano (2012)]. ferent studies havefollicular suggested thatT macrophages [Groot et al. (2008)] and follicular helper T cells [Perreau et al. (2013)] are good candidates forthat supporting residual[Groot virus ferent studies have suggested macrophages et al. (2008)] and follicular helper T cells [Perreau et al. (2013)] are good candidates for supporting residual virus replication, but it is not clear whether these[Perreau infectedet cells et al. (2008)] and follicular helper T cells al. (2013)] are good supporting replication, but itcandidates is not clearfor whether theseresidual infectedvirus cells can survive for it several in supporting patients onresidual optimalvirus an(2013)] are good candidates replication, but is not years clearfor whether these infected cells can survive for several years in on optimal antiretroviral [Eisele Siliciano (2012)]. replication, but is not clearand whether these infected can survivetreatment for it several years in patients patients on optimalcells antiretroviral treatment [Eisele and Siliciano (2012)]. can survivetreatment for several[Eisele years and in patients on optimal antiretroviral Siliciano (2012)]. Latently infected resting memory CD4+ T cells represent tiretroviral treatment [Eisele and Siliciano (2012)]. Latently infected resting CD4+ T represent the only cell type in whichmemory it has been clearly demonstrated Latently infected resting memory CD4+ T cells cells represent the only cell type in which it has been clearly demonstrated that replication-competent virus canclearly persist for several Latently infected resting CD4+ T cells represent the cell type in whichmemory it has been demonstrated thatonly replication-competent virus can persist for several years in cell patients al.been (2014); Chomont et al. the only type in[Buzon which itethas clearly demonstrated that replication-competent virus can persist for several years in patients [Buzon et al. (2014); Chomont et al. (2009); et al.[Buzon (2015);etDeeks et al.persist (2012); that replication-competent virus can forJaafoura several years inCohn patients al. (2014); Chomont et al. Fig. 1. Multi-type Markov branching process framework. (2009); et al. (2015); et al. Jaafoura 1. Multi-type Markov branching process framework. et al. (2014)]. In direction, recent studies years inCohn patients etDeeks al. (2014); Chomont et al. Fig. (2009); Cohn et this al.[Buzon (2015); Deeks et experimental al. (2012); (2012); Jaafoura mathematical model describes the dynamics of Fig. The 1. Multi-type Markov branching process framework. et al. (2014)]. In this direction, recent experimental studies mathematical model describes the dynamics of have aimed latency reversal can Fig. The (2009); CohntoIn etdevelop al. (2015); Deeks et experimental al. agents (2012);which Jaafoura et al. (2014)]. this direction, recent studies HIV (V ); productively infected cells (I);dynamics stem (LSof ), 1. Multi-type Markov branching process framework. The mathematical model describes the have aimed to develop latency reversal agents which can HIV (V ); productively infected cells (I); stem (L ), S purge HIV reservoirs thus contributing to the which viral supet al. aimed (2014)]. this direction, recent experimental studies have toIndevelop latency reversal agents can central (Lproductively effector (LE )cells memory CD4+ T The mathematical describes the HIV (V ); infected (I);dynamics stem (LSSof ), C ), and model purge HIV reservoirs thus contributing to the viral supC E central (L ), and effector (L ) memory CD4+ T pression or viral eradication [Archin et al. (2012); Archin have aimed to develop latency reversal agents which can C ), and E )cells purge HIV reservoirs thus contributing to the viral supcells. (V LS); , L at rate , γC(Land HIV productively infected (I); γstem ), central (L (L memory C and Leffector E proliferate SCD4+ ST C E pression or viral eradication [Archin et al. (2012); Archin cells. LSS , LC C and LE E proliferate at rate γS S , γC C and and Margolis (2014); Laird et[Archin al. (2015)]. Although differpurge HIV reservoirs thus contributing to the viral suppression or viral eradication et al. (2012); Archin γ respectively. Furthermore, the three subsets are central (L ), and effector (L ) memory CD4+ T cells. L , L and L proliferate at rate γ , γ and E C E S C E S C are and Margolis (2014); Laird et al. Although differγE Furthermore, the three subsets ent LRA candidates are available, their efficacy in clinical pression or viral eradication et al. (2012); Archin E respectively. and Margolis (2014); Laird et[Archin al. (2015)]. (2015)]. Although differassumed d and the become cells. LS , to LCdie andat Lrate atthree rateproductively γsubsets γ Furthermore, are E proliferate S , γC and E respectively. ent LRA candidates are available, their efficacy in clinical assumed to die at rate d and become productively trials is still fragmented. and LRA Margolis (2014);are Laird et al. (2015)]. Although different candidates available, their efficacy in clinical infected cells (I) at the activation rate a. Productively γ Furthermore, the three subsets are assumed to die rate d and become productively E respectively. trials is still fragmented. infected cells (I) at the activation rate a. Productively ent LRA candidates are available, their efficacy in clinical trials is still fragmented. cells canat either new a. virus particles assumedcells to die rateactivation drelease and become productively infected the Productively Mathematical modeling has given an important contribuinfected cells(I) can either release rate new virus particles trials is still fragmented. Mathematical modeling has given an important contribu(V ) at rate p or die at rate δ. New particles infected cells (I) at the activation rate a. Productively cells can either release new virus tion to understanding treatment e.g. [Conway and Mathematical modelingHIV has given an important contribu(V ) at rate p or die at rate δ. New virus particles tion to understanding HIV treatment e.g. [Conway and can) promote infection of latent infected cellspnew can either release new virus reservoir (V at rate or cycles die atofrate δ. New particles Coombs (2011); FerreiraHIV et al. (2011); Hernandez-Vargas Mathematical modeling has given an important contribution to understanding treatment e.g. [Conway and can cycles infection of latent Coombs (2011); Ferreira et (2011); Hernandez-Vargas via the target cells at the rate fvirus (1 − reservoir )kT or (V ) promote at rate pnew or die(Tat)of rate δ. New particles can promote new cycles of infection of latent reservoir et al. to (2014, 2013); Hill et al. (2014); Ke et al. [Conway (2015); Laird tion understanding HIV treatment e.g. and Coombs (2011); Ferreira et al. al. (2011); Hernandez-Vargas via the target cells (T ) at the rate f (1 − )kT or et al. (2014, 2013); Hill et al. (2014); Ke et al. (2015); Laird alternatively productively infected cells (I), at rate can promote new cycles of infection of latent reservoir via the target cells (T ) at the rate f (1 − or al.(2014, (2015); Perelson (1997); al. Coombs (2011); Ferreira etal. al. (2011); Hernandez-Vargas et 2013); Hill etet (2014); Ke Rosenbloom et al. (2015); et Laird alternatively productively infected cells (I), )kT at rate et al. al. (2015); Perelson etal.al. (1997); Rosenbloom et al. (1 − f )(1 − )kT . Virus is cleared at rate c. via the target cells (T ) at the rate f (1 − )kT or alternatively productively infected cells (I), at rate (2012)]. In the context the LRAs, first(2015); attempt to al. 2013); Hill etof (2014); Kethe et al. Laird et al.(2014, (2015); Perelson etal. al. (1997); Rosenbloom et al. (1 − f )(1 − )kT . Virus is cleared at rate c. (2012)]. In the context of the LRAs, the first attempt to alternatively productively infected cells (I), at rate (1 − f )(1 − )kT . Virus is cleared at rate c. quantify the clinical goal for HIV eradication was done by et al. (2015); Perelson et al. (1997); Rosenbloom et al. (2012)]. In the context of the LRAs, the first attempt to found out that a 2000 fold reduction of the latent reservoir quantify the clinical goal for HIV was done by (1out − fthat )(1 −a )kT . fold Virus is cleared at rate c. reservoir the computational work Hill eteradication al. (2014). authors (2012)]. In context the LRAs, the firstThe attempt to found 2000 of the latent quantify thethe clinical goalofof for HIV was done by decreases the probability ofreduction viral rebound in 1 year and a the computational work of Hill eteradication al. (2014). The authors found out that a 2000 fold reduction of the latent reservoir the probability of viral rebound in 1 year and a quantify the clinicalwork goal of forHill HIV eradication was done by decreases the computational etAlfons al. (2014). The authors This 10000 fold could eradicate HIV. In another modeling work,a found out that a 2000 fold reduction of the latent reservoir decreases the probability of viral rebound in 1 year and work was supported by the und Gertrud Kassel This 10000 fold could eradicate HIV. In another modeling work, the computational work of Hill et al. (2014). The authors work was supported by the Alfons und Gertrud Kassel Ke et al. (2015) elucidated the effect of the LRA vorinostat Stiftung. Corresponding author: [email protected] decreases the probability of viral rebound in 1 year and 10000 fold could eradicate HIV. In another modeling work,a This work was supported by the Alfons und Gertrud KasselKe et al. (2015) elucidated the effect of the LRA vorinostat Stiftung. Corresponding author: [email protected] 10000 fold couldelucidated eradicate HIV. In another modeling work, Ke et al. (2015) the effect of the LRA vorinostat Stiftung. Corresponding author: This work was supported [email protected] the Alfons und Gertrud KasselKe et al. (2015) elucidated the effect of the LRA vorinostat Stiftung. Corresponding author: [email protected] 2405-8963 © 2018, IFACConference (International Proceedings, 2nd IFAC onFederation of Automatic Control) 85 Hosting by Elsevier Ltd. All rights reserved. Proceedings, 2nd IFAC Conference on 85 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference on 85 Modelling, Identification and Control of Nonlinear 10.1016/j.ifacol.2018.07.259 Systems Modelling, Identification and Controlon of Nonlinear Proceedings, 2nd IFAC Conference 85 Systems Guadalajara, Mexico, June 20-22, 2018 Systems Identification Modelling, and Control of Nonlinear
2018 IFAC MICNON 86 Guadalajara, Mexico, June 20-22, 2018 Alessandro Boianelli et al. / IFAC PapersOnLine 51-13 (2018) 85–90
on the latent reservoir dynamics which results in a transient activation of the whole latent subset. However, the main limitation of these works is that the latent reservoirs are considered as a unique cell pool with homogeneous dynamics. On the contrary, latest results from Jaafoura et al. (2014) revealed a progressive reduction of latent reservoirs around a core of less-differentiated memory subsets, that is central memory T cells (TCM ), effector memory T cells (TEM ), and stem central memory T cells (TSCM ). This process appears to be driven by the differences in initial sizes and decay rates between latently infected memory subsets. A complete review in modeling HIV cure can be found in Hernandez-Vargas (2017).
Note that the multi-type Markov branching process has the following fundamental properties: (H1) each cell action is independent from others; (H2) each cell offspring generates its own branching process; and (H3) at each time point t, the cell has no memory about previous time steps. In this fashion, Li (t) is a random variable vector representing the number of TSCM , TCM and TEM cells respectively at the time point t. In the Markov branching process formulation, each reservoir has a life time exponentially distributed with a parameter λi = γi + d + a. At the end of its life, a singular type cell in each subset independently from the others can produce its progeny according to the probability generating function (pgf) defined as:
The dynamics of viral populations are usually governed by deterministic phenomena when large numbers are observed. When a very small number of virus particles and infected cells are present, the stochastic fate could be the main driver of the clearance timing and eradication [Hawkins et al. (2007)]. In HIV patients under long ARTs, the stochastic variability could lead ultimately to post– treatment control or to the viral eradication. These outcomes could arise from the processes of proliferation, apoptosis, differentiation, and activation of the different stem, central and effector subsets. Understanding the stochastic fate decisions and dynamic events are relevant to develop optimal purging strategies. A suitable stochastic framework is the Markov branching process, which has been applied to predict immune response dynamics e.g., CD8+ T cell clonal expansion [Buchholz et al. (2013)], HIV dynamics [Conway and Coombs (2011); Hill et al. (2014); Laird et al. (2015)] and CD4+ T cell responses to vaccination [Boianelli et al. (2015); Pettini et al. (2013)].
fi (xi ) =
∞
pik xki ,
(1)
k=0
where pik is the probability of generating k offspring cells, while xi is an auxiliary variable. The probability pik is computed as the ratio between the rate of having k offspring cells to the sum of the rates of the all possible events. This stochastic model named birth, death and activation process leads to the simplification of (1) in the following form: fi (xi ) = pi0 + pi2 x2i = (d + a + γi x2i )λ−1 i , pi0
pi2
(2) pi0
= (d + a)/λi and = γi /λi , with + where pi2 = 1. For the complete description of the birth and death/activation process, we define the global probability generating function for one latent reservoir cell (xi ): Fi (xi , t) =
This paper proposes a Markov branching process model describing the dynamics of latent reservoirs (Fig.1) using different settings of the ART-LRA treatment combination. The main difference respect to previous works in HIV modeling research, this paper provides for the first time a qualitative dynamic behaviour of the virus under the combination of ART and LRA treatment considering different reservoirs dynamics. Although the results are from a computational study, they provide preliminary insights that need to be assessed with in vivo studies in view of improving therapeutic treatments for an HIV cure.
∞
P (Li (t) = k|Li (0) = 1)xki .
(3)
k=0
With P (Li (t) = k|Li (0) = 1), we denote the conditional probability of having Li (t) = k cells at the time t starting from Li (0) = 1. The temporal evolution of Fi (xi , t) is governed by the backward Kolmogorov equation in its general form: ∂Fi (xi , t) = ui (Fi (xi , t)), ∂t
(4)
where: ui (xi ) = λi (Fi (xi , t) − xi ), with the initial condition Fi (xi , 0) = xi . This initial condition means that the process is initiated from one single stem, central and effector CD4+ T cell. The temporal description of the probability generating functions Fi (xi , t) is provided by the backward Kolmogorov equation which presents the form as follows:
2. HIV RESERVOIRS DYNAMICS IN ISOLATION In this section, we distinguish three key reservoirs such as the stem cell (TSCM ), the central memory (TCM ) and the effector memory (TEM ). We assume that the three HIV reservoir subsets undergo only proliferation, death and activation. To understand the reservoir dynamics in isolation, in this section, we did not include viral dynamics and consequently new cycles of infection, see red circle in Fig.1. This can represent the case when patients are under long ARTs. The three sub-reservoirs (TSCM , TCM and TEM ) are abbreviated in the form of Ti with i = 3. Reservoirs proliferate at the rate γi . The reservoirs die at the rate di . Each subset can be activated at the rate ai respectively. For the sake of the analysis, we assume that the three reservoirs have the same activation rate ai = a as well as the same death rate di = d.
∂Fi = γi (Fi (xi , t)2 − Fi (xi , t)) + (d + a)(1 − Fi (xi , t)), ∂t This differential equation system can be solved analytically with the method of characteristics, providing a unique solution: Fi (xi , t) = 86
(d + a)(xi − 1) − e(d+a−γi )t (γi xi − d + a) . γi (xi − 1) − e(d+a−γi )t (γi xi − d + a)
2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Alessandro Boianelli et al. / IFAC PapersOnLine 51-13 (2018) 85–90
144 months
*
TEM half life
88 months
a Activation rate 0.001 day−1 d Reservoir death rate 0.004 day−1 LS (0) TSCM initial size [1 1000] LC (0) TCM initial size [1 1000] LE (0) TEM initial size [1 1000] * Jaafoura et al. (2014) ** Conway and Perelson (2015)
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Fig. 2. Extinction probabilities. Pi (t) surfaces for the different initial reservoir values (Li (0)) were considered with activation rate values of (A) a = 0.001 d−1 , (B) a = 0.002 d−1 , (C) a = 0.005 d−1 , (D) a = 0.01 d−1 . The colorbar on the right side indicates the value of extinction probabilities. only after very long time periods (see Fig. 2A). More specifically, the TSCM reservoir has shorter probability of eradication regions than TCM and TEM cells. This can be observed with negligible extinction probability PS (t) before 1000 months. This could imply that although the values for LS (0) are lower compared to the other subsets, the TSCM has a great stability and can significantly contribute to the viral rebound when ART treatment is interrupted. Conversely for the TEM and TCM subsets, the extinction probabilities of PE (t) and PC (t) reach high values for larger regions of initial conditions (LE (0) and LC (0)).
Table 1. Model parameter values. Reference *
600
200
To evaluate the effect of model parameters, we compute the median extinction time for different initial conditions of latent reservoirs subsets [Conway and Coombs (2011)]. In the first case, we assume the normal activation rate value a = 0.001 day−1 and then two, five and ten folds increase of the normal activation rate. We perform the numerical simulations of the extinction probabilities Pi (t) for different initial size (Li (0)) in a time horizon of 1000 months, approximately 83 years. The death rates are assumed equal among the different reservoirs. The half-life of each reservoir class as well as the experimental ranges of initial latent reservoir size for the different subsets were taken from previous estimates in patients on prolonged ART [Jaafoura et al. (2014)]. For the nominal value of the activation rate (a), we assume the estimate previously considered in Conway and Perelson (2015). A summary of model parameters used in our simulations is presented in Table 1. Note that the reservoir proliferation rates γi can i be calculated by γi = a + d − ln(2)/T1/2 . Value (Units) 277 months
600
LE(0)
S
L (0)
B
2.1 Simulation setting
Biological meaning TSCM half life
600
0
Li (0) (d + a) − (d + a)e(d+a−γi )t .(5) Pi (t) = Fi (0, t) = γi − (d + a)e(d+a−γi )t The extinction probability for each reservoir subset is a function of the rate parameters, initial conditions Li (0) and time point t.
Parameter TS 1/2
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. Fi (xi , t) = Fi (xi , t) Assuming the initial condition Li (0), the probabilities of extinction, i.e. the probabilities of having no Ti offspring at time point t, are defined as follows:
STEM RESERVOIRS
1000
LC(0)
A
LC(0)
Considering the independence property (H1) of the branching process [Harris (2002)], we can obtain:
87
* ** ** * * *
Assuming 2 fold increase activation rate (a = 0.002 d−1 ), a clear increase of the extinction probabilities of the three subsets is observed in Fig. 2B. In particular, the activation rate (a) has a drastic effect on the extinction probability of the TSCM cells, resulting in high value for every initial condition. This qualitative behaviour is also conserved for the extinction probabilities in the other reservoirs TCM and TEM . Intuitively, the progressive extinction of the three reservoir compartments is even more pronounced with a 5 folds increase of the activation rate (Fig. 2C), where an effective eradication of the TSCM , TCM and TEM reservoirs can be achieved within 250 months (approximately 21 years) for any initial size of the reservoir subsets. A possible eradication of the latent reservoirs could be achieved increasing 10 folds the activation rate a, whereas the extinction probabilities, Pi (t) = 1, can be achieved within 50 months (Fig. 2D).
2.2 Simulation Results We test initially the scenario where the reservoirs subsets present the normal activation rate value a = 0.001 d−1 . The extinction probabilities are computed for the reservoirs initial values reported in Table 1. Consequently, we consider for the same initial conditions, different scenarios for increasing values of the activation rate (a) promoted by LRAs were 0.002, 0.005, and 0.01 d−1 . Fig. 2 shows the extinction probabilities surfaces for all these different scenarios. For the normal activation rate (a = 0.001 d−1 ), the latent reservoirs would present a high probability of eradication 87
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3. MODELING HIV RESERVOIRS WITH VIRAL DYNAMICS AND NEW CYCLES OF INFECTION
uniform distribution spanning from 1 to 49 copies/mL. This is a more realistic assumption as the viral load within a host on prolonged ART is undetectable. For the initial condition of productively infected cells, I(0) can be set to the quasi-steady state approximation I(0) = V (0)c/p, because the productively infected cells and virus present faster dynamics than latent reservoirs. Parameters and results were re-scaled over the entire blood volume, approximately 5 liters.
In the previous sections as well as in the most of the stochastic models of HIV, the viral dynamics and the infection process are omitted. In order to evaluate qualitatively the latent reservoirs progression and their influence on the viral dynamics we built a comprehensive stochastic Markov branching process [Harris (2002); Karlin and Taylor (1981)] describing HIV infection mechanisms, see Fig.1. Our framework is composed by the uninfected CD4+ T cells T , the three different latent reservoir subsets (Li ), the productively infected cells (I) and the virus (V ). The uninfected cells are replenished at a rate ST and die with a rate dT per cell. The newly infected cells can become latent reservoirs at a rate f (1-)k, where k is the infection rate, f is the fraction of the newly infected cells that become latently infected cells and is the ART efficacy. Alternatively the uninfected cells can transit in the productively infected cells state I with a rate (1f )(1-)k. Productively infected cells can either release HIV virions at a rate p or die with a rate δ due to cytopathic effects. Virus particles (V ) are cleared with a rate c. 3.1 Simulation setting We assume that the in silico treatment protocol presents 4.5 years on ART, followed by 6 months on a combination of ART and LRA. After this 6 months, both LRA and ART are interrupted and the stochastic model dynamics are recorded for 5 years, scheme presented in Fig. 3. Parameters values used for simulations are presented in Table 2. Table 2. Model parameter values.
Fig. 3. Combined ART+LRA simulation scenario. The first 4 years and half are considered on ART treatment (green panel) followed by a combination on ART and LRA treatment for 6 months (blue planel). Consequently both treatments are interrupted (TI) in 5 years period to evaluate the possibility of viral rebound (red line) or suppression (green line).
Parameter Biological meaning Value (Units) Reference ST Cell production T (0)dT d−1 dT Cell death 0.01 d−1 * T (0) Cell initial size 500 µL−1 ** k Infection rate 5×10−8 mLd−1 ** f Latent cell fraction 10−6 ** Treatment efficacy 1 p Viral replication 1000 d−1 ** c Viral clearance 23 d−1 ** LS (0) Initial TSCM 560 * LC (0) Initial TCM 2600 * LE (0) Initial TEM 800 * I(0) Initial infected cells V (0)c p−1 * Jaafoura et al. (2014) ** Conway and Perelson (2015)
As the proposed model in this section presents a high reaction number, all the stochastic simulations are performed with the τ leaping method [Cao et al. (2006)], which results particularly faster than the stochastic simulation algorithm despite the accuracy in the chemical master equation solution. A crucial factor in the accuracy determination is the τ parameter defined as the time interval where one or more reactions can happen allowing the stochastic system to move in the next state. The value τ = 0.002 represents a good trade-off for the approximation of the Markov branching process solution.
Different scenarios with variations in the activation rate (a) were explored. Note that the initial values of latent reservoirs in Table 2 are calculated considering the total number of latent reservoir equal to 1 per 106 CD4+ T cells per mL as in other computational works e.g. Hill et al. (2014). The latent reservoir size provided by the measurement assay is prone to errors and present high variability among different in vivo studies [Bruner et al. (2015)]. Instead of considering the latent reservoir ranges from these studies, we assume valid the estimation of 20 per 106 CD4+ T cells found in Crooks et al. (2015). Furthermore, instead of assuming constant the initial viral load V (0), this quantity is sampled randomly from a
We perform 20 stochastic simulations with the initial size of the three latent reservoir subsets [Jaafoura et al. (2014)]. When the LRAs are introduced assuming a 10 fold activation rate for 6 months before treatment interruption (TI), we observe the same qualitative behaviour for the viral load (Fig. 4B) and productively infected cells I. In this scenario, although the post-LRA treatment reduction of LS , LC , and LE was 92%, 95% and 93% respectively (Table 3), the three subsets still persist. Among them, LS is the only subsets that increase after ART-LRA treatment, see Table 3. In the case of 20 fold activation (Fig. 4C), a large variability appears in the viral rebound time as well as in the productively infected cells.
3.2 Stochastic Simulations
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89
Table 3. Median values of 20 stochastic simulations. Activation rate (a) Fold increase 10 20 30
TI 102 2 0
ART 528 534 521
LE ART+LRA 35 2 0
TI 33 3 0
Coupling viral dynamics and new cycles of infections into a more realistic scenario, we evaluated the HIV reservoirs whit a combination of 5 years ART and 6 months LRA treatment. Results pointed out that LRAs of 30 fold activation is required in order to avoid viral rebound in all the stochastic simulations after 5 years of ART interruption. The necessary condition for the viral suppression may be obtained by eradicating strategically the LS , LC and LE subsets. In fact, this framework shows that even after a significant reduction of the latent reservoirs (99%), these sub-reservoirs could promote either a viral rebound immediately or 4 years after ART-LRA interruption. Additionally, the post-LRA treatment level of the latent reservoir reveals that the TSCM cell pool could be the main obstacle for HIV eradication. Further incorporation of control strategies could serve as a critical tool to refine purging strategies and facilitate HIV eradication.
LC ART+LRA 125 5 0
rate, the extinction probabilities of all reservoir subsets are observed within 50 months (approximately 4 years).
ART 1990 2044 2065
TI 35 5 0
LS ART+LRA 29 1 0
ART 470 441 412
Fig. 4. HIV dynamics during the ART and LRA treatment. 20 stochastic simulations are performed with (A) normal activation, (B) 10 fold, (C) 20 fold, and (D) 30 fold activation (parameter a) of the latent reservoirs.
REFERENCES Archin, N.M., Liberty, a.L., Kashuba, a.D., Choudhary, S.K., Kuruc, J.D., Crooks, a.M., Parker, D.C., Anderson, E.M., Kearney, M.F., Strain, M.C., Richman, D.D., Hudgens, M.G., Bosch, R.J., Coffin, J.M., Eron, J.J., Hazuda, D.J., and Margolis, D.M. (2012). Administration of vorinostat disrupts HIV-1 latency in patients on antiretroviral therapy. Nature, 487(7408), 482–485. Archin, N. and Margolis, D. (2014). Emerging strategies to deplete the HIV reservoir. Current opinion in infectious diseases, 27(1), 29–35. Boianelli, A., Pettini, E., Prota, G., Medaglini, D., and Vicino, A. (2015). A stochastic model for CD4+ T cell proliferation and dissemination network in primary immune response. PloS one, 10(8), e0135787. Bruner, K.M., Hosmane, N.N., and Siliciano, R.F. (2015). Towards an HIV-1 cure: measuring the latent reservoir. Trends in microbiology, 23(4), 192–203. Buchholz, V.R., Flossdorf, M., Hensel, I., Kretschmer, L., Weissbrich, B., Gr¨af, P., Verschoor, A., Schiemann, M., H¨ofer, T., and Busch, D.H. (2013). Disparate individual fates compose robust CD8+ T cell immunity. Science, 340(6132), 630–635. Buzon, M.J., Sun, H., Li, C., Shaw, A., Seiss, K., Ouyang, Z., Martin-Gayo, E., Leng, J., Henrich, T.J., Li, J.Z., Pereyra, F., Zurakowski, R., Walker, B.D., Rosenberg, E.S., Yu, X.G., and Lichterfeld, M. (2014). HIV-1 persistence in CD4+ T cells with stem cell-like properties. Nature Medicine, 20(2), 139–42. doi:10.1038/nm.3445. Cao, Y., Gillespie, D.T., and Petzold, L.R. (2006). Efficient step size selection for the tau-leaping simulation method. The Journal of chemical physics, 124(4), 044109. Chomont, N., El-Far, M., Ancuta, P., Trautmann, L., Procopio, F.a., Yassine-Diab, B., Boucher, G., Boulassel,
For some simulations after the treatment interruption, the virus can rebind rapidly as in the case of 10 fold activation, while in the best scenario the viral rebound appears approximately within 5 years upon treatment termination. The factors that lead to this dichotomy could be attributed to the latent reservoirs size and stochastic events. In fact, even the LS , LC and LE subsets are present at extremely low size due to a reduction of their 99% with respect to pre-LRA treatment level, sub-reservoirs could still become activated and produce HIV virions within early months after treatment interruption. In particular, examining the dynamics of the LS compartment through the observation period, LS reservoirs gradually oversize the LC and LE subsets in the late stage of the treatment interruption. From this result, it turns out that even a very small subset of LS could have a direct implication in HIV eradication. Viral suppression is clearly visible in the case of 30 fold activation (Fig. 4D), where the viral load has values below the limit of detection (50 copies/mL), which is a consequence of the concomitant 100% latent reservoir subsets clearance (Table 3). 4. CONCLUSIONS In this paper, a multi-type Markov branching process model is applied in order to understand the dynamics of the different latent reservoir pools under different settings of the ART-LRA treatment combination. Assuming a reduced model with HIV reservoirs subsets in isolation, the temporal profile of the extinction probabilities reveal that the stem cell reservoirs although representing the smallest pool can persist longer than other HIV reservoirs. Additionally, with a 10 fold increase in the activation 89
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Hill, A.L., Rosenbloom, D.I., Fu, F., Nowak, M.A., and Siliciano, R.F. (2014). Predicting the outcomes of treatment to eradicate the latent reservoir for HIV1. Proceedings of the National Academy of Sciences, 111(37), 13475–13480. Jaafoura, S., de Herve, M.G.d.G., Hernandez-Vargas, E.A., Hendel-Chavez, H., Abdoh, M., Mateo, M., Krzysiek, R., Merad, M., Seng, R., Tardieu, M., et al. (2014). Progressive contraction of the latent HIV reservoir around a core of less-differentiated CD4+ memory T cells. Nature communications, 5. Karlin, S. and Taylor, H.E. (1981). A second course in stochastic processes. Elsevier. Ke, R., Lewin, S.R., Elliott, J.H., and Perelson, A.S. (2015). Modeling the effects of vorinostat in vivo reveals both transient and delayed HIV transcriptional activation and minimal killing of latently infected cells. PLoS Pathog, 11(10), e1005237. Laird, G.M., Bullen, C.K., Rosenbloom, D.I., Martin, A.R., Hill, A.L., Durand, C.M., Siliciano, J.D., and Siliciano, R.F. (2015). Ex vivo analysis identifies effective HIV-1 latency–reversing drug combinations. The Journal of clinical investigation, 125(5), 1901–1912. Perelson, A.S., Essunger, P., Cao, Y., Vesanen, M., Hurley, A., Saksela, K., Markowitz, M., and Ho, D.D. (1997). Decay characteristics of HIV-1-infected compartments during combination therapy. Nature. Perreau, M., Savoye, A.L., De Crignis, E., Corpataux, J.M., Cubas, R., Haddad, E.K., De Leval, L., Graziosi, C., and Pantaleo, G. (2013). Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production. The Journal of experimental medicine, 210(1), 143–56. Pettini, E., Prota, G., Ciabattini, A., Boianelli, A., Fiorino, F., Pozzi, G., Vicino, A., and Medaglini, D. (2013). Vaginal immunization to elicit primary T-cell activation and dissemination. PloS one, 8(12), e80545. Rosenbloom, D., Hill, A., and Rabi, S. (2012). Antiretroviral dynamics determines HIV evolution and predicts therapy outcome. Nature medicine, 18(9), 1378–1385.
M.R., Ghattas, G., Brenchley, J.M., Schacker, T.W., Hill, B.J., Douek, D.C., Routy, J.P., Haddad, E.K., and S´ekaly, R.P. (2009). HIV reservoir size and persistence are driven by T cell survival and homeostatic proliferation. Nature medicine, 15(8), 893–900. Cohn, L.B., Silva, I.T., Oliveira, T.Y., Rosales, R.A., Parrish, E.H., Learn, G.H., Hahn, B.H., Czartoski, J.L., McElrath, M.J., Lehmann, C., Klein, F., Caskey, M., Walker, B.D., Siliciano, J.D., Siliciano, R.F., Jankovic, M., and Nussenzweig, M.C. (2015). HIV-1 integration landscape during latent and active infection. Cell, 160(3), 420–32. Conway, J.M. and Coombs, D. (2011). A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients. PLoS Comput Biol, 7(4), e1002033. Conway, J.M. and Perelson, A.S. (2015). Post-treatment control of HIV infection. Proceedings of the National Academy of Sciences, 112(17), 5467–5472. Crooks, A.M., Bateson, R., Cope, A.B., Dahl, N.P., Griggs, M.K., Kuruc, J.D., Gay, C.L., Eron, J.J., Margolis, D.M., Bosch, R.J., et al. (2015). Precise quantitation of the latent HIV-1 reservoir: implications for eradication strategies. Journal of Infectious Diseases, 212(9), 1361–1365. Deeks, S.G., Autran, B., Berkhout, B., Benkirane, M., Cairns, S., Chomont, N., Chun, T.W., Churchill, M., Mascio, M.D., Katlama, C., Lafeuillade, A., Landay, A., Lederman, M., Lewin, S.R., Maldarelli, F., Margolis, D., Markowitz, M., Martinez-Picado, J., Mullins, J.I., Mellors, J., Moreno, S., O’Doherty, U., Palmer, S., Penicaud, M.C., Peterlin, M., Poli, G., Routy, J.P., Rouzioux, C., Silvestri, G., Stevenson, M., Telenti, A., Lint, C.V., Verdin, E., Woolfrey, A., Zaia, J., and Barr´eSinoussi, F. (2012). Towards an HIV cure: a global scientific strategy. Nature Reviews Immunology, 12(8), 607–614. Eisele, E. and Siliciano, R.F. (2012). Redefining the viral reservoirs that prevent HIV-1 eradication. Immunity, 37(3), 377–88. Ferreira, J., Hernandez-Vargas, E.A., and Middleton, R.H. (2011). Computer simulation of structured treatment interruption for HIV infection. Computer methods and programs in biomedicine, 104(2), 50–61. Groot, F., Welsch, S., and Sattentau, Q.J. (2008). Efficient HIV-1 transmission from macrophages to T cells across transient virological synapses. Blood, 111(9), 4660–3. Harris, T.E. (2002). The theory of branching processes. Courier Corporation. Hawkins, E.D., Turner, M.L., Dowling, M.R., Van Gend, C., and Hodgkin, P.D. (2007). A model of immune regulation as a consequence of randomized lymphocyte division and death times. Proceedings of the National Academy of Sciences, 104(12), 5032–5037. Hernandez-Vargas, E. (2017). Modeling kick-kill strategies toward HIV cure. Frontiers in Immunology, 8. Hernandez-Vargas, E., Colaneri, P., and Middleton, R.H. (2014). Switching Strategies to Mitigate HIV Mutation. IEEE Transactions on Control Systems Technology, (1), 1–6. Hernandez-Vargas, E.A., Colaneri, P., and Middleton, R.H. (2013). Optimal therapy scheduling for a simplified HIV infection model. Automatica, 49(9), 2874–2880. 90
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A stochastic Markov branching process A stochastic Markov branching process A stochastic Markov branching process framework to evaluate HIV eradication A stochastic Markov branching process HIV eradication framework to evaluate framework to evaluate HIV eradication ∗ ∗∗ framework to evaluate HIV eradication Alessandro Boianelli Esteban A. Hernandez–Vargas ∗ ∗∗ ∗ ∗∗
Alessandro Boianelli ∗ Esteban A. Hernandez–Vargas ∗∗ Alessandro Boianelli Esteban A. Hernandez–Vargas ∗Alessandro Boianelli ∗ Esteban A. Hernandez–Vargas ∗∗ Department of Drug Discovery Sciences, Boehringer Ingelheim ∗ ∗ of Drug Discovery Sciences, Boehringer Ingelheim ∗ Department Pharma GmbH Co KG, 88397, Biberach, Germany. Department of Drug& Sciences, Boehringer Ingelheim Pharma GmbH & Discovery Co KG, 88397, Biberach, Germany. ∗∗ ∗ Frankurt Institute for Advanced Studies (FIAS), Department of Drug Sciences, Boehringer Ingelheim Pharma GmbH & Discovery Co KG, 88397, Biberach, Germany. ∗∗ ∗∗ for Advanced Studies (FIAS), ∗∗ Frankurt Ruth-Moufang-Strasse Frankfurt am Main, Germany. Pharma GmbHInstitute & 1, Co60438 KG, 88397, Biberach, Germany. Frankurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. ∗∗
Frankurt Institute for Advanced (FIAS), 1, 60438 FrankfurtStudies am Main, Germany. Ruth-Moufang-Strasse Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, HIV can persist within a small pool of long-lived resting memory CD4+ T cells. In concert Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, HIV can persist within a small pool of long-lived resting memory CD4+ T cells. In concert with ART, different latency reversal agents (LRAs) are under development tocells. activate latent Abstract: Although antiretroviral therapies (ARTs) reduce viral load to undetectable levels, HIV can persist within a small pool of long-lived resting memory CD4+ T In concert with ART, different latency reversal agents (LRAs) are under development to activate latent reservoirs thus reducing viral persistence. Based on aresting stochastic Markov branching process with HIV can persist within a small pool agents of long-lived memory CD4+ Ttocells. In concert with ART, different latency reversal (LRAs) are under development activate reservoirs thus reducing viral persistence. Based on a stochastic Markov branching processlatent with HIV reservoirs dynamics in isolation, simulations suggest folds increase the activation with ART,thus different latency reversal agents (LRAs) are that under10Markov development toinactivate latent reservoirs reducing viral persistence. Based on a stochastic branching process with HIV reservoirs dynamics in isolation, simulations suggest that 10 folds increase in the activation rate from latently to actively infectedsimulations could reduce extinction time for all in reservoirs to 50 reservoirs thus reducing viral persistence. Based on athe stochastic Markov branching process with HIV reservoirs dynamics in isolation, suggest that 10 folds increase the activation rate from latently to actively infected could reduce the extinction time for all reservoirs to 50 months. However, when viral dynamics and reduce newsuggest infection are incorporated a to more HIV reservoirs dynamics in isolation, simulations thatcycles 10 folds increase in the in activation rate from latently to actively infected could the extinction time for all reservoirs 50 months. However, when viral dynamics and new infection cycles are incorporated in a more realistic simulations point outinfection that LRAs would least 30to fold rate fromscenario, latently stochastic to actively infected could reduce the extinction timerequire for all at reservoirs 50 months. However, when viral dynamics and new cycles are incorporated in a more realistic scenario, stochastic simulations point out that LRAs would require at least 30 fold increase in the activation ratesimulations to induce concomitant eradication of are the require differentatsub-reservoirs. months. However, when viral dynamics and new infection cycles incorporated in a30more realistic scenario, stochastic point out that LRAs would least fold increase in the activation rate to induce concomitant eradication of the different sub-reservoirs. realistic scenario, stochastic point out that LRAs would least 30 fold increase in the activation ratesimulations to induce concomitant eradication of the require differentatsub-reservoirs. © 2018, IFAC Federation of Automatic Control) Hosting byofcure, Elsevier Ltd. Allsub-reservoirs. rights reserved. increase in Stochastic the(International activation rate to Markov induce concomitant eradication the different Keywords: Modeling, Branching Process, HIV Reservoirs Keywords: Stochastic Modeling, Markov Branching Process, HIV cure, Reservoirs Keywords: Stochastic Modeling, Markov Branching Process, HIV cure, Reservoirs Keywords: Stochastic Modeling, Markov Branching Process, HIV cure, Reservoirs 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Although antiviral1.therapy potently suppresses the virus, Although antiviral therapy potently suppresses the virus, a functional or eventherapy sterile HIV cure suppresses remains a challenge. Although antiviral potently the virus, a functional or sterile HIV cure remains a HIVantiviral infection, a subset of latently cells Although potently suppresses the virus, aUpon functional or even eventherapy sterile HIV cure remainsinfected a challenge. challenge. Upon HIV infection, a of latently infected cells carrying inactive integrated proviral DNA (HIV a functional or even sterile HIV cure remains a reservoirs) challenge. Upon HIV infection, a subset subset of latently infected cells carrying inactive integrated proviral DNA reservoirs) is rapidly established and (2012)]. DifUpon HIV infection, a[Eisele subset of Siliciano latently infected cells carrying integrated proviral DNA (HIV (HIV reservoirs) is rapidlyinactive established [Eisele and Siliciano (2012)]. Different studies have suggested thatSiliciano macrophages [Groot carrying inactive integrated proviral DNA (HIV reservoirs) is rapidly established [Eisele and (2012)]. Different studies have suggested that macrophages [Groot et al. (2008)] and cells [Perreau etDifal. is rapidly established [Eiselehelper and Siliciano (2012)]. ferent studies havefollicular suggested thatT macrophages [Groot et al. (2008)] and follicular helper T cells [Perreau et al. (2013)] are good candidates forthat supporting residual[Groot virus ferent studies have suggested macrophages et al. (2008)] and follicular helper T cells [Perreau et al. (2013)] are good candidates for supporting residual virus replication, but it is not clear whether these[Perreau infectedet cells et al. (2008)] and follicular helper T cells al. (2013)] are good supporting replication, but itcandidates is not clearfor whether theseresidual infectedvirus cells can survive for it several in supporting patients onresidual optimalvirus an(2013)] are good candidates replication, but is not years clearfor whether these infected cells can survive for several years in on optimal antiretroviral [Eisele Siliciano (2012)]. replication, but is not clearand whether these infected can survivetreatment for it several years in patients patients on optimalcells antiretroviral treatment [Eisele and Siliciano (2012)]. can survivetreatment for several[Eisele years and in patients on optimal antiretroviral Siliciano (2012)]. Latently infected resting memory CD4+ T cells represent tiretroviral treatment [Eisele and Siliciano (2012)]. Latently infected resting CD4+ T represent the only cell type in whichmemory it has been clearly demonstrated Latently infected resting memory CD4+ T cells cells represent the only cell type in which it has been clearly demonstrated that replication-competent virus canclearly persist for several Latently infected resting CD4+ T cells represent the cell type in whichmemory it has been demonstrated thatonly replication-competent virus can persist for several years in cell patients al.been (2014); Chomont et al. the only type in[Buzon which itethas clearly demonstrated that replication-competent virus can persist for several years in patients [Buzon et al. (2014); Chomont et al. (2009); et al.[Buzon (2015);etDeeks et al.persist (2012); that replication-competent virus can forJaafoura several years inCohn patients al. (2014); Chomont et al. Fig. 1. Multi-type Markov branching process framework. (2009); et al. (2015); et al. Jaafoura 1. Multi-type Markov branching process framework. et al. (2014)]. In direction, recent studies years inCohn patients etDeeks al. (2014); Chomont et al. Fig. (2009); Cohn et this al.[Buzon (2015); Deeks et experimental al. (2012); (2012); Jaafoura mathematical model describes the dynamics of Fig. The 1. Multi-type Markov branching process framework. et al. (2014)]. In this direction, recent experimental studies mathematical model describes the dynamics of have aimed latency reversal can Fig. The (2009); CohntoIn etdevelop al. (2015); Deeks et experimental al. agents (2012);which Jaafoura et al. (2014)]. this direction, recent studies HIV (V ); productively infected cells (I);dynamics stem (LSof ), 1. Multi-type Markov branching process framework. The mathematical model describes the have aimed to develop latency reversal agents which can HIV (V ); productively infected cells (I); stem (L ), S purge HIV reservoirs thus contributing to the which viral supet al. aimed (2014)]. this direction, recent experimental studies have toIndevelop latency reversal agents can central (Lproductively effector (LE )cells memory CD4+ T The mathematical describes the HIV (V ); infected (I);dynamics stem (LSSof ), C ), and model purge HIV reservoirs thus contributing to the viral supC E central (L ), and effector (L ) memory CD4+ T pression or viral eradication [Archin et al. (2012); Archin have aimed to develop latency reversal agents which can C ), and E )cells purge HIV reservoirs thus contributing to the viral supcells. (V LS); , L at rate , γC(Land HIV productively infected (I); γstem ), central (L (L memory C and Leffector E proliferate SCD4+ ST C E pression or viral eradication [Archin et al. (2012); Archin cells. LSS , LC C and LE E proliferate at rate γS S , γC C and and Margolis (2014); Laird et[Archin al. (2015)]. Although differpurge HIV reservoirs thus contributing to the viral suppression or viral eradication et al. (2012); Archin γ respectively. Furthermore, the three subsets are central (L ), and effector (L ) memory CD4+ T cells. L , L and L proliferate at rate γ , γ and E C E S C E S C are and Margolis (2014); Laird et al. Although differγE Furthermore, the three subsets ent LRA candidates are available, their efficacy in clinical pression or viral eradication et al. (2012); Archin E respectively. and Margolis (2014); Laird et[Archin al. (2015)]. (2015)]. Although differassumed d and the become cells. LS , to LCdie andat Lrate atthree rateproductively γsubsets γ Furthermore, are E proliferate S , γC and E respectively. ent LRA candidates are available, their efficacy in clinical assumed to die at rate d and become productively trials is still fragmented. and LRA Margolis (2014);are Laird et al. (2015)]. Although different candidates available, their efficacy in clinical infected cells (I) at the activation rate a. Productively γ Furthermore, the three subsets are assumed to die rate d and become productively E respectively. trials is still fragmented. infected cells (I) at the activation rate a. Productively ent LRA candidates are available, their efficacy in clinical trials is still fragmented. cells canat either new a. virus particles assumedcells to die rateactivation drelease and become productively infected the Productively Mathematical modeling has given an important contribuinfected cells(I) can either release rate new virus particles trials is still fragmented. Mathematical modeling has given an important contribu(V ) at rate p or die at rate δ. New particles infected cells (I) at the activation rate a. Productively cells can either release new virus tion to understanding treatment e.g. [Conway and Mathematical modelingHIV has given an important contribu(V ) at rate p or die at rate δ. New virus particles tion to understanding HIV treatment e.g. [Conway and can) promote infection of latent infected cellspnew can either release new virus reservoir (V at rate or cycles die atofrate δ. New particles Coombs (2011); FerreiraHIV et al. (2011); Hernandez-Vargas Mathematical modeling has given an important contribution to understanding treatment e.g. [Conway and can cycles infection of latent Coombs (2011); Ferreira et (2011); Hernandez-Vargas via the target cells at the rate fvirus (1 − reservoir )kT or (V ) promote at rate pnew or die(Tat)of rate δ. New particles can promote new cycles of infection of latent reservoir et al. to (2014, 2013); Hill et al. (2014); Ke et al. [Conway (2015); Laird tion understanding HIV treatment e.g. and Coombs (2011); Ferreira et al. al. (2011); Hernandez-Vargas via the target cells (T ) at the rate f (1 − )kT or et al. (2014, 2013); Hill et al. (2014); Ke et al. (2015); Laird alternatively productively infected cells (I), at rate can promote new cycles of infection of latent reservoir via the target cells (T ) at the rate f (1 − or al.(2014, (2015); Perelson (1997); al. Coombs (2011); Ferreira etal. al. (2011); Hernandez-Vargas et 2013); Hill etet (2014); Ke Rosenbloom et al. (2015); et Laird alternatively productively infected cells (I), )kT at rate et al. al. (2015); Perelson etal.al. (1997); Rosenbloom et al. (1 − f )(1 − )kT . Virus is cleared at rate c. via the target cells (T ) at the rate f (1 − )kT or alternatively productively infected cells (I), at rate (2012)]. In the context the LRAs, first(2015); attempt to al. 2013); Hill etof (2014); Kethe et al. Laird et al.(2014, (2015); Perelson etal. al. (1997); Rosenbloom et al. (1 − f )(1 − )kT . Virus is cleared at rate c. (2012)]. In the context of the LRAs, the first attempt to alternatively productively infected cells (I), at rate (1 − f )(1 − )kT . Virus is cleared at rate c. quantify the clinical goal for HIV eradication was done by et al. (2015); Perelson et al. (1997); Rosenbloom et al. (2012)]. In the context of the LRAs, the first attempt to found out that a 2000 fold reduction of the latent reservoir quantify the clinical goal for HIV was done by (1out − fthat )(1 −a )kT . fold Virus is cleared at rate c. reservoir the computational work Hill eteradication al. (2014). authors (2012)]. In context the LRAs, the firstThe attempt to found 2000 of the latent quantify thethe clinical goalofof for HIV was done by decreases the probability ofreduction viral rebound in 1 year and a the computational work of Hill eteradication al. (2014). The authors found out that a 2000 fold reduction of the latent reservoir the probability of viral rebound in 1 year and a quantify the clinicalwork goal of forHill HIV eradication was done by decreases the computational etAlfons al. (2014). The authors This 10000 fold could eradicate HIV. In another modeling work,a found out that a 2000 fold reduction of the latent reservoir decreases the probability of viral rebound in 1 year and work was supported by the und Gertrud Kassel This 10000 fold could eradicate HIV. In another modeling work, the computational work of Hill et al. (2014). The authors work was supported by the Alfons und Gertrud Kassel Ke et al. (2015) elucidated the effect of the LRA vorinostat Stiftung. Corresponding author: [email protected] decreases the probability of viral rebound in 1 year and 10000 fold could eradicate HIV. In another modeling work,a This work was supported by the Alfons und Gertrud KasselKe et al. (2015) elucidated the effect of the LRA vorinostat Stiftung. Corresponding author: [email protected] 10000 fold couldelucidated eradicate HIV. In another modeling work, Ke et al. (2015) the effect of the LRA vorinostat Stiftung. Corresponding author: This work was supported [email protected] the Alfons und Gertrud KasselKe et al. (2015) elucidated the effect of the LRA vorinostat Stiftung. Corresponding author: [email protected] 2405-8963 © 2018, IFACConference (International Proceedings, 2nd IFAC onFederation of Automatic Control) 85 Hosting by Elsevier Ltd. All rights reserved. Proceedings, 2nd IFAC Conference on 85 Control. Peer reviewIdentification under responsibility of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference on 85 Modelling, Identification and Control of Nonlinear 10.1016/j.ifacol.2018.07.259 Systems Modelling, Identification and Controlon of Nonlinear Proceedings, 2nd IFAC Conference 85 Systems Guadalajara, Mexico, June 20-22, 2018 Systems Identification Modelling, and Control of Nonlinear
2018 IFAC MICNON 86 Guadalajara, Mexico, June 20-22, 2018 Alessandro Boianelli et al. / IFAC PapersOnLine 51-13 (2018) 85–90
on the latent reservoir dynamics which results in a transient activation of the whole latent subset. However, the main limitation of these works is that the latent reservoirs are considered as a unique cell pool with homogeneous dynamics. On the contrary, latest results from Jaafoura et al. (2014) revealed a progressive reduction of latent reservoirs around a core of less-differentiated memory subsets, that is central memory T cells (TCM ), effector memory T cells (TEM ), and stem central memory T cells (TSCM ). This process appears to be driven by the differences in initial sizes and decay rates between latently infected memory subsets. A complete review in modeling HIV cure can be found in Hernandez-Vargas (2017).
Note that the multi-type Markov branching process has the following fundamental properties: (H1) each cell action is independent from others; (H2) each cell offspring generates its own branching process; and (H3) at each time point t, the cell has no memory about previous time steps. In this fashion, Li (t) is a random variable vector representing the number of TSCM , TCM and TEM cells respectively at the time point t. In the Markov branching process formulation, each reservoir has a life time exponentially distributed with a parameter λi = γi + d + a. At the end of its life, a singular type cell in each subset independently from the others can produce its progeny according to the probability generating function (pgf) defined as:
The dynamics of viral populations are usually governed by deterministic phenomena when large numbers are observed. When a very small number of virus particles and infected cells are present, the stochastic fate could be the main driver of the clearance timing and eradication [Hawkins et al. (2007)]. In HIV patients under long ARTs, the stochastic variability could lead ultimately to post– treatment control or to the viral eradication. These outcomes could arise from the processes of proliferation, apoptosis, differentiation, and activation of the different stem, central and effector subsets. Understanding the stochastic fate decisions and dynamic events are relevant to develop optimal purging strategies. A suitable stochastic framework is the Markov branching process, which has been applied to predict immune response dynamics e.g., CD8+ T cell clonal expansion [Buchholz et al. (2013)], HIV dynamics [Conway and Coombs (2011); Hill et al. (2014); Laird et al. (2015)] and CD4+ T cell responses to vaccination [Boianelli et al. (2015); Pettini et al. (2013)].
fi (xi ) =
∞
pik xki ,
(1)
k=0
where pik is the probability of generating k offspring cells, while xi is an auxiliary variable. The probability pik is computed as the ratio between the rate of having k offspring cells to the sum of the rates of the all possible events. This stochastic model named birth, death and activation process leads to the simplification of (1) in the following form: fi (xi ) = pi0 + pi2 x2i = (d + a + γi x2i )λ−1 i , pi0
pi2
(2) pi0
= (d + a)/λi and = γi /λi , with + where pi2 = 1. For the complete description of the birth and death/activation process, we define the global probability generating function for one latent reservoir cell (xi ): Fi (xi , t) =
This paper proposes a Markov branching process model describing the dynamics of latent reservoirs (Fig.1) using different settings of the ART-LRA treatment combination. The main difference respect to previous works in HIV modeling research, this paper provides for the first time a qualitative dynamic behaviour of the virus under the combination of ART and LRA treatment considering different reservoirs dynamics. Although the results are from a computational study, they provide preliminary insights that need to be assessed with in vivo studies in view of improving therapeutic treatments for an HIV cure.
∞
P (Li (t) = k|Li (0) = 1)xki .
(3)
k=0
With P (Li (t) = k|Li (0) = 1), we denote the conditional probability of having Li (t) = k cells at the time t starting from Li (0) = 1. The temporal evolution of Fi (xi , t) is governed by the backward Kolmogorov equation in its general form: ∂Fi (xi , t) = ui (Fi (xi , t)), ∂t
(4)
where: ui (xi ) = λi (Fi (xi , t) − xi ), with the initial condition Fi (xi , 0) = xi . This initial condition means that the process is initiated from one single stem, central and effector CD4+ T cell. The temporal description of the probability generating functions Fi (xi , t) is provided by the backward Kolmogorov equation which presents the form as follows:
2. HIV RESERVOIRS DYNAMICS IN ISOLATION In this section, we distinguish three key reservoirs such as the stem cell (TSCM ), the central memory (TCM ) and the effector memory (TEM ). We assume that the three HIV reservoir subsets undergo only proliferation, death and activation. To understand the reservoir dynamics in isolation, in this section, we did not include viral dynamics and consequently new cycles of infection, see red circle in Fig.1. This can represent the case when patients are under long ARTs. The three sub-reservoirs (TSCM , TCM and TEM ) are abbreviated in the form of Ti with i = 3. Reservoirs proliferate at the rate γi . The reservoirs die at the rate di . Each subset can be activated at the rate ai respectively. For the sake of the analysis, we assume that the three reservoirs have the same activation rate ai = a as well as the same death rate di = d.
∂Fi = γi (Fi (xi , t)2 − Fi (xi , t)) + (d + a)(1 − Fi (xi , t)), ∂t This differential equation system can be solved analytically with the method of characteristics, providing a unique solution: Fi (xi , t) = 86
(d + a)(xi − 1) − e(d+a−γi )t (γi xi − d + a) . γi (xi − 1) − e(d+a−γi )t (γi xi − d + a)
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144 months
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a Activation rate 0.001 day−1 d Reservoir death rate 0.004 day−1 LS (0) TSCM initial size [1 1000] LC (0) TCM initial size [1 1000] LE (0) TEM initial size [1 1000] * Jaafoura et al. (2014) ** Conway and Perelson (2015)
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Fig. 2. Extinction probabilities. Pi (t) surfaces for the different initial reservoir values (Li (0)) were considered with activation rate values of (A) a = 0.001 d−1 , (B) a = 0.002 d−1 , (C) a = 0.005 d−1 , (D) a = 0.01 d−1 . The colorbar on the right side indicates the value of extinction probabilities. only after very long time periods (see Fig. 2A). More specifically, the TSCM reservoir has shorter probability of eradication regions than TCM and TEM cells. This can be observed with negligible extinction probability PS (t) before 1000 months. This could imply that although the values for LS (0) are lower compared to the other subsets, the TSCM has a great stability and can significantly contribute to the viral rebound when ART treatment is interrupted. Conversely for the TEM and TCM subsets, the extinction probabilities of PE (t) and PC (t) reach high values for larger regions of initial conditions (LE (0) and LC (0)).
Table 1. Model parameter values. Reference *
600
200
To evaluate the effect of model parameters, we compute the median extinction time for different initial conditions of latent reservoirs subsets [Conway and Coombs (2011)]. In the first case, we assume the normal activation rate value a = 0.001 day−1 and then two, five and ten folds increase of the normal activation rate. We perform the numerical simulations of the extinction probabilities Pi (t) for different initial size (Li (0)) in a time horizon of 1000 months, approximately 83 years. The death rates are assumed equal among the different reservoirs. The half-life of each reservoir class as well as the experimental ranges of initial latent reservoir size for the different subsets were taken from previous estimates in patients on prolonged ART [Jaafoura et al. (2014)]. For the nominal value of the activation rate (a), we assume the estimate previously considered in Conway and Perelson (2015). A summary of model parameters used in our simulations is presented in Table 1. Note that the reservoir proliferation rates γi can i be calculated by γi = a + d − ln(2)/T1/2 . Value (Units) 277 months
600
LE(0)
S
L (0)
B
2.1 Simulation setting
Biological meaning TSCM half life
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0
Li (0) (d + a) − (d + a)e(d+a−γi )t .(5) Pi (t) = Fi (0, t) = γi − (d + a)e(d+a−γi )t The extinction probability for each reservoir subset is a function of the rate parameters, initial conditions Li (0) and time point t.
Parameter TS 1/2
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. Fi (xi , t) = Fi (xi , t) Assuming the initial condition Li (0), the probabilities of extinction, i.e. the probabilities of having no Ti offspring at time point t, are defined as follows:
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A
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Considering the independence property (H1) of the branching process [Harris (2002)], we can obtain:
87
* ** ** * * *
Assuming 2 fold increase activation rate (a = 0.002 d−1 ), a clear increase of the extinction probabilities of the three subsets is observed in Fig. 2B. In particular, the activation rate (a) has a drastic effect on the extinction probability of the TSCM cells, resulting in high value for every initial condition. This qualitative behaviour is also conserved for the extinction probabilities in the other reservoirs TCM and TEM . Intuitively, the progressive extinction of the three reservoir compartments is even more pronounced with a 5 folds increase of the activation rate (Fig. 2C), where an effective eradication of the TSCM , TCM and TEM reservoirs can be achieved within 250 months (approximately 21 years) for any initial size of the reservoir subsets. A possible eradication of the latent reservoirs could be achieved increasing 10 folds the activation rate a, whereas the extinction probabilities, Pi (t) = 1, can be achieved within 50 months (Fig. 2D).
2.2 Simulation Results We test initially the scenario where the reservoirs subsets present the normal activation rate value a = 0.001 d−1 . The extinction probabilities are computed for the reservoirs initial values reported in Table 1. Consequently, we consider for the same initial conditions, different scenarios for increasing values of the activation rate (a) promoted by LRAs were 0.002, 0.005, and 0.01 d−1 . Fig. 2 shows the extinction probabilities surfaces for all these different scenarios. For the normal activation rate (a = 0.001 d−1 ), the latent reservoirs would present a high probability of eradication 87
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3. MODELING HIV RESERVOIRS WITH VIRAL DYNAMICS AND NEW CYCLES OF INFECTION
uniform distribution spanning from 1 to 49 copies/mL. This is a more realistic assumption as the viral load within a host on prolonged ART is undetectable. For the initial condition of productively infected cells, I(0) can be set to the quasi-steady state approximation I(0) = V (0)c/p, because the productively infected cells and virus present faster dynamics than latent reservoirs. Parameters and results were re-scaled over the entire blood volume, approximately 5 liters.
In the previous sections as well as in the most of the stochastic models of HIV, the viral dynamics and the infection process are omitted. In order to evaluate qualitatively the latent reservoirs progression and their influence on the viral dynamics we built a comprehensive stochastic Markov branching process [Harris (2002); Karlin and Taylor (1981)] describing HIV infection mechanisms, see Fig.1. Our framework is composed by the uninfected CD4+ T cells T , the three different latent reservoir subsets (Li ), the productively infected cells (I) and the virus (V ). The uninfected cells are replenished at a rate ST and die with a rate dT per cell. The newly infected cells can become latent reservoirs at a rate f (1-)k, where k is the infection rate, f is the fraction of the newly infected cells that become latently infected cells and is the ART efficacy. Alternatively the uninfected cells can transit in the productively infected cells state I with a rate (1f )(1-)k. Productively infected cells can either release HIV virions at a rate p or die with a rate δ due to cytopathic effects. Virus particles (V ) are cleared with a rate c. 3.1 Simulation setting We assume that the in silico treatment protocol presents 4.5 years on ART, followed by 6 months on a combination of ART and LRA. After this 6 months, both LRA and ART are interrupted and the stochastic model dynamics are recorded for 5 years, scheme presented in Fig. 3. Parameters values used for simulations are presented in Table 2. Table 2. Model parameter values.
Fig. 3. Combined ART+LRA simulation scenario. The first 4 years and half are considered on ART treatment (green panel) followed by a combination on ART and LRA treatment for 6 months (blue planel). Consequently both treatments are interrupted (TI) in 5 years period to evaluate the possibility of viral rebound (red line) or suppression (green line).
Parameter Biological meaning Value (Units) Reference ST Cell production T (0)dT d−1 dT Cell death 0.01 d−1 * T (0) Cell initial size 500 µL−1 ** k Infection rate 5×10−8 mLd−1 ** f Latent cell fraction 10−6 ** Treatment efficacy 1 p Viral replication 1000 d−1 ** c Viral clearance 23 d−1 ** LS (0) Initial TSCM 560 * LC (0) Initial TCM 2600 * LE (0) Initial TEM 800 * I(0) Initial infected cells V (0)c p−1 * Jaafoura et al. (2014) ** Conway and Perelson (2015)
As the proposed model in this section presents a high reaction number, all the stochastic simulations are performed with the τ leaping method [Cao et al. (2006)], which results particularly faster than the stochastic simulation algorithm despite the accuracy in the chemical master equation solution. A crucial factor in the accuracy determination is the τ parameter defined as the time interval where one or more reactions can happen allowing the stochastic system to move in the next state. The value τ = 0.002 represents a good trade-off for the approximation of the Markov branching process solution.
Different scenarios with variations in the activation rate (a) were explored. Note that the initial values of latent reservoirs in Table 2 are calculated considering the total number of latent reservoir equal to 1 per 106 CD4+ T cells per mL as in other computational works e.g. Hill et al. (2014). The latent reservoir size provided by the measurement assay is prone to errors and present high variability among different in vivo studies [Bruner et al. (2015)]. Instead of considering the latent reservoir ranges from these studies, we assume valid the estimation of 20 per 106 CD4+ T cells found in Crooks et al. (2015). Furthermore, instead of assuming constant the initial viral load V (0), this quantity is sampled randomly from a
We perform 20 stochastic simulations with the initial size of the three latent reservoir subsets [Jaafoura et al. (2014)]. When the LRAs are introduced assuming a 10 fold activation rate for 6 months before treatment interruption (TI), we observe the same qualitative behaviour for the viral load (Fig. 4B) and productively infected cells I. In this scenario, although the post-LRA treatment reduction of LS , LC , and LE was 92%, 95% and 93% respectively (Table 3), the three subsets still persist. Among them, LS is the only subsets that increase after ART-LRA treatment, see Table 3. In the case of 20 fold activation (Fig. 4C), a large variability appears in the viral rebound time as well as in the productively infected cells.
3.2 Stochastic Simulations
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Table 3. Median values of 20 stochastic simulations. Activation rate (a) Fold increase 10 20 30
TI 102 2 0
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LE ART+LRA 35 2 0
TI 33 3 0
Coupling viral dynamics and new cycles of infections into a more realistic scenario, we evaluated the HIV reservoirs whit a combination of 5 years ART and 6 months LRA treatment. Results pointed out that LRAs of 30 fold activation is required in order to avoid viral rebound in all the stochastic simulations after 5 years of ART interruption. The necessary condition for the viral suppression may be obtained by eradicating strategically the LS , LC and LE subsets. In fact, this framework shows that even after a significant reduction of the latent reservoirs (99%), these sub-reservoirs could promote either a viral rebound immediately or 4 years after ART-LRA interruption. Additionally, the post-LRA treatment level of the latent reservoir reveals that the TSCM cell pool could be the main obstacle for HIV eradication. Further incorporation of control strategies could serve as a critical tool to refine purging strategies and facilitate HIV eradication.
LC ART+LRA 125 5 0
rate, the extinction probabilities of all reservoir subsets are observed within 50 months (approximately 4 years).
ART 1990 2044 2065
TI 35 5 0
LS ART+LRA 29 1 0
ART 470 441 412
Fig. 4. HIV dynamics during the ART and LRA treatment. 20 stochastic simulations are performed with (A) normal activation, (B) 10 fold, (C) 20 fold, and (D) 30 fold activation (parameter a) of the latent reservoirs.
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For some simulations after the treatment interruption, the virus can rebind rapidly as in the case of 10 fold activation, while in the best scenario the viral rebound appears approximately within 5 years upon treatment termination. The factors that lead to this dichotomy could be attributed to the latent reservoirs size and stochastic events. In fact, even the LS , LC and LE subsets are present at extremely low size due to a reduction of their 99% with respect to pre-LRA treatment level, sub-reservoirs could still become activated and produce HIV virions within early months after treatment interruption. In particular, examining the dynamics of the LS compartment through the observation period, LS reservoirs gradually oversize the LC and LE subsets in the late stage of the treatment interruption. From this result, it turns out that even a very small subset of LS could have a direct implication in HIV eradication. Viral suppression is clearly visible in the case of 30 fold activation (Fig. 4D), where the viral load has values below the limit of detection (50 copies/mL), which is a consequence of the concomitant 100% latent reservoir subsets clearance (Table 3). 4. CONCLUSIONS In this paper, a multi-type Markov branching process model is applied in order to understand the dynamics of the different latent reservoir pools under different settings of the ART-LRA treatment combination. Assuming a reduced model with HIV reservoirs subsets in isolation, the temporal profile of the extinction probabilities reveal that the stem cell reservoirs although representing the smallest pool can persist longer than other HIV reservoirs. Additionally, with a 10 fold increase in the activation 89
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