Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models

Bulletin of Mathematical Biology (2008) 70: 785–799 DOI 10.1007/s11538-007-9279-9 O R I G I N A L A RT I C L E

Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models Hulin Wua,∗ , Haihong Zhua , Hongyu Miaoa , Alan S. Perelsonb a

Department of Biostatistics and Computational Biology, University of Rochester School of Medicine and Dentistry, 601 Elmwood Avenue, Box 630, Rochester, NY 14642, USA b Theoretical Biology and Biophysics Group, MS-K710, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received: 31 January 2007 / Accepted: 19 September 2007 / Published online: 5 February 2008 © Society for Mathematical Biology 2007

Abstract We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and λ) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems. ∗ Corresponding author.

E-mail address: [email protected] (Hulin Wu).

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Keywords Identifiability · Inverse problem · Statistical estimation · Viral dynamics 1. Introduction Mathematical models have been developed to describe HIV and its interaction with CD4+ T cells. These models have made a substantial impact on the understanding of HIV infection and treatment (Ho et al., 1995; Wei et al., 1995; Perelson et al., 1996, 1997; Perelson and Nelson, 1999; Wu et al., 1999; Nowak and May, 2000; Tan and Wu, 2005). HIV dynamic models are formulated as a system of nonlinear differential equations. To use HIV dynamic models as a tool for treatment decisions, it is essential to determine the parameters in the model for each individual patient. Some parameter estimation methods have been developed for longitudinal clinical data (Wu and Ding, 1999). Huang and Wu (2006) and Huang et al. (2006) recently proposed a Bayesian approach to estimate the parameters in HIV dynamic models for a population of patients. Their method is highly dependent on informative priors for most of the parameters and is computationally intensive, which may limit its use. Xia (2003), Xia and Moog (2003), and Jeffrey and Xia (2005) investigated the identifiability of HIV dynamic models, but the solution of the identifiability problem was not provided for the case where only measurements of viral load are available. In this work, we study the parameter identifiability of a popular three-dimensional (3D) HIV/AIDS dynamic model based on measurements of viral load level. We introduce the concepts of an identification function and an identification equation. An alternative method is proposed to form the identification function that is shown to have some advantages over the existing method. We also discuss the effect of initial values of state variables on the identifiability of unknown parameters. Simulation studies are performed to validate our theoretical analysis results. In Section 2, we investigate the identifiability of the popular 3D HIV/AIDS model based on measurements of viral load and present our main results. In Section 3, the least squares method is proposed to estimate the parameters in the 3D HIV dynamic model based on measurements of viral load, and simulation studies are performed to validate the identifiability analysis results. In Section 4, we apply the proposed methods to estimate HIV dynamic parameters in HIV-1 infected individuals receiving antiretroviral treatment in a clinical study. We conclude the paper with some discussions in Section 5.

2. Identifiability of HIV dynamic models Xia and Moog (2003) and Jeffrey and Xia (2005) studied the identifiability of many HIV/AIDS models. They assumed that the measurements of viral load and T cells (either the number of uninfected CD4+ T cells or the total number of CD4+ T cells) are available. In clinical practice, it is difficult to measure the number of uninfected CD4+ T cells. The measurement of total CD4+ T cells in blood may not be representative of the total body pool of CD4+ T cells that are involved in HIV infection. In addition, the variation of total CD4+ T cell counts in blood is large. Thus, the total CD4+ T cell count data are not reliable, although the use of CD4+ T cell data may help to resolve the identifiability problem. Here we focus on the case that only the measurements of viral load are available.

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Consider the following three dimensional HIV/AIDS model: dT = λ − ρT − βT V , T (0) = T0 , dt dT ∗ = βT V − δT ∗ , T ∗ (0) = T0∗ , dt dV = N δT ∗ − cV , V (0) = V0 , dt

(1) (2) (3)

where T , T ∗ and V represent the number of uninfected cells, infected cells and virions, respectively, λ is the regeneration rate of T cells, ρ is the death rate of T cells, β is the rate at which T cells become infected, δ is the death rate of infected T cells, N is the number of new virions produced by each infected cell during its life-time, and c is the clearance rate of free virions. Let θ = (λ, δ, ρ, β, N, c) denote all the system parameters. Here we assume that only the measurements of V are available. Identifiability is a basic system property and determines whether all parameters can be uniquely estimated based on measured outputs. For a formal definition of identifiability of a dynamic system, see Conte et al. (1999) and Glad (1997). For the identifiability of nonlinear systems, see Tunali and Tarn (1987), Diop and Fliess (1991), Ljung and Glad (1994). Xia and Moog (2003) and Jeffrey and Xia (2005) investigated the algebraic identifiability of various HIV/AIDS dynamic models. The basic idea of algebraic identifiability is to allow one to identify parameters by solving algebraic equations based only on the initial values and the measurements of output variables. The system (1)–(3) is said to be algebraically identifiable if there exists a time t ∗ , a positive integer k, and a function Φ : R 6 × R 6(k+1) → R 6 such that det

∂Φ = 0 ∂θ

(4)

and   Φ θ, V , V˙ , . . . , V (k) = 0

(5)

hold on [0, t ∗ ], where V˙ , . . . , V (k) are the derivatives of V with respect to t . We call Eq. (5) the identification equation and the function Φ(·) the identification function. To formulate an identification function, we may need to eliminate the unobserved (latent) state variables from the original system by taking higher order derivatives of the output (observation) variables. For our 3D HIV dynamic model, based on Eqs. (2) and (3), we have for the second derivative V¨ = N δ T˙ ∗ − cV˙ = N δ(βT V − δT ∗ ) − cV˙ = N δβT V − δcV − (δ + c)V˙ . Continuing to take the 3rd-order derivative and combining it with Eq. (1), we obtain V (3) = N δβ(T˙ V + T V˙ ) − δcV˙ − (δ + c)V¨   = N δβT V˙ − ρV − βV 2 + N δβλV − δcV˙ − (δ + c)V¨

(6)

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   = V −1 V˙ − ρ − βV V¨ + (δ + c)V˙ + δcV + N λδβV − δcV˙ − (δ + c)V¨ ,

(7)

an equation that does not depend on the unobservable (latent) state variables T and T ∗ . A close examination of Eq. (7) reveals that the parameters λ and N only appear in the equation as N λ. This fact suggests that λ and N are indistinguishable and only the product N λ can be identified when only the measurements of viral load V are available. We reparameterize the system parameters as β = β, ρ = ρ, ν = δc, μ = δ + c, η = N λβδ. There is a one-to-one mapping between (β, δ, ρ, c, N λ) and θ ∗ = (β, ρ, ν, μ, η). Denote the right side of (7) as f (t, θ ∗ , V , V˙ , V¨ ), that is,   f (t, θ ∗ , V , V˙ , V¨ ) = V −1 V˙ − ρ − βV (V¨ + μV˙ + νV ) + ηV − ν V˙ − μV¨ .

(8)

Then, we have ∂f = −V (V¨ + μV˙ + νV ), ∂β ∂f = −(V¨ + μV˙ + νV ), ∂ρ ∂f = V, ∂η   ∂f = V˙ V −1 V˙ − ρ − βV − V¨ , ∂μ

(9)

∂f = −ρV − βV 2 . ∂ν To identify the five parameters θ ∗ = (β, ρ, ν, μ, η) or equivalently (δ, ρ, β, c, N λ), five identification equations are needed. Two methods, which will be introduced in the following subsections, can be employed to construct the identification function based on Eqs. (7) and (8). 2.1. Higher-order derivative (HOD) method The basic idea of the method of Xia and Moog (2003) and Jeffrey and Xia (2005) is to eliminate all latent (unobservable) state variables by taking the higher order derivatives of the output variables, so that a set of identification functions of unknown parameters can be generated. For the model (1)–(3) considered here, following the procedure described in Xia and Moog (2003) and Jeffrey and Xia (2005), we construct   Φ0 = V (3) − f, V (4) − f˙, . . . , V (7) − f (4)  = 0. Then, if det

∂Φ0 = 0, ∂θ ∗

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θ ∗ = (β, ρ, ν, μ, η) can be identified according to the implicit function theorem. That is, if ⎛  Rank

∂Φ0 ∂θ ∗



∂f ∂β

⎜ (1) ⎜ ∂f ⎜ ∂β ⎜ (2) ⎜ = Rank ⎜ ∂f∂β ⎜ (3) ⎜ ∂f ⎜ ∂β ⎝ ∂f (4) ∂β

∂f ∂ρ

∂f ∂η

∂f ∂μ

∂f ∂ν

∂f (1) ∂ρ

∂f (1) ∂η

∂f (1) ∂μ

∂f (1) ∂ν

∂f (2) ∂ρ

∂f (2) ∂η

∂f (2) ∂μ

∂f (2) ∂ν

∂f (3) ∂ρ

∂f (3) ∂η

∂f (3) ∂μ

∂f (3) ∂ν

∂f (4) ∂ρ

∂f (4) ∂η

∂f (4) ∂μ

∂f (4) ∂ν

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ = 5, ⎟ ⎟ ⎟ ⎠

(10)

we can identify the five parameters in the model. The identification function Φ0 involves the 7th order derivative of V which requires at least 8 measurements of V to evaluate. When this method is applied to a high dimensional parameter space like the one we are 0 considering here, the matrix ∂Φ usually becomes very complicated to calculate and its ∂θ ∗ rank is difficult to evaluate. For example, one element in the matrix above is,  ∂f (4) = V (−5) βV (5) V 6 + 10β V¨ V (3) V 5 + 5β V˙ V (4) V 5 + ρV (5) V 5 + V (6) V 5 ∂μ  2 − 6 V (3) V 4 − 8V¨ V (4) V 4 − 2V˙ V (5) V 4 + 12V¨ 3 V 3 + 44V˙ V¨ V (3) V 3  + 9V˙ 2 V (4) V 3 − 78V˙ 2 V¨ 2 V 2 − 32V˙ 3 V (3) V 2 + 84V˙ 4 V¨ V − 24V˙ 6 . Thus, it is not easy to evaluate the rank of the above matrix. To avoid such complexity and evaluation of the high order derivatives, an alternative method for formulating the identification functions Φ(·) is proposed below. 2.2. Multiple time points (MTP) method Suppose we have the quantities (V , V˙ , V¨ , V (3) ) at five distinct time points t1 , . . . , t5 . Denote the values of (V , V˙ , V¨ ) at t = ti as (Vi , V˙i , V¨i ) for i = 1, . . . , 5. Let f1 = f (t1 , θ ∗ , V1 , V˙1 , V¨1 ), . . . , f5 = f (t5 , θ ∗ , V5 , V˙5 , V¨5 ). By (7) and (8), we have   Φ1 = V1(3) − f1 , . . . , V5(3) − f5  = 0.

(11)

If ⎛ ∂f1 ⎜ ⎜ ⎜ ∂Φ1 ⎜ det ∗ = det ⎜ ⎜ ∂θ ⎜ ⎝

∂β ∂f2 ∂β ∂f3 ∂β ∂f4 ∂β ∂f5 ∂β

∂f1 ∂ρ ∂f2 ∂ρ ∂f3 ∂ρ ∂f4 ∂ρ ∂f5 ∂ρ

∂f1 ∂η ∂f2 ∂η ∂f3 ∂η ∂f4 ∂η ∂f5 ∂η

∂f1 ∂μ ∂f2 ∂μ ∂f3 ∂μ ∂f4 ∂μ ∂f5 ∂μ

∂f1 ∂ν ∂f2 ∂ν ∂f3 ∂ν ∂f4 ∂ν ∂f5 ∂ν

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ = 0, ⎟ ⎟ ⎠

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by the implicit function theorem, there is a unique solution of θ ∗ to Eq. (11). Assuming that β = 0 and using the derivatives (9), some algebra shows that the rank of (∂Φ1 /∂θ ∗ ) is equal to the rank of ⎛

V1 (V¨1 + μV˙1 ) ⎜ V (V¨ + μV˙ ) ⎜ 2 2 2 ⎜ ¨ ˙ Σ =⎜ V ( V + μ V 3) ⎜ 3 3 ⎜ ¨ ˙ ⎝ V4 (V4 + μV4 ) V5 (V¨5 + μV˙5 )

V¨1 + μV˙1 V¨2 + μV˙2

V1

V12

V2

V¨3 + μV˙3 V¨4 + μV˙4

V22

V3

V32

V4

V¨5 + μV˙5

V42

V5

V52

⎞ V˙1 (V1−1 V˙1 − ρ − βV1 ) − V¨1 V˙2 (V2−1 V˙2 − ρ − βV2 ) − V¨2 ⎟ ⎟ ⎟ V˙3 (V3−1 V˙3 − ρ − βV3 ) − V¨3 ⎟ ⎟ . (12) ⎟ −1 ˙ ˙ ¨ V4 (V4 V4 − ρ − βV4 ) − V4 ⎠ V˙5 (V5−1 V˙5 − ρ − βV5 ) − V¨5

As long as det(Σ) = 0, we have det(∂Φ1 /∂θ ∗ ) = 0. Note that for evaluating V (3) at one time point, at least four measurements of V are needed. In order to form the five identification equations (11), at least eight measurements of V are necessary. This conclusion is consistent with that of the HOD method. Note that this model is more likely to be locally identifiable instead of globally identifiable since Σ also contains unknown parameters. Compared to the HOD method, the multiple time points (MTP) method is less computationally intensive and easier to implement since only the lower-order derivatives (the 3rd or lower order derivatives in our case) of V need to be evaluated. In addition, whether the matrix Σ is of full rank cannot be judged by direct observation or simple algebraic operations on the matrix (12). One practical method is to use computer simulations to numerically evaluate the rank of Σ . We may simulate the output variable V (t) at a sequence of time points from the dynamic model (1)–(3) with fixed parameters. The higher order derivatives of V (t) can be estimated using local polynomial or other smoothing methods based on the simulated data of V (t). Then we can numerically calculate the determinant of Σ from (12) to check whether it is of full rank for given parameter values. If we repeat this calculation over a grid of parameter values in biologically reasonable ranges and show that Σ is of full rank, then we may claim that Σ is of full rank and the associated parameters are identifiable, at least locally, in the range of parameter values used for the simulations. Although this is an ad hoc method to check the rank of Σ , it is practical and useful. Similarly, we may examine the rank of the matrix in (10) for the HOD method, but the higher order derivatives need to be estimated and evaluated, which is computationally tedious and expensive. Jeffrey and Xia (2005) also studied the identifiability of the 3D HIV model (1)–(3), but they assumed that both viral load (V ) and total CD4+ T cell count (T + T ∗ ) are observed. In this case, they showed that all six parameters are identifiable. We summarize our results together with those of Jeffrey and Xia (2005) in Table 1. 2.3. Effect of initial values of state variables on parameter identifiability In the above discussion, the initial value of the output variables (V or T + T ∗ ) is assumed to be known and is the first available measurement. The initial values of unobservable (latent) state variables are assumed to be unknown. However, in practice, we may know the initial states exactly for some of the unobservable (latent) state variables. In this case, the additional information of the initial values of unobservable (latent) state variables may help to identify more parameters. For example, in our 3D HIV dynamic model with

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Table 1 Algebraic identifiability of the 3D HIV model Fixed parameters

Identifiable parameters

Variables and minimum # of measurements required

N λ None None

(β, ρ, δ, λ, c) (β, ρ, δ, c, N ) (β, ρ, δ, c, N λ) (β, ρ, δ, c, N , λ)

V V V V V

:n=8 :n=8 :n=8 : n = 5 and T + T ∗ : n = 4 or : n = 4 and T + T ∗ : n = 5

the only output variable V , we can identify five parameters. However, if we know the initial value of the unobservable (latent) state variable T ∗ , say, T ∗ (0) = T0∗ , we are able to identify all six parameters in the model. To see this, assume that the initial values of the output variable and its derivative are known, i.e., V (0) = V0 and V˙ (0) = V˙0 . From Eq. (3), we have V˙0 = N δT0∗ − cV0 . By combining this equation with the other five identification equations of the model, we can identify all the six parameters β, ρ, δ, c, N and λ. We need to acknowledge that there is a difference between mathematical (algebraic) identifiability and statistical identifiability in the presence of measurement error. In particular, when the initial values of (observable) output variables are available but measured with error, the true initial values of the output variables are still unknown and need to be estimated from the noisy data. In this case, the unknown initial values of the (observable) output variables do not affect the mathematical (algebraic) identifiability of the unknown parameters, and we may treat the unknown initial values as additional parameters that can be estimated from the data. However, since the initial values are measured with noise, more data are required to get the desired estimation accuracy of the unknown parameters compared to the case when the initial values are exactly known. For a dynamic system with unobservable (latent) state variables such as the case of the 3D HIV dynamic model (1)–(3), when the initial values of the unobservable (latent) state variables are unknown, we may not be able to directly use the original dynamic system to estimate the identifiable but unknown parameters. In this case, we may need to employ the identification equations to obtain parameter estimates. For example, we can use Eq. (7) to deal with this problem for the 3D HIV dynamic system (1)–(3) when only viral load (V ) is observable. To do this, we rewrite Eq. (7) as dV = V˙ , V (0) = V0 , dt d V˙ = V¨ , V˙ (0) = V˙0 , dt    d V¨ = V −1 V˙ − ρ − βV V¨ + (δ + c)V˙ + δcV dt + N λβδV − δcV˙ − (δ + c)V¨ , V¨ (0) = V¨0 .

(13) (14)

(15)

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This system is solvable if the initial values of the output (observable) variable V and its derivatives are available. Thus, the identifiable parameters can be estimated from this system of equations and the initial values of the unobservable (latent) state variables (T and T ∗ ) are not needed. If the initial values V0 , V˙0 and V¨0 are not known exactly, they can be treated as additional parameters to be estimated from the data.

3. Statistical estimation and simulation validation In the section above, we studied the mathematical (algebraic) identifiability of the dynamic model (1)–(3). The minimum number of measurements of the output variables was determined in order to identify the unknown parameters in the system (Table 1). Only in the ideal case with no measurement error, the minimum numbers of measurements are sufficient to determine the identifiable unknown parameters uniquely. In practice, all measurements are measured with error. Consequently, more data are needed to circumvent the measurement noise to estimate the unknown parameters with desired accuracy. A standard and simple estimation method is to use the least squares principle to estimate the unknown parameters in the dynamic model. For our 3D HIV dynamic model (1)–(3) with the only output variable V , the observation equation is y(ti ) = log10 V (ti ; θ ) + i ,

i = 1, . . . , n,

(16)

where the log10 -scale is conventionally used in AIDS clinical studies, V (ti ; θ ) is the solution to Eqs. (1)–(3) at time ti which is a function of unknown parameters, and y(ti ) is the corresponding measurement of log10 V (ti ; θ ). The measurement errors i ’s are usually assumed to be Gaussian and independent with mean 0 and variance σ 2 . In this case, the maximal likelihood estimate of the parameters is the same as the nonlinear least squares (NLS) estimate, which can be obtained by minimizing the following objective function SSR =

n  

2 y(ti ) − log10 V (ti ; θ ) .

(17)

i=1

To minimize and evaluate the above objective function, we need to solve the initial value problem (1)–(3). As we have discussed at the end of Section 2.3, this is not doable since the initial values of the unobserved (latent) state variables T and T ∗ are not available. Instead, we can use the system (13)–(15). The initial values of V , V˙ and V¨ can be obtained by smoothing the data of y(t) using local polynomial or other smoothing methods. We can also treat the initial values, V0 , V˙0 and V¨0 , as additional unknown parameters that can be estimated from the data. As we have discussed above, this does not affect the identifiability of other dynamic parameters. In the following, we perform computer simulation studies to validate the identifiability results from the theoretical analysis in the previous section. First, we simulate the observed data from the 3D HIV dynamic model (1)–(3). We choose parameter values based on previous HIV dynamic studies (Huang et al., 2003, 2006), i.e., (β, ρ, δ, c, N, λ) = (0.00002, 0.15, 0.55, 5.5, 900, 80). The initial values of state variables are taken as (T0 , T0∗ , V0 ) = (180, 20, 50000). A sequence of viral load data are generated at time points t = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 2, 4, 6, . . . , 60 days. We add

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Table 2 Estimation results for the case with known initial conditions and σ = 0.01 Parameters

Est. value

Std. error

True value

log(β) log(ρ) log(δ) log(c) log(N ) log(λ)

−10.83 −2.02 −0.67 1.70 6.85 4.27

0.022 0.090 0.049 0.015 0.032 0.070

−10.82 −1.90 −0.60 1.70 6.80 4.38

Residual standard error: 0.0089 on 30 degrees of freedom

Gaussian measurement noise to the generated data based on the output model (16). To guarantee the positivity of the parameters, we reparameterize all the parameters as ˜ λ˜ ) ˜ ρ, ˜ c, (β, ˜ δ, ˜ N,   = log(β), log(ρ), log(δ), log(c), log(N ), log(λ) = (−10.81978, −1.89712, −0.597837, 1.704748, 6.802395, 4.382027). Case 1: T0 , T0∗ and V0 Known. First we assume that the initial conditions T0 and T0∗ as well as V0 are exactly known. In this case, as we have discussed above, all the six parameters are identifiable. The noise level is set to σ = 0.01. Then we apply the least squares method to the model (1)–(3) by numerically solving the differential equations and obtain the parameter estimates. We plot the model fitting results in Fig. 1, which displays the generated noisy viral load data (circles), the generated true viral load trajectory (solid line), and the fitted viral load curve (dashed line). The parameter estimation results are reported in Table 2. From Fig. 1 and Table 2, we see, as expected, that the model fitting is very good and all six unknown parameters are well identified and accurately estimated. Here we used 36 data points and the measurement noise was set to be small. For comparison, we increase the data noise level, i.e., σ = 0.10, in the generated data, and repeat the least squares model fitting and parameter estimation procedure. We present the results for this case in Fig. 2 and Table 3. In this case, the estimated viral load curve still reasonably follows the true viral load trajectory, but Table 3 shows that the variations (standard error) of parameter estimates are significantly larger compared to the case with the lower noise. The convergence of the parameter estimates is still very good in this large data noise case. Case 2: T0 and T0∗ Unknown but V0 Known. Now we assume that the initial conditions T0 and T0∗ are unknown, but V0 is exactly known. In this case, only five out of the six original dynamic parameters can be identified. Parameters N and λ cannot be distinguished and only their product (N λ) can be identified. We fix the parameter N = 450. Since the initial values of the latent state variables T0 and T0∗ are unknown, we cannot use the original dynamic model (1)–(3), instead we have to use Eqs. (13)–(15) to solve for V . Here we also assume that V˙0 and V¨0 are unknown and will be estimated from the ˙ ˙ data. To stabilize the estimates, we reparameterize V˙0 and V¨0 as V˜˙ 0 = (eV0 − e−V0 )/2 ¨ ¨ and V˜¨ 0 = (eV0 − e−V0 )/2. We set the noise level as σ = 0.01. We apply the least squares

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Fig. 1 Initial values T0 , T0∗ and V0 are known and σ = 0.01. The circles represent observed viral load data on log10 -scale, the solid line represents true viral load trajectory (solution to the HIV dynamic model), and the dashed line represents the estimated viral load curve.

Fig. 2 Initial values T0 , T0∗ and V0 are known and σ = 0.10. The circles represent observed viral load data on log10 -scale, the solid line represents true viral load trajectory (solution to the HIV dynamic model), and the dashed line represents the estimated viral load curve.

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Table 3 Estimation results for the case with known initial conditions and σ = 0.10 Parameters

Est. value

Std. error

True value

log(β) log(ρ) log(δ) log(c) log(N ) log(λ)

−11.03 −4.05 −1.41 1.62 7.40 3.00

0.23 2.97 0.45 0.14 0.34 0.73

−10.82 −1.90 −0.60 1.70 6.80 4.38

Residual standard error: 0.10 on 30 degrees of freedom

Table 4 Estimation results for the case with unknown initial conditions and σ = 0.01 Parameters

Est. value

Std. error

True value

V˜˙ 0 V˜¨ 0 log(β) log(ρ) log(δ) log(c) log(λ)

−13.18 14.94 −10.58 −2.24 −0.95 1.73 4.81

0.015 0.033 0.72 0.79 0.81 0.051 0.60

−10.82 −1.90 −0.60 1.70 4.38

Residual standard error: 0.0091 on 29 degrees of freedom

estimation method to fit model (13)–(15) to the noisy viral load data. The model fitting results are presented in Fig. 3. The fitted viral load curve (dashed line) tracks the true viral load trajectory (solid line) very well. The parameter estimation results are reported in Table 4 from which we can see that the variations (standard error) of the parameter estimates are much larger than those for the case with the known initial values but with the same measurement noise level (Table 2). This is consistent with our discussion in Section 2.3 since it costs more to estimate the unknown initial values in addition to the unknown parameters. But all five identifiable parameters and the unknown initial values are well estimated as we have expected based on the above theoretical analysis and discussions. Although the estimates listed in Tables 2–4 are from a single simulation run, our further simulation studies (data not shown) indicate that the mean estimates converge to the true values rapidly as the sample size increases (e.g., for 100 simulation runs with σ = 0.10, the average estimation error is less than 3.3% compared to the true values). Note that the estimate of λ depends on the values of N . But the estimates of parameters β, ρ, δ and c are not sensitive to the fixed value of N . That is, when we fix N to different values, the change in the estimates of β, ρ, δ and c is very small. We also tried to increase the measurement noise level and refit the model. But in this case with unknown initial conditions, convergence is hard to achieve. This indicates that it is not easy to identify the four parameters β, ρ, δ, c and the product (N λ) in practice with noisy data when the initial conditions of the state variables are unknown, although they are theoretically identifiable.

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Fig. 3 Initial values T0 and T0∗ are unknown and σ = 0.01. The circles represent observed viral load data on log10 -scale, the solid line represents true viral load trajectory (solution to the HIV dynamic model), and the dashed line represents the estimated viral load curve.

4. Model fitting to clinical data The 3D model (1)–(3) has been used to fit the data from subjects with HIV primary infection by Stafford et al. (2000) to estimate HIV dynamic parameters. For comparisons, we fitted the model (1)–(3) to clinical data set from HIV-1 infected patients with antiretroviral treatments. In this clinical study, a four-drug antiretroviral regimen along or in combination with an immune-based therapy were administered to HIV-1 infected subjects who were antiretroviral naive. Frequent viral load measurements were performed for the first week after initiating the antiretroviral regimen: 14 measurements during the first day, 5 measurements during the second day, 4 measurements during the third day, and then days 6 and 10, weeks 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 24, 28, 32, 36, 40, 44 and 48. Notice that the long-lived infected cells or latently infected cells (Perelson et al., 1997; and Perelson and Nelson, 1999) are not considered in the model (1)–(3). Thus, we only used the viral load data for the first week to fit the model (1)–(3). To compare our results to those of Stafford et al. (2000), we used the same assumption c = 3. In addition, the proliferation rate (λ) of uninfected T cells was assumed to be known since it cannot be distinguished from parameter N . We set λ to be the median estimate from Stafford et al. (2000). The four other parameters (ρ, β, δ, N ) were estimated from the viral load data with unknown initial values of T and T ∗ . Stafford et al. (2000) employed the Nelder–Mead simplex method to estimate parameters, which is sensitive to starting values (Adanu, 2006). For ordinary differential equation models, the least square objective function may have multiple local minima, in which the simplex method can be easily trapped. Global search methods such as the differential evolution method (Storn and Price, 1997) are more suitable for such problems. The advantages of the differential evolution method compared to the simplex method were also

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Table 5 Estimated parameter values for 6 HIV-1 infected patients with antiretroviral treatment. The same assumption about the virus clearance rate (c = 3) as in Stafford et al. (2000) was used. The proliferation rate of uninfected T cells (λ) was assumed to be the median estimate in Stafford et al. (2000) Patient

ρ

β

δ

N

1 2 3 4 5 6 Median Sample SD

0.127 0.659 0.286 0.439 0.542 0.698 0.459 0.221

9.14E−05 1.29E−05 1.25E−05 8.06E−05 4.04E−05 1.43E−05 4.20E−05 3.58E−05

0.517 0.666 0.550 0.405 0.218 0.745 0.517 0.188

753 629 728 523 456 435 587 137

discussed by Adanu (2006), and the differential evolution method was used in fitting our data. Also notice that, for the differential evolution method, the constraints on parameters and initial values can be easily incorporated into the minimization procedure. The model fitting results are reported in Table 5. Comparing our results to those in Stafford et al. (2000), we found that the death rate of uninfected T cells (ρ) was one order of magnitude greater than the estimate in Stafford et al. (2000). However, the infection rate (β) was one order of magnitude smaller than that of Stafford’s estimates. In addition, our estimate of the number of virions (N ) produced by each infected T cell during its life-time was also much smaller than the estimates obtained by Stafford et al. (2000) and the estimated death rate of infected cells (δ) was larger from our study compared to that in Stafford et al. (2000). These results may indicate that the turnover rates of both uninfected cells and infected cells are faster in HIV infected individuals with antiretroviral treatment compared to those subjects with primary HIV infection without treatment. Moreover, due to antiviral treatment effect, the infection rate (β) and the number of virions (N ) produced by each infected cell are also smaller in patients with antiviral treatment compared to subjects with primary HIV infection. However, we also need to notice the limitations of our estimates that include the number of subjects is quite small (only 6 patients), the frequent viral load measurements are subject to large noise and several critical assumptions were imposed regarding the model and parameter values. 5. Discussion and conclusion Stafford et al. (2000) also studied the identifiability of the 3D HIV dynamic model using a simple variable transformation argument. Similarly, they found that λ and N cannot be identified simultaneously. The method they used was to let T = T (t), T ∗ = T ∗ (t), and V = V (t) be the solution to Eqs. (1)–(3). Then for any positive α, they let T˜ = αT , T˜ ∗ = αT ∗ , and V˜ = V , and showed that (T˜ , T˜ ∗ , V ) is the solution to the following initial value problem d T˜ = λ˜ − ρ T˜ − β T˜ V , T˜ (0) = αT0 , dt d T˜ ∗ = β T˜ V − δ T˜ ∗ , T˜ ∗ (0) = αT0∗ , dt

(18) (19)

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dV = N˜ δ T˜ ∗ − cV , dt

V (0) = V0 ,

(20)

where λ˜ = αλ, and N˜ = N/α. This is the exact same model as model (1)–(3). Thus, if only the viral load (V ) is measurable, parameters λ and N cannot be identified uniquely and only their product (λN ) can be identified from the viral load data. In addition, we also see from this argument that the latent state variables T and T ∗ cannot be uniquely identified. This simple variable transformation argument can be used to evaluate the identifiability of some specific dynamic models, but it is not generalizable to systematically study the identifiability of nonlinear dynamic models. In this paper, we have used a technique from engineering (Xia and Moog, 2003; Jeffrey and Xia, 2005) to investigate the algebraic identifiability of a popular threedimensional HIV/AIDS dynamic model. We found that not all six parameters in the model can be identified if only the viral load is measured—the parameters N and λ cannot be distinguished and only their product (λN ) is identifiable. This conclusion is consistent with an earlier ad hoc argument by Stafford et al. (2000). We have used a notion of an identification function and an identification equation, and proposed the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, 2003; Jeffrey and Xia, 2005). The newly proposed MTP method has some advantages over the HOD method in practical implementation. We also studied the effect of initial values of state variables on the identifiability of unknown parameters. We found that the initial values of the output (observable) variables are part of the data required to identify the unknown parameters, and they do not affect the identifiability of parameters if the exact initial values are not known but measured with error. However, these noisy initial values may increase the estimation error of the unknown parameters. The exact initial values of the output (observable) variables can be treated as additional parameters that can be estimated from data. If the initial values of the latent state variables are exactly known, the additional parameters may be identified. To validate the identifiability analysis results, we performed simulation studies to estimate the unknown parameters and initial values from noisy data. The simulation results confirmed our theoretical identifiability analysis. Although our analysis has focused on a 3D HIV dynamic model, the basic ideas and the proposed methodologies are generally applicable to any nonlinear dynamic systems. Finally, the 3D HIV model was fitted to a clinical data set to estimate the dynamic parameters in HIV-1 infected individuals administered with antiretroviral treatment. The death rates of uninfected T cells (ρ) and infected T cells(δ), the infection rate (β), and the number of virions produced by each infected T cell (N ) were estimated. We compared our estimates with those in HIV infected subjects with primary HIV infection and without treatment (Stafford et al., 2000). We found that the turnover rates of uninfected and infected cells were faster, and the infection rate and the number of virions produced by each infected T cell were smaller for subjects with antiretroviral treatment compared to those with primary HIV infection and without antiviral treatment (Stafford et al., 2000). However, more studies with careful design may be needed to confirm these results. Acknowledgement This work was partially supported by NIAID/NIH research grants AI052765, AI055290, AI27658 and AI 50020.

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