fitting polyexponentials and quasipolynomials to hiv viral ... · fitting polyexponentials and...

Fitting Polyexponentials and Quasipolynomials to HIV Viral Load Data

L.W. Huson 1 , J. Chung 2, M. Salgo 2 Biostatistics and Clinical Science Groups, F.Hoffman-La Roche, Welwyn, UK1 and Nutley, NJ2

Introduction Measurements of HIV1-RNA plasma concentrations are an important method of assessing patient response to anti-HIV1 treatments, and in most clinical trials of such treatments these plasma concentrations are assessed at regular intervals of time. HIV1-RNA plasma levels in successfully treated patients tend to follow a standard pattern, which shows a relatively rapid early decline in viral load, followed by a period of much slower continuing decline or a steady level ( see e.g. Havlir and Richman, 1996; Huang et. al. 2001; Hatzakis et. al., 2001; ) There is much theoretical work on the dynamics of HIV1 infections which may provide explanations of why the decline in HIV1-RNA plasma concentrations exhibits this type of pattern – often referred to as a “biphasic decline” (e.g. Perelson and Nelson, 1999; Wu and Ding, 1999; Ding and Wu, 1999). Fitting nonlinear regression models to these patterns of declining HIV1-RNA levels can be of value in comparing different treatment regimes and in predicting treatment outcome. Simple exponential-decline models can give an adequate fit to the typical pattern of decreasing HIV1-RNA plasma levels, but we have explored the extent to which curve-fitting can be improved by using some less commonly encountered nonlinear regression models. Specifically, we describe here the fitting of polyexponential, multiple polyexponential, and quasipolynomial regression models to longitudinal HIV1-RNA plasma data collected in two recent trials of the novel anti-HIV1 treatment Fuzeon®. Description of Trials Studies T20-301 and T20-302 (also know as TORO 1 and TORO 2, respectively) were randomized, open-label, active-controlled, parallel-group, multicenter Phase III studies designed to assess the efficacy and safety of the HIV1 fusion-inhibitor enfuvirtide (also known as Fuzeon®) in combination with an optimized background (OB) regimen of anti-retroviral agents. In both studies, patients were randomized to one of two treatment groups - ENFUVIRTIDE+OB or OB alone - in a 2:1 ratio. Randomization was stratified according to the patients’ screening values of HIV1-RNA ( ≥40,000 copies/ml, <40,000 copies/ml) and whether or not the patient was taking investigative anti-HIV1 drugs. The choice of OB regimen was individualized by the investigator and the patient prior to randomization

based on genotypic and phenotypic resistance tests. The two studies were essentially identical in design; T20- 301 recruited patients primarily from the USA, Canada, Brazil and Mexico, and T20-302 from Western Europe and Australia. Full details of the two studies are reported by Lalezari et. al. (2003) and Lazzarin et. al. (2003). Data A total of 995 patients were recruited into the two studies, and 661 of these were treated with Fuzeon®. All regression analyses reported here used data from the 661 Fuzeon®-treated patients, pooled across the two studies. Each patient had HIV1-RNA plasma concentrations measured at 2, 4 and 8 week intervals over the first 48 weeks of the trials, or until withdrawal from the trial, if earlier. A total of 7177 individual HIV1-RNA plasma concentrations were used in the regression modeling reported here. HIV1-RNA plasma values were log-transformed for all regression analyses. Stretched Exponential Model The pattern of HIV1 data observed in the two TORO trials (pooled) is illustrated in Figure 1, which shows mean values of HIV1-RNA plasma concentrations across all patients at each assessment time. Clearly, a suitable initial approximation to the observed data can be made using a simple stretched-exponential model of the form:

yt = y0 * exp((-t/c1)**c2) ) where y0 is the log10 HIV1 value at time zero, yt is the log10 HIV1 value at time t, and c1 and c2 are constants. This fitted curve is shown in Figure 1. The fitted curve is:

logHIVt = logHIV0 * exp((-t/76.07)**0.327) Fitting of this standard model is easily accomplished using non-linear regression software: for this and all other regression models described here, we used SAS® PROC NLIN. The fit of this simple stretched-exponential model is adequate, but we wished to explore whether a better fit to the observed pattern of HIV1 plasma levels could be obtained using some less well known forms of nonlinear regression model – multiple polyexponentials

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Proceedings of the American Statistical Association 2005, Biopharmaceutical Section 559

and quasipolynomials. These model types are little used in general nonlinear regression analyses, though their flexible structures make them potentially suitable for a wide range of applications. We therefore fitted a selection of such models to the observed HIV1 data, and this report compares these models and comments on the practicalities of fitting them using standard software. We compare the fit of the models using a variety of criteria, and we study the sensitivity of the model solutions to the values of initial parameter estimates. Polyexponential models Simple polyexponential models are regression models of the form:

yt = ∑i wi * y0 * exp(-cit) where, in our application, y0 is the log10 HIV1 value at time zero, yt is the log10 HIV1 value at time t, the ci are constants and the wi are constant weights i.e. the dependent variable is modeled as a weighted sum of exponentials. The number of exponential terms included in these models can be varied, and the weights varied, in order to improve fit, subject to the usual considerations of over-parameterization. Models of this form are also referred to in the literature as sum-of-exponential models, and are widely used in pharmacokinetic studies in the investigation of compartmental systems for drug metabolism. They are not, however, often used as general nonlinear regression models, though work based on the population dynamics of HIV suggests that models based on specific forms of sums-of-exponentials have some theoretical justification (e.g. Perelson and Nelson, 1999; Wu and Ding, 1999, Ding and Wu, 1999). Such specific models have been applied to HIV data by, amongst others, Notermans et. al. (1998), Melvin et. al. (1998), Luzuriaga et. al. (2000), Lindback et. al. (2000), Nelson et. al. (2001), and Markovitz et. al. (2003). Multiple polyexponential models are defined here as simple extensions to the above: functions of several predictor variables in addition to the values of t and y0. They are in this sense analogous to standard multiple regression models. The general form of a multiple polyexponential is:

yt = ∑i wi * y0 * exp(-ci xi t) where the xi are independent predictor variables. Models of this form appear to have been little-used in nonlinear regression applications, and have not previously been applied to HIV data.

Quasipolynomial models Quasipolynomials are variously defined in the mathematical literature, depending upon the field of application. In the present context, we define a quasipolynomial as a generalization of the above exponential construction, using the form:

yt = (y0 * exp(-c1 t)) + ( c2*t* exp(-c3 t)) + ( c4*t2* exp(-c5 t)) +…

where, in our application, y0 is the log10 HIV1 value at time zero, yt is the log10 HIV1 value at time t, and the ci are constants i.e. the successive exponential terms in the sum are multiplied by polynomial terms in time. This definition accords, for example, with the quasipolynomial forms used in different contexts (not regression modeling) by Boltyanski and Gorelikova (1996), by Vale (2004), and by Szekelyhidi (2004). Other definitions of the quasipolynomial which appear in the mathematical literature include polynomial terms multiplied by cyclical functions of an independent variable, and polynomial terms multiplied by any arbitrary function of an independent variable. Again, models of this form appear to have been little-used in nonlinear regression applications, and have not before been applied to HIV data. Strategy for Model Fitting Although polyexponential forms are very flexible, and hence potentially widely applicable, they are nonlinear forms of a type which are known to be difficult to fit (see, e.g. Seber & Wild (2003); Petersson & Holmstrom (1998)), and care is required in estimating the parameters of these models. Since the quasipolynomials we fitted are of a similar general structure to the polyexponentials, we also anticipated possible problems in fitting these model forms. All of the nonlinear models described in this report were fitted to our HIV1-RNA data using SAS® PROC NLIN, using the GAUSS estimation algorithm provided as the default by PROC NLIN. We selected 10 different sets of initial parameter values for each curve type, and examined the solutions found for each set of starting values. Initial parameter values were selected partly empirically, by finding values which yielded curves broadly fitting the observed mean HIV1-RNA values, and partly using the exponential-peeling technique (see Seber & Wild (2003) p. 407). For each model, we compared the solutions found in terms of the residual mean squared error (MSE) of the fitted regression. Any initial parameter values which resulted in a failure to

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converge were re-run using the MARQUARDT algorithm option in PROC NLIN. We assessed model fit using a variety of criteria, as described below. Fitting two and three part polyexponentials Our first attempt to improve on the fit of the simple stretched-exponential model involved fitting polyexponentials with two and with three exponential terms (i.e. biexponentials and triexponentials). Figure 2 shows the predicted mean values derived from these fitted polyexponential curves, and it is visually clear that the 3-term polyexponential is a reasonable fit to the mean HIV1-RNA values, whereas the two-term polyexponential is not. The fitted curves are: biexponential:

logHIVt = 0.2842 * logHIV0 * exp(-1.5191*t) + 0.7119 * logHIV0 * exp(-0.0139*t)

triexponential:

logHIVt = 0.2309 * logHIV0 * exp(-2.7220*t) + 0.1167 * logHIV0 * exp(0.0202*t) + 0.6484 * logHIV0 * exp(-0.0330*t)

Fitting a multiple polyexponential Another option for fitting the HIV1-RNA data is to use a polyexponential model which includes additional independent variables which are known to be related to HIV1-RNA plasma levels. This amounts to a multiple regression analysis using a nonlinear model form. The covariates chosen for inclusion in our multiple polyexponential model were those which have, in previous analyses, been found to be statistically significant predictors of HIV1-RNA plasma levels: the baseline CD4+ cell count (BLCD4), the number of anti-HIV1 drugs to which the patient is sensitive (PSSCOV), the number of anti-HIV1 drugs previously taken by the patient (PARVN), and the total number of positive prognostic factors for the patient (NPPF) (Lalezari et. al. 2002; Lange et. al. 2002). The four component exponentials in the model were given equal weights in our analysis. Figure 3 shows the predicted mean values derived from the fitted multiple polyexponential, which appears visually to fit the observed mean values at least as well as the triexponential curve shown in Figure 2. The fitted equation is:

logHIVt = 0.25 * logHIV0 * exp(-0.002*BLCD4*t) + 0.25 * logHIV0 * exp(-0.0316*PSSCOV*t) + 0.25 * logHIV0 * exp(-0.2096*PARVN*t) + 0.25 * logHIV0 * exp(0.00128*NPPF*t)

Fitting a quasipolynomial The final nonlinear model which we studied was the quasipolynomial form. We selected this since we believed that such a form might give a better fit to nonlinear curves which have unusual features, and we were interested particularly in improving the fit of our model to the form of the observed HIV1-RNA data between weeks 2 and 10, where the decline in HIV1-RNA mean values first stabilizes for short time, and then subsequently shows a more marked decrease, producing a complex pattern which is not adequately fitted at this point by the polyexponential or multiple polyexponential forms. The mean values derived from our fitted quasipolynomial are shown in Figure 4, and visually it can be seen that this curve appears to fit the portion of the data between weeks 2 and 10 much better than any of the polyexponential forms, whilst still fitting well to the rest of the curve. The fitted quasipolynomial is:

logHIVt = logHIV0 * exp(-0.0171*t) - 1.8805 * t * exp(-0.5554*t) - 0.0445 * t2 * exp(-0.1397*t)

Measures of Fit All models described here were fitted to individual patient data, and in order to formally compare the fit of the curves to individual data, we first computed the Akaike Information Criterion (AIC) and Bayes Information Criterion (BIC) for each of the nonlinear models we fitted. However, the graphical representations of the fitted curves shown here display mean values of actual data and of predicted values from the fitted curves, and predictions of these mean values was our main objective. Since best prediction of individual patient values does not necessarily correspond with best prediction of mean values, we also assessed model fit by computing the sum of squared deviations between actual means and predicted means, to see which of the nonlinear forms best predicted mean values. Finally, we split the data at random into 441 patients (4798 data points) whose data were used to fit the curves, and 220 patients (2379 data points) for which we used the curves to predict HIV1-RNA values, and to measure model fit we computed the sum of squared deviations between actual and predicted values for this “prediction” data set. All of the results are summarized in Table 1. The AIC and BIC indicate the same pattern of fit of the curves: the multiple polyexponential provides the best fit to the individual patient data. Fitting the multiple polyexponential to a “fitting” subset of data and then using the fitted curve to predict for the “prediction”

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subset also gave the best result. For these three measures of fit, the quasipolynomial form is a close second best to the multiple polyexponential, despite not using the information provided by the additional covariates. However, measuring the ability to predict mean values across patients, the quasipolynomial form fits best, with the multiple polyexponential second best. The triexponential curve is the next best fit on all measures, and the simple stretched exponential form provides a better fit than the biexponential on all goodness-of-fit measures. Notes on Fitting When fitting the 2-term polyexponential, 2 of the 10 sets of initial parameter values resulted in failure to converge with the GAUSS algorithm. All of the 8 starting sets which did converge yielded the same solution. The two starting set failures both converged with the MARQUARDT algorithm, one to the same solution as the other 8 starting sets, but the other converging to a different and slightly sub-optimal solution, as measured by the mean squared error. Using a “good” initial parameter set (i.e. one that converges to the lowest MSE), both the GAUSS and MARQUARDT algorithms converged to the same solution. In fitting the 3-term polyexponential, 3 of the 10 sets of initial parameter values failed to converge with the GAUSS algorithm. Of the other 7 sets of starting values, 5 converge to same “best” solution, but 2 converge to a slightly sub-optimal solution. With the MARQUARDT algorithm, one of the 3 failures converges to the “best” solution, and other 2 converge to slightly sub-optimal solutions. With the “good“ initial parameter values, both the GAUSS and MARQUARDT algorithms converge to the “best” solution. Of the 10 sets of initial parameter values, 9 converge in fitting the multiple polyexponential, but only one of these converges to the “best” solution. Seven of the 10 converge to the same slightly sub-optimal fit, and one further starting set converges to a slightly different sub-optimal fit. The one starting set that does not converge with the GAUSS algorithm converges successfully with the MARQUARDT algorithm, but to the same slightly sub-optimal fit found by 7 of the starting sets with the GAUSS algorithm. For the “best” initial starting set, both the GAUSS and MARQUARDT algorithms converge to same “best” solution. For the quasipolynomial form, 6 of 10 starting sets converge, but only one of these converges to the “best” solution. Three of the others converge to the same slightly sub-optimal solution, and 2 to a different and slightly sub-optimal solution. With the 4 initial parameter sets which fail to converge, the

MARQUARDT algorithm converges, but in each case to a different slightly sub-optimal solution. However, the initial parameter values which produce the “best” solution when fitted to the full data set produce poor results when used as starting values with the “fitting-data” subset. The SSD for prediction set with these initial parameter values is 2061.053. The initial values which, with the full data, produced the second-best fit, were then also applied to the “fitting-data” subset, and produced much better predictions for the “prediction-data” subset – the SSD is 1994.224. Hence starting parameter values which were optimal for all data were not optimal when used on a subset of data. Using the “best” initial parameter value set, the GAUSS and MARQUARDT algorithms converge to slightly different solutions – the GAUSS algorithm producing the “best” solution in terms of mean squared error. With the “second-best” initial parameter value set (which works best for the “fitting/prediction” split) – the GAUSS and MARQUARDT algorithms converge to the same solution – (“second best” for the full set). Discussion Although standard exponential forms of nonlinear regression models can provide an adequate fit to data which show a decline over time, in practice there are also many patterns of decline seen in biological systems which are more complicated. For these more complex patterns of decline, other non-linear models may provide an improved fit to data. Lee et. al. (2001) have shown, using simulated data, how the stretched-exponential form often provides a good fit to data which are genuinely polyexponential in nature, and the fit of this form to our data is certainly adequate; however, we found that a 3 term polyexponential form, a multiple polyexponential form, and a quasipolynomial form all gave a better fit to our HIV1-RNA plasma level data, assessed both visually and in terms of formal goodness-of-fit criteria, than the simple stretched-exponential model. The polyexponential and quasipolynomial forms have been little used in nonlinear regression, but their inherent flexibility suggests that they should be more widely considered. However, fitting of these models requires some care. We found that, especially for the multiple polyexponential and quaispolynomial forms, several solutions with very similar goodness-of-fit could be found, depending on the initial parameter estimates supplied to the convergence algorithms. This is an example of the problem known generally as “parameter-redundancy”, which is discussed at some length by Seber & Wild (2003) and Petersson & Holmstrom (1998). Essentially, this phenomenon manifests itself as the existence of a number of different

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model solutions which all fit the data similarly well, with only very small differences in mean squared error. Seber & Wild (2003) give a number of examples illustrating the parameter redundancy problem, and define a number of quantities which can be used to measure the degree of parameter redundancy. These are difficult to compute, however, and in practice the problem is generally readily identified when the estimated standard errors of parameters in fitted models are very large. However, for our HIV1-RNA data, although we found a number of different solutions which fitted our data similarly well, in all cases the estimated standard errors of the model parameters had reasonable values. We found also that some of the algorithms used by SAS® PROC NLIN would converge for certain initial parameter values, whilst others would not. We therefore recommend that, when fitting these models, a variety of estimation algorithms should be tried whenever possible. In general, fitting of nonlinear models should be done cautiously, and the advice offered, for example, by Motulsky & Ransnas (1987) and Smyth (2002) is of value. Polyexponentials have been extensively used as models in kinetic studies (see, e.g. Jacquez (1972); Landaw & DiStefano (1984)), but not widely applied outside of that field, although Petersson & Holmstrom (1998) refer to their occasional use in economic modelling. General applications in the life sciences include Grossman and Koops (2001), who fitted a polyexponential model to describe egg production in chickens, Knisley & Glen (1996) using polyexponential models in the context of firing of neurons, Mukherjee et. al. (2001), who use polyexponential models in the study of brain maturation, and Taft et. al. (2005), who fit a sum of two exponentials to data on cell division rates. Work on population dynamics of HIV in immune system cells has suggested that certain specific polyexponential forms have some theoretical justification (e.g. Perelson and Nelson, 1999; Wu and Ding, 1999, Ding and Wu, 1999), and models deriving from this theoretical work have been fitted to HIV data by , for example, Notermans et. al. (1998), Melvin et. al. (1998), Luzuriaga et. al. (2000), Lindback et. al. (2000), Nelson et. al. (2001), and Markovitz et. al. (2003).

These authors, however, do not comment on the practical problems of fitting such models using nonlinear regression software. Quasipolynomial forms appear to have been used only rarely as nonlinear regression models, and we can find no report of the use of our particular quasipolynomial model form in the literature. However, terminology in respect of this type of functional form is variable in the literature and these models may be referred to by other terms. Some applications have been reported using the more general definition of a quasipolynomial as a product of polynomial terms and arbitrary functions of a dependent variable. Rohac et. al. (2002), for example, describe the fitting of such an arbitrary model to data on heat capacities of phthalate esters, and Vilupuru & Glasser (2002) use such a model to describe visual accommodation in Rhesus monkeys. Maes et. al. (1998) and Glasbey (1980) have also fitted nonlinear model forms which are similar to quasipolynomials. An alternative nonlinear model form which also has good flexibility is the fractional-polynomial model proposed by Royston & Altman (1994). Specifically in the context of modeling HIV plasma levels, the fractional-polynomial was used by Gray et. al. (2004) to model HIV1 levels in children who acquire an HIV infection during gestation or delivery. Garcia et. al. (2004) also use fractional polynomials to model the pattern of CD4+ cell count changes over time in HIV-infected patients. Other nonlinear regression models used in the context of data from HIV-infected patients include the hyperbolic model, used by Gieshke et. al. (1999) to model the relationship between HIV-RNA plasma levels and AUC of antiretroviral drug levels. Sasomsin et. al. (2002) used both hyperbolic and exponential models to describe the relationship between P24 antigenaemia and AUC of antiretroviral drug levels. In summary, we found that multiple polyexponential and quasipolynomial models provided a good fit to our empirical HIV1-RNA data, and the flexibility of the these model forms suggests that they are potentially of value in nonlinear regression applications.

Table 1. Measures of Fit for the five nonlinear models

Curve No. of

parameters AIC BIC SSD

Mean Prediction

SSD Prediction Set

Stretched Exponential 3 -842.47 -821.85 0.2317 2050.95 Biexponential 5 -793.34 -758.97 0.4235 2064.11 Triexponential 7 -901.04 -852.92 0.1322 2041.00 Multiple polyexponential 5 -1325.74 -1291.38 0.1081 1862.32 Quasipolynomial 6 -1076.51 -1035.27 0.0410 1994.22

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References Boltyanski V, Gorelikova S (1997). Optimal synthesis for non-oscillatory controlled objects. Journal of Applied Analysis 3(1): 1-21. Ding AA, Wu H (1999) Relationships between Antiviral Treatment Effects and Biphasic Viral Decay Rates in Modeling HIV Dynamics. Mathematical Biosciences (1999) 160(1): 63-82. García F, de Lazzar Ei, Plana M, Castro P, Mestre G, Nomdedeu M, Fumero E, Martínez E, Mallolas J, Blanco JL, Miró JM, Pumarola T, Gallart T, Gatel JM (2004) Long-Term CD4+ T-Cell Response to Highly Active Antiretroviral Therapy According to Baseline CD4+ T-Cell Count. J Acquir Immune Defic Syndr. 36 (2): 702-713. Gieschke R, Fotteler B, Buss N, Steimer J-L (1999) Relationships Between Exposure to Saquinavir Monotherapy and Antiviral Response in HIV-Positive Patients. Clin Pharmacokinetics 37 (1): 75-86. Glasbey CA (1980) Nonlinear regression with autoregressive time series errors. Biometrics 36: 135-140. Gray L, Cortina-Borja M, Newell M-L (2004). Modelling HIV-RNA viral load in vertically infected children. Statistics in Medicine 23: 769-781. Grossman M, Koops WJ (2001) A model for individual egg production in chickens. Poultry Science 80: 859-867. Hatzakis A, Gargalianos P, Kiosses V, Lazanas M, Sypsa V, Anastassopoulou C, Vigklis V, Sambatakou H, Botsi C, Paraskevis D, Stalgis C (2001) Low-Dose IFN-a Monotherapy in Treatment-Naïve Individuals with HIV-1 Infection: Evidence of Potent Suppression of Viral Replication. Journal of Interferon and Cytokine Research 21:861–869. Havlir DV, Richman DD (1996) Viral Dynamics of HIV: Implications for Drug Development and Therapeutic Strategies. Annals of Internal Medicine 124 (11): 984 -994. Huang W, De Gruttola V. Fischl M, Hammer S, Richman D, Havlir D, Gulick R, Squires K, Mellors J (2001). Patterns of Plasma Human Immunodeficiency Virus Type 1 RNA Response to Antiretroviral Therapy. The Journal of Infectious Diseases 2001;183:1455–65 Jacquez JA (1972) Compartmental Analysis in Biology and Medicine. Elsevier. New York.

Jowett IG, Rochardson J (1990) Microhabitat preferences of benthic invertebrates in a New Zealand river and the development of in-stream flow-habitat models for Deleatidium spp.. New Zealand Journal of Marine and Freshwater Research 24: 19-30. Knisley JR, Glen LL (1996) A linear method for the curve-fitting of multiexponentials. Journal of Neuroscience Methods 67: 177-183. Lalezari JP, Henry K, O’Hearn M (2002). Enfuvirtide n combination with an optimized background (OB) regimen vs. OB alone: week 24 response among categories of treatment experience and baseline HIV antiretroviral (ARV) resistance. 42nd Interscience Conference on Antimicrobial Agents and Chemotherapy (ICAAC), San Diego CA. Abstract H-1074. Lalezari JP, Henry K, O’Hearn M, et al.. (2003) Enfuvirtide, an HIV-1 fusion inhibitor, for drug-resistant HIV infection in North and South America. New England Journal of Medicine 348(22): 2175-2185. Landaw E, DiStefano JJ (1984) Multiexponential, Multicompartmental and Noncompartmental Modeling: Physiological Data Analysis and Statistical Considerations. American Journal of Physiology, 246:R665- R677, 1984. Lange J, Lazzarin A, Clotet B (2002). Enfuvirtide in combination with an optimized background (OB) regimen vs. OB alone: week 24 response among categories of baseline demographics, treatment experience, and HIV anti-retroviral (ARV) resistance. 6th International Congress on Drug Therapy in HIV Infection., Glasgow, Scotland. Poster BL 14.3. Lazzarin A, Clotet B, Cooper D, et. al.. (2003). Efficacy of Enfuvirtide in patients infected with drug-resistant HIV1 in Europe and Australia. New England Journal of Medicine 348(22), 2186-2195. Lee KCB, Siegel J, Webb SED, Leveque-Fort S, Cole MJ, Jones R, Dowling K, Lever MJ, French PMW (2001) Application of the Stretched Exponential Function to Fluorescence Lifetime Imaging. Biophysical Journal 81:1265–1274 Lindback S, Karlsson AC, Mittler J, Blaxhult A, Carlsson M, Briheim G, Sonnerborg A, Gaines H (2000) Viral dynamics in primary HIV- infection. AIDS 2000, 14:2283-2291.

565


Luzuriaga K, Wu H, MCManus M, Britto P, Borkowsky W, Burchett S, Smith B, Mofenson L, Sullivan JL (1999) Dynamics of Human Immunodeficiency Virus Type 1 Replication in Vertically Infected Infants. Journal Of Virology 73(1): 362–367 Maes BD, Mys G, Geypens BJ, Evenepoel P, Ghoos YF, Rutgeerts PJ (1998) Gastric emptying flow curves separated from carbon-labeled octanoic acid breath test results. Am. J. Physiol. 275 (Gastrointest. Liver Physiol. 38): G169–G175. Markowitz M, Louie M, Hurley A, Sun E, Di Mascio M, Perelson AS, Ho DD (2003). A Novel Antiviral Intervention Results in More Accurate Assessment of Human Immunodeficiency Virus Type 1 Replication Dynamics and T-Cell Decay In Vivo. Journal Of Virology 77(8): 5037–5038 Melvin AJ, Rodrigo AG, Mohan KM, Lewis PA, Manns-Arcuino L, Coombs RW, Mullins JI, Frenkel LM (1999) HIV-1 Dynamics in Children. J Acquir Immune Defic Syndr 20(5): 468-473. Motulsky HJ, Ransnas LA (1987) Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB J: 1: 365-374. Mukerjee P, Miller JH, Shimony JS, Conturo TE, Lee BCP, Almli CR, McKinstry RC (2001) Normal Brain Maturation During Childhood: Developmental Trends Characterized with Diffusion-Tensor MR Imaging. Radiology 221: 349-358. Nelson PW, Mittler J E, Perelson AS (2001) Effect of Drug Efficacy and the Eclipse Phase of the Viral Life Cycle on Estimates of HIV Viral Dynamic Parameters. J Acquir Immune Defic Syndr 26(5): 405-412.

Notermans DW, Goudsmit J, Danner SA, de Wolf F, Perelson AS, Mittler J (1998) Rate of HIV-1 decline following antiretroviral therapy is related to viral load at baseline and drug regimen. AIDS 12:1483–1490 Petersson J, Holmstrom K (1998) A Review of the Parameter Estimation Problem of Fitting Positive Exponential Sums to Empirical Data. Malardalen University Center of Mathematical Modelling Technical Report IMa-TOM-1997-08. Perelson, AS, Nelson PW (1999) Mathematical Analysis of HIV-1 Dynamics in Vivo. SIAM Review 41(1): 3-44.

Rohac V, Fulem M, Schmidt HG, Ruzicka V, Ruzicka K, Wolf G (2002). Heat Capacities of Some Phthalate Esters. Journal of Thermal Analysis and Calorimetry 70: 455-466. Royston P, Altman GD (1994). Regression using fractional polynomials of continuous covariates: parsimonious parameter modelling. Applied Statistics 43:429–467. Sasomsin P, Mentre F, Diquet B, Simon F, Brun-Vezinet F (2002) Relationship between exposure to zidovudine and decrease of P24 antigenemia in HIV-infected patients in monotherapy. Fundamental & Clinical Pharmacology 16: 347–352. Seber, GAF, Wild CJ (2003) Nonlinear Regression. Wiley, NJ. ISBN 0-471-47135-6. Smyth, GK. (2002) Nonlinear Regression. In Encylopedia of Environmetrics Eds. El-Shaarawi AH, Piegorsch WW. Wiley ISBN 0471-899976. Szekelyhidi L (2004) Spectral Analysis and Spectral Synthesis On Polynomial Hypergroups. International Conference on Probability Theory on Groups and Related Structures, Budapest, Hungary. Taft R, Moore D, Yount G (2005) Time-lapse analysis of potential cellular responsiveness to Johrei, a Japanese healing technique. BMC Complementary and Alternative Medicine 2005, 5:2 Vale JR (2004) The Human Immune System: A Challenging Control Problem. Master of Applied Science Thesis, University of Waterloo, Canada. Vilupuru AS, Glasser A (2002) Dynamic Accommodation in Rhesus Monkeys. Vision Research 42: 125-141. West SA, Flanagan KE, Godfray HCJ (1999) Sex allocation and clutch size in parasitoid wasps that produce single-sex broods. Animal Behaviour 57: 265-275. Wu H, Ding AA (1999). Population HIV1 dynamics in vivo: applicable models and inferential tools for virological data from AIDS clinical trials. Biometrics 55: 410-418.

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