Quantitative Methods for HIV/AIDS Research Quantitative Methods for HIV/AIDS Research Edited by Cliburn Chan Michael G Hudgens Shein-Chung Chow Cover credit: Peter Hraber, Thomas B Kepler, Hua-Xin Liao, Barton F Haynes Adapted from Liao et al (2013) Nature 496: 469 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper International Standard Book Number-13: 978-1-4987-3423-3 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in 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by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Title: Quantitative methods for HIV/AIDS research / Cliburn Chan, Michael G Hudgens, Shein-Chung Chow Description: Boca Raton : Taylor & Francis, 2017 | “A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.” | Includes bibliographical references and index Identifiers: LCCN 2017008215| ISBN 9781498734233 (hardback) | ISBN 9781315120805 (e-book) Subjects: LCSH: HIV infections Research Methodology Popular works | AIDS (Disease) Research Methodology Popular works Classification: LCC RC606.64 Q36 2017 | DDC 616.97/920072 dc23 LC record available at https://lccn.loc.gov/2017008215 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xi Contributors xvii Section I Quantitative Methods for Clinical Trials and Epidemiology Statistical Issues in HIV Non-Inferiority Trials Mimi Kim Sample Size for HIV-1 Vaccine Clinical Trials with Extremely Low Incidence Rate 17 Shein-Chung Chow, Yuanyuan Kong, and Shih-Ting Chiu Adaptive Clinical Trial Design 41 Shein-Chung Chow and Fuyu Song Generalizing Evidence from HIV Trials Using Inverse Probability of Sampling Weights .63 Ashley L Buchanan, Michael G Hudgens, and Stephen R Cole Statistical Tests of Regularity among Groups with HIV Self-Test Data 87 John Rice, Robert L Strawderman, and Brent A Johnson Section II Quantitative Methods for Analysis of Laboratory Assays Estimating Partial Correlations between Logged HIV RNA Measurements Subject to Detection Limits 109 Robert H Lyles Quantitative Methods and Bayesian Models for Flow Cytometry Analysis in HIV/AIDS Research 135 Lin Lin and Cliburn Chan The Immunoglobulin Variable-Region Gene Repertoire and Its Analysis 157 Thomas B Kepler and Kaitlin Sawatzki ix 260 Quantitative Methods for HIV/AIDS Research Perelson model as the “null hypothesis.” Additional details on the approach can be found in [17] Assuming constant per capita decay rates, the decline (or growth) of a population is exponential, and depends on the size of the population in a linear way, that is, the decay in the population is proportional (via a constant) to the size of the population, and can be described with linear differential equations In a density-dependent decay model, this proportionality is variable: the per capita decay rate depends on the size of the population, resulting in a model described by nonlinear differential equations which governs the population dynamics Population dynamics models of this type are often used in population biology [18] as an alternative to long-term exponential growth or decay: density dependent homeostatic mechanisms are described for lymphocyte populations in mice [19] and humans [12,20,21]; timedependent decay of a single infected-cell compartment was suggested as a possible alternative explanation for the biphasic decay pattern observed in HIV-1 decline by [8] and [22] In order to assess the accuracy of the assumption of simple exponential decay of infected cell populations, a parameterized model for HIV-1 plasma RNA decline after HAART was developed [17] In this model, a single parameter can be tested with statistical methods to determine if densitydependent decay is a factor in the biphasic pattern that has been observed in HIV-1 RNA decay after treatment Incorporating the density-dependent decay mechanism for both the short-lived and long-lived infected cells described in [4] can dramatically alter conclusions about the rate at which infected cells are decaying and the associated estimates of time to eradication of both short-lived and long-lived infected cells This model also has parameters which can be tested to determine if, in addition to density-dependent decay, more than one population of infected cells is contributing to the overall population of viral RNA The possibility that only one population of infected cells is contributing to the total viral load in the presence of density-dependent decay was suggested in [8] and [22], that is, that what has previously been described as biphasic decay is the result of densitydependent decay of this single population It should be noted that the alternative model evaluated is designed to test the assumption of simple exponential decay of infected cell populations, and does not specify the mechanisms; possible mechanisms include differential activity by the immune system or natural cellular homeostasis responsible for this nonlinear decay Data used for analysis was collected from HIV-1-infected children initiating treatment with HAART consisting of at least three agents, at least one of which was a protease inhibitor Blood samples were taken prior to starting therapy and at multiple time points afterwards, with data collected so that both first- and second-phase decay could be estimated Adherence to the prescribed drug regimen was assessed by direct observation and parent interviews These data have been Precision in ODE Specification and Parameter Estimation 261 presented previously [9,17] where additional details on study design and methods are provided The following model for decay of HIV-1 RNA after initiation of HAART is used to test the assumption of constant log linear decay of infected cell compartments Here, X represents the population of short-lived infected cells, Y the population of long-lived infected cells, and V the population of HIV-1 RNA dX = − δXr dt (12.3) dY = − μYr dt (12.4) dV = px X + py Y − cV dt (12.5) These equations extend the model described in [4] by allowing for density-dependent decay of infected cell populations Note that the righthand sides of Equations 12.3 and 12.4 can be written as (δXr−1) X and (µYr−1) Y, respectively, so that the per capita decay rate for the short-lived infected cells is δXr−1 and for long-lived infected cells is µYr−1 If r is not equal to one, the per capita decay rate of these two populations depends on the density of the decaying population Note that if the parameter r is equal to one, this model reduces to the model presented in [4] which is referred to as the constant decay model for viral decay after initiation of HAART The parameter px represents the contribution to the viral RNA population from short-lived infected cells and py represents the contribution to the viral RNA population from long-lived infected cells To test whether or not density-dependent decay is a factor in the decline of the infected cell populations X and Y, the primary hypothesis of interest is H0 : r = Other relevant estimates and tests are whether px = and py = 0, that is, whether at least two populations of infected cells contribute to the total viral load, and whether δ = and µ = 0, that is, whether there is significant decay in the infected cell populations which are producing viral RNA in the presence of density-dependent decay To obtain the model solutions for plasma viremia, short- and long-lived infected cells, we solved Equations 12.3 and 12.4 under the assumption that r to obtain Xtị = ẵ ðr − 1Þ Ã t + x10 − r 1 r Ytị = ẵ r 1ị Ã t + y10 − r 1 − r (12.6) 262 Quantitative Methods for HIV/AIDS Research and substituted these solutions into Equation 12.5 to obtain 1 dV = px ½δ Ã ðr − 1Þ Ã t + x10 − r 1 − r + py ½μ Ã ðr − 1Þ Ã t + y01 − r 1 − r − cV dt (12.7) If r = the solutions to Equations 12.3 and 12.4 are simple first-order exponential decay curves and the model is identical to the one presented in [4] Equation 12.6 describes the model-predicted densities of short-lived infected and long-lived infected cells after drug therapy and the solution of Equation 12.7 describes the model-predicted plasma viremia density after drug therapy We used numerical solutions to Equation 12.7 in the Marquart nonlinear least squares algorithm to estimate δ, µ, r, px, and py All analyses allowed used observed initial viral load, and both infected cell populations for each child; assuming that, conditional on initial viral load and infected cell compartments, the viral load trajectories are independent and conducted various diagnostics to verify this assumption Profile bootstrap methods were used to calculate the 95% confidence intervals for the parameters of interest Additional details on methods for parameters estimation and inference, as well as other details of the study design and methods can be found on [9,17] The estimates of time to eradication of the short-lived infected and longlived infected cell populations under the density-dependent decay model were obtained by using Equation 12.6 with the estimated parameters to determine the time required for the initial populations to decay to one infected cell For the constant decay model, we used first-order exponential trajectories for the infected cell populations Data from all six children were used to fit both the density-dependent and constant models The estimates of δ, µ, r, px, and py from the density-dependent and constant decay models and associated 95% confidence intervals are shown in Table 12.1 For all but one child (Child 4), the confidence interval for the estimate of the parameter r does not contain one, indicating that density-dependent decay plays a role in the decline of infected cell populations after initiation of treatment in this data set In other words, the assumption of simple exponential decay for both short- and long-lived infected cells is violated for the data we analyzed Also px and py, the viral production rates for short-lived and longlived infected cells, are both significantly different than zero, indicating that at least two populations of infected cells are contributing to the viral pool in the presence of density-dependent decay Finally, our estimates of both δ and µ, the decay coefficients for the short-lived and long-lived infected cell populations, are significantly different than zero for all but one child using the constant decay model In contrast, the estimated coefficient, µ, is not significantly different than zero for five of the six children using the density-dependent model, suggesting that there may be no overall decline in the long-lived infected cell compartment Both the estimated density-dependent and constant decay model trajectories along with the observed plasma viremia data are shown in Figure 12.1 Projections based on the constant and density-dependent decay models for time to viral eradication (viral load