Dynamical models of biomarkers and clinical progression for personalized medicine: the HIV context.

Mechanistic models, based on ordinary differential equation systems, can exhibit very good predictive abilities that will be useful to build treatment monitoring strategies. In this review, we present the potential and the limitations of such models for guiding treatment (monitoring and optimizing) in HIV-infected patients. In the context of antiretroviral therapy, several biological processes should be considered in addition to the interaction between viruses and the host immune system: the mechanisms of action of the drugs, their pharmacokinetics and pharmacodynamics, as well as the viral and host characteristics. Another important aspect to take into account is clinical progression, although its implementation in such modelling approaches is not easy. Finally, the control theory and the use of intrinsic properties of mechanistic models make them very relevant for dynamic treatment adaptation. Their implementation would nevertheless require their evaluation through clinical trials.

Estimation of the distribution of infection times using longitudinal serological markers of HIV: implications for the estimation of HIV incidence.

In the last decade, interest has been focused on human immunodeficiency virus (HIV) antibody assays and testing strategies that could distinguish recent infections from established infection in a single serum sample. Incidence estimates are obtained by using the relationship between prevalence, incidence, and duration of recent infection (window period). However, recent works demonstrated limitations of this approach due to the use of an estimated mean "window period." We propose an alternative approach that consists in estimating the distribution of infection times based on serological marker values at the moment when the infection is first discovered. We propose a model based on the repeated measurements of virological markers of seroconversion for the marker trajectory. The parameters of the model are estimated using data from a cohort of HIV-infected patients enrolled during primary infection. This model can be used for estimating the distribution of infection times for newly HIV diagnosed subjects reported in a HIV surveillance system. An approach is proposed for estimating HIV incidence from these results.

Practical identifiability of HIV dynamics models.

We study the practical identifiability of parameters, i.e., the accuracy of the estimation that can be hoped, in a model of HIV dynamics based on a system of non-linear Ordinary Differential Equations (ODE). This depends on the available information such as the schedule of the measurements, the observed components, and the measurement precision. The number of patients is another way to increase it by introducing an appropriate statistical "population" framework. The impact of each improvement of the experimental condition is not known in advance but it can be evaluated via the Fisher Information Matrix (FIM). If the non-linearity of the biological model, as well as the complex statistical framework makes computation of the FIM challenging, we show that the particular structure of these models enables to compute it as precisely as wanted. In the HIV model, measuring HIV viral load and total CD4+ count were not enough to achieve identifiability of all the parameters involved. However, we show that an appropriate statistical approach together with the availability of additional markers such as infected cells or activated cells should considerably improve the identifiability and thus the usefulness of dynamical models of HIV.

Maximum likelihood estimation in dynamical models of HIV.

The study of dynamical models of HIV infection, based on a system of nonlinear ordinary differential equations (ODE), has considerably improved the knowledge of its pathogenesis. While the first models used simplified ODE systems and analyzed each patient separately, recent works dealt with inference in non-simplified models borrowing strength from the whole sample. The complexity of these models leads to great difficulties for inference and only the Bayesian approach has been attempted by now. We propose a full likelihood inference, adapting a Newton-like algorithm for these particular models. We consider a relatively complex ODE model for HIV infection and a model for the observations including the issue of detection limits. We apply this approach to the analysis of a clinical trial of antiretroviral therapy (ALBI ANRS 070) and we show that the whole algorithm works well in a simulation study.

Analysis of the effect of aluminum in drinking water and transferrin C2 allele on Alzheimer's disease.

Although highly controversial, the hypothesis of a link between aluminum (Al) in drinking water and Alzheimer's disease (AD) has been supported by several epidemiological studies. Transferrin (Tf) is a major transport protein for both iron and Al. Moreover, it has been demonstrated that defective binding of iron and Al to the Tf variant C2 could be present in AD. Individuals carrying the Tf C2 allele might therefore be at greater risk of developing AD. We investigated whether the Tf C2 allele might be responsible for susceptibility to AD in a sample of 292 subjects (with 55 AD) aged > or = 75 years from south-west France, some exposed to high levels of Al in tap water (n = 181 subjects) and others to low levels of Al (n = 111 subjects). We also examined the combined genetic effects of Tf C2 and epsilon4 allele of apolipoprotein E gene (ApoE). Logistic regression analysis showed that neither Tf C2 nor its interaction with Al or with the epsilon4 allele of the ApoE were significantly associated with the risk of AD.

Incidence and mortality of Alzheimer's disease or dementia using an illness-death model.

We present an illness-death model for studying the incidence and the prevalence of Alzheimer's disease or dementia. We argue that the illness-death model is better than a survival model for this purpose. In this model the best choice for the basic time-scale is age. Then we present extensions of this model for incorporating covariates and taking account of a possible effect of calendar time. Calendar time is introduced via a proportional intensity model. We give the likelihood for a mixed discrete-continuous observation pattern from this model: clinical status is observed at discrete visit-times while the date of death is observed exactly or right-censored. The penalized likelihood approach allows to non-parametrically estimate the transition intensities. Application on the data of the Paquid study allows to produce estimates of the age-specific incidence of dementia together with mortality rates of both demented and non-demented subjects. Then the effect of calendar time and educational level are studied. Low educational level increases the risk of dementia. The risk of dementia increases with calendar time while the mortality of demented subjects decreases. The most likely explanation of this result seems to be in a shift in the diagnosis of dementia towards earlier stages of the disease prompted by a change in the perception of dementia and the arrival of new drugs.

Bootstrap choice of estimators in parametric and semiparametric families: an extension of EIC.

Ishiguro, Sakamoto, and Kitagawa (1997, Annals of the Institute of Statistical Mathematics 49, 411-434) proposed EIC as an extension of Akaike criterion (AIC); the idea leading to EIC is to correct the bias of the log-likelihood, considered as an estimator of the Kullback-Leibler information, using bootstrap. We develop this criterion for its use in multivariate semiparametric situations, and argue that it can be used for choosing among parametric and semiparametric estimators. A simulation study based on aregression model shows that EIC is better than its competitors although likelihood cross-validation performs nearly as well except for small sample size. Its use is illustrated by estimating the mean evolution of viral RNA levels in a group of infants infected by HIV.

A latent process model for joint modeling of events and marker.

The paper formulates joint modeling of a counting process and a sequence of longitudinal measurements, governed by a common latent stochastic process. The latent process is modeled as a function of explanatory variables and a Brownian motion process. The conditional likelihood given values of the latent process at the measurement times, has been drawn using Brownian bridge properties; then integrating over all possible values of the latent process at the measurement times leads to the desired joint likelihood. An estimation procedure using joint likelihood and a numerical optimization is described. The method is applied to the study of cognitive decline and Alzheimer's disease.

Inference for multi-state models from interval-censored data.

Clinical statuses of subjects are often observed at a finite number of visits. This leads to interval-censored observations of times of transition from one state to another. The likelihood can still easily be written in terms of both transition probabilities and transition intensities. In homogeneous Markov models, transition probabilities can be expressed simply in terms of transition intensities, but this is not the case in more general multi-state models. In addition, inference in homogeneous Markov models is easy because these are parametric models. Non-parametric approaches to non-homogeneous Markov models may follow two paths: one is the completely non-parametric approach and can be seen as a generalisation of the Turnbull approach; the other implies a restriction to smooth intensities models. In particular, the penalized likelihood method has been applied to this problem. This paper gives a review of these topics.

Mortality with dementia: results from a French prospective community-based cohort.

Despite the magnitude of the problem, little is known about the duration of dementia. Survival and risk factors of mortality with dementia and the impact of dementia on the risk of death were investigated using the Personnes Agées Quid (PAQUID) prospective population-based cohort study between 1988 and 1998. Statistical models dealing with interval censoring were performed. Among 3,675 participants aged 65 years or older and initially nondemented, 2,923 have been followed up for 8 years. Of these, 281 persons with incident dementia were actively diagnosed. The mean age of onset of dementia was 82.3 years. In the total population, the relative risk of dying after developing dementia was estimated to be 1.82 (95% confidence interval (CI): 1.77, 2.68) when adjusted for sociodemographic variables and comorbidity. Deaths from cerebrovascular diseases and respiratory diseases were particularly increased among persons with dementia, compared with those without. The median survival time of the persons with dementia was estimated to be 4.5 years. Women with dementia had a longer survival than did men with dementia, particularly for Alzheimer-type dementia (relative risk = 0.47, 95% CI: 0.27, 0.83). Educational level was not significantly associated with survival in persons with dementia. These results provide further evidence of the malignancy of dementia, which will be a challenge for the 21st century.

Correction of the P-value after multiple coding of an explanatory variable in logistic regression.

We propose a method and a program to determine a significance level for a series of codings of an explanatory variable in logistic regression. Dichotomous and Box-Cox transformations are considered. Three methods of correcting the significance level are studied: the Bonferroni method; Efron's method, which uses the correlation between successive tests, and the exact calculation by numerical integration using all correlations. A simulation study has led to a strategy for the choice and number of the different codings of the variable. This method is illustrated using the data of a study of the relation between cholesterol and dementia.

MKVPCI: a computer program for Markov models with piecewise constant intensities and covariates.

We present a computer program for fitting Markov models with piecewise constant intensities and for estimating the effect of covariates on transition intensities. The basic idea of the proposed approach is to introduce artificial time-dependent covariates in the data to represent the time dependence of the transition intensities, and to use a modified time-homogeneous Markov model to estimate the baseline transition intensities and the regression coefficients. The program provides the maximum likelihood estimates of the parameters together with their estimated standard errors, and allows testing various statistical hypotheses. To illustrate the use of the program, we present a three-state model for analyzing the smoking habits of school children.

A cognitive screening battery for dementia in the elderly.

The objective of this study is to propose a screening instrument for dementia based on a reduced number of neuropsychological tests. The sample consists in the pooled data of the five follow-up visits of the Paquid cohort study on cerebral aging: the estimation sample included 2792 subjects (8830 observations) and the validation sample included 985 subjects (2643 observations). Among scores significantly associated with dementia, we retained only those that increased the specificity of the model for a sensitivity of one. Seven neuropsychological tests and the MMSE subscores were considered. The most discriminant combination of tests included the MMSE and the subscores "orientation to time" and "recall three objects," the Benton Visual Retention Test, and Isaacs' Set Test of verbal fluency. The specificity of this screening instrument was 0. 77 for a sensitivity of 1.

Intake of flavonoids and risk of dementia.

It has been postulated that oxidative stress may play a key role in dementia. This is substantiated by the recent discovery of the protective effect of wine. In wine, the flavonoids--powerful antioxidant substances also contained in tea, fruits and vegetables--have been thought to offer such protection. We investigated whether flavonoid intake could be associated with a lower incidence of dementia in a cohort of 1367 subjects above 65 years of age (Paquid). A questionnaire was used to evaluate their intake of flavonoids and subjects were followed-up for 5 years between 1991 and 1996: 66 incident cases of dementia were observed. We estimated the relative risk (RR) of dementia according to tertiles of flavonoid intake using a Cox model. The age-adjusted RR of dementia was 0.55 for the two highest tertiles compared to the lowest (95% CI: 0.34-0.90; p = 0.02). After additional adjustment for gender, education, weight and vitamin C intake, the RR was 0.49 (95% CI: 0.26-0.92; p = 0.04). We conclude that the intake of antioxidant flavonoids is inversely related to the risk of incident dementia.

Standardized martingale residuals applied to grouped left truncated observations of dementia cases.

The use of martingale residuals have been proposed for model checking and also to get a non-parametric estimate of the effect of an explanatory variable. We apply this approach to an epidemiological problem which presents two characteristics: the data are left truncated due to delayed entry in the cohort; the data are grouped into geographical units (parishes). This grouping suggests a natural way of smoothing the graph of residuals which is to compute the sum of the residuals for each parish. It is also natural to present a graph with standardized residuals. We derive the variances of the estimated residuals for left truncated data which allows computing the standardized residuals. This method is applied to the study of dementia in a cohort of old people, and to the possible effect of the concentration of aluminum and silica in drinking water on the risk of developing dementia.

Relation between aluminum concentrations in drinking water and Alzheimer's disease: an 8-year follow-up study.

To investigate the effect of aluminum and silica in drinking water on the risk of dementia and Alzheimer's disease, the authors analyzed data from a large prospective cohort (Paquid), including 3,777 subjects aged 65 years and over living at home in 75 civil parishes in Gironde and Dordogne in southwestern France in 1988-1989. The subjects were followed for up for 8 years with an active search for incident cases of dementia or Alzheimer's disease. Mean exposure to aluminum and silica in drinking water was estimated in each area. The sample studied included 2,698 nondemented subjects at baseline, for whom components of drinking water and covariates were available. A total of 253 incident cases of dementia (with 17 exposed to high levels of aluminum), including 182 Alzheimer's disease (with 13 exposed to high aluminum levels), were identified. The relative risk of dementia adjusted for age, gender, educational level, place of residence, and wine consumption was 1.99 (95 percent CI: 1.20, 3.28) for subjects exposed to an aluminum concentration greater than 0.1 mg/liter. This result was confirmed for Alzheimer's disease (adjusted relative risk = 2.14, 95 percent CI: 1.21, 3.80). However, no dose-response relation was found. Inversely, the adjusted relative risk of dementia for subjects exposed to silica (> or = 11.25 mg/liter) was 0.74 (95 percent CI: 0.58, 0.96). These findings support the hypothesis that a high concentration of aluminum in drinking water may be a risk factor for Alzheimer's disease.

Analysis of left-censored longitudinal data with application to viral load in HIV infection.

The classical model for the analysis of progression of markers in HIV-infected patients is the mixed effects linear model. However, longitudinal studies of viral load are complicated by left censoring of the measures due to a lower quantification limit. We propose a full likelihood approach to estimate parameters from the linear mixed effects model for left-censored Gaussian data. For each subject, the contribution to the likelihood is the product of the density for the vector of the completely observed outcome and of the conditional distribution function of the vector of the censored outcome, given the observed outcomes. Values of the distribution function were computed by numerical integration. The maximization is performed by a combination of the Simplex algorithm and the Marquardt algorithm. Subject-specific deviations and random effects are estimated by modified empirical Bayes replacing censored measures by their conditional expectations given the data. A simulation study showed that the proposed estimators are less biased than those obtained by imputing the quantification limit to censored data. Moreover, for models with complex covariance structures, they are less biased than Monte Carlo expectation maximization (MCEM) estimators developed by Hughes (1999) Mixed effects models with censored data with application to HIV RNA Levels. Biometrics 55, 625-629. The method was then applied to the data of the ALBI-ANRS 070 clinical trial for which HIV-1 RNA levels were measured with an ultrasensitive assay (quantification limit 50 copies/ml). Using the proposed method, estimates obtained with data artificially censored at 500 copies/ml were close to those obtained with the real data set.

Marital status and risk of Alzheimer's disease: a French population-based cohort study.

OBJECTIVE: To analyze the relationship between marital status and risk of AD or dementia. METHODS: This study was carried out from the Personnes Agées QUID (PAQUID) cohort, an epidemiologic study on normal and pathologic aging after age 65 years. The PAQUID cohort began in 1988. Individuals were followed up at 1, 3, and 5 years, with an active detection of dementia. Marital status was divided into four categories: widowed, never married, divorced or separated, and the reference category, married or cohabitant. The longitudinal relationship between marital status and risk of incident AD or dementia was analyzed by a Cox model with delayed entry. RESULTS: Among the 3,675 individuals initially not demented, 2,106 were married or cohabitants, 1,287 were widowers, 179 were never married, and 103 were divorced or separated. Among the 2,881 individuals reevaluated at least once for the risk of dementia during the 5-year follow-up, 190 incident cases of dementia were identified, including 140 with AD. The relative risks (RRs) of dementia (RR = 1.91, p = 0.018) and of AD (RR = 2.68, p<0.001) were increased for the never-married individuals compared with those who were married or cohabitants. This excess of risk was specifically associated with AD. Adjustment for other risk factors of dementia (education, wine consumption), or for factors reflecting social environment, leisure activities, and depression, did not modify the risk of AD for never-married individuals (RR = 2.31, p = 0.02). CONCLUSIONS: We confirmed an association between marital status and AD, with an excess risk observed among never-married individuals. This association may provide clues about the pathogenesis of AD.

Multipoint linkage analysis using the weighted-pairwise correlation statistic.

The weighted pairwise correlation (WPC) approach provides simple and flexible non-parametric tests for genetic linkage which may be adapted to qualitative, quantitative or age-dependent traits. Although this statistic has often been used with identity-by-state (IBS) information, there is much to gain by using multipoint identity-by-descent (IBD) information. The purpose of this paper is to use the unified multipoint approach of Kruglyak et al. [1996] with the WPC statistic. We used the simulated data of a complex disease proposed in GAW11 (problem 2) to validate this approach and to compare the multipoint WPC to the IBS-WPC and to the multilocus approach using the conventional non-parametric statistics S(pairs) and S(all). The results suggest that the multipoint WPC may be the most powerful approach.