Dynamic modeling of multivariate dimensions and their temporal relationships using latent processes: Application to Alzheimer's disease.

Alzheimer's disease gradually affects several components including the cerebral dimension with brain atrophies, the cognitive dimension with a decline in various functions, and the functional dimension with impairment in the daily living activities. Understanding how such dimensions interconnect is crucial for Alzheimer's disease research. However, it requires to simultaneously capture the dynamic and multidimensional aspects and to explore temporal relationships between dimensions. We propose an original dynamic structural model that accounts for all these features. The model defines dimensions as latent processes and combines a multivariate linear mixed model and a system of difference equations to model trajectories and temporal relationships between latent processes in finely discrete time. Dimensions are simultaneously related to their observed (possibly multivariate) markers through nonlinear equations of observation. Parameters are estimated in the maximum likelihood framework enjoying a closed form for the likelihood. We demonstrate in a simulation study that this dynamic model in discrete time benefits the same causal interpretation of temporal relationships as models defined in continuous time as long as the discretization step remains small. The model is then applied to the data of the Alzheimer's Disease Neuroimaging Initiative. Three longitudinal dimensions (cerebral anatomy, cognitive ability, and functional autonomy) measured by six markers are analyzed, and their temporal structure is contrasted between different clinical stages of Alzheimer's disease.

Adaptive protocols based on predictions from a mechanistic model of the effect of IL7 on CD4 counts.

In human immunodeficiency virus-infected patients, antiretroviral therapy suppresses the viral replication, which is followed in most patients by a restoration of CD4+ T cells pool. For patients who fail to do so, repeated injections of exogenous interleukin 7 (IL7) are experimented. The IL7 is a cytokine that is involved in the T cell homeostasis and the INSPIRE study has shown that injections of IL7 induced a proliferation of CD4+ T cells. Phase I/II INSPIRE 2 and 3 studies have evaluated a protocol in which a first cycle of three IL7 injections is followed by a new cycle at each visit when the patient has less than 550 CD4 cells/µL. Restoration of the CD4 concentration has been demonstrated, but the long-term best adaptive protocol is yet to be determined. A mechanistic model of the evolution of CD4 after IL7 injections has been developed, which is based on a system of ordinary differential equations and includes random effects. Based on the estimation of this model, we use a Bayesian approach to forecast the dynamics of CD4 in new patients. We propose four prediction-based adaptive protocols of injections to minimize the time spent under 500 CD4 cells/µL for each patient, without increasing the number of injections received too much. We show that our protocols significantly reduce the time spent under 500 CD4 over a period of two years, without increasing the number of injections. These protocols have the potential to increase the efficiency of this therapy.

Dealing with death when studying disease or physiological marker: the stochastic system approach to causality.

The stochastic system approach to causality is applied to situations where the risk of death is not negligible. This approach grounds causality on physical laws, distinguishes system and observation and represents the system by multivariate stochastic processes. The particular role of death is highlighted, and it is shown that local influences must be defined on the random horizon of time of death. We particularly study the problem of estimating the effect of a factor V on a process of interest Y, taking death into account. We unify the cases where Y is a counting process (describing an event) and the case where Y is quantitative; we examine the case of observations in continuous and discrete time and we study the issue of whether the mechanism leading to incomplete data can be ignored. Finally, we give an example of a situation where we are interested in estimating the effect of a factor (blood pressure) on cognitive ability in elderly.

cytometree: A binary tree algorithm for automatic gating in cytometry analysis.

Flow cytometry is a powerful technology that allows the high-throughput quantification of dozens of surface and intracellular proteins at the single-cell level. It has become the most widely used technology for immunophenotyping of cells over the past three decades. Due to the increasing complexity of cytometry experiments (more cells and more markers), traditional manual flow cytometry data analysis has become untenable due to its subjectivity and time-consuming nature. We present a new unsupervised algorithm called "cytometree" to perform automated population identification (aka gating) in flow cytometry. cytometree is based on the construction of a binary tree, the nodes of which are subpopulations of cells. At each node, the marker distributions are modeled by mixtures of normal distributions. Node splitting is done according to a model selection procedure based on a normalized difference of Akaike information criteria between two competing models. Post-processing of the tree structure and derived populations allows us to complete the annotation of the populations. The algorithm is shown to perform better than the state-of-the-art unsupervised algorithms previously proposed on panels introduced by the Flow Cytometry: Critical Assessment of Population Identification Methods project. The algorithm is also applied to a T-cell panel proposed by the Human Immunology Project Consortium (HIPC) program; it also outperforms the best unsupervised open-source available algorithm while requiring the shortest computation time. © 2018 International Society for Advancement of Cytometry.

Lasso regularization for left-censored Gaussian outcome and high-dimensional predictors.

BACKGROUND: Biological assays for the quantification of markers may suffer from a lack of sensitivity and thus from an analytical detection limit. This is the case of human immunodeficiency virus (HIV) viral load. Below this threshold the exact value is unknown and values are consequently left-censored. Statistical methods have been proposed to deal with left-censoring but few are adapted in the context of high-dimensional data. METHODS: We propose to reverse the Buckley-James least squares algorithm to handle left-censored data enhanced with a Lasso regularization to accommodate high-dimensional predictors. We present a Lasso-regularized Buckley-James least squares method with both non-parametric imputation using Kaplan-Meier and parametric imputation based on the Gaussian distribution, which is typically assumed for HIV viral load data after logarithmic transformation. Cross-validation for parameter-tuning is based on an appropriate loss function that takes into account the different contributions of censored and uncensored observations. We specify how these techniques can be easily implemented using available R packages. The Lasso-regularized Buckley-James least square method was compared to simple imputation strategies to predict the response to antiretroviral therapy measured by HIV viral load according to the HIV genotypic mutations. We used a dataset composed of several clinical trials and cohorts from the Forum for Collaborative HIV Research (HIV Med. 2008;7:27-40). The proposed methods were also assessed on simulated data mimicking the observed data. RESULTS: Approaches accounting for left-censoring outperformed simple imputation methods in a high-dimensional setting. The Gaussian Buckley-James method with cross-validation based on the appropriate loss function showed the lowest prediction error on simulated data and, using real data, the most valid results according to the current literature on HIV mutations. CONCLUSIONS: The proposed approach deals with high-dimensional predictors and left-censored outcomes and has shown its interest for predicting HIV viral load according to HIV mutations.

Dynamic models for estimating the effect of HAART on CD4 in observational studies: Application to the Aquitaine Cohort and the Swiss HIV Cohort Study.

Highly active antiretroviral therapy (HAART) has proved efficient in increasing CD4 counts in many randomized clinical trials. Because randomized trials have some limitations (e.g., short duration, highly selected subjects), it is interesting to assess the effect of treatments using observational studies. This is challenging because treatment is started preferentially in subjects with severe conditions. This general problem had been treated using Marginal Structural Models (MSM) relying on the counterfactual formulation. Another approach to causality is based on dynamical models. We present three discrete-time dynamic models based on linear increments models (LIM): the first one based on one difference equation for CD4 counts, the second with an equilibrium point, and the third based on a system of two difference equations, which allows jointly modeling CD4 counts and viral load. We also consider continuous-time models based on ordinary differential equations with non-linear mixed effects (ODE-NLME). These mechanistic models allow incorporating biological knowledge when available, which leads to increased statistical evidence for detecting treatment effect. Because inference in ODE-NLME is numerically challenging and requires specific methods and softwares, LIM are a valuable intermediary option in terms of consistency, precision, and complexity. We compare the different approaches in simulation and in illustration on the ANRS CO3 Aquitaine Cohort and the Swiss HIV Cohort Study.

Repeated Cycles of Recombinant Human Interleukin 7 in HIV-Infected Patients With Low CD4 T-Cell Reconstitution on Antiretroviral Therapy: Results of 2 Phase II Multicenter Studies.

BACKGROUND: Phase I/II studies in human immunodeficiency virus (HIV)-infected patients receiving antiretroviral therapy have shown that a single cycle of 3 weekly subcutaneous (s/c) injections of recombinant human interleukin 7 (r-hIL-7) is safe and improves immune CD4 T-cell restoration. Herein, we report data from 2 phase II trials evaluating the effect of repeated cycles of r-hIL-7 (20 µg/kg) with the objective of restoring a sustained CD4 T-cell count >500 cells/µL. METHODS: INSPIRE 2 was a single-arm trial conducted in the United States and Canada. INSPIRE 3 was a 2 arm trial with 3:1 randomization to r-hIL-7 versus control conducted in Europe and South Africa. Participants with plasma HIV RNA levels <50 copies/mL during antiretroviral therapy and with CD4 T-cell counts between 101 and 400 cells/µL were eligible. A repeat cycle was administered when CD4 T-cell counts fell to <550 cells/µL. RESULTS: A total of 107 patients were treated and received 1 (n = 107), 2 (n = 74), 3 (n = 14), or 4 (n = 1) r-hIL-7 cycles during a median follow-up of 23 months. r-hIL-7 was well tolerated. Four grade 4 events were observed, including 1 case of asymptomatic alanine aminotransferase elevation. After the second cycle, anti-r-hIL-7 binding antibodies developed in 82% and 77% of patients in INSPIRE 2 and 3, respectively (neutralizing antibodies in 38% and 37%), without impact on the CD4 T-cell response. Half of the patients spent >63% of their follow-up time with a CD4 T-cell count >500 cells/µL. CONCLUSIONS: Repeated cycles of r-hIL-7 were well tolerated and achieved sustained CD4 T-cell restoration to >500 cells/µL in the majority of study participants. CLINICAL TRIALS REGISTRATION: INSPIRE II: clinicaltrials.gov (NCT01190111) and INSPIRE III: EudraCT (No. 2010-019773-15) and clinicaltrials.gov (NCT01241643).

A Method to Estimate the Size and Characteristics of HIV-positive Populations Using an Individual-based Stochastic Simulation Model.

It is important not only to collect epidemiologic data on HIV but to also fully utilize such information to understand the epidemic over time and to help inform and monitor the impact of policies and interventions. We describe and apply a novel method to estimate the size and characteristics of HIV-positive populations. The method was applied to data on men who have sex with men living in the UK and to a pseudo dataset to assess performance for different data availability. The individual-based simulation model was calibrated using an approximate Bayesian computation-based approach. In 2013, 48,310 (90% plausibility range: 39,900-45,560) men who have sex with men were estimated to be living with HIV in the UK, of whom 10,400 (6,160-17,350) were undiagnosed. There were an estimated 3,210 (1,730-5,350) infections per year on average between 2010 and 2013. Sixty-two percent of the total HIV-positive population are thought to have viral load <500 copies/ml. In the pseudo-epidemic example, HIV estimates have narrower plausibility ranges and are closer to the true number, the greater the data availability to calibrate the model. We demonstrate that our method can be applied to settings with less data, however plausibility ranges for estimates will be wider to reflect greater uncertainty of the data used to fit the model.

Quantifying and predicting the effect of exogenous interleukin-7 on CD4+ T cells in HIV-1 infection.

Exogenous Interleukin-7 (IL-7), in supplement to antiretroviral therapy, leads to a substantial increase of all CD4+ T cell subsets in HIV-1 infected patients. However, the quantitative contribution of the several potential mechanisms of action of IL-7 is unknown. We have performed a mathematical analysis of repeated measurements of total and naive CD4+ T cells and their Ki67 expression from HIV-1 infected patients involved in three phase I/II studies (N = 53 patients). We show that, besides a transient increase of peripheral proliferation, IL-7 exerts additional effects that play a significant role in CD4+ T cell dynamics up to 52 weeks. A decrease of the loss rate of the total CD4+ T cell is the most probable explanation. If this effect could be maintained during repeated administration of IL-7, our simulation study shows that such a strategy may allow maintaining CD4+ T cell counts above 500 cells/µL with 4 cycles or fewer over a period of two years. This in-depth analysis of clinical data revealed the potential for IL-7 to achieve sustained CD4+ T cell restoration with limited IL-7 exposure in HIV-1 infected patients with immune failure despite antiretroviral therapy.

State selection in Markov models for panel data with application to psoriatic arthritis.

Markov multistate models in continuous-time are commonly used to understand the progression over time of disease or the effect of treatments and covariates on patient outcomes. The states in multistate models are related to categorisations of the disease status, but there is often uncertainty about the number of categories to use and how to define them. Many categorisations, and therefore multistate models with different states, may be possible. Different multistate models can show differences in the effects of covariates or in the time to events, such as death, hospitalisation, or disease progression. Furthermore, different categorisations contain different quantities of information, so that the corresponding likelihoods are on different scales, and standard, likelihood-based model comparison is not applicable. We adapt a recently developed modification of Akaike's criterion, and a cross-validatory criterion, to compare the predictive ability of multistate models on the information which they share. All the models we consider are fitted to data consisting of observations of the process at arbitrary times, often called 'panel' data. We develop an implementation of these criteria through Hidden Markov models and apply them to the comparison of multistate models for the Health Assessment Questionnaire score in psoriatic arthritis. This procedure is straightforward to implement in the R package 'msm'.

Clonally diverse T cell homeostasis is maintained by a common program of cell-cycle control.

Lymphopenia induces T cells to undergo cell divisions as part of a homeostatic response mechanism. The clonal response to lymphopenia is extremely diverse, and it is unknown whether this heterogeneity represents distinct mechanisms of cell-cycle control or whether a common mechanism can account for the diversity. We addressed this question by combining in vivo and mathematical modeling of lymphopenia-induced proliferation (LIP) of two distinct T cell clonotypes. OT-I T cells undergo rapid LIP accompanied by differentiation that superficially resembles Ag-induced proliferation, whereas F5 T cells divide slowly and remain naive. Both F5 and OT-I LIP responses were most accurately described by a single stochastic division model where the rate of cell division was exponentially decreased with increasing cell numbers. The model successfully identified key biological parameters of the response and accurately predicted the homeostatic set point of each clone. Significantly, the model was successful in predicting interclonal competition between OT-I and F5 T cells, consistent with competition for the same resource(s) required for homeostatic proliferation. Our results show that diverse and heterogeneous clonal T cell responses can be accounted for by a single common model of homeostasis.

The stochastic system approach for estimating dynamic treatments effect.

The problem of assessing the effect of a treatment on a marker in observational studies raises the difficulty that attribution of the treatment may depend on the observed marker values. As an example, we focus on the analysis of the effect of a HAART on CD4 counts, where attribution of the treatment may depend on the observed marker values. This problem has been treated using marginal structural models relying on the counterfactual/potential response formalism. Another approach to causality is based on dynamical models, and causal influence has been formalized in the framework of the Doob-Meyer decomposition of stochastic processes. Causal inference however needs assumptions that we detail in this paper and we call this approach to causality the "stochastic system" approach. First we treat this problem in discrete time, then in continuous time. This approach allows incorporating biological knowledge naturally. When working in continuous time, the mechanistic approach involves distinguishing the model for the system and the model for the observations. Indeed, biological systems live in continuous time, and mechanisms can be expressed in the form of a system of differential equations, while observations are taken at discrete times. Inference in mechanistic models is challenging, particularly from a numerical point of view, but these models can yield much richer and reliable results.

[Mathematical dynamical models for personalized medicine]. / Modèles mathématiques dynamiques pour la médecine personnalisée.

One of the necessary conditions to perform any personalized medicine is to obtain good individual predictions. In addition to the numerous markers available (omics data), the methods used to analyze the data are very important too. We are presenting an example of mathematical dynamical mechanistic model that could be used for adapting the antiretroviral treatment in patients infected by the human immunodeficiency virus. The interest of this type of approach is to build a model based on biological knowledge about the interaction between markers and therefore to allow for a better predictive power.

Prevalence projections of chronic diseases and impact of public health intervention.

The estimation of future prevalences of chronic diseases is essential for public health policy. Using incidence estimates from cohort data and demographic projections for general mortality and population sizes, we propose a method based on a general illness-death model to make prevalence projections for chronic diseases. In contrast to previously published methods, we account for differences between global mortality and mortality of healthy subjects and compare two assumptions regarding the secular trend for mortality of diseased subjects. Then we develop a methodology to estimate changes in future disease prevalences resulting from prevention campaign to reduce the frequency or the excess risk associated with a risk factor. The methods are applied for estimating dementia prevalence in France between 2010 and 2030.

20-Year prevalence projections for dementia and impact of preventive policy about risk factors.

Incidence of dementia increases sharply with age and, because of the increase in life expectancy, the number of dementia cases is expected to rise dramatically over time. Some studies suggest that controlling some modifiable risk factors for dementia like diabetes or hypertension could lower its incidence. However, as treating these vascular factors would also reduce mortality risk, the actual impact of such public-health intervention on dementia prevalence is not known. Accounting for the impact of dementia and risk factors on mortality, the aim of this work was (1) to compute projections of age- and-sex specific prevalence of dementia in France from 2010 to 2030, (2) to evaluate how public-health interventions targeting risk factors for dementia could change these projections. Age-and-sex specific incidence of dementia and mortality of demented subjects were estimated from the Paquid population-based cohort using a semi-parametric illness-death model. Future global mortality rates and population sizes were obtained from national demographic projections. Under the assumption that life expectancy will increase by 3.5 years for men and 2.8 years for women by 2030, the number of subjects with dementia was estimated to increase by about 75% from 2010 to 2030 with a 200% increase after 90 years of age. Therapeutic intervention on the whole population reducing high blood pressure prevalence would lead to a decrease in both dementia incidence rates and mortality and would have a modest impact on the number of dementia cases. On the other hand, a preventive dementia treatment targeting ApoE4 carriers would probably not improve survival and hence would decrease dementia prevalence by 15-25%.

Evidence synthesis through a degradation model applied to myocardial infarction.

We propose an evidence synthesis approach through a degradation model to estimate causal influences of physiological factors on myocardial infarction (MI) and coronary heart disease (CHD). For instance several studies give incidences of MI and CHD for different age strata, other studies give relative or absolute risks for strata of main risk factors of MI or CHD. Evidence synthesis of several studies allows incorporating these disparate pieces of information into a single model. For doing this we need to develop a sufficiently general dynamical model; we also need to estimate the distribution of explanatory factors in the population. We develop a degradation model for both MI and CHD using a Brownian motion with drift, and the drift is modeled as a function of indicators of obesity, lipid profile, inflammation and blood pressure. Conditionally on these factors the times to MI or CHD have inverse Gaussian ([Formula: see text]) distributions. The results we want to fit are generally not conditional on all the factors and thus we need marginal distributions of the time of occurrence of MI and CHD; this leads us to manipulate the inverse Gaussian normal distribution ([Formula: see text]) (an [Formula: see text] whose drift parameter has a normal distribution). Another possible model arises if a factor modifies the threshold. This led us to define an extension of [Formula: see text] obtained when both drift and threshold parameters have normal distributions. We applied the model to results published in five important studies of MI and CHD and their risk factors. The fit of the model using the evidence synthesis approach was satisfactory and the effects of the four risk factors were highly significant.

Treatment monitoring of HIV-infected patients based on mechanistic models.

For most patients, the HIV viral load can be made undetectable by highly active antiretroviral treatments highly active antiretroviral therapy: the virus, however, cannot be eradicated. Thus, the major problem is to try to reduce the side effects of the treatment that patients have to take during their life time. We tackle the problem of monitoring the treatment dose, with the aim of giving the minimum dose that yields an undetectable viral load. The approach is based on mechanistic models of the interaction between virus and the immune system. It is shown that the "activated cells model," allows making good predictions of the effect of dose changes and, thus, could be a good basis for treatment monitoring. Then, we use the fact that in dynamical models, there is a nontrivial equilibrium point, that is with a virus load larger than zero, only if the reproductive number R(0) is larger than one. For reducing side effects, we may give a dose just above the critical dose corresponding to R(0) equal to 1. A prior distribution of the parameters of the model can be taken as the posterior arising from the analysis of previous clinical trials. Then the observations for a given patient can be used to dynamically tune the dose so that there is a high probability that the reproductive number is below one. The advantage of the approach is that it does not depend on a cost function, weighing side effects and efficiency of the drug. It is shown that it is possible to approach the critical dose if the model is correct. A sensitivity analysis assesses the robustness of the approach.

Choice of prognostic estimators in joint models by estimating differences of expected conditional Kullback-Leibler risks.

Prognostic estimators for a clinical event may use repeated measurements of markers in addition to fixed covariates. These measurements can be linked to the clinical event by joint models that involve latent features. When the objective is to choose between different prognosis estimators based on joint models, the conventional Akaike information criterion is not well adapted and decision should be based on predictive accuracy. We define an adapted risk function called expected prognostic cross-entropy. We define another risk function for the case of right-censored observations, the expected prognostic observed cross-entropy (EPOCE). These risks can be estimated by leave-one-out cross-validation, for which we give approximate formulas and asymptotic distributions. The approximated cross-validated estimator CVPOL (a) of EPOCE is studied in simulation and applied to the comparison of several joint latent class models for prognosis of recurrence of prostate cancer using prostate-specific antigen measurements.

Estimating survival of dental fillings on the basis of interval-censored data and multi-state models.

We aim to compare the life expectancy of a filling in a primary tooth between two types of treatments. We define the probabilities that a dental filling survives without complication until the permanent tooth erupts from beneath (exfoliation). We relate the time to exfoliation of the tooth to the age of the child and the time to failure of the filling to the duration since the treatment. We followed up fillings at repeated examinations where information is collected regarding the filling and the tooth. Several fillings can be placed in the same mouth, possibly by the same dentist. To deal with all these particularities, we propose to use a parametric four-state model with three random effects to take into account the hierarchical cluster structure. For inference, right and interval censoring as well as left truncation have to be dealt with. With the proposed approach, we can conclude that the estimated probability that a filling survives without complication until exfoliation is larger for one treatment than for the other, for all ages of the child at the time of treatment.

Joint modeling of the clinical progression and of the biomarkers' dynamics using a mechanistic model.

Joint models are used to rigorously explore the relationship between the dynamics of biomarkers and clinical events. In the context of HIV infection, where the multivariate dynamics of HIV-RNA and CD4 are complex, a mechanistic approach based on a system of nonlinear differential equations naturally takes into account the correlation between the biomarkers. Using data from a randomized clinical trial comparing dual antiretroviral therapy to a single drug regimen, a full maximum likelihood approach is proposed to explore the relationship between the evolution of the biomarkers and the time to a clinical event. The role of each marker as an independent predictor of disease progression is assessed. We show that the joint dynamics of HIV-RNA and CD4 captures the effect of antiretroviral treatment; the CD4 dynamics alone is found to capture most but not all of the treatment effect.