A Bounded Integer Model for Rating and Composite Scale Data

Modeling of Parkinson's Disease Progression and Implications for Detection of Disease Modification in Treatment Trials

Background

Objectively measuring Parkinson's disease (PD) signs and symptoms over time is critical for the successful development of treatments aimed at halting the disease progression of people with PD.

Objective

To create a clinical trial simulation tool that characterizes the natural history of PD progression and enables a data-driven design of randomized controlled studies testing potential disease-modifying treatments (DMT) in early-stage PD.

Methods

Data from the Parkinson's Progression Markers Initiative (PPMI) were analyzed with nonlinear mixed-effect modeling techniques to characterize the progression of MDS-UPDRS part I (non-motor aspects of experiences of daily living), part II (motor aspects of experiences of daily living), and part III (motor signs). A clinical trial simulation tool was built from these disease models and used to predict probability of success as a function of trial design.

Results

MDS-UPDRS part III progresses approximately 3 times faster than MDS-UPDRS part II and I, with an increase of 3 versus 1 points/year. Higher amounts of symptomatic therapy is associated with slower progression of MDS-UPDRS part II and III. The modeling framework predicts that a DMT effect on MDS-UPDRS part III could precede effect on part II by approximately 2 to 3 years.

Conclusions

Our clinical trial simulation tool predicted that in a two-year randomized controlled trial, MDS-UPDRS part III could be used to evaluate a potential novel DMT, while part II would require longer trials of a minimum duration of 3 to 5 years underscoring the need for innovative trial design approaches including novel patient-centric measures.

Employing zero-inflated beta distribution in an exposure-response analysis of TYK2/JAK1 inhibitor brepocitinib in patients with plaque psoriasis

Brepocitinib is an oral selective dual TYK2/JAK1 inhibitor and based on its cytokine inhibition profile is expected to provide therapeutic benefit in the treatment of plaque psoriasis. Efficacy data from a completed Phase 2a study in patients with moderate-to-severe plaque psoriasis were utilized to develop a population exposure-response model that can be employed to inform dose selection decisions for further clinical development. A modeling approach that employs the zero-inflated beta distribution was used to account for the bounded nature and distributional characteristics of the Psoriasis Area and Severity Index (PASI) score data. The developed exposure-response model provided an adequate description of the observed PASI scores across all the treatment arms tested and across both the induction and maintenance dosing periods of the study. In addition, the developed model exhibited a good predictive capacity with regard to the derived responder metrics (e.g., 75%/90%/100% improvement in PASI score [PASI75/90/100]). Clinical trial simulations indicated that the induction/maintenance dosing paradigm explored in this study does not offer any advantages from an efficacy perspective and that doses of 10, 30, and 60 mg once-daily may be suitable candidates for clinical evaluation in subsequent Phase 2b studies.

Bayesian compositional models for ordinal response

Ordinal response is commonly found in medicine, biology, and other fields. In many situations, the predictors for this ordinal response are compositional, which means that the sum of predictors for each sample is fixed. Examples of compositional data include the relative abundance of species in microbiome data and the relative frequency of nutrition concentrations. Moreover, the predictors that are strongly correlated tend to have similar influence on the response outcome. Conventional cumulative logistic regression models for ordinal responses ignore the fixed-sum constraint on predictors and their associated interrelationships, and thus are not appropriate for analyzing compositional predictors.To solve this problem, we proposed Bayesian Compositional Models for Ordinal Response to analyze the relationship between compositional data and an ordinal response with a structured regularized horseshoe prior for the compositional coefficients and a soft sum-to-zero restriction on coefficients through the prior distribution. The method was implemented with R package rstan using efficient Hamiltonian Monte Carlo algorithm. We performed simulations to compare the proposed approach and existing methods for ordinal responses. Results revealed that our proposed method outperformed the existing methods in terms of parameter estimation and prediction. We also applied the proposed method to a microbiome study HMP2Data, to find microorganisms linked to ordinal inflammatory bowel disease levels. To make this work reproducible, the code and data used in this paper are available at https://github.com/Li-Zhang28/BCO.

Moving Beyond Boundaries: Utilization of Longitudinal Exposure-Response Model for Bounded Outcome Score to Inform Decision Making in the Accelerated Drug Development Paradigm

BACKGROUND AND OBJECTIVES

As drug development scientists strive to accelerate availability of therapies for patients, model-informed drug development (MIDD) plays an important role in contextualizing existing information and facilitating decision making. This paper describes an example of MIDD, where modeling and simulation informed decision making in the circumstance of a combined phase 2b and single pivotal study for ritlecitinib (JAK3/TEC family kinases inhibitor).

METHODS

Longitudinal exposure-response (ER) modeling was conducted to describe ritlecitinib efficacy in alopecia areata patients. The Severity of Alopecia Tool (SALT) score (a continuous bounded outcome [CBO] score [0-100]) was used as the efficacy response. The average concentration during the time interval between two adjacent SALT scores was used as the exposure metric driving efficacy.

RESULTS

The developed model well described the longitudinal SALT profile of ritlecitinib as well as the frequency of boundary data. The CBO model indicated tested doses in the phase 2b/3 clinical trial are in the ascending region of ER and contextualized a loading dose effect that impacted onset of efficacy without long-term benefit. It also identified disease severity as the only covariate impacting efficacy. The model-based simulation further informed impact of treatment interruption on the loss of efficacy in the absence of a dedicated treatment withdrawal study. Results indicated temporary treatment interruption ≤ 6 weeks is not expected to result in significant loss of efficacy.

CONCLUSION

The CBO modeling approach and simulation supported the single pivotal trial strategy and guided dose selection in the accelerated drug development program of ritlecitinib, which can be applied to many indications where efficacy is measured on a bounded scale.

A disease model predicting placebo response and remission status of patients with ulcerative colitis using modified Mayo score

The Mayo Clinical Score is used in clinical trials to describe the clinical status of patients with ulcerative colitis (UC). It comprises four subscores: rectal bleeding (RB), stool frequency (SF), physician's global assessment, and endoscopy (ENDO). According to recent US Food and Drug Administration guidelines (Ulcerative colitis: developing drugs for treatment, Guidance Document, https://www.fda.gov/regulatory-information/s. 2022), clinical response and remission should be based on modified Mayo Score (mMS) relying on RB, SF, and ENDO. Typically, ENDO is performed at the beginning and end of each phase, whereas RB and SF are more frequently available. Item response theory (IRT) models allow the shared information to be used for prediction of all subscores at each observation time; therefore, it leverages information from RB and SF to predict ENDO. A UC disease IRT model was developed based on four etrolizumab phase III studies to describe the longitudinal mMS subscores, placebo response, and remission at the end of induction and maintenance. For each subscore, a bounded integer model was developed. The placebo response was characterized by a mono-exponential function acting on all mMS subscores similarly. The final model reliably predicted longitudinal mMS data. In addition, remission was well-predicted by the model, with only 5% overprediction at the end of induction and 3% underprediction at the end of maintenance. External evaluation of the final model using placebo arms from five different studies indicated adequate performance for both longitudinal mMS subscores and remission status. These results suggest utility of the current disease model for informed decision making in UC clinical development, such as assisting future clinical trial designs and evaluations.

Optimized infusion rates for N,N-dimethyltryptamine to achieve a target psychedelic intensity based on a modeling and simulation framework

N,N-dimethyltryptamine (DMT) is a psychedelic compound that is being studied as a therapeutic option in various psychiatric disorders. Due to its short half-life, continuous infusion of DMT has been proposed to extend the psychedelic experience and potential therapeutic effects. The primary aim of this work was to design an infusion protocol for DMT based on a desired level of psychedelic intensity using population pharmacokinetic/pharmacodynamic modeling. As a secondary aim, the impact of choosing a continuous variable or a bounded integer pharmacokinetic/pharmacodynamic model to inform such an infusion protocol was investigated. A previously published continuous variable model and two newly developed bounded integer models were used to assess optimal doses for achieving a target response. Simulations were performed to identify an optimal combination of a bolus dose and an infusion rate. Based on the simulations, optimal doses to achieve intensity ratings between 7 and 9 (possible range = 0-10) were a bolus dose of 16 mg DMT fumarate followed by an infusion rate of 1.4 mg/min based on the continuous variable model and 14 mg with 1.2 mg/min for the two bounded integer models. However, the proportion within target was low (<53%) for all models, indicating that individual dose adjustments would be necessary. Furthermore, some differences between the models were observed. The bounded integer models generally predicted lower proportions within a target of 7-9 with higher proportions exceeding target compared with the continuous variable model. However, results varied depending on target response with the major differences observed at the boundaries of the scale.

Plan E, Weber B. Bounded integer model-based analysis of psoriasis area and severity index in patients with moderate-to-severe plaque psoriasis receiving BI 730357

BI 730357 is investigated as an oral treatment of plaque psoriasis. We analyzed the impact of three dosage regimens on the Psoriasis Area and Severity Index (PASI) response with modeling based on phase I and II data from 109 healthy subjects and 274 patients with moderate-to-severe plaque psoriasis. The pharmacokinetics (PK) was characterized by a two-compartment model with dual absorption paths and a first-order elimination. Higher baseline C-reactive protein was associated with lower clearance and patients generally had lower clearance compared with healthy subjects. A bounded integer PK/pharmacodynamic model characterized the effect on the observed PASI. The maximum drug effect was largest for patients with no prior biologic use, smaller for patients with prior use of non-interleukin-17 inhibitors, and smallest for patients with prior interleukin-17 inhibitor use. The models allowed robust simulation of large patient populations, predicting a plateau in PASI outcomes for BI 730357 exposure above 2000 nmol/L.

An Item Response Theory-Informed Strategy to Model Total Score Data from Composite Scales

Composite scale data is widely used in many therapeutic areas and consists of several categorical questions/items that are usually summarized into a total score (TS). Such data is discrete and bounded by nature. The gold standard to analyse composite scale data is item response theory (IRT) models. However, IRT models require item-level data while sometimes only TS is available. This work investigates models for TS. When an IRT model exists, it can be used to derive the information as well as expected mean and variability of TS at any point, which can inform TS-analyses. We propose a new method: IRT-informed functions of expected values and standard deviation in TS-analyses. The most common models for TS-analyses are continuous variable (CV) models, while bounded integer (BI) models offer an alternative that respects scale boundaries and the nature of TS data. We investigate the method in CV and BI models on both simulated and real data. Both CV and BI models were improved in fit by IRT-informed disease progression, which allows modellers to precisely and accurately find the corresponding latent variable parameters, and IRT-informed SD, which allows deviations from homoscedasticity. The methodology provides a formal way to link IRT models and TS models, and to compare the relative information of different model types. Also, joint analyses of item-level data and TS data are made possible. Thus, IRT-informed functions can facilitate total score analysis and allow a quantitative analysis of relative merits of different analysis methods.

Bounded Integer Modeling of Symptom Scales Specific to Lower Urinary Tract Symptoms Secondary to Benign Prostatic Hyperplasia

The International Prostate Symptom Score (IPSS), the quality of life (QoL) score, and the benign prostatic hyperplasia impact index (BII) are three different scales commonly used to assess the severity of lower urinary tract symptoms associated with benign prostatic hyperplasia (BPH-LUTS). Based on a phase II clinical trial including 403 patients with moderate to severe BPH-LUTS, the objectives of this study were to (i) develop traditional pharmacometric and bounded integer (BI) models for the IPSS, QoL score, and BII endpoints, respectively; (ii) compare the power and type I error in detecting drug effects of BI modeling with traditional methods through simulation; and (iii) obtain quantitative translation between scores on the three abovementioned scales using a BI modeling framework. All developed models described the data adequately. Pharmacometric modeling using a continuous variable (CV) approach was overall found to be the most robust in terms of type I error and power to detect a drug effect. In most cases, BI modeling showed similar performance to the CV approach, yet severely inflated type I error was generally observed when inter-individual variability (IIV) was incorporated in the BI variance function (g()). BI modeling without IIV in g() showed greater type I error control compared to the ordered categorical approach. Lastly, a multiple-scale BI model was developed and estimated the relationship between scores on the three BPH-LUTS scales with overall low uncertainty. The current study yields greater understanding of the operating characteristics of the novel BI modeling approach and highlights areas potentially requiring further improvement.

Two-part models for repeatedly measured ordinal data with "don't know" category

Ordinal data (eg, "low," "medium," "high"; graded response on a Likert scale) with an additional "don't know" category are frequently encountered in the medical, social, and behavioral science literature. The handling of a "don't know" option presents unique challenges as it often "destroys" the ordinal nature of the data. Commonly, nominal models are employed which ignore the partial ordering and have a complicated interpretation, especially in situations with repeatedly measured outcomes. We propose two-part models that easily accommodate longitudinal partially ordered (semiordinal) data. The most easily interpretable formulation consists of a random effect logistic submodel for "don't know" vs all the other categories combined, and a random effect ordinal submodel for the ordered categories. Correlated random effects account for statistical dependence within individual. An extension allowing for nonproportionality of odds for the predictor effects in the ordinal submodel is also considered. Maximum likelihood estimation is performed using adaptive Gaussian quadrature in SAS PROC NLMIXED. A simulation study is performed to evaluate the performance of the estimation algorithm in terms of bias and efficiency, and to compare the results of joint and separate models of the two parts, and of proportional and nonproportional model formulations. The methods are motivated and illustrated on a dataset from a study of adolescents' perceptions of nicotine strength of JUUL e-cigarettes. Using the proposed approach we show that adolescents perceive 5% nicotine content as relatively low, a misconception more pronounced among past month nonusers than among past month users of JUUL e-cigarettes.

Comparison of Precision and Accuracy of Five Methods to Analyse Total Score Data

Total score (TS) data is generated from composite scales consisting of several questions/items, such as the Movement Disorder Society-Unified Parkinson's Disease Rating Scale (MDS-UPDRS). The analysis method that most fully uses the information gathered is item response theory (IRT) models, but these are complex and require item-level data which may not be available. Therefore, the TS is commonly analysed with standard continuous variable (CV) models, which do not respect the bounded nature of data. Bounded integer (BI) models do respect the data nature but are not as extensively researched. Mixed models for repeated measures (MMRM) are an alternative that requires few assumptions and handles dropout without bias. If an IRT model exists, the expected mean and standard deviation of TS can be computed through IRT-informed functions-which allows CV and BI models to estimate parameters on the IRT scale. The fit, performance on external data and parameter precision (when applicable) of CV, BI and MMRM to analyse simulated TS data from the MDS-UPDRS motor subscale are investigated in this work. All models provided accurate predictions and residuals without trends, but the fit of CV and BI models was improved by IRT-informed functions. The IRT-informed BI model had more precise parameter estimates than the IRT-informed CV model. The IRT-informed models also had the best performance on external data, while the MMRM model was worst. In conclusion, (1) IRT-informed functions improve TS analyses and (2) IRT-informed BI models had more precise IRT parameter estimates than IRT-informed CV models.

Improved numerical stability for the bounded integer model

This article highlights some numerical challenges when implementing the bounded integer model for composite score modeling and suggests an improved implementation. The improvement is based on an approximation of the logarithm of the error function. After presenting the derivation of the improved implementation, the article compares the performance of the algorithm to a naive implementation of the log-likelihood using both simulations and a real data example. In the simulation setting, the improved algorithm yielded more precise and less biased parameter estimates when the within-subject variability was small and estimation was performed using the Laplace algorithm. The estimation results did not differ between implementations when the SAEM algorithm was used. For the real data example, bootstrap results differed between implementations with the improved implementation producing identical or better objective function values. Based on the findings in this article, the improved implementation is suggested as the new default log-likelihood implementation for the bounded integer model.

A Time-to-Event Model Relating Integrated Craving to Risk of Smoking Relapse Across Different Nicotine Replacement Therapy Formulations

Smoking increases the risk of cancer and other diseases, causing an estimated 7 million deaths per year. Nicotine replacement therapy (NRT) reduces craving for smoking, therefore, increasing an individual's probability to remain abstinent. In this work, we for the first time quantitatively described the relationship between craving and smoking abstinence, using retrospectively collected data from 19 studies, including 3 NRT formulations (inhaler, mouth spray, and patch) and a combination of inhaler and patch. Smokers motivated to quit were included in the NRT or placebo arms. Integrated craving (i.e., craving over a period of time) was assessed with 4-category, 5-category, or 100-mm visual analogue scale. The bounded integer model was used to assess latent craving from all scales. A time-to-event model linked predicted integrated craving to the hazard of smoking relapse. Available data included 9,323 adult subjects, observed for 3 weeks up to 2 years. At the study end, 9% (11% for NRT and 5% for placebo), on average, remained abstinent according to the protocol definition. A Gompertz-Makeham hazard best described the data, with a hazard of smoking relapse decreasing over time. Latent integrated craving was positively related to the hazard of smoking relapse, through a sigmoidal maximum effect function. For the same craving, being on NRT was found to reduce the hazard of relapse by an additional 30% compared with placebo. This work confirmed that low craving is associated with a high probability of remaining smoking abstinent and that NRT, in addition to reducing craving, increases the probability of remaining smoking abstinent.

A phase I-II design based on periodic and continuous monitoring of disease status and the times to toxicity and death

A Bayesian phase I-II dose-finding design is presented for a clinical trial with four coprimary outcomes that reflect the actual clinical observation process. During a prespecified fixed follow-up period, the times to disease progression, toxicity, and death are monitored continuously, and an ordinal disease status variable, including progressive disease (PD) as one level, is evaluated repeatedly by scheduled imaging. We assume a proportional hazards model with piecewise constant baseline hazard for each continuous variable and a longitudinal multinomial probit model for the ordinal disease status process and include multivariate patient frailties to induce association among the outcomes. A finite partition of the nonfatal outcome combinations during the follow-up period is constructed, and the utility of each set in the partition is elicited. Posterior mean utility is used to optimize each patient's dose, subject to a safety rule excluding doses with an unacceptably high rate of PD, severe toxicity, or death. A simulation study shows that, compared with the proposed design, a simpler design based on commonly used efficacy and toxicity outcomes obtained by combining the four variables described above performs poorly and has substantially smaller probabilities of correctly choosing truly optimal doses and excluding truly unsafe doses.

Relating Nicotine Plasma Concentration to Momentary Craving Across Four Nicotine Replacement Therapy Formulations

Tobacco use is a major health concern. To assist smoking cessation, nicotine replacement therapy (NRT) is used to reduce nicotine craving. We quantitatively described the relationship between nicotine pharmacokinetics (PKs) from NRTs and momentary craving, linking two different pharmacodynamic (PD) scales for measuring craving. The dataset comprised retrospective data from 17 clinical studies and included 1,077 adult smokers with 39,802 craving observations from four formulations: lozenge, gum, mouth spray, and patch. A PK/PD model was developed that linked individual predicted nicotine concentrations with the categorical and visual analogue PD scales through a joint bounded integer model. A maximum effect model, accounting for acute tolerance development, successfully related nicotine concentrations to momentary craving. Results showed that all formulations were similarly effective in reducing craving, albeit with a fourfold lower potency for the patch. Women were found to have a higher maximal effect of nicotine to reduce craving, compared with men.

On the Comparison of Methods in Analyzing Bounded Outcome Score Data

Clinical trial endpoints often take the form of bounded outcome scores (BOS) which report a discrete set of values on a finite range. Conceptually such endpoints are ordered categorical in nature, but in practice they are often analyzed as continuous variables, which may result in data range violations and difficulties to handle data skewness. Analysis methods dedicated for BOS data have been proposed; however, much confusion exists among pharmacometricians on how to compare the possible methods. This commentary reviews the main methods used in pharmacometrics applications and discusses their theoretical and practical comparisons. The expected performance of some conceptually appealing methods in different situations is discussed, and a guideline is provided on selecting analysis methods in practice.