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Pharmacokinetic (PK) studies in children are usually small and have ethical constraints due to the medical complexities of drawing blood in this special population. Often, population PK models for the drug(s) of interest are available in adults, and these models can be extended to incorporate the expected deviations seen in children. As a consequence, there is increasing interest in the use of optimal design methodology to design PK sampling schemes in children that maximize information using a small sample size and limited number of sampling times per dosing period. As a case study, we use the novel tuberculosis drug delamanid, and show how applications of optimal design methodology can result in highly efficient and model-robust designs in children for estimating PK parameters using a limited number of sampling measurements. Using developed population PK models based on available data from adults living with and without HIV, and limited data on children without HIV, competing designs for children living with HIV were derived and assessed based on robustness to model uncertainty.

Modern randomization methods in clinical trials are invariably adaptive, meaning that the assignment of the next subject to a treatment group uses the accumulated information in the trial. Some of the recent adaptive randomization methods use mathematical programming to construct attractive clinical trials that balance the group features, such as their sizes and covariate distributions of their subjects. We review some of these methods and compare their performance with common covariate-adaptive randomization methods for small clinical trials. We introduce an energy distance measure that compares the discrepancy between the two groups using the joint distribution of the subjects covariates. This metric is more appealing than evaluating the discrepancy between the groups using their marginal covariate distributions. Using numerical experiments, we demonstrate the advantages of the mathematical programming methods under the new measure. In the supplementary material, we provide R codes to reproduce our study results and facilitate comparisons of different randomization procedures.

Purpose of reviewInnovative clinical trial designs for glioblastoma (GBM) are needed to expedite drug discovery. Phase 0, window of opportunity, and adaptive designs have been proposed, but their advanced methodologies and underlying biostatistics are not widely known. This review summarizes phase 0, window of opportunity, and adaptive phase I-III clinical trial designs in GBM tailored to physicians.Recent findingsPhase 0, window of opportunity, and adaptive trials are now being implemented for GBM. These trials can remove ineffective therapies earlier during drug development and improve trial efficiency. There are two ongoing adaptive platform trials: GBM Adaptive Global Innovative Learning Environment (GBM AGILE) and the INdividualized Screening trial of Innovative GBM Therapy (INSIGhT). The future clinical trials landscape in GBM will increasingly involve phase 0, window of opportunity, and adaptive phase I-III studies. Continued collaboration between physicians and biostatisticians will be critical for implementing these trial designs.

Nature-inspired swarm-based algorithms are increasingly applied to tackle high-dimensional and complex optimization problems across disciplines. They are general purpose optimization algorithms, easy to implement and assumption-free. Some common drawbacks of these algorithms are their premature convergence and the solution found may not be a global optimum. We propose a general, simple and effective strategy, called heterogeneous Perturbation-Projection (HPP), to enhance an algorithm's exploration capability so that our sufficient convergence conditions are guaranteed to hold and the algorithm converges almost surely to a global optimum. In summary, HPP applies stochastic perturbation on half of the swarm agents and then project all agents onto the set of feasible solutions. We illustrate this approach using three widely used nature-inspired swarm-based optimization algorithms: particle swarm optimization (PSO), bat algorithm (BAT) and Ant Colony Optimization for continuous domains (ACO). Extensive numerical experiments show that the three algorithms with the HPP strategy outperform the original versions with 60-80% the times with significant margins.

BackgroundIdiopathic pulmonary fibrosis (IPF) is a progressive, irreversible, and usually fatal lung disease of unknown reasons, generally affecting the elderly population. Early diagnosis of IPF is crucial for triaging patients' treatment planning into anti-fibrotic treatment or treatments for other causes of pulmonary fibrosis. However, current IPF diagnosis workflow is complicated and time-consuming, which involves collaborative efforts from radiologists, pathologists, and clinicians and it is largely subject to inter-observer variability.PurposeThe purpose of this work is to develop a deep learning-based automated system that can diagnose subjects with IPF among subjects with interstitial lung disease (ILD) using an axial chest computed tomography (CT) scan. This work can potentially enable timely diagnosis decisions and reduce inter-observer variability.MethodsOur dataset contains CT scans from 349 IPF patients and 529 non-IPF ILD patients. We used 80% of the dataset for training and validation purposes and 20% as the holdout test set. We proposed a two-stage model: at stage one, we built a multi-scale, domain knowledge-guided attention model (MSGA) that encouraged the model to focus on specific areas of interest to enhance model explainability, including both high- and medium-resolution attentions; at stage two, we collected the output from MSGA and constructed a random forest (RF) classifier for patient-level diagnosis, to further boost model accuracy. RF classifier is utilized as a final decision stage since it is interpretable, computationally fast, and can handle correlated variables. Model utility was examined by (1) accuracy, represented by the area under the receiver operating characteristic curve (AUC) with standard deviation (SD), and (2) explainability, illustrated by the visual examination of the estimated attention maps which showed the important areas for model diagnostics.ResultsDuring the training and validation stage, we observe that when we provide no guidance from domain knowledge, the IPF diagnosis model reaches acceptable performance (AUC±SD = 0.93±0.07), but lacks explainability; when including only guided high- or medium-resolution attention, the learned attention maps are not satisfactory; when including both high- and medium-resolution attention, under certain hyperparameter settings, the model reaches the highest AUC among all experiments (AUC±SD = 0.99±0.01) and the estimated attention maps concentrate on the regions of interests for this task. Three best-performing hyperparameter selections according to MSGA were applied to the holdout test set and reached comparable model performance to that of the validation set.ConclusionsOur results suggest that, for a task with only scan-level labels available, MSGA+RF can utilize the population-level domain knowledge to guide the training of the network, which increases both model accuracy and explainability.

MotivationModeling single-cell gene expression trends along cell pseudotime is a crucial analysis for exploring biological processes. Most existing methods rely on nonparametric regression models for their flexibility; however, nonparametric models often provide trends too complex to interpret. Other existing methods use interpretable but restrictive models. Since model interpretability and flexibility are both indispensable for understanding biological processes, the single-cell field needs a model that improves the interpretability and largely maintains the flexibility of nonparametric regression models.ResultsHere, we propose the single-cell generalized trend model (scGTM) for capturing a gene's expression trend, which may be monotone, hill-shaped or valley-shaped, along cell pseudotime. The scGTM has three advantages: (i) it can capture non-monotonic trends that are easy to interpret, (ii) its parameters are biologically interpretable and trend informative, and (iii) it can flexibly accommodate common distributions for modeling gene expression counts. To tackle the complex optimization problems, we use the particle swarm optimization algorithm to find the constrained maximum likelihood estimates for the scGTM parameters. As an application, we analyze several single-cell gene expression datasets using the scGTM and show that scGTM can capture interpretable gene expression trends along cell pseudotime and reveal molecular insights underlying biological processes.Availability and implementationThe Python package scGTM is open-access and available at https://github.com/ElvisCuiHan/scGTM.Supplementary informationSupplementary data are available at Bioinformatics online.

The key aim of this paper is to suggest a more quantitative approach to designing a dose-response experiment, and more specifically, a concentration-response experiment. The work proposes a departure from the traditional experimental design to determine a dose-response relationship in a developmental toxicology study. It is proposed that a model-based approach to determine a dose-response relationship can provide the most accurate statistical inference for the underlying parameters of interest, which may be estimating one or more model parameters or pre-specified functions of the model parameters, such as lethal dose, at maximal efficiency. When the design criterion or criteria can be determined at the onset, there are demonstrated efficiency gains using a more carefully selected model-based optimal design as opposed to an ad-hoc empirical design. As an illustration, a model-based approach was theoretically used to construct efficient designs for inference in a developmental toxicity study of sea urchin embryos exposed to trimethoprim. This study compares and contrasts the results obtained using model-based optimal designs versus an ad-hoc empirical design.

BackgroundIdiopathic pulmonary fibrosis (IPF) is a fatal interstitial lung disease characterized by an unpredictable decline in lung function. Predicting IPF progression from the early changes in lung function tests have known to be a challenge due to acute exacerbation. Although it is unpredictable, the neighboring regions of fibrotic reticulation increase during IPF's progression. With this clinical information, quantitative characteristics of high-resolution computed tomography (HRCT) and a statistical learning paradigm, the aim is to build a model to predict IPF progression.DesignA paired set of anonymized 193 HRCT images from IPF subjects with 6-12 month intervals were collected retrospectively. The study was conducted in two parts: (1) Part A collects the ground truth in small regions of interest (ROIs) with labels of "expected to progress" or "expected to be stable" at baseline HRCT and develop a statistical learning model to classify voxels in the ROIs. (2) Part B uses the voxel-level classifier from Part A to produce whole-lung level scores of a single-scan total probability's (STP) baseline.MethodsUsing annotated ROIs from 71 subjects' HRCT scans in Part A, we applied Quantum Particle Swarm Optimization-Random Forest (QPSO-RF) to build the classifier. Then, 122 subjects' HRCT scans were used to test the prediction. Using Spearman rank correlations and survival analyses, we ascertained STP associations with 6-12 month changes in quantitative lung fibrosis and forced vital capacity.ConclusionThis study can serve as a reference for collecting ground truth, and developing statistical learning techniques to predict progression in medical imaging.

PurposeDomain knowledge (DK) acquired from prior studies is important for medical diagnosis. This paper leverages the population-level DK using an optimality design criterion to train a deep learning model in an end-to-end manner. In this study, the problem of interest is at the patient level to diagnose a subject with idiopathic pulmonary fibrosis (IPF) among subjects with interstitial lung disease (ILD) using a computed tomography (CT). IPF diagnosis is a complicated process with multidisciplinary discussion with experts and is subject to interobserver variability, even for experienced radiologists. To this end, we propose a new statistical method to construct a time/memory-efficient IPF diagnosis model using axial chest CT and DK, along with an optimality design criterion via a DK-enhanced loss function of deep learning.MethodsFour state-of-the-art two-dimensional convolutional neural network (2D-CNN) architectures (MobileNet, VGG16, ResNet-50, and DenseNet-121) and one baseline 2D-CNN are implemented to automatically diagnose IPF among ILD patients. Axial lung CT images are retrospectively acquired from 389 IPF patients and 700 non-IPF ILD patients in five multicenter clinical trials. To enrich the sample size and boost model performance, we sample 20 three-slice samples (triplets) from each CT scan, where these three slices are randomly selected from the top, middle, and bottom of both lungs respectively. Model performance is evaluated using a fivefold cross-validation, where each fold was stratified using a fixed proportion of IPF vs non-IPF.ResultsUsing DK-enhanced loss function increases the model performance of the baseline CNN model from 0.77 to 0.89 in terms of study-wise accuracy. Four other well-developed models reach satisfactory model performance with an overall accuracy >0.95 but the benefits brought on by the DK-enhanced loss function is not noticeable.ConclusionsWe believe this is the first attempt that (a) uses population-level DK with an optimal design criterion to train deep learning-based diagnostic models in an end-to-end manner and (b) focuses on patient-level IPF diagnosis. Further evaluation of using population-level DK on prospective studies is warranted and is underway.

D-optimal designs are frequently used in controlled experiments to obtain the most accurateestimate of model parameters at minimal cost. Finding them can be a challenging task, especially whenthere are many factors in a nonlinear model. As the number of factors becomes large and interact withone another, there are many more variables to optimize and the D-optimal design problem becomes highdimensionaland non-separable. Consequently, premature convergence issues arise. Candidate solutions gettrapped in local optima and the classical gradient-based optimization approaches to search for the D-optimaldesigns rarely succeed. We propose a specially designed version of differential evolution (DE) which is arepresentative gradient-free optimization approach to solve such high-dimensional optimization problems.The proposed specially designed DE uses a new novelty-based mutation strategy to explore the variousregions in the search space. The exploration of the regions will be carried out differently from the previouslyexplored regions and the diversity of the population can be preserved. The proposed novelty-based mutationstrategy is collaborated with two common DE mutation strategies to balance exploration and exploitationat the early or medium stage of the evolution. Additionally, we adapt the control parameters of DE as theevolution proceeds. Using logistic models with several factors on various design spaces as examples, oursimulation results show our algorithm can find D-optimal designs efficiently and the algorithm outperformsits competitors. As an application, we apply our algorithm and re-design a 10-factor car refueling experimentwith discrete and continuous factors and selected pairwise interactions. Our proposed algorithm was able toconsistently outperform the other algorithms and find a more efficient D-optimal design for the problem.

Optimization of clinical trial designs can help investigators achieve higher qualityresults for the given resource constraints. The present paper gives an overviewof optimal designs for various important problems that arise in different stages ofclinical drug development, including phase I dose–toxicity studies; phase I/II studiesthat consider early efficacy and toxicity outcomes simultaneously; phase IIdose–response studies driven by multiple comparisons (MCP), modeling techniques(Mod), or their combination (MCP–Mod); phase III randomized controlled multiarmmulti-objective clinical trials to test difference among several treatment groups;and population pharmacokinetics–pharmacodynamics experiments. We find thatmodern literature is very rich with optimal design methodologies that can be utilizedby clinical researchers to improve efficiency of drug development.

Several common general purpose optimization algorithms are compared for findingA- and D-optimal designs for different types of statistical models of varying complexity,including high dimensional models with five and more factors. The algorithms of interestinclude exact methods, such as the interior point method, the Nelder–Mead method, theactive set method, the sequential quadratic programming, and metaheuristic algorithms,such as particle swarm optimization, simulated annealing and genetic algorithms.Several simulations are performed, which provide general recommendations on theutility and performance of each method, including hybridized versions of metaheuristicalgorithms for finding optimal experimental designs. A key result is that general-purposeoptimization algorithms, both exact methods and metaheuristic algorithms, perform wellfor finding optimal approximate experimental designs.

Idiopathic pulmonary fibrosis (IPF) is a fatal lung disease characterized by an unpredictable progressive declinein lung function. Natural history of IPF is unknown and the prediction of disease progression at the time ofdiagnosis is notoriously difficult. High resolution computed tomography (HRCT) has been used for the diagnosisof IPF, but not generally for monitoring purpose. The objective of this work is to develop a novel predictivemodel for the radiological progression pattern at voxel-wise level using only baseline HRCT scans. Mainly, thereare two challenges: (a) obtaining a data set of features for region of interest (ROI) on baseline HRCT scans andtheir follow-up status; and (b) simultaneously selecting important features from high-dimensional space, andoptimizing the prediction performance. We resolved the first challenge by implementing a study design andhaving an expert radiologist contour ROIs at baseline scans, depending on its progression status in follow-upvisits. For the second challenge, we integrated the feature selection with prediction by developing an algorithmusing a wrapper method that combines quantum particle swarm optimization to select a small number of featureswith random forest to classify early patterns of progression. We applied our proposed algorithm to analyzeanonymized HRCT images from 50 IPF subjects from a multi-center clinical trial. We showed that it yields aparsimonious model with 81.8% sensitivity, 82.2% specificity and an overall accuracy rate of 82.1% at the ROIlevel. These results are superior to other popular feature selections and classification methods, in that ourmethod produces higher accuracy in prediction of progression and more balanced sensitivity and specificity witha smaller number of selected features. Our work is the first approach to show that it is possible to use onlybaseline HRCT scans to predict progressive ROIs at 6 months to 1year follow-ups using artificial intelligence.

Lot Quality Assurance Sampling (LQAS) plans are widely used for health monitoring purposes. We propose a systematic approach to design multiple-objective LQAS plans that meet user-specified type 1 and 2 error rates and targets for selected diagnostic accuracy metrics. These metrics may include sensitivity, specificity, positive predictive value, and negative predictive value in high or low anticipated prevalence rate populations. We use Mixed Integer Nonlinear Programming (MINLP) tools to implement our design methodology. Our approach is flexible in that it can directly generate classic LQAS plans that control error rates only and find optimal LQAS plans that meet multiple objectives in terms of diagnostic metrics. We give examples, compare results with the classic LQAS and provide an application using a malaria outcome indicator survey in Mozambique.

Objective Patients with SLE have an increased risk ofatherosclerosis (ATH) that is not adequately explainedby traditional risk factors. We previously described thePredictors of Risk for Elevated Flares, Damage Progression,and Increased Cardiovascular disease in PaTients withSLE (PREDICTS) atherosclerosis-risk panel, which includesproinflammatory HDL (piHDL), leptin, soluble tumournecrosis factor-like weak inducer of apoptosis (sTWEAK)and homocysteine, as well as age and diabetes. A highPREDICTS score confers 28-fold increased odds forfuture atherosclerosis in SLE. The aim of this study is todetermine whether PREDICTS biomarkers are modifiable bycommon lupus therapies.Methods This prospective observational study includedSLE subjects started on new lupus treatments. Leptin,sTWEAK, homocysteine and antioxidant function of HDLwere measured at baseline (prior to drug initiation), 6weeks and 12 weeks.Results 16 subjects started mycophenolate (MMF), 18azathioprine (AZA) and 25 hydroxychloroquine (HCQ).In MMF-treated subjects, HDL function progressivelyimproved from 2.23 ± 1.32 at baseline to 1.37±0.81at 6 weeks (p=0.02) and 0.93±0.54 at 12 weeks(p=0.009). sTWEAK levels also improved in MMF-treatedsubjects from 477.5±447.1 to 290.3±204.6 pg/mLafter 12 weeks (p=0.04), but leptin and homocysteinelevels were not significantly changed. In HCQ-treatedsubjects, only HDL function improved from 1.80±1.29 atbaseline to 1.03±0.74 after 12 weeks (p=0.05). Therewere no changes in the AZA group. MMF treatmentwas still associated with significant improvements inHDL function after accounting for potential confounderssuch as total prednisone dose and changes in diseaseactivity. Overall, the mean number of high-risk PREDICTSbiomarkers at week 12 significantly decreased in theentire group of patients started on a new lupus therapy(2.1±0.9 to 1.8±0.9, p=0.02) and in the MMF-treatedgroup (2.4±0.8 vs 1.8±0.9, p=0.003), but not in theAZA or HCQ groups. In multivariate analysis, the odds ofhaving a high PREDICTS atherosclerosis risk score at 12weeks were lower with MMF treatment (OR 0.002, 95%CI 0.000 to 0.55, p=0.03).Conclusions 12 weeks of MMF therapy improves theoverall PREDICTS atherosclerosis biomarker profile.Further studies will determine whether biomarkerchanges reflect decreases in future cardiovascularevents.

Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studyingtheir properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariablyapplicable to the particular problem only.We propose a systematic approach to construct optimal exact designs by incorporatingthe Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. Asexamples, we apply the methodology to find D- and A-optimal exact designs for linear and nonlinear models using global orlocal optimizers. Our examples include design problems with constraints on the locations or the number of replicates at theoptimal design points.

Thresholding variable plays a crucial role in subgroup identification for personalizedmedicine. Most existing partitioning methods split the sample basedon one predictor variable. In this paper, we consider setting the splitting rulefrom a combination of multivariate predictors, such as the latent factors, principlecomponents, and weighted sum of predictors. Such a subgrouping methodmay lead to more meaningful partitioning of the population than using a singlevariable. In addition, our method is based on a change point regression modeland thus yields straight forward model-based prediction results. After choosinga particular thresholding variable form, we apply a two-stage multiple changepoint detection method to determine the subgroups and estimate the regressionparameters. We show that our approach can produce two or more subgroupsfrom the multiple change points and identify the true grouping with high probability.In addition, our estimation results enjoy oracle properties. We design asimulation study to compare performances of our proposed and existing methodsand apply them to analyze data sets from a Scleroderma trial and a breastcancer study.

Finding Bayesian optimal designs for nonlinear models is a difficult task because the optimality criteriontypically requires us to evaluate complex integrals before we perform a constrained optimization. Wepropose a hybridized method where we combine an adaptive multidimensional integration algorithm anda metaheuristic algorithm called imperialist competitive algorithm to find Bayesian optimal designs. Weapply our numerical method to a few challenging design problems to demonstrate its efficiency. Theyinclude finding D-optimal designs for an item response model commonly used in education, Bayesianoptimal designs for survivalmodels, and Bayesian optimal designs for a four-parameter sigmoid Emax doseresponse model. Supplementary materials for this article are available online and they contain an R packagefor implementing the proposed algorithm and codes for reproducing all the results in this paper.

When a model-based approach is appropriate, an optimal design can guide how tocollect data judiciously for making reliable inference at minimal cost. However, findingoptimal designs for a statistical model with several possibly interacting factors can beboth theoretically and computationally challenging, and this issue is rarely discussed inthe literature. We propose nature-inspired metaheuristic algorithms, like particle swarmoptimization (PSO) and its variants, to solve such optimization problems. Wedemonstrate that such techniques, which are easy to implement, can find differenttypes of optimal designs for models with several factors efficiently. To facilitate use ofsuch algorithms, we provide computer codes to generate tailor made optimal designsand evaluate efficiencies of competing designs. As applications, we apply PSO and findBayesian optimal designs for Exponential models useful in HIV studies and re-design acar-refuelling study for a Logistic model with ten factors and some interacting factors.

Forecasting and predictive inference are fundamental data analysis tasks. Most studies employ parametricapproaches making strong assumptions about the data generating process. On the other hand, while nonparametric modelsare applied, it is sometimes found in situations involving low signal to noise ratios or large numbers of covariates that theirperformance is unsatisfactory. We propose a new varying-coefficient semiparametric model averaging prediction (VC-SMAP)approach to analyze large data sets with abundant covariates. Performance of the procedure is investigated with numericalexamples. Even though model averaging has been extensively investigated in the literature, very few authors have consideredaveraging a set of semiparametric models. Our proposed model averaging approach provides more flexibility than parametricmethods, while being more stable and easily implemented than fully multivariate nonparametric varying-coefficient models.We supply numerical evidence to justify the effectiveness of our methodology.

Lot Quality Assurance Sampling (LQAS) plans are widely used for health monitoringpurposes. We propose a systematic approach to design multiple-objective LQAS plans that meet userspecifiedtype 1 and 2 error rates and targets for selected diagnostic accuracy metrics. These metricsmay include sensitivity, specificity, positive predictive value, and negative predictive value in high orlow anticipated prevalence rate populations. We use Mixed Integer Nonlinear Programming (MINLP)tools to implement our design methodology. Our approach is flexible in that it can directly generateclassic LQAS plans that control error rates only and find optimal LQAS plans that meet multipleobjectives in terms of diagnostic metrics. We give examples, compare results with the classic LQASand provide an application using a malaria outcome indicator survey in Mozambique.

Identifying optimal designs for generalized linear models with a binary response can be a challengingtask, especially when there are both discrete and continuous independent factors in the model. Theoreticalresults rarely exist for such models, and for the handful that do, they usually come with restrictive assumptions.In this article, we propose the d-QPSO algorithm, a modified version of quantum-behaved particleswarm optimization, to find a variety of D-optimal approximate and exact designs for experiments withdiscrete and continuous factors and a binary response. We show that the d-QPSO algorithm can efficientlyfind locally D-optimal designs even for experiments with a large number of factors and robust pseudo-Bayesian designs when nominal values for the model parameters are not available. Additionally, we investigaterobustness properties of the d-QPSO algorithm-generated designs to variousmodel assumptions andprovide real applications to design a bio-plastics odor removal experiment, an electronic static experiment,and a 10-factor car refueling experiment. Supplementary materials for the article are available online.

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models wherethe design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets.Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When theregression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtainan equivalent semidefinite program.When there are two or more factors in the models, we apply a moment relaxation techniqueand approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When therelaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and itsoptimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples.

Linear mixed models have become popular in many statistical applications duringrecent years. However design issues for multi-response linear mixed models are rarelydiscussed. Themain purpose of this paper is to investigate D-optimal designs formultiresponselinear mixed models. We provide two equivalence theorems to characterizethe optimal designs for the estimation of the fixed effects and the prediction of randomeffects, respectively. Two examples of the D-optimal designs formulti-response linearmixed models are given for illustration.

Hormesis has been widely observed and debated in a variety of context in biomedicineand toxicological sciences. Detecting its presence can be an important problem withwide ranging implications. However, there is little work on constructing an efficientexperiment to detect its existence or estimate the threshold dose.We use optimal designtheory to develop a variety of locally optimal designs to detect hormesis, estimatethe threshold dose and the zero-equivalent point (ZEP) for commonly used modelsin toxicology and risk assessment. To facilitate use of more efficient designs todetect hormesis, estimate threshold dose and estimate the ZEP in practice, we implementcomputer algorithms and create a user-friendly web site to help the biomedicalresearcher generate different types of optimal designs. The online tool facilitates theuser to evaluate robustness properties of a selected design to various model assumptionsand compare designs before implementation.