Analysis of Potato Canopy Coverage as Assessed Through Digital Imagery by Nonlinear Mixed Effects Models

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

Different mixed-effects models were compared to evaluate the population dose-response and relative potency of two albuterol inhalers. Bronchodilator response was measured after ascending doses of each inhaler in 37 asthmatic patients. A linear mixed-effects model was developed based on the approach proposed by Finney for the evaluation of bioassay data. A nonlinear mixed-effects (Emax) model with interindividual and interoccasion variability (IOV) in the different pharmacodynamic parameters was also fit to the data. Both methods produced a similar estimate of relative potency. However, the estimate of relative potency was 22% lower with the nonlinear mixed-effects model if IOV was not taken into account. Monte Carlo simulations based on a similar study design demonstrated that more biased and variable estimates of ED50 and relative potency were obtained when the nonlinear mixed-effects model ignored the presence of IOV in the data. Furthermore, the linear mixed-effects model that did not account for IOV produced confidence intervals for relative potency that were too narrow and thus could lead to erroneous conclusions. These problems were avoided when the estimation model could account for IOV. Results of the simulations were consistent with those of the experimental data. Although the linear or the nonlinear mixed-effects model may be used to evaluate population dose-response and relative potency, there are important differences in the assumptions made by each method.

Animal growth models are used to identify alternative strategies to improve the efficiency of livestock production and to estimate daily nutrient requirements for the animal of different age and sex group. In this study, the efficiency of nonlinear mixed effects models were explored and a comparison was made between the predictive power of fixed effects models and the mixed effects models. Three hundred body weight (BW) data including male and female pigs were used for model fitting. One pig specific random effect was included in each of the models. The random function was a random deviation of mature BW of the individual from average mature BW of its genotype. Logistic, Gompertz and Von-Bertalanffy fixed and mixed models were explored for these data. It was found that Logistic mixed effects model performed better than the other nonlinear mixed effects models based on mean square error (MSE) and root mean square error (RMSE).

The aim of this study was to investigate the use of nonlinear mixed effects (NLME) models in a real bioequivalence study and compare it to noncompartmental analysis (NCA) which is proposed by regulatory agencies. NCA requires few hypotheses but a large number of samples per subject. On the other hand, NLME approach is more complex than NCA but it has some advantages such as it requires few samples per subject. A real data application was provided for the study, which was get from Ege University Drug Development and Pharmacokinetics Research Center. For NLME, we used the stochastic approach to expectation-maximization (SAEM) algorithm, whereas linear trapezoidal rule was used for NCA. We estimate pharmacokinetic parameters, area under the curve (AUC0-∞) and maximum concentration (Cmax), and perform a bioequivalence tests using NCA and NLME. According to real data analysis, NLME approach has smaller within subject error, narrower confidence intervals than non-compartmental analysis. However, NLME models have some limitations because of increasing type I error. Therefore, caution is needed for small sample size and data with high variability.

This work is motivated by a meta-analysis case study on antipsychotic medications. The Michaelis-Menten curve is employed to model the nonlinear relationship between the dose and D2 receptor occupancy across multiple studies. An intraclass correlation coefficient (ICC) is used to quantify the heterogeneity across studies. To interpret the size of heterogeneity, an accurate estimate of ICC and its confidence interval is required. The goal is to apply a recently proposed generic beta-approach for construction the confidence intervals on ICCs for linear mixed effects models to nonlinear mixed effects models using four estimation methods. These estimation methods are the maximum likelihood, second-order generalized estimating equations and two two-step procedures. The beta-approach is compared with a large sample normal approximation (delta method) and bootstrapping. The confidence intervals based on the delta method and the nonparametric percentile bootstrap with various resampling strategies failed in our settings. The beta-approach demonstrates good coverages with both two-step estimation methods and consequently, it is recommended for the computation of confidence interval for ICCs in nonlinear mixed effects models for small studies.

In drug development, pharmacometric approaches consist in identifying via a model selection (MS) process the model structure that best describes the data. However, making predictions using a selected model ignores model structure uncertainty, which could impair predictive performance. To overcome this drawback, model averaging (MA) takes into account the uncertainty across a set of candidate models by weighting them as a function of an information criterion. Our primary objective was to use clinical trial simulations (CTSs) to compare model selection (MS) with model averaging (MA) in dose finding clinical trials, based on the AIC information criterion. A secondary aim of this analysis was to challenge the use of AIC by comparing MA and MS using five different information criteria. CTSs were based on a nonlinear mixed effect model characterizing the time course of visual acuity in wet age-related macular degeneration patients. Predictive performances of the modeling approaches were evaluated using three performance criteria focused on the main objectives of a phase II clinical trial. In this framework, MA adequately described the data and showed better predictive performance than MS, increasing the likelihood of accurately characterizing the dose-response relationship and defining the minimum effective dose. Moreover, regardless of the modeling approach, AIC was associated with the best predictive performances.

The stand basal area, closely related to age, site quality, and stand density, is an important factor for predicting forest growth and yield. The accurate estimation of site quality is especially a key component in the stand basal area model. We utilized sample plots with Picea asperata Mast. as the dominant species in the multi-period National Forest Inventory (NFI) dataset to establish a site index (SI) model including climate effects through the difference form of theoretical growth equations and mixed-effects models. We combined the SI calculated from the SI model, stand age, and stand density index to construct a basal area growth model for Picea asperata Mast. stands. The results show that the Korf model is the best SI base model for Picea asperata Mast. The mean temperatures in summer and winter precipitation were used as the fixed parameters to construct a nonlinear model. Ultimately, elevation, origin, and region, as random effects, were incorporated into the mixed-effects model. The coefficients (R2) of determination of the base model, the nonlinear model including climate, and the nonlinear mixed-effects model are 0.869, 0.899, and 0.921, with root-mean-square errors (RMSEs) of 1.320, 1.315, and 1.301, respectively. Among the basal area models, the Richards model has higher precision. And the basal area model including an SI incorporating climatic factors had a higher determination coefficient (R2) of 0.918 than that of the model including an SI without considering climatic effects. The mixed-effects model incorporating climatic and topographic factors shows a better fitting performance of SI, resulting in a higher precision of the basal area model. This indicates that in the development of forest growth models, both biophysical and climatic factors should be comprehensively considered.

Some bootstrap methods in nonlinear mixed-effect models

IntroductionDiameter at breast height (DBH) is a key parameter for assessing tree growth, carbon storage, and ecological functions. Traditional ground surveys are inefficient, labor-intensive, and terrain-limited, making them unsuitable for large-scale monitoring. Airborne LiDAR, as an advanced remote sensing tool, provides an efficient and non-destructive method for DBH estimation. However, most existing LiDAR-based models overlook the influence of genotype differences, limiting prediction accuracy.MethodsIn this study, we used data from 2,899 Catalpa bungei trees of different genotypes to develop a nonlinear mixed-effects (NLME) model that incorporates genotype as a random effect. This approach improved model generalizability by using LiDAR-derived tree height (LH) and LiDAR-derived crown diameter (LCD) as core predictors. Multiple sampling strategies were also evaluated to assess their impact on model performance.ResultsThe results showed that, considering genotype effects, the proposed NLME model outperformed both traditional regression models and dummy-variable models (R2 = 0.8624, RMSE = 1.1330, TRE = 3.9555), demonstrating the important role of genotype differences in improving model accuracy. Random sampling further improved prediction accuracy while effectively reducing measurement costs.DiscussionThis research introduces a new framework for integrating genotype variability into DBH prediction models and offers valuable insights for future LiDAR-based studies in genetically heterogeneous plantations. The findings provide technical support for forest management and ecosystem monitoring, as well as a methodological foundation for predicting tree growth under varying site and genetic conditions.

IntroductionAccurate estimation of individual tree aboveground biomass (AGB) is essential for tree species selection, carbon accounting, and precision forestry. Unmanned aerial vehicle (UAV) LiDAR provides rapid access to detailed tree structural information, offering a promising tool for high-frequency biomass assessment.MethodsIn this study, a nonlinear mixed-effects (NLME) model integrating UAV LiDAR and field measurements was developed to quantify the influence of genetic heterogeneity and environmental factors on AGB estimation of Catalpa bungei. Data from 2,941 trees across 79 genotypes were collected in Henan Province, including LiDAR-derived tree height (LH), LiDAR-derived crown diameter (LCD), and AGB. By incorporating genotype as a random effect and planting density as a dummy variable, the NLME model significantly outperformed traditional dummy-variable models.ResultsGenotype effects explained significant AGB variation, achieving high accuracy (R²=0.7916, RMSE = 3.7095) and reducing TRE by 23.29% compared to the basic power function model. Leave-one-genotype-out cross-validation confirmed robustness. Calibration with the four largest trees yielded the best performance (TRE = 13.09%), while a simplified scheme using only two trees per genotype maintained high accuracy (TRE = 13.24%), markedly reducing field effort.DiscussionThese results highlight the superiority of NLME AGB models over linear approaches and demonstrate that accounting for genotype effects is critical for reliable biomass estimation. The proposed framework provides an efficient and cost-effective solution for biomass monitoring, tree breeding, carbon sink assessment, and precision forestry.

An individual-tree diameter growth model was developed for Cunninghamia lanceolata in Fujian province, southeast China. Data were obtained from 72 plantation-grown China-fir trees in 24 single-species plots. Ordinary non-linear least squares regression was used to choose the best base model from among 5 theoretical growth equations; selection criteria were the smallest absolute mean residual and root mean square error and the largest adjusted coefficient of determination. To account for autocorrelation in the repeated-measures data, we developed one-level and nested two-level nonlinear mixed-effects (NLME) models, constructed on the selected base model; the NLME models incorporated random effects of the tree and plot. The best random-effects combinations for the NLME models were identified by Akaike's information criterion, Bayesian information criterion and −2 logarithm likelihood. Heteroscedasticity was reduced with two residual variance functions, a power function and an exponential function. The autocorrelation was addressed with three residual autocorrelation structures: a first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and a compound symmetry structure (CS). The one-level (tree) NLME model performed best. Independent validation data were used to test the performance of the models and to demonstrate the advantage of calibrating the NLME models.

The population pharmacokinetics of gentamicin in neonates was determined using a nonlinear, mixed-effects model (NONMEM). The final regression equations derived to estimate clearance (Cl) and volume of distribution (Vd) were Cl = 0.120 * (WT/2.4)1.36 L/hr and Vd = 0.429 * (WT) L. The interindividual variability (% CV) for clearance was 26.2% and for volume of distribution 15.9%. Intraindividual variability was 11.0%. In a separate group of 30 neonates, the predictive ability of the NONMEM-generated population variables was compared to the predictions from a standard two-stage population analysis. The trough concentrations predicted using NONMEM-generated parameters were significantly less biased and more precise; there were no significant differences between the methods in predicting peaks. NONMEM is a useful tool for determining population pharmacokinetics and appears to be consistent across populations using routine clinical data and limited observation.

Using ultrashort echo time (UTE) magnetic resonance imaging (MRI), T2∗ can be measured and used to evaluate cartilage degradation. In general, high-quality MRI is required for accurate T2∗ measurement. However, such high-quality MRI requires long scan times which make the clinical utility of T2∗ measurement infeasible. Thus, a method that can produce satisfactory T2∗ maps within reasonable scan times (i.e. limited data) is required. Non-linear mixed-effects model (NLME) is a powerful tool for analyzing incomplete longitudinal data. Since NLME contains both fixed and random effects, it is preferred over more traditional methods. The aim of this study is to investigate whether T2∗ map can be generated using NLME. The simulation results show that NLME is a feasible method for T2∗ measurement. Moreover, NLME can improve the accuracy and precision of parameter estimates compared to the traditional non-linear least square fit.

Research Highlights: Determination of merchantable wood volume is one of the key preconditions for sustainable forest management. This study explores accuracy of calibrated predictions of merchantable wood volume of Norway spruce (Picea abies (L.) H. Karst.) using stem taper curves (STC) in a form of a mixed model. Background and Objectives: The study is devoted to the determination of merchantable wood volume (over bark) of individual standing stems based on the integration of an STC model calibrated using upper diameter measurements. Various options of upper diameter measurement were tested and their impact on the accuracy of merchantable wood volume prediction was evaluated. Materials and Methods: To model stem taper curves, a Kozak 02 function was applied in a form of a nonlinear, mixed effects model. Accuracies of calibrated merchantable wood volume predictions obtained through remote (optical) upper diameter measurements were compared to accuracies corresponding to contact measurements by a caliper. The performance of two alternative methods used in the Czech National Forest Inventory (NFI) and forestry practice, involving diameter at breast height and total tree height as the only predictors, were also tested. The contact measurements were performed at identical stem positions after felling the respective sample tree. The calibration was done in order to account for factors inherent in particular location, and, optionally, also in a particular sample stem (within the respective location). Input data was sourced as part of a dedicated survey involving the entire territory of the Czech Republic. In total, 716 individual spruce trees were measured, felled and analysed at 169 locations. Results: In general, the best merchantable volume predictions were obtained by integrating the STC fitted (and calibrated) by minimising errors of stem cross-sectional areas instead of diameters. In terms of calibrated predictions, using single-directional, caliper measurement of upper diameter at 7 m (after felling) led to the best accuracy. In this case, the observed mean bias of merchantable volume prediction was only 0.63%, indicating underestimation. The best optical calibration strategy involved upper diameter measurements at two heights (5 and 7 m) simultaneously. Bias of this volume prediction approach was estimated at 2.1%, indicating underestimation. Conclusions: Concerning the prediction of merchantable stem volume of standing Norway spruce trees, STC calibration using two optical upper diameter measurements (at 5 and 7 m) was found to be practically applicable, provided a bias up to 3.7% can be accepted. This method was found to be more accurate than the existing national alternatives using diameter at breast height and the total tree height as the only predictors.