A geometric characterization of cc-optimal designs for heteroscedastic regression

We consider the common nonlinear regression model where the variance, as well as the mean, is a parametric function of the explanatory variables. The $c$-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of $c$-optimal designs in this context is presented, which generalizes the classical result of Elfving [Ann. Math. Statist. 23 (1952) 255--262] for $c$-optimal designs. As in Elfving's famous characterization, $c$-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking, the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The $c$-optimal designs are characterized as representations of the points where the line in direction of the vector $c$ intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects.Comment: Published in at http://dx.doi.org/10.1214/09-AOS708 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A geometric characterization of c-optimal designs for heteroscedastic regression

We consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables. The c-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of c-optimal designs in this context is presented, which generalizes the classical result of Elfving (1952) for c-optimal designs. As in Elfving's famous characterization c-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The c-optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects. --c-optimal design,heteroscedastic regression,Elfving's theorem,pharmacokinetic models,random effects,locally optimal design,geometric characterization

Optimal designs for random effect models with correlated errors with applications in population pharmacokinetics

We consider the problem of constructing optimal designs for population pharmacokinetics which use random effect models. It is common practice in the design of experiments in such studies to assume uncorrelated errors for each subject. In the present paper a new approach is introduced to determine efficient designs for nonlinear least squares estimation which addresses the problem of correlation between observations corresponding to the same subject. We use asymptotic arguments to derive optimal design densities, and the designs for finite sample sizes are constructed from the quantiles of the corresponding optimal distribution function. It is demonstrated that compared to the optimal exact designs, whose determination is a hard numerical problem, these designs are very efficient. Alternatively, the designs derived from asymptotic theory could be used as starting designs for the numerical computation of exact optimal designs. Several examples of linear and nonlinear models are presented in order to illustrate the methodology. In particular, it is demonstrated that naively chosen equally spaced designs may lead to less accurate estimation.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS324 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bestimmung c-optimaler Versuchspläne in Modellen mit zufälligen Effekten, mit Anwendungen in der Pharmakokinetik

Medizinische Studien im Bereich der Pharmakokinetik basieren in vielen Fällen auf speziellen Modellen mit zufälligen Effekten, den sogenannten Populationsmodellen. In solchen Studien werden jeweils mehrere Messungen an einer Anzahl verschiedener Patienten durchgeführt. Aus dem Blickwinkel der optimalen Versuchsplanung führt dies zu methodischen Schwierigkeiten, da der zufällige Effekt sowohl eine parameterabhängige Varianz der Beobachtungen als auch teilweise korrelierte Daten zur Folge hat. Auf diese Weise sind zwei der Schlüsselannahmen klassischer Versuchsplanungsliteratur verletzt. Es ist das Ziel dieser Arbeit, die bestehende Methodik so anzupassen und zu ergänzen, dass auch diese Situationen betrachtet werden können. Da die wichtigsten zu schätzenden Kenngrößen in der Pharmakokinetik bestimmte Summengrößen der Parameter sind (z.B. die Fläche unter der Konzentrationskurve eines Präparates) wird der Schwerpunkt dieser Arbeit auf c-optimalen Designs liegen, die die optimale Schätzung solcher Größen erlauben. Im einzelnen wird zunächst die geometrische Repräsentation optimaler Designs nach Elfving (1952) so verallgemeinert, dass die beschriebene Situation abgedeckt ist. Im zweiten Schritt wird die Äquivalenztheorie nach Kiefer (1974) und Pukelsheim (1993) auf das spezifische Modell angewendet. Dritter Schritt ist die Anpassung multplikativer Algorithmen zur numerischen Bestimmung optimaler Designs in dieser Situation, und als letztes Ergebnis wird das Konzept asymptotisch optimaler Designs auf einen speziellen Fall korrelierter Beobachtungen angewendet

A geometric characterization of c-optimal designs for regression models with correlated observations

We consider the problem of optimal design of experiments for random effects models, especially population models, where a small number of correlated observations can be taken on each individual, while the observations corresponding to different individuals can be assumed to be uncorrelated. We focus on c-optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of Elfving (1952) from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients. The theory is demonstrated in a linear model with correlated observations and a nonlinear random effects population model, which is commonly used in pharmacokinetics

Semi-automatic 3D-volumetry of liver metastases from neuroendocrine tumors to improve combination therapy with 177Lu-DOTATOC and 90Y-DOTATOC

PURPOSEPatients with neuroendocrine tumors (NET) often present with disseminated liver metastases and can be treated with a number of different nuclides or nuclide combinations in peptide receptor radionuclide therapy (PRRT) depending on tumor load and lesion diameter. For quantification of disseminated liver lesions, semi-automatic lesion detection is helpful to determine tumor burden and tumor diameter in a time efficient manner. Here, we aimed to evaluate semi-automated measurement of total metastatic burden for therapy stratification.METHODSNineteen patients with liver metastasized NET underwent contrast-enhanced 1.5 T MRI using gadolinium-ethoxybenzyl diethylenetriaminepentaacetic acid. Liver metastases (n=1537) were segmented using Fraunhofer MEVIS Software for three-dimensional (3D) segmentation. All lesions were stratified according to longest 3D diameter >20 mm or ≤20 mm and relative contribution to tumor load was used for therapy stratification.RESULTSMean count of lesions ≤20 mm was 67.5 and mean count of lesions >20 mm was 13.4. However, mean contribution to total tumor volume of lesions ≤20 mm was 24%, while contribution of lesions >20 mm was 76%.CONCLUSIONSemi-automatic lesion analysis provides useful information about lesion distribution in predominantly liver metastasized NET patients prior to PRRT. As conventional manual lesion measurements are laborious, our study shows this new approach is more efficient and less operator-dependent and may prove to be useful in the decision making process selecting the best combination PRRT in each patient

Increased x-ray attenuation in malignant vs. benign mediastinal nodes in an orthotopic model of lung cancer

PURPOSEStaging of lung cancer is typically performed with fluorodeoxyglucose-positron emission tomography-computed tomography (FDG-PET/CT); however, false positive PET scans can occur due to inflammatory disease. The CT scan is used for anatomic registration and attenuation correction. Herein, we evaluated x-ray attenuation (XRA) within nodes on CT and correlated this with the presence of malignancy in an orthotopic lung cancer model in rats.METHODS1×106 NCI-H460 cells were injected transthoracically in six National Institutes of Health nude rats and six animals served as controls. After two weeks, animals were sacrificed; lymph nodes were extracted and scanned with a micro-CT to determine their XRA prior to histologic analysis.RESULTSMedian CT density in malignant lymph nodes (n=20) was significantly higher than benign lymph nodes (n=12; P = 0.018). Short-axis diameter of metastatic lymph nodes was significantly different than benign nodes (3.4 mm vs. 2.4 mm; P = 0.025). Area under the curve for malignancy was higher for density-based lymph node analysis compared with size measurements (0.87 vs. 0.7).CONCLUSIONXRA of metastatic mediastinal lymph nodes is significantly higher than benign nodes in this lung cancer model. This suggests that information on nodal density may be useful when used in combination with the results of FDG-PET in determining the likelihood of malignant adenopath

The CYP2J2 G-50T polymorphism and myocardial infarction in patients with cardiovascular risk profile

<p>Abstract</p> <p>Background</p> <p>Cytochrome P450 (CYP) enzyme 2J2, an epoxygenase predominantly expressed in the heart, metabolises arachidonic acid to biologically active eicosanoids. One of the CYP2J2 products, 11, 12-epoxyeicosatrienoic acid, has several vasoprotective effects. The CYP2J2-G-50T-promotor polymorphism decreases gene expression and is associated with coronary artery disease. This association supports the vascular protective role of CYP-derived eicosanoids in cardiovascular disease. In the present study, we investigated the influence of this polymorphism on survived myocardial infarction in two study groups of patients with on average high cardiovascular risk profile.</p> <p>Methods</p> <p>The CYP2J2 polymorphism was genotyped in two groups of patients that were collected with the same method of clinical data collection. Data from 512 patients with sleep apnoea (group: OSA) and on average high cardiovascular risk profile and from another 488 patients who were admitted for coronary angiography (CAR-group) were evaluated for a potential correlation of the CYP2J2 polymorphism G-50T and a history of myocardial infarction. The G-50T polymorphism of the CYP2J2 gene was genotyped by allele specific restriction and light cycler analysis.</p> <p>Results</p> <p>The T-allele of the polymorphism was found in 111 (11.1%; CAR-group: N = 65, 13.3%; OSA: N = 46, 9.0%). 146 patients had a history of myocardial infarction (CAR: N = 120, 24.6%; OSA: N = 26, 5.1%). Cardiovascular risk factors were equally distributed between the different genotypes of the CYP2J2 G-50T polymorphism. In the total group of 1000 individuals, carriers of the T-allele had significantly more myocardial infarctions compared to carriers of the wild type (T/T or G/T: 21.6%; G/G: 13.7%; p = 0.026, odds ratio 1.73, 95%-CI [1.06–2.83]). In the multivariate logistic regression analysis the odds ratio for a history of myocardial infarction in carriers of the T-allele was 1.611, 95%-CI [0.957–2.731] but this trend was not significant (p = 0.073).</p> <p>Conclusion</p> <p>In presence of other risk factors, the CYP2J2 G-50T failed to show a significant role in the development of myocardial infarction. However, since our result is close to the border of significance, this question should be clarified in larger, prospective studies in the future.</p