Word problem of the Perkins semigroup via directed acyclic graphs

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkinsmonoid B½. Let ( f B½ n ) denote the free spectrum of B½, let an be the number of distinct alternation word digraphs on words whose alphabet is contained in {x1, . . . , xn}, and let pn denote the number of distinct labeled posets on {1, . . . , n}. The word problem for the Perkins semigroup B½ is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over B½ if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: pn ≤ an ≤ f B½ n ≤ 2na2 2n. A result of the second author in a companion paper states that (log an) ∈ O(n3), from which it follows that (log f B½ n ) ∈ O(n3) as well. Alternation word digraphs are of independent interest combinatorially. It is shown here that the computational complexity problem that has as instance {u, v} where u, v are words of finite length, and question “Is Alt(u) = Alt(v)?”, is co-NP-complete. Additionally, alternation word digraphs are acyclic, and certain of them are natural extensions of posets; each realizer of a finite poset determines an extension by an alternation word digraph

Population pharmacokinetics/pharmacodynamics of docetaxel in phase II studies in patients with cancer

PURPOSE The population pharmacokinetic/pharmacodynamic (PK/PD) approach was prospectively integrated in the clinical development of docetaxel to assess the PK profile in a large population of patients and investigate systemic exposure as a prognostic factor for clinical outcome. PATIENTS AND METHODS PK analysis was performed at first course in 24 phase II studies of docetaxel monotherapy using four randomized limited-sampling schedules. Bayesian estimates of clearance (CL), area under the concentration-time curve (AUC), and peak and duration of plasma levels greater than threshold levels were used as measures of exposure. PD data included for efficacy, response rate, time to first response, and time to progression (TTP) in breast cancer and non-small-cell lung cancer (NSCLC), and for toxicity, grade 4 neutropenia, and febrile neutropenia at first course and time to onset of fluid retention. PK/PD analysis was conducted using logistic and Cox multivariate regression models. RESULTS PK protocol implementation was successful. Most of the patients registered (721 of 936, 77%) were sampled and 68% were assessable for PK (640 patients). First-course docetaxel AUC was a significant predictor (P = .0232) of TTP in NSCLC (n = 151). Docetaxel CL was a strong independent predictor (P &lt; .0001) of both grade 4 neutropenia and febrile neutropenia (n = 582). Cumulative dose was the strongest predictor (P &lt; .0001) of the time to onset of fluid retention (n = 631). However, the duration of exposure over 0.20 micromol/L (0.16 microg/mL) at first course was an independent predictor (P = .0029). Few patients (n = 25, 4%) received the recommended dexamethasone premedication. CONCLUSION First-course docetaxel PK is a predictor of first-course hematologic toxicity, but also of fluid retention, which is cumulative in nature. Patients with elevated hepatic enzymes have a 27% reduction in docetaxel CL and are at a higher risk of toxicity. A starting dose of 75 mg/m2 is currently being evaluated in this population. Prospective implementation of large-scale population PK/PD evaluation is feasible in early drug development and this approach generates clinically relevant findings. </jats:sec