The original pharmacokinetics (PK) two-compartment model illustrates the change law of the drug in the central and peripheral chambers after administration, respectively. Although this deterministic model has derived many practical conclusions, it is based on simplifications and neglects noise, which is inherent to pharmacological processes. The actual pharmacological processes are always influenced by factors that cannot be entirely understood or modeled explicitly.Modeling without considering these phenomena may impact the accuracy and the related conclusions. A stochastic type of model can capture such noise. We proposed a novel two-compartment PK model concerning drug administration through intravenous route by combining optimal control theory and stochastic analysis. The selection of the objective function was based on the goal of obtaining the best possible therapeutic effect with the least possible drug dosage.Firstly, we extended the original PK two-compartment model based on optimal control and proved the existence and uniqueness of the switch in the control.Moreover, considering the possible uncertain factors, we added disturbances to the distance between the drug concentration and equilibrium point of the dynamic system and extended the model to a stochastic differential equation model. Qualitative and quantitative analyses showed that optimal controls were bang-bang, that is, alternating the drug dosages at a full dose with rest-periods in-between. Our analysis provided a schedule for optimal dosage and timing.The solutions of the model provided estimates of the drug concentration at any given time. Finally, we simulated the model using R and showed that the numerical method is stable.
First-order 1-compartment pharmacokinetic model for extravascular administered drugs can be used to derive many useful quantities by comparing the predicted values with actual data. However, less research has been done in actually formulating them as optimal control problems. Moreover, real pharmacological processes are always exposed to influences that are not completely understood or not feasible to model explicitly. Ignoring these phenomena in the modeling may affect the estimation of PK/PD models' (pharmacokinetic/pharmacodynamic models') parameters and the derived conclusions.Therefore there is an increasing need to extend the deterministic models to models including a stochastic component. In our study, we modify the 1-compartment pharmacokinetic model to a stochastic differential equation model based on an optimal control problem. A schedule of optimal dosing and timing has been given from our proposed model. INDEX TERMSStochastic differential equation, optimal control, pharmacokinetics, stability, E-M method.
ObjectivesThis study aimed to investigate the association of α-linolenic acid (ALA; 18:3 ω-3) dietary intake with very short sleep duration (<5 h) in adults based on the CDC's National Health and Nutrition Examination Survey data.MethodsMultinomial logistic regression was used to explore the association of ALA intake with very short sleep. To make the estimation more robust, bootstrap methods of 1,000 replications were performed. Rolling window method was used to investigate the trend of the odds ratios of very short sleep with age. A Kruskal–Wallis test was applied to estimate the differences in the ORs of very short sleep between genders and different age groups.ResultsCompared with the first tertile, the ORs of very short sleep and the corresponding 95% CIs for the second and the third tertile of dietary ALA intake in males were 0.618 (0.612, 0.624) and 0.544 (0.538, 0.551), respectively, and in females were 0.575 (0.612, 0.624) and 0.432 (0.427, 0.437). In most cases, the differences between different ages were more significant than those between different sexes. Men's very short sleep odds ratios for the second tertile of ALA intake increased linearly with age before 60.ConclusionsThe risk of a very short sleep duration was negatively related to the dietary intake of ALA. The effect of ALA on very short sleep is significantly different among groups of different genders and ages.
<abstract><p>In this paper, we propose a beta kernel estimator to measure functional dependence (MFD). The MFD not only can measure the strength of linear or monotonic relationships, but it is also suitable for more complicated functional dependence. We derive the asymptotic distribution of the proposed estimator and then use several simulated examples to compare our estimator with the traditional measures. Our simulation results demonstrate that beta kernel provides high accuracy in estimation. A real data example is also given to illustrate one possible application of the new estimator.</p></abstract>
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