An optimization framework is presented to support the model builder in elucidating compartmental models that plausibly describe data obtained during experimentation. Here, one specifies a priori the maximum number of compartments and type of flows to contemplate during the optimization. The mathematical model follows a ‘superstructure’ approach, which inherently considers the different feasible flows between any pair of compartments. The model activates those flows/compartments that provide the optimal fit for a given set of experimental data. A regularized log-likelihood function is formulated as the performance metric. To deal with the resulting set of differential equations orthogonal collocation on finite elements is employed. A case study related to pharmacokinetics of an oncological agent demonstrates the advantages and limitations of the proposed approach. Numerical results show that the proposed approach can provide 33% smaller mean square prediction error in comparison with a compartmental model previously suggested in the literature.

Markov chain Monte Carlo approaches have been widely used for Bayesian inference. The drawback of these methods is that they can be computationally prohibitive especially when complex models are analyzed. In such cases, variational methods may provide an efficient and attractive alternative. However, the variational methods reported to date are applicable to relatively simple models and most are based on a factorized approximation to the posterior distribution. Here, we propose a variational approach that is capable of handling models that consist of a system of differential-algebraic equations and whose posterior approximation can be represented by a multivariate distribution. Under the proposed approach, the solution of the variational inference problem is decomposed into three steps: a maximum a posteriori optimization, which is facilitated by using an orthogonal collocation approach, a preprocessing step, which is based on the estimation of the eigenvectors of the posterior covariance matrix, and an expected propagation optimization problem. To tackle multivariate integration, we employ quadratures derived from the Smolyak rule (sparse grids). Examples are reported to elucidate the advantages and limitations of the proposed methodology. The results are compared to the solutions obtained from a Markov chain Monte Carlo approach. It is demonstrated that significant computational savings can be gained using the proposed approach. This article has supplementary material online.

In this work an optimization framework is presented to support the model builder in postulating compartmental models that plausibly describe data that is obtained during experimentation. In the proposed approach, one specifies a priori the maximum number of compartments and the type of flows (e. g., zero order, first order, second order rate flows) to contemplate. With this input, the mathematical model follows a "flexible" approach, which inherently considers all feasible flows between any pair of compartments. The model activates those flows/compartments that provide the optimal fit for a given set of experimental data. A regularized log-likelihood function is formulated as performance metric in order to handle parameter over-fitting. To deal with the resulting set of differential equations orthogonal collocation on finite elements is employed. A case study related to the pharmacokinetics of an oncological agent is reported to demonstrate the advantages and limitations of the proposed approach. Numerical results show that the proposed approach can provide 33 % smaller mean prediction errors in comparison with a compartmental model previously suggested in the literature that employs a larger number of parameters.

Study ObjectiveVariable metabolism, dose-dependent efficacy, and a narrow therapeutic target of cyclophosphamide (CY) suggest that dosing based on individual pharmacokinetics (PK) will improve efficacy and minimize toxicity. Real-time individualized CY dose adjustment was previously explored using a maximum a posteriori (MAP) approach based on a five serum-PK sampling in patients with hematologic malignancy undergoing stem cell transplantation. The MAP approach resulted in an improved toxicity profile without sacrificing efficacy. However, extensive PK sampling is costly and not generally applicable in the clinic. We hypothesize that the assumption-free Bayesian approach (AFBA) can reduce sampling requirements, while improving the accuracy of results.DesignRetrospective analysis of previously published CY PK data from 20 patients undergoing stem cell transplantation. In that study, Bayesian estimation based on the MAP approach of individual PK parameters was accomplished to predict individualized day-2 doses of CY. Based on these data, we used the AFBA to select the optimal sampling schedule and compare the projected probability of achieving the therapeutic end points.Measurements and Main ResultsBy optimizing the sampling schedule with the AFBA, an effective individualized PK characterization can be obtained with only two blood draws at 4 and 16 hours after administration on day 1. The second-day doses selected with the AFBA were significantly different than the MAP approach and averaged 37% higher probability of attaining the therapeutic targets.ConclusionsThe AFBA, based on cutting-edge statistical and mathematical tools, allows an accurate individualized dosing of CY, with simplified PK sampling. This highly accessible approach holds great promise for improving efficacy, reducing toxicities, and lowering treatment costs.

e18555 Background: Many elderly or frail myeloma patients ineligible for autologous stem cell transplant (ASCT) receive non-myeloablative therapies such as the melphalan/prednisone/thalidomide (MPT) combination. These agents have variable pharmacokinetics (PK), and hence variable anti-tumor responses and toxicities. Yet, their dosing is uniform, partly because no convenient procedure exists to individualize dosage-schedules (costs and logistical difficulties with multiple blood samplings for a given patient). We herein propose a novel PK methodology that allows for individualized dosing of these agents. Methods: The MPT regimen is commonly used for older patients and consists of six cycles of 4mg/m 2 M, 40mg/m 2 P and 100mg T administered for seven days followed by a three-week rest period. To compare the peripheral blood concentrations between individuals, a marker combining the level and duration of exposure is the time weighted “average concentration” defined by CC=(Cumulative Area under the concentration time curve (AUC))/(Elapsed time from start of treatment). Using pattern recognition software, an individual’s PK profile was simulated from the population PK models using a single blood draw 2h after administration. The AUC for the individual is then used to adjust the starting dose to match the CC target profile based on the average population response. Results: Using PK data for the individual patients used to construct the PK population model, the simulated adjusted doses for initializing the therapy vary as follows: M 2.5-6mg/m 2 ; P 12.5-140mg/m 2 ; and T 76-130mg which is substantially different from the "standard" uniform dosing. Conclusions: The wide variability in dose levels obtained with the pattern recognition software may explain the marked differences in response and toxicities to the “standard” dosing protocol. An individualized starting dose can be applied using the proposed methodology which requires only a single blood draw after a lower “test dose” of these agents “prior” to commencing actual therapy.

Clinical trials and health care studies generate an enormous amount of data. This data is used by pharmaceutical companies during new drug development processes to characterize patient populations and determine a standardized dosage regimen for new patients, make commercial decisions, and gain approval from regulatory agencies. Nevertheless, the knowledge embedded in such data is rarely further exploited for customized patient care. In most cases, there is a significant difference between the pharmacokinetic profile of patients in a population, yet these differences are not reflected in the standardized dosage regimen. Here, a Bayesian methodology is proposed to individualize dosage regimens by combining the pharmacokinetic data collected from a patient population during clinical trials and additional data coming from a minimal number of serum samples from the new patient. In the Bayesian sense, the distribution of pharmacokinetic parameters from the population data is treated as prior information, and the posterior patient specific distribution of pharmacokinetic parameters is calculated. Then, such a posterior distribution is used to obtain dosage regimens that result in drug concentrations that are kept within the therapeutic window at a target confidence level for that patient. Moreover, a methodology is presented to suggest the sampling schedule for new patients so as to reduce the number of samples required to obtain well characterized individual pharmacometric parameters. Available pharmacokinetic data for Gabapentin, a therapeutic agent for epilepsy and neuropathic pain, is used to illustrate the concepts underlying the proposed strategy and the benefits of an individualized regimen over a standardized dosage regimen.

The introduction of large amounts of wind power into the electricity system raises potential reliability issues for the grid due to the intermittent nature of wind power. Wind power cannot be scheduled in advance like conventional generation units and thus forecasts of the wind power that will be produced in future hours are used to schedule the amount of wind power available. Any improvements in wind power forecasting have the potential to reduce the amount of reserves necessary in systems with significant amounts of wind power, and eventually lower the cost of electricity in such systems. In this work we examine the ability of statistical time series analysis tools, namely autoregressive integrative moving average (ARIMA) models, to forecast future wind power output from historical data. A systematic approach to determine the best values for the assortment of variables associated with the models, such as training period length and model orders, has been developed and applied. The ability of the models to outperform a standard forecasting benchmark has been examined at a number of different forecast period lengths. The application of the tools to total power output of the many wind farms that may be present within the territory of a single independent system operator is studied. Finally, a case study involving wind farm data from Ontario, Canada is used to show how the improvements that these statistical techniques offer may be beneficial for the independent system operator.

In this paper, a simulation-based optimization approach is proposed to study design of life-support systems, i.e., systems that provide basic life-support elements such as potable water and oxygen, for manned space missions. The method integrates deterministic mathematical programming models, which set tactical control parameters, stochastic discrete-event simulation, which accounts for the uncertainty, and a time-series data-mining approach, which calculates strategic set points for tactical control. Total cost distributions of the system, probabilistic capacities of the technologies in the system and the basic life-support element amounts that are needed to support crew life and activities for the duration of a given mission are determined using the proposed framework. The methodology is demonstrated using four different technology levels for a mission scenario at which a crew of six spends 600 days on the Martian surface.

Recent advances in statistical procedures, coupled with the availability of high performance computational resources and the large mass of data generated from high throughput screening, have enabled a new paradigm for building mathematical models of the kinetic behavior of catalytic reactions. A Bayesian approach is used to formulate the model building problem, estimate model parameters by Monte Carlo based methods, discriminate rival models, and design new experiments to improve the discrimination and fidelity of the parameter estimates. The methodology is illustrated with a typical, model building problem involving three proposed Langmuir-Hinshelwood rate expressions. The Bayesian approach gives improved discrimination of the three models and higher quality model parameters for the best model selected as compared to the traditional methods that employ linearized statistical tools. This paper describes the methodology and its capabilities in sufficient detail to allow kinetic model builders to evaluate and implement its improved model discrimination and parameter estimation features.

The objective of this research program is to optimize drug dose regimen for an individual, using minimally invasive clinical testing, in order to reduce both the total cost of treatment and the risk for over or under-medication using a Bayesian modeling approach. The challenge is to extract the PharmacoKinetic/PharmacoDynamic(PK/PD) parameters for an individual from population level plasma concentration information gathered in clinical trials along with one or two plasma samples from an individual and use these personalized parameters in determining most appropriate dose regimen for a specific patient. In this study we illustrate the plausibility of our methodology through a proof-of-concept study with simulated data.

The current world energy system is highly complex and is rapidly evolving to incorporate emerging energy technologies. An understanding of how these technologies will be incorporated into the existing system is needed in order to make intermediate and long term research and infrastructure decisions. We propose an agent-based modeling and simulation approach for energy system analysis that can be used to investigate the mechanisms by which changes occur within the system. The approach has been applied to the Indiana state energy system.

In the petrochemical, chemical and pharmaceutical industries, supply chains typically consist of multiple stages of production facilities, warehouse/distribution centers, logistical subnetworks and end customers. Supply chain performance in the face of various market and technical uncertainties is usually measured by service level, that is, the expected fraction of demand that the supply chain can satisfy within a predefined allowable delivery time window. Safety stock is introduced into supply chains as an important hedge against uncertainty in order to provide customers with the promised service level. Although a higher safety stock level guarantees a higher service level, it does increase the supply chain operating cost and thus these levels must be suitably optimized. The complexities in safety stock management for multi-stage supply chain with multiple products and production capacity constraints arise from: (1) the nonlinear performance functions that relate the service level, expected inventory with safety stock control variables at each site; (2) the interdependence of the performances of different sites; and (3) finally the margin by which production capacity exceeds the uncertain demand. Given the complexities, the integrated management of safety stocks across the supply chain imposes significant computational challenges. In this research, we propose an approach in which the evaluation of the performance functions and the decision on safety stock related variables are decomposed into two separate computational frameworks. For evaluating the performance functions, off-line computation using a discrete event simulation model is proposed. A linear programming based safety stock management model is developed, in which the safety stock control variables (the target inventory levels used in production planning and scheduling models, base-stock levels for the base-stock policy at the warehouses) and service levels at both plant stage and warehouse stages are used as important decision variables. In the linear programming model, the nonlinear performance functions, interdependence of the performances, and the safety production capacity limits in safety stock management are properly represented. To demonstrate the effectiveness of the proposed safety stock management model, a case study of a realistically scaled polymer supply chain problem is presented. In the case problem, the supply chain is composed of two geographically separated production sites and 3–8 warehouses supplying 10 final products to 30 sales regions.

Managing a pharmaceutical R&D pipeline is a complex undertaking that involves an inter-play of strategic and tactical decision-making around portfolio selection, activity scheduling and resource allocation across multiple projects along a multi-year scope. In this paper we focus on improving the quality of pharmaceutical resource management decisions and practices. Pharmaceutical resource management is broadly comprised of decisions relating to scheduling and allocation of limited resources across development activities of multiple drug projects that are subject to technological and market uncertainties, long development cycle times as well as work process constraints. These complexities only magnify the risk of sub-optimal resource management. Notwithstanding this risk, the literature on joint optimization of scheduling and resource allocation decisions in the context of pharmaceutical R&D pipelines is limited. In this paper, a framework—SIM-OPT is proposed as an integrated resource management tool with the goals of maximizing the portfolio's expected net present value (ENPV), controlling risk and reducing drug development cycle times. The framework includes three key components: (1) a stochastic simulation of the pharmaceutical work flow process, (2) a ‘resource manager’ based on a mixed integer linear programming formulation that schedules and allocates resources as a function of demands from the simulated work process and (3) a ‘strategy learner’ that evaluates the impact of various resource strategies on the financial and cycle time performance of the simulated pipeline and draws key learnings. The output is a recommended set of resource management strategies and their impacts on expected return, risk and cycle time metrics. The effectiveness of the framework is demonstrated via its application to two industrial case studies drawn from a major pharmaceutical corporation. This approach offers potential benefits in terms of its ability to transform key learnings into efficient and reliable resource management practices that are well aligned with the overall business strategy.

Mathematical models of physicochemical systems are usually built in an iterative fashion during the course of an experimental investigation. In this paper, a novel Bayesian approach to model building is presented. This approach is now feasible because of breakthroughs in Monte Carlo sampling procedures and high performance computing, that make it possible to deal directly with the nonlinear mathematical models themselves instead of their linear approximations. By including an error model for experimental data, it is further possible to use nonlinear statistical concepts to test a given model for adequacy against experimental data and prior knowledge, and to place realistic confidence limits on the resulting model parameters. In this paper a model building work flow that takes advantage of these recent advances to enable high fidelity mathematical modeling is proposed. A set of models and their parameters are needed to initiate the process. Probability distributions for the models and their parameters based on available quantitative and subjective information must also be supplied. Finally, an error model describing the heteroscedasticity in the data along with probability distributions for the error model parameters must be generated from exploratory data. Then experiments are designed and data collected. Using Bayes’ theorem, Monte Carlo (MC) or Markov Chain Monte Carlo (MCMC) methods are used to generate a sequence of samples of parameter values for each postulated model. These sets of samples are then used to discriminate among the models using the criteria introduced in this paper. Once discrimination is achieved, a global lack of fit test is introduced to determine model adequacy. After a single adequate model is selected, highest probability density (HPD) intervals are determined for the individual parameters and HPD density regions are constructed for all model parameter pairs. Experiments are then designed to reduce the uncertainty in the joint posterior probability HPD regions. Finally, a sampling procedure is described to properly represent uncertainties in predictions made from the model. The proposed approach is demonstrated by an illustrative problem where three simple models are discriminated and the parameters in the most suitable ones are estimated rigorously.

To stay ahead of their competition, pharmaceutical firms must make effective use of their new product development (NPD) capabilities by efficiently allocating its analytical, clinical testing and manufacturing resources across various drug development projects. The resulting project scheduling problems involve coordinating hundreds of testing and manufacturing activities over a period of several quarters. Most conventional integer programming approaches are computationally impractical for problems of this size, while priority rule-driven heuristics seldom provide consistent solution quality. We propose a Lagrangian decomposition (LD) heuristic that exploits the special structure of these problems. Some resources (typically manpower) are shared across all on-going projects while others (typically equipment) are specific to individual project categories. Our objective function is a weighted discounted cost expressed in terms of activity completion times. The LD heuristics were subjected to a comprehensive experimental study based on typical operational instances. While the conventional “Reward–Risk” priority rule heuristic generates duality gaps between 47–58%, the best LD heuristic achieves duality gaps between 10–20%. The LD heuristics also yield makespan reductions of over 30% over the Reward–Risk priority rule.

The superior activity and stability of Au/TS-1 catalysts has allowed the first comprehensive kinetic analysis of the propylene epoxidation system in the absence of significant deactivation. A unique design of experiments combining the best features of factorial experiments with one-at-a-time experimentation over the nonflammable range was used to collect kinetic information, from which a power rate law was extracted using the statistical software package JMP. Explaining the resultant fractional reactant orders (O2=0.31±0.04, H2=0.60±0.03, and C3H6=0.18±0.04) requires a sequence of elementary kinetic steps having a minimum of two active sites participating in the rate-determining step. A reaction sequence is proposed that accounts for the experimentally determined reaction orders and is consistent with DFT calculations and other results from the literature. This mechanism suggests that titanium and gold sites must generate and use the epoxidation oxidant simultaneously rather than sequentially, as previously suggested in the literature.