A pharmacokinetic model of filgrastim and pegfilgrastim application in normal mice and those with cyclophosphamide-induced granulocytopaenia

Objectives:  Recombinant human granulocyte colony-stimulating factor (rhG-CSF) is widely used as treatment for granulocytopaenia during cytotoxic chemotherapy; however, optimal scheduling of this pharmaceutical is unknown. Biomathematical models can help to pre-select optimal application schedules but precise pharmacokinetic properties of the pharmaceuticals are required at first. In this study, we have aimed to construct a pharmacokinetic model of G-CSF derivatives filgrastim and pegfilgrastim in mice.
Methods:  Healthy CD-1 mice and those with cyclophosphamide-induced granulocytopaenia were studied after administration of filgrastim and pegfilgrastim in different dosing and timing schedules. Close meshed time series of granulocytes and G-CSF plasma concentrations were determined. An ordinary differential equations model of pharmacokinetics was constructed on the basis of known mechanisms of drug distribution and degradation.
Results:  Predictions of the model fit well with all experimental data for both filgrastim and pegfilgrastim. We obtained a unique parameter setting for all experimental scenarios. Differences in pharmacokinetics between filgrastim and pegfilgrastim can be explained by different estimates of model parameters rather than by different model mechanisms. Parameter estimates with respect to distribution and clearance of the drug derivatives are in agreement with qualitative experimental results.
Conclusion:  Dynamics of filgrastim and pegfilgrastim plasma levels can be explained by the same pharmacokinetic model but different model parameters. Beause of a strong clearance mechanism mediated by granulocytes, granulocytotic and granulocytopaenic conditions must be studied simultaneously to construct a reliable model. The pharmacokinetic model will be extended to a murine model of granulopoiesis under chemotherapy and G-CSF application.     

Optimal control approach to termination of re-entry waves in cardiac electrophysiology

This work proposes an optimal control approach for the termination of re-entry waves in cardiac electrophysiology. The control enters as an extracellular current density into the bidomain equations which are well established model equations in the literature to describe the electrical behavior of the cardiac tissue. The optimal control formulation is inspired, in part, by the dynamical systems behavior of the underlying system of differential equations. Existence of optimal controls is established and the optimality system is derived formally. The numerical realization is described in detail and numerical experiments, which demonstrate the capability of influencing and terminating reentry phenomena, are presented.     

Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics

An optimal control problem for cancer chemotherapy is considered that includes immunological activity. In the objective a weighted average of several quantities that describe the effectiveness of treatment is minimized. These terms include (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), (iii) the overall amount of cytotoxic agents given as a measure for the side effects of treatment and (iv) a small penalty on the terminal time that limits the overall therapy horizon which is assumed to be free. This last term is essential in obtaining a well-posed problem formulation. Employing a Gompertzian growth model for the cancer cells, for various scenarios optimal controls and corresponding responses of the system are calculated. Solutions initially follow a full dose treatment, but then at one point switch to a singular regimen that only applies partial dosages. This structure is consistent with protocols that apply an initial burst to reduce the tumor volume and then maintain a small volume through lower dosages. Optimal controls end with either a prolonged period of no dose treatment or, in a small number of scenarios, this no dose interval is still followed by one more short burst of full dose treatment.     

Optimal control of drug therapy: melding pharmacokinetics with viral dynamics

Pharmacokinetics were melded with a viral dynamical model to design an optimal drug administration regimen such that the basic reproductive number for the virus was minimized. One-compartmental models with two kinds of drug delivery routes, intravenous and extravascular with multiple dosages, and two drug elimination rates, first order and Michaelis-Menten rates, were considered. We defined explicitly the basic reproductive number for the viral dynamical model melded with pharmacokinetics. When the average plasma drug concentration was constant, intravenous administration of the drug with small dosages applied frequently minimized the basic reproductive number. For extravascular administration, the basic reproductive number initially decreases to a trough point and then increases as the drug dosage increases. When a therapeutic window is considered, numerical studies indicate that the wider the window, the smaller the basic reproductive number. Once the width of the therapeutic window is fixed, the basic reproductive number monotonously declines as the minimum therapeutic level increases. The findings suggest that the existence of drug dosage and drug administration interval that minimize the basic reproductive number could help design the optimal drug administration regimen.     

Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls

We investigate a mathematical model of tumor-immune interactions with chemotherapy, and strategies for optimally administering treatment. In this paper we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss numerical results of the optimal strategies. The form of the model allows us to test and compare various optimal control strategies, including a quadratic control, a linear control, and a state-constraint. We establish the existence of the optimal control, and solve for the control in both the quadratic and linear case. In the linear control case, we show that we cannot rule out the possibility of a singular control. An interesting aspect of this paper is that we provide a graphical representation of regions on which the singular control is optimal.     

A mechanistic framework for a priori pharmacokinetic predictions of orally inhaled drugs

The fate of orally inhaled drugs is determined by pulmonary pharmacokinetic processes such as particle deposition, pulmonary drug dissolution, and mucociliary clearance. Even though each single process has been systematically investigated, a quantitative understanding on the interaction of processes remains limited and therefore identifying optimal drug and formulation characteristics for orally inhaled drugs is still challenging. To investigate this complex interplay, the pulmonary processes can be integrated into mathematical models. However, existing modeling attempts considerably simplify these processes or are not systematically evaluated against (clinical) data. In this work, we developed a mathematical framework based on physiologically-structured population equations to integrate all relevant pulmonary processes mechanistically. A tailored numerical resolution strategy was chosen and the mechanistic model was evaluated systematically against data from different clinical studies. Without adapting the mechanistic model or estimating kinetic parameters based on individual study data, the developed model was able to predict simultaneously (i) lung retention profiles of inhaled insoluble particles, (ii) particle size-dependent pharmacokinetics of inhaled monodisperse particles, (iii) pharmacokinetic differences between inhaled fluticasone propionate and budesonide, as well as (iv) pharmacokinetic differences between healthy volunteers and asthmatic patients. Finally, to identify the most impactful optimization criteria for orally inhaled drugs, the developed mechanistic model was applied to investigate the impact of input parameters on both the pulmonary and systemic exposure. Interestingly, the solubility of the inhaled drug did not have any relevant impact on the local and systemic pharmacokinetics. Instead, the pulmonary dissolution rate, the particle size, the tissue affinity, and the systemic clearance were the most impactful potential optimization parameters. In the future, the developed prediction framework should be considered a powerful tool for identifying optimal drug and formulation characteristics.     

Metabolic scaling of antibiotics in reptiles: Basis and limitations

The allometric equation y = a · x^b has been used to scale many morphological and physiological attributes relative to body mass. For instance, in eutherian mammals, the equation P_met = 70M_b^0.75 has been used to describe the relationship between metabolic rate (P_met) and body mass (M_b). Similar equations have been derived for squamate reptiles. Recently, this relationship between metabolic rate and body mass has been used in determining appropriate dosages and dosing intervals of antibiotics both intraspecifically for different sized reptiles and interspecifically for those reptiles in which antibiotic pharmacokinetic studies have not been performed. Although this is a simple mathematical process, a number of problems surface when this approach is examined closely. First, the mass constant (a) in reptiles varies from 1–5 for snakes and 6–10 for lizards. No such information is available for chelonians or crocodilians. Unless the mass constant for the unknown species approximates that of the known species, inappropriate dosages and intervals of administration will be calculated. Second, pharmacokinetic differences may exist between widely divergent species, independent of metabolic rate. Third, all available pharmacokinetic studies and metabolic allometric equations are derived from clinically healthy reptiles. Differences more than likely exist between healthy and ill reptiles in regard to uptake, distribution, and elimination of drugs and overall metabolism. While metabolic scaling of antibiotics is a potentially useful and practical tool in drug dosing, these limitations must be considered when dosing an ill reptile. Until more scientifically derived information is available for demonstrating the accuracy of metabolic scaling of antibiotics in reptiles, the clinician will need to understand the limitations of this approach. © 1996 Wiley-Liss, Inc.     

On optimal delivery of combination therapy for tumors

A mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent is formulated. Conditions are given that allow tumor eradication under constant infusion therapies. Then the optimal scheduling of a vessel disruptive agent in combination with a cytotoxic drug is considered as an optimal control problem. Both theoretical and numerical results on the structure of optimal controls are presented.     

Clinical development of aromatase inhibitors for the treatment of breast and prostate cancer

Numerous aromatase inhibitors are under development for breast cancer treatment. The major aims are to obtain a drug which at its dose of maximum efficacy has no effect on other endocrine systems, has no clinical side-effects and its convenient to administer. During the early clinical stages of development detailed endocrine and pharmacokinetic analyses are a valuable aid in the establishment of a drug's selectivity and its optimum dose, route and frequency of administration. The optimal dose may be defined as the minimum that will achieve maximal and sustained suppression of aromatase activity. This has generally been measured indirectly by comparing the suppression of plasma oestrogen levels at a selection of dosages. This approach has major advantages in speeding dose selection for therapeutic clinical trials. However, it also has some disadvantages including the unproven assumption that clinical response has a direct relationship with the degree of oestrogen suppression. In addition there are technical difficulties of analysis, of wide variability in endocrine response between patients and of demonstrating oestrogen suppression to be equivalent between doses (necessary to show maximal suppression). The direct measurement of aromatase inhibition in vivo by isotopic infusion analysis provides support to these indirect estimates. Its value is shown by our recent results with CGS16949A. The additional value of collating pharmacokinetic and endocrine measurements is apparent from our investigations of 4-hydroxyandrostenedione (4-OHA) and pyridoglutethimide. A consideration of our experience with these inhibitors may be helpful in directing the development of future agents.

Whilst the value of aromatase inhibition in breast cancer is established its value in prostatic cancer is in doubt: we have found that 4-OHA is only poorly efficacious in advanced prostatic cancer.     

Bayesian Population Physiologically-Based Pharmacokinetic (PBPK) Approach for a Physiologically Realistic Characterization of Interindividual Variability in Clinically Relevant Populations

Interindividual variability in anatomical and physiological properties results in significant differences in drug pharmacokinetics. The consideration of such pharmacokinetic variability supports optimal drug efficacy and safety for each single individual, e.g. by identification of individual-specific dosings. One clear objective in clinical drug development is therefore a thorough characterization of the physiological sources of interindividual variability. In this work, we present a Bayesian population physiologically-based pharmacokinetic (PBPK) approach for the mechanistically and physiologically realistic identification of interindividual variability. The consideration of a generic and highly detailed mechanistic PBPK model structure enables the integration of large amounts of prior physiological knowledge, which is then updated with new experimental data in a Bayesian framework. A covariate model integrates known relationships of physiological parameters to age, gender and body height. We further provide a framework for estimation of the a posteriori parameter dependency structure at the population level. The approach is demonstrated considering a cohort of healthy individuals and theophylline as an application example. The variability and co-variability of physiological parameters are specified within the population; respectively. Significant correlations are identified between population parameters and are applied for individual- and population-specific visual predictive checks of the pharmacokinetic behavior, which leads to improved results compared to present population approaches. In the future, the integration of a generic PBPK model into an hierarchical approach allows for extrapolations to other populations or drugs, while the Bayesian paradigm allows for an iterative application of the approach and thereby a continuous updating of physiological knowledge with new data. This will facilitate decision making e.g. from preclinical to clinical development or extrapolation of PK behavior from healthy to clinically significant populations.     

Costs versus benefits: best possible and best practical treatment regimens for HIV

Current HIV therapy, although highly effective, may cause very serious side effects, making adherence to the prescribed regimen difficult. Mathematical modeling may be used to evaluate alternative treatment regimens by weighing the positive results of treatment, such as higher levels of helper T cells, against the negative consequences, such as side effects and the possibility of resistance mutations. Although estimating the weights assigned to these factors is difficult, current clinical practice offers insight by defining situations in which therapy is considered “worthwhile”. We therefore use clinical practice, along with the probability that a drug-resistant mutation is present at the start of therapy, to suggest methods of rationally estimating these weights. In our underlying model, we use ordinary differential equations to describe the time course of in-host HIV infection, and include populations of both activated CD4⁺ T cells and CD8⁺ T cells. We then determine the best possible treatment regimen, assuming that the effectiveness of the drug can be continually adjusted, and the best practical treatment regimen, evaluating all patterns of a block of days “on” therapy followed by a block of days “off” therapy. We find that when the tolerance for drug-resistant mutations is low, high drug concentrations which maintain low infected cell populations are optimal. In contrast, if the tolerance for drug-resistant mutations is fairly high, the optimal treatment involves periods of reduced drug exposure which consequently boost the immune response through increased antigen exposure. We elucidate the dependence of the optimal treatment regimen on the pharmacokinetic parameters of specific antiviral agents.     

Optimal pharmacokinetic delivery of infused drugs: application to the treatment of cardiac arrhythmias

A common clinical goal of infusing drugs is to attain therapeutic steady-state concentrations as rapidly as possible. The desire is to closely approximate a step function in plasma concentration of the therapeutic agent. We have developed a novel approach to achieve this goal by using the principles of systems and compartmental analyses. The approach is to build a pharmacokinetic model for the disposition of the drug and then calculate backwards from the desired output function to derive the optimal input infusion function. We applied this technique to the infusion of Lidocaine, an antiarrhythmic agent which is often difficult to control. An optimal infusion function, used to drive a servo-controlled infusion pump, was derived to closely approximate a step-function response of drug levels. The efficacy of this infusion function was verified experimentally in dogs.     

Estimation and inference for a spline-enhanced population pharmacokinetic model

This article is motivated by an application where subjects were dosed three times with the same drug and the drug concentration profiles appeared to be the lowest after the third dose. One possible explanation is that the pharmacokinetic (PK) parameters vary over time. Therefore, we consider population PK models with time-varying PK parameters. These time-varying PK parameters are modeled by natural cubic spline functions in the ordinary differential equations. Mean parameters, variance components, and smoothing parameters are jointly estimated by maximizing the double penalized log likelihood. Mean functions and their derivatives are obtained by the numerical solution of ordinary differential equations. The interpretation of PK parameters in the model and its flexibility are discussed. The proposed methods are illustrated by application to the data that motivated this article. The model's performance is evaluated through simulation.     

Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method

In this paper, we present a distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. The dynamic model comprises system of three coupled reaction-diffusion models, involving the tumor cells, the normal tissues and the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameters method based on the application of variational calculus is used on an integral-Hamiltonian, which is then used to obtain an optimal coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure as well as to facilitate computation. Finally a three-dimensional test case is considered and partitioned into a set of spherical finite elements, using tri-linear basis functions, except for the elements affected by singularities of polar and azimuthal angles, as well as the origin.     

Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

Pulse HIV Vaccination: Feasibility for Virus Eradication and Optimal Vaccination Schedule

We modify the classical virus dynamics model by incorporating an immune response with fixed or fluctuating vaccination frequencies and dosages to obtain a system of impulsive differential equations for the virus dynamics of both the wild-type and mutant strains. This model framework permits us to obtain precise conditions for the virus elimination, which are much more feasible compared with existing results, which require frequent vaccine administration with large dosage. We also consider the corresponding impulsive optimal control problem to describe when and how much of the vaccine should be administered in order to maximize levels of healthy CD4⁺ T cells and immune response cells. A gradient-based optimization method is applied to obtain the optimal schedule numerically. For a case study when the CTL vaccine is administered in a period of one year, our numerical studies support the optimal vaccination schedule consisting of vaccine administration three times, with the first dosage strong (to boost the immune system), followed by a second dosage shortly after (to strengthen the immune response) and then the third and final dosage long after (to ensure the immune system can handle viruses rebound).     

Optimal dosage regimen calculation program based on the remaining drug concentrations in plasma

A program is presented to calculate the optimal drug doses to make the plasma drug concentration at a constant level throughout the therapy. The package requires data input of experimentally determined pharmacokinetic parameters such as As and ramdas, and provides the optimal intravenous drug doses at every minute. This program is applicable to any drug in a given individual, provided that the single-dose plasma concentration curve equation of the first dose of the therapy in this inidividual is known.     

The matrix representation of pharmacokinetic models.

An equation is developed from the matrix of rate constants which describes the behaviour of linear pharmacokinetic models for any initial condition as a function of time. This general matrix equation is then used to derive analogous expressions for drug distribution after a period of infusion, at the steady state, or during a multiple constant-dosage regimen. Matrix expressions are also derived for areas under drug concentration curves for any compartment after single doses or during multiple dosing. General matrix equations are shown to yield loading dosage schedules to achieve plateau concentrations throughout any open system.It is suggested that matrix methods have advantages over previously used mathematical techniques in pharmacokinetics in the simplicity of the algebraic expressions, and their ease of manipulation. An algebraic example of an open two-compartment model is worked to indicate the applicability of the general expressions.     

A network-based approach for resistance transmission in bacterial populations

Horizontal transfer of mobile genetic elements (conjugation) is an important mechanism whereby resistance is spread through bacterial populations. The aim of our work is to develop a mathematical model that quantitatively describes this process, and to use this model to optimize antimicrobial dosage regimens to minimize resistance development. The bacterial population is conceptualized as a compartmental mathematical model to describe changes in susceptible, resistant, and transconjugant bacteria over time. This model is combined with a compartmental pharmacokinetic model to explore the effect of different plasma drug concentration profiles. An agent-based simulation tool is used to account for resistance transfer occurring when two bacteria are adjacent or in close proximity. In addition, a non-linear programming optimal control problem is introduced to minimize bacterial populations as well as the drug dose. Simulation and optimization results suggest that the rapid death of susceptible individuals in the population is pivotal in minimizing the number of transconjugants in a population. This supports the use of potent antimicrobials that rapidly kill susceptible individuals and development of dosage regimens that maintain effective antimicrobial drug concentrations for as long as needed to kill off the susceptible population. Suggestions are made for experiments to test the hypotheses generated by these simulations.