Pharmacokinetic modelling of digoxin and theophylline: Effect of volume of distribution and plasma concentration

INTRODUCTION

Pharmaceutical technology is a field focused on the development, production and quality control of pharmaceutical products. This discipline includes formulation science, which involves designing drug products such as tablets, capsules and injections that ensure stability, efficacy and patient compliance. It encompasses various processes such as compounding, blending, granulation, encapsulation and coating to transform active pharmaceutical ingredients into safe and effective dosage forms.^[1-3] It also plays a critical role in developing controlled-release formulations, which gradually release drugs over time, and targeted delivery systems, which direct drugs to specific areas of the body.^[4] Recent advances, such as nanotechnology, biotechnology and automation, have enabled the development of more sophisticated drug delivery systems that improve bioavailability and therapeutic outcomes. As a whole, pharmaceutical technology bridges scientific discoveries with practical, high-quality products that meet regulatory standards and patient needs.^[5,6] Pharmaceutical technology and pharmacology are deeply interconnected.^[6] While pharmacology focuses on understanding the mechanism of action of a drug, therapeutic effects and interactions within the body, pharmaceutical technology applies this knowledge to design and manufacture formulations that maximise efficacy and safety.^[7] Insights from pharmacology about absorption, metabolism and potential side effects of a drug guide the development of controlled-release or targeted delivery systems, ensuring that the drug reaches its action site at the right concentration and for the appropriate duration. In turn, advances in pharmaceutical technology, such as nanoparticle carriers or extended-release formulations, enhance pharmacological effects by improving bioavailability, prolonging action and reducing side effects. Together, these fields ensure that medications are not only effective at a cellular level but are also stable, convenient and tailored to patient needs, transforming laboratory research into clinical application.^[8-10]

Pharmacology is the branch of medicine and biology that examines how drugs interact with living organisms to produce therapeutic or toxic effects. It includes studying drug mechanisms, therapeutic effects, side effects and interactions within the body.^[11,12] Pharmacology is divided into two main areas: Pharmacodynamics and pharmacokinetics. Pharmacodynamics focuses on the biochemical and physiological effects of drugs on the body and how they influence cellular receptors, enzymes and signal pathways.^[13] Pharmacokinetics, a subfield of pharmacology, studies how the body absorbs, distributes, metabolises and excretes drugs (often summarised as ADME).^[14] Key pharmacokinetic parameters include the rate of absorption, bioavailability, volume of distribution (V_d) and clearance (Cl) rate. These parameters are crucial for developing effective dosing strategies, minimising adverse effects and maximising the therapeutic efficacy of drugs.^[15,16]

Digoxin is a cardiac glycoside commonly used to treat heart failure and atrial fibrillation. It strengthens heart contractions by inhibiting the sodium-potassium ATPase enzyme, which increases intracellular calcium in heart muscle cells. It also helps control heart rate in patients with atrial fibrillation by influencing electrical impulse conduction.^[17-19] However, digoxin has a narrow therapeutic window, so plasma levels require careful monitoring to avoid toxicity, which can lead to severe side effects such as nausea, confusion and potentially fatal arrhythmias. Its complex pharmacokinetics, influenced by V_d and plasma concentration, affect the maintenance dose (MD) required to achieve therapeutic levels while avoiding toxicity.^[20,21] In addition, factors such as body weight and renal function contribute to the variability in digoxin distribution among patients.^[22] In contrast, theophylline is a bronchodilator primarily used to manage chronic respiratory conditions such as asthma and chronic obstructive pulmonary disease (COPD). It works by inhibiting phosphodiesterase, relaxing bronchial smooth muscles and dilating airways, thus improving airflow and relieving breathing difficulties.^[23] Theophylline also has mild anti-inflammatory effects, making it beneficial in managing chronic respiratory inflammation. Like digoxin, theophylline has a narrow therapeutic range, and high plasma levels that cause toxicity, with symptoms such as nausea, tremors and seizures. Its pharmacokinetics, including V_d and plasma concentration, play a crucial role in determining effective MDs.^[24] Factors such as hepatic function and metabolic differences contribute to inter-patient variability in theophylline Cl, presenting challenges in maintaining therapeutic levels without adverse effects.^[25]

Digoxin and theophylline require careful management due to their narrow therapeutic windows and variable pharmacokinetic profiles. Although both drugs have been extensively studied in clinical settings, there is limited research on simulation-based studies exploring the combined effects of plasma concentration and V_d on MD adjustments. Most pharmacokinetic studies rely on real-world patient data, which may overlook the insights provided by simulation-based research in predicting and optimising dosing regimens. In addition, the combined impact of V_d and plasma concentration on determining MDs is underexplored in non-clinical modelling contexts. While substantial clinical research has focused on dosing for these medications, simulation-based studies that examine the interaction between plasma concentration, V_d and MD adjustments remain scarce. Most existing studies emphasise real-world data and case studies rather than controlled simulations, which could offer valuable insights for dose adjustments. This study addresses this gap using non-clinical simulation modelling to examine how plasma concentration and V_d influence maintenance dosing for digoxin and theophylline. This approach provides a controlled method for predicting dosing needs that can later be adapted for individualised therapy.

The primary aim of this study is to simulate the effect of plasma concentration and V_d on the maintenance dosing of digoxin and theophylline, providing insights to guide dosing strategies for these drugs with narrow therapeutic windows. The specific objectives are (i) to develop a 3-D surface model that quantifies the relationship between plasma concentration, V_d and MD for digoxin and theophylline, and (ii) to compare the pharmacokinetic profiles of digoxin and theophylline within the surface plot to identify similarities and differences in their response to varying plasma concentrations and V_d.

MATERIALS AND METHODS

An online registered version of the Matrix Laboratory from Mathworks, Inc., USA, was used for the surface plot. Pharmacokinetic data of digoxin and theophylline were collected from an online source.^[26] This online resource was initially developed within the framework of a ‘Swiss virtual Campus’ project, in close collaboration with the Center of the University of Lausanne. The content was subsequently transferred to this platform and complemented by the addition of several new pages. This work was coordinated by the eLearning coordinator of the Faculty of Biology and Medicine, University of Lausanne.

Pharmacokinetic parameters are essential measurements used to understand the fate of a drug within the body, including how it is absorbed, distributed, metabolised and excreted. These parameters help clinicians determine optimal dosing regimens and predict the concentration of the drug at its target site over time, ensuring therapeutic efficacy while minimising toxicity.^[27] The absorption rate (k_a) refers to how quickly a drug moves from its site of administration (such as the gastrointestinal tract) into the bloodstream. A faster absorption rate leads to a quicker onset of action, while a slower rate may extend the duration of the effects of drug. Bioavailability (F) is the fraction of the administered dose that reaches the systemic circulation in an active form. For orally administered drugs, bioavailability is often <100%, due to factors such as the first-pass effect in the liver and incomplete absorption. A loading dose (LD) is an initial, often higher, dose of a drug given to rapidly achieve a target plasma concentration, especially for drugs with long half-lives. It helps bring drug levels into the therapeutic range quickly, allowing for faster onset of action.

V_d indicates the extent to which a drug distributes into body tissues relative to plasma. A high V_d suggests widespread distribution into body tissues, while a low V_d indicates the drug remains primarily in the bloodstream. V_d is crucial for understanding dosing requirements and tissue penetration. Half-life (t_½) is the time required for the plasma concentration of a drug to decrease by half. It reflects how long the drug stays in the body and helps determine dosing frequency. Drugs with shorter half-lives need more frequent dosing, while drugs with longer half-lives require less frequent administration. Cl is a measure of the body’s ability to eliminate a drug from the bloodstream, usually through the liver and kidneys. It represents the volume of plasma cleared of the drug per unit time and influences dosing frequency. Drugs with high Cl are eliminated quickly, requiring more frequent doses. The elimination rate constant (k_e) describes the proportion of the drug removed from the body per unit of time. It is closely related to both Cl and half-life, playing a key role in determining how long a drug stays at therapeutic levels. Peak plasma concentration (C_max) represents the highest concentration of the drug in the bloodstream after administration. It is important for understanding the intensity of the effects of the drug and assessing the risk of toxicity, as higher C_max values can increase the likelihood of adverse effects. Time to peak concentration (T_max) is the time it takes for the drug to reach its maximum plasma concentration, which is useful for estimating how quickly the drug begins to take effect.

Area under the curve represents the total drug exposure over time, derived from the plasma concentration-time curve. It is used to assess drug absorption, efficacy and overall exposure. Steady-state concentration is the drug concentration in the bloodstream when the rate of drug administration equals the rate of elimination. This concentration is achieved after multiple doses and is important for long-term therapies where a consistent therapeutic effect is needed. The dosing rate is the amount of drug needed per unit time to maintain steady-state plasma levels. The dosing interval (τ) refers to the time between successive doses, which is determined based on the drug’s half-life and therapeutic window. A shorter dosing interval helps maintain more stable plasma levels, while a longer interval may allow fluctuations in concentration. Finally, the MD is the regular dose needed to keep the plasma concentration within the therapeutic range over time, ensuring that the drug stays at effective levels without causing toxicity. Together, these pharmacokinetic parameters provide critical information for tailoring drug therapy to individual patients. They are especially important for drugs with narrow therapeutic windows, where precise control over plasma levels is necessary to achieve the desired therapeutic effect while avoiding adverse outcomes.

RESULTS

Effect of plasma concentration and V_d on MD for digoxin

Figure 1 represents the relationship between three pharmacokinetic variables: V_d (l) on the x-axis, plasma concentration (ng/mL) on the y-axis and the LD of digoxin (mg) on the z-axis, which is indicated by both the height of the surface and the colour gradient. Analysing this plot helps provide insights into dosing strategies for digoxin, a medication commonly used to treat heart failure and arrhythmias.

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Figure 1:

Effect of plasma concentration and volume of distribution on maintenance dose for digoxin.

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Effect of plasma concentration and V_d on MD for theophylline

Figure 2 provides a valuable visualisation of how theophylline, a bronchodilator often used to treat asthma and COPD, is distributed within the body based on plasma concentration and the V_d. In pharmacokinetics, understanding these relationships is essential for optimising drug dosage to achieve therapeutic effectiveness while minimising side effects.

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Figure 2:

Effect of plasma concentration and volume of distribution on maintenance dose for theophylline.

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DISCUSSION

The V_d, shown on the x-axis, reflects how extensively digoxin distributes throughout the tissues of the body. A higher V_d suggests that digoxin is distributed into various body tissues rather than remaining solely in the bloodstream. The plot indicates that as the V_d increases, the required MD of digoxin also increases. This is expected, as a larger V_d generally means that the drug disperses into more tissue compartments, necessitating a higher initial dose to achieve the desired therapeutic concentration in the plasma. Since digoxin has a narrow therapeutic index, its V_d varies based on factors such as age, weight, kidney function and overall body composition, which significantly influence its dosing requirements.

Plasma concentration, represented on the y-axis, is another critical factor for digoxin dosing. It refers to the amount of digoxin in the blood and is a key measure in ensuring therapeutic efficacy while avoiding toxicity. Since digoxin has a narrow therapeutic range, maintaining plasma concentrations within this range is crucial for both effectiveness and safety. In this plot, as plasma concentration increases, there is a corresponding increase in the required MD. This trend suggests that higher plasma concentrations require higher doses, though careful monitoring is essential to avoid concentrations that could lead to toxicity.

The MD of digoxin (mg), shown on the z-axis, represents the regular dose required to keep the plasma concentration of the drug within the therapeutic range over time, maintaining a steady state without toxicity or subtherapeutic levels. The colour gradient on the surface plot ranges from blue (indicating lower doses) to yellow (indicating higher doses), illustrating the combined impact of V_d and plasma concentration on the LD. As both the V_d and plasma concentration increase, the MD of digoxin increases proportionally. This relationship is essential for drugs with a narrow therapeutic index, such as digoxin, to reach therapeutic levels quickly without overshooting into toxic levels.^[19]

From a clinical perspective, this plot provides useful insights into tailoring digoxin LDs based on individual patient characteristics. For example, a patient with a larger V_d due to higher body mass or tissue volume may require a higher MD to achieve the target plasma concentration. Conversely, patients with a smaller V_d might need a lower MD to avoid excessive plasma concentrations that could lead to adverse effects. This is particularly important for digoxin because factors such as renal function, age and body composition can significantly impact both its V_d and plasma Cl.^[20] In total, this surface plot helps illustrate the delicate balance required in dosing digoxin, where achieving the right plasma concentration is critical for efficacy and safety. Clinicians are encouraged to consider both the V_d and the target plasma concentration when calculating the LD, as well as individual patient factors that may alter these parameters. This approach supports precise, patient-specific dosing strategies that maximise therapeutic benefits while minimising the risk of toxicity, which is especially valuable for a drug with the therapeutic sensitivity of digoxin.

The V_d, represented on the x-axis, is a pharmacokinetic parameter that indicates the extent to which theophylline distributes into body tissues relative to the plasma. A higher Vd generally suggests that more of the drug moves into tissues rather than remaining in the blood. This factor is influenced by various physiological attributes, such as body composition and the affinity of drug for fat or water. In this plot, as the V_d increases, the amount of theophylline in the body also rises, reflecting that larger distribution volumes typically mean a larger reservoir of the drug within the body. This characteristic is critical for drugs like theophylline, as it affects how long and how widely the drug acts within the body.

Plasma concentration, shown on the y-axis, is another crucial variable in the pharmacokinetic profile of a drug. It indicates the amount of theophylline that is present in the blood at any given time. Higher plasma concentrations are typically associated with higher doses or situations where drug Cl is impaired. In clinical practice, maintaining an optimal plasma concentration is vital since theophylline has a narrow therapeutic window; too low a concentration may be ineffective, while too high could lead to toxicity. In the plot, as plasma concentration increases, there is a corresponding rise in the amount of drug in the body. This relationship highlights the importance of careful dosing and monitoring to keep plasma concentrations within a therapeutic range.

The amount of theophylline in the body, represented by the colour gradient and height of the surface, combines the effects of plasma concentration and V_d. The clear linear increase in this variable with respect to both plasma concentration and V_d suggests a direct proportional relationship, consistent with the pharmacokinetics. This linear trend shows that any increase in either plasma concentration or V_d leads to a proportional increase in the total amount of theophylline within the body. Clinically, this relationship aids in predicting how much drug will be present in the system based on administered doses and individual patient factors, such as body composition, which influence the V_d.

From a therapeutic standpoint, this plot is useful for understanding how theophylline behaves within different physiological conditions. For instance, in patients with a larger V_d, often seen in individuals with higher body fat or altered body water compartments, the drug may spread out more widely within tissues, potentially requiring adjusted dosing to achieve therapeutic plasma levels.^[28] Conversely, in patients with decreased V_d, higher plasma concentrations might be reached with standard doses, increasing the risk of toxicity. Therefore, this plot assists clinicians in adjusting dosage to meet individual patient needs and ensure safety. In summary, this surface plot illustrates the interconnectedness of V_d, plasma concentration and the overall amount of theophylline in the body. By examining these relationships visually, healthcare professionals could better anticipate drug behaviour in the body and make more informed decisions about dosing regimens. Understanding these dynamics is essential for drugs like theophylline, where precise control over plasma levels is necessary due to the risk of adverse effects associated with even slight deviations from the therapeutic range.^[29]

Figures 1 and 2 highlight the critical influence of pharmacokinetic parameters, V_d, plasma concentration and LD/MD, on achieving therapeutic success with digoxin and theophylline. Both drugs, due to their narrow therapeutic indices, require precise dose adjustments based on individual factors affecting V_d and plasma levels. For digoxin, calculating the correct MD is essential to rapidly attain effective plasma levels, while dosing of theophylline is fine-tuned to avoid adverse effects due to its broad tissue distribution. These visualisations underscore the importance of patient-specific dosing strategies, particularly in drugs with narrow therapeutic windows.

CONCLUSION

The pharmacokinetic analysis of digoxin and theophylline underscores the critical role of V_d and plasma concentration in dose optimisation for drugs with narrow therapeutic indices. For digoxin, a drug primarily used in heart failure and arrhythmia treatment, higher volumes of distribution necessitate increased LDs to achieve therapeutic plasma levels without risking toxicity. In theophylline, used in respiratory conditions, a similar relationship dictates maintenance dosing to sustain effective plasma levels. The models emphasise that understanding patient-specific pharmacokinetic parameters such as V_d affected by body composition or renal function, is essential for tailoring dosing strategies. This approach enhances therapeutic efficacy while minimising adverse effects, underscoring the importance of individualised medicine in pharmacotherapy for drugs such as digoxin and theophylline.