Pharmacokinetic model

One of the central methodological pillars is the use of compartmental PK models, which provide a mathematically rigorous yet physiologically interpretable framework to describe the dynamics of drugs in the body [2,35]. This methodology provides a structured and systematic framework for modeling the dynamic distribution of substances within an organism, encompassing absorption, distribution, metabolism, and excretion processes (ADME) [35]. At its core, compartmental analysis conceptualizes the organism as a system of discrete compartments, each representing a distinct physiological, biochemical, or anatomical region with relatively uniform properties [36]. To quantitatively describe these processes, ordinary differential equations are used. These equations capture the rate of exchange between compartments, which is typically dictated by concentration gradients and transport kinetics [2,3,37].

For the purposes of optimal treatment design, compartmental PK modeling offers several advantages. It enables a clear distinction between pharmacokinetic parameters that can be modified through drug formulation (design)—such as the rate at which the drug is released—and parameters that reflect intrinsic biological properties of the patient, such as metabolic clearance (elimination) or absorption, which cannot be altered by pharmaceutical design. Compartmental PK models are also mathematically well-structured and interpretable through the system and control sciences.

Consider an -compartment PK model which jointly describes the drug liberation, absorption, transport, and elimination mechanisms through the dynamics of state variables (drug amounts) using linear differential equations and model parameters corresponding to the PK response to the -th dose such that

(1)

where [1/h] is a negative-definite Metzler matrix describing the causal relation between the states (compartments), [mg] is the input gain vector (including the dose size), [1/h] is the input representing the unit rate of the drug administration via the oral route, [mg] is the state vector describing the drug amounts in individual compartments (gastrointestinal tract, blood circulation, peripheral blood) after the administration of the -th dose. The parameter vector of adjustable parameters to be optimized includes the dose size and chosen drug characteristics, which can be technologically influenced/modified. The parameters remain constant between the consecutive dosing injections (discrete events). The other PK parameters affecting and , such as the elimination rates and inter-compartmental transport remain fixed and immutable.

The dose size [mg], as part of the parameter vector , will be incorporated into the input gain vector rather than directly into the input signal , such that the PK response is linear with respect to via . This convention allows us to jointly optimize the dose along with the other drug technological parameters (liberation rate) affecting the dynamics via .

The assumption of linear pharmacokinetics constitutes an important modeling restriction of the present framework. The methodology is therefore primarily applicable to drugs operating within concentration ranges where absorption, distribution, and elimination processes remain approximately first-order and non-saturating.

This excludes drugs exhibiting strong nonlinearities such as saturable metabolism, concentration-dependent clearance, transporter saturation, autoinduction, target-mediated drug disposition, or nonlinear protein binding. Nevertheless, linear compartmental models remain highly relevant in practical pharmacokinetics because many therapeutics operate predominantly within approximately linear exposure ranges under standard dosing conditions.

Considering the repeated dosing with the dosing period [h], we will model the administration rate (input) [1/h] corresponding to the -th dose in terms of Dirac function as

(2)

The Dirac impulse defined in Eq (2) provides a convenient abstraction for oral drug dosing, representing discrete, intermittent administration. In the present study, these doses are assumed to occur at fixed equidistant intervals. This assumption is necessary for mathematical tractability: it ensures that the response to each individual dose can be represented as a vector of known length and starting time, which allows stacking and superposition of the response vectors required for the nonlinear least squares problem. While exact adherence to dosing times may vary, small deviations from prescribed times generally have a limited effect on therapeutic outcomes for drugs with moderate half-lives and gradual pharmacodynamic [2,3]. Allowing fully variable dosing times would transform the problem into a more general hybrid optimal control problem with variable event timing, substantially increasing computational complexity and altering the structure of the sensitivity equations.

Each administration introduces the prescribed dose into the gastrointestinal compartment over a small time interval that is negligible compared to the characteristic pharmacokinetic time scales, i.e., , where are the first-order elimination or transfer rates of the compartments in the PK model. This ensures that each dose can be accurately approximated as an instantaneous (impulsive) input for the purpose of PK–PD modeling and optimization. From a physiological standpoint, this ensures that the drug is effectively “available” in the gastrointestinal compartment before significant redistribution or elimination occurs. Consequently, the framework is primarily intended for intermittent oral or bolus-like administration and is not directly applicable to prolonged infusion therapies or continuously varying administration profiles.

All modifiable parameters of dosing events can be stacked into the joined parameter vector as

(3)

The overall effect of multiple dose administration can be expressed in terms of the superposition principle applied to Eq (1) as

(4)

where is the number of administered doses up to time .

The solution of the system state for no previous treatment, represented by , then gets [38,39]

(5)

The partial contribution of the -th dose gets

(6)

The drug concentration [mg/ml] in the drug receptor (target) compartment is related to the drug amount such as

(7)

where the structure of vector depends on which compartment (state variable) contains the drug receptors and on the corresponding distribution volume [ml] such that

(8)

Pharmacodynamic model and therapeutic effect profile

While pharmacokinetics describes how a drug is absorbed, distributed, metabolized, and eliminated, it does not by itself determine the therapeutic outcome. Clinical efficacy depends on how drug concentrations interact with molecular targets and translate into biological effects, as described by pharmacodynamics. As a result, PK-only models are insufficient: dosing regimens with similar concentration–time profiles can produce markedly different therapeutic responses when the concentration–effect relationship is nonlinear or saturable. Incorporating pharmacodynamics is therefore essential to link drug exposure to clinical effect and ensure that optimized dosing strategies are pharmacologically meaningful [2,3,35].

One of the most widely used mathematical representations is the Hill function, which describes the relative nonlinear relation between the drug concentration and its therapeutic effect in the interval as [2,3]:

(9)

where [mg/ml] is the half-saturation constant (the value of at which ) and is the Hill coefficient. The parameters , represent intrinsic biochemical characteristics of the drug–receptor interaction and therefore cannot be modified through dose design or therapeutic scheduling. The half-saturation constant quantifies the affinity between the drug molecule and its target receptor, reflecting the equilibrium between the binding and unbinding events at the molecular level. This affinity is determined exclusively by the drug’s chemical structure and the receptor’s binding pocket. Similarly, the Hill coefficient captures the cooperativity of binding, which depends on structural features of the receptor complex, the conformational changes induced by ligand binding, and the molecular arrangement of binding sites. These properties are fixed for a given drug-target pair and are not affected by dosage form, administration rate, or formulation-based modifications.

In the context of repeated drug dosing, the Hill function becomes particularly powerful: it transforms the concentration–time profile into an effect trajectory that can be shaped through dose size and modifications to pharmacokinetic parameters (PK parameters).

Consequently, the framework does not presently capture dynamic pharmacodynamic phenomena such as receptor turnover, delayed signal transduction, tolerance development, hysteresis, indirect response mechanisms, or adaptive biological feedback. The static formulation was selected intentionally to isolate the influence of pharmacokinetic shaping on the therapeutic effect trajectory while preserving analytical differentiability of the optimization problem.

The derivative of the drug effect given by Eq (9) (Hill function) with respect to the concentration gets

(10)

According to Eq (7), the gradient of the drug effect with respect to the state vector (drug amounts) can be derived based on the chain rule as

(11)

Furthermore, the gradient of the drug effect with respect to the parameter vector can be derived based on the chain rule as

(12)

where is the state-parameter Jacobian matrix defining the sensitivity of the -th state variable to the variations of the -th parameter such that

(13)

Solution of the sensitivity model

Consider the vector (sensitivity function) , which quantifies the sensitivity of the state response with respect to the changes of the -th parameter in the parameter subvector corresponding to the -th dose in terms of Eq (3) such as [40].

(14)

The sensitivity function can be understood as a quantitative measure of how strongly a change in a particular PK parameter influences the resulting drug amounts in each compartment over time. In practical terms, it answers a simple but clinically meaningful question: “If we slightly modify this parameter how much will the concentration–time profile change?”

The full sensitivity matrix defined by Eq (13) is then formed as

(15)

The solution of the sensitivity function can be obtained by utilizing its time derivative which can be derived as

(16)

where is the gradient of the differential equation of the state response to the -th dose with respect to and is the Jacobian matrix of the state dynamics.

In the case of the linear state-space model given by Eq (1), we can write as

(17)

while the Jacobian is

(18)

Therefore, the differential equation for the sensitivity function of the linear state space model given by Eq (1) gets

(19)

Finally, to obtain the sensitivities for all parameters across doses, we will solve the augmented linear models such that

(20)

for , subject to initial conditions .

According to Eq (12), the derivative of with respect to then gets

(21)

Optimization of the therapeutic response

In clinical practice, every therapeutic intervention implicitly targets a desired profile of the drug effect—a trajectory that reflects how strong the therapeutic action should be. For antibiotics, physicians often aim to maintain the drug effect above the minimum inhibitory concentration for a sufficient duration to suppress bacterial growth without promoting resistance [41,42]. Similarly, analgesic regimens are designed to maintain pain relief above a minimum acceptable threshold while preventing high peaks that could induce sedation or respiratory depression [43,44].

The central aim of this work is to systematically formalize and optimize drug treatments so that patients receive exactly the therapeutic effect they need. Achieving such precision requires finding not only an appropriate sequence of doses but also choosing the pharmacokinetic properties of the drug formulation for each dose—primarily the liberation rate—so that the drug’s concentration–effect dynamics produces the desired response.

Consider given time points of interest such that , and the corresponding desired therapeutic effect profile arranged in a vector as

(22)

Vector encapsulates the therapeutic goals that clinicians strive to achieve.

The therapeutic (optimization horizon) is chosen to cover the patient response to dosing events (dosing periods), for example 48 hours.

The model-based prediction of the actual therapeutic profile can be grouped into a vector as

(23)

where each subvector corresponds to the therapeutic profile during the -th dosing period.

To find the optimal parameters of all doses , we want to fit the resulting pharmacodynamic effect (therapeutic profile) to the desired profile in terms of minimizing the sum of squared deviations . Additionally, to ensure minimal dose size and minimal technological modifications of the drug, the Tikhonov regularization term will be included.

To this end, consider the cost function such that

(24)

where , , is the regularization matrix and are the nominal parameters of treatment for the given drug.

In simple terms, the goal of optimization is to choose the drug dose and formulation parameters so that the actual therapeutic effect over time follows as closely as possible the effect profile we want. By the cost function , we measure how far the achieved effect is from the desired one. At the same time, we also include the regularization term – a safeguard that discourages unnecessarily large doses or excessive deviations from the drug’s standard formulation.

From an optimization standpoint, it is important to emphasize that the drug effect profile follows the nonlinear Hill function given by Eq (9), which is a compound function of the drug concentration and the state as a nonlinear function of the parameters as given by Eq (5). Therefore, it can be inferred that is nonlinear with respect to , hence the optimization problem (24) must be treated as the nonlinear least squares problem [45].

The gradient of the cost function given by Eq (24) then gets

(25)

where is the Jacobian matrix of therapeutic profile .

The condition for unconstrained optimality to be satisfied is

(26)

It is obvious that the solution to the optimality condition given by Eq (26) cannot be found analytically. Instead, we opt for iterative numeric optimization. For the approximate calculation of Hessian matrix , we will locally linearize around the current guess in terms of Taylor series such that

(27)

The approximate Hessian matrix of the cost function given by Eq (24) therefore gets

(28)

The optimization problem can now be solved in terms of the Gauss-Newton iterative algorithm. The update of the optimized variables can be obtained as the solution of the linear equation system [46]:

(29)

The Gauss–Newton method is particularly suitable in this setting because it exploits the structure of least-squares problems by approximating the Hessian using only first-order derivative information (Jacobian).

To evaluate the Gauss-Newton iteration given by Eq (29), the Jacobian matrix of is required, which has the following block matrix structure

(30)

According to Eq (12) and Eq (14), the submatrices of Jacobian corresponding to the individual doses can be conveniently written using the derivatives of the Hill function given by Eq (10) and the sensitivity functions introduced earlier such that:

(31)

It is important to emphasize that the proposed sensitivity analysis is local in nature. Consequently, the Jacobian matrices characterize only local parameter influence around the current operating point and parameter estimate.

For strongly nonlinear PK–PD systems or large parameter deviations, local sensitivities may not fully capture global parameter interactions or multimodal optimization structure. Nevertheless, for the considered class of compartmental PK–PD models operating within clinically relevant concentration ranges, local sensitivity information provides an efficient and computationally tractable approximation enabling structured Gauss–Newton optimization.

The inclusion of the Tikhonov regularization term in Eq (24) is important not only from a therapeutic standpoint, but also from a numerical inverse-problem perspective. Besides penalizing excessively aggressive dosing strategies, the regularization improves conditioning of the approximate Hessian matrix and reduces sensitivity of the optimization to parameter correlations and local Jacobian degeneracies.

Compartmental model

For validation of the proposed methodology, we will consider a particular four-compartment pharmacokinetic model describing the drug liberation, absorption, elimination, and the transport mechanisms via four state variables in terms of general linear state-space model given by Eq (1) such that

(32)

where , , , , and the state variable [mg] represents the amount of the unliberated drug in the stomach, [mg] is the liberated drug amount in the gastrointestinal tract, [mg] represents the amount of absorbed drug in the central compartment (blood circulation), and [mg] is the drug amount in the peripheral compartment (site of drug action where receptors are present), [mg] is the dose size, while [1/h] is given by Eq (2). This model has been inspired by our previous works [4,5,47], and modified by adding the drug liberation dynamics.

The parameters , , , , and [1/h] are real positive constants. The parameter represents the liberation rate, and the parameter quantifies the rate of drug absorption from the gastrointestinal tract. The parameters , affect the rates of the drug elimination from the compartments and , and determines the rate of the inter-compartment drug transport (flow) from to .

The first two compartments describe the liberation of the drug from its formulation and its presence in the gastrointestinal tract, where absorption occurs. The liberation process is strictly unidirectional: once liberated, the drug cannot return to its initial bound state, reflecting a physically irreversible transition.

The liberated drug enters the second compartment, which represents the accessible drug mass in the gastrointestinal tract. From this compartment, two distinct physiological pathways emerge. First, the absorption flow governed by transports drug molecules across the intestinal membrane into the systemic circulation. The second term captures small but physiologically plausible losses (e.g., degradation or insufficient absorption).

The third compartment represents the central systemic circulation, and it plays a pivotal causal role in determining therapeutic exposure. All drug that reaches this compartment is considered bioavailable, and its magnitude directly influences both downstream receptor engagement and systemic elimination. The inflow into the third compartment is entirely determined by the absorption rate, creating a clear and physiologically grounded causal dependency: the faster the absorption from the gastrointestinal tract, the more rapid the rise in systemic concentration. The drug molecules in the central compartment then undergo two competing processes. The first is inter-compartmental transport toward the receptor compartment. This flow reflects the distribution into peripheral tissues or specific receptor sites, depending on the drug’s mechanism of action. The second competing process is systemic elimination. Although not explicitly included in the transport term, central elimination is effectively modeled within the flow structure to reflect physiological clearance mechanisms such as hepatic metabolism or renal excretion.

The fourth compartment corresponds to the receptor-site compartment. It receives drug exclusively through the inter-compartmental flow. This strict causal direction embodies the pharmacological principle that receptor-site exposure depends entirely on systemic availability. Once in the receptor compartment, the drug can either bind to receptors or undergo elimination, with the latter modeled as a first-order process.

The optimized PK parameters include the dose size and its liberation rate , hence and the parameter vector gets

(33)

This choice is strongly motivated by established clinical practice and contemporary pharmaceutical technology. In real-world oral drug administration, each dose can be individually tailored with respect to both amount and formulation. Dose magnitude is the principal control variable clinicians use to regulate systemic drug exposure, and individualized dose titration is a well-established cornerstone of clinical pharmacotherapy [2,3].

Equally important, the liberation rate is increasingly recognized as a tunable parameter due to advancements in formulation science. Modern drug delivery technologies—including controlled-release matrices, enteric coatings, multilayer tablets, microencapsulation, and polymer-based delivery systems [48,49] —enable precise manipulation of dissolution and release profiles for each administered dose [50]. As a result, distinct doses of the same drug can be engineered to exhibit specific release kinetics tailored to therapeutic goals such as delay of onset, extension of exposure, or reduction of peak-related adverse effects.

From a modeling standpoint, optimizing these parameters directly influences the entire causal flow of the PK chain: the dose size scales the amplitude of the system response, whereas the liberation rate governs the temporal profile of drug availability for absorption, effectively acting as the inverse of a time constant.

In contrast to the modifiable parameters, the remaining pharmacokinetic parameters—namely the absorption rate, the elimination rates, and the inter-compartment transfer rate, cannot be considered optimization variables within a therapeutic design framework. These parameters represent intrinsic physiological or biochemical processes that are fundamentally determined by the patient’s biology and the molecular properties of the drug. For example, the absorption rate depends on factors such as intestinal membrane permeability, gastrointestinal motility, transporter expression, and local blood flow—all of which are governed by the patient’s physiology and cannot be manipulated on a per-dose basis. Similarly, elimination constants reflect hepatic metabolism, renal clearance, and systemic enzymatic degradation. The inter-compartmental transfer rate is likewise determined by the physicochemical properties of the molecule—lipophilicity, binding affinity, tissue permeability, and receptor-site kinetics.

Experiment setup

The values of the fixed PK parameters are adopted from [4,37,47] as 1/h, 1/h, 1/h, 1/h.

The distribution volume of the peripheral compartment in Eq (8) was assumed as ml, hence vector gets

(34)

Considering the model given by Eq (32) and the parameter vector given by Eq (33), derivatives , in Eq (17) are as follows

(35)

The half-saturation constant and the Hill coefficient in Eq (9) were set to mg/ml, .

The half-saturation constant is a value consistent with mid-micromolar potencies commonly observed for small-molecule agents. The Hill coefficient was set to , representing a moderately steep and cooperative concentration–effect relationship that is consistent with many therapeutic agents exhibiting partial receptor cooperativity or nonlinear amplification in their downstream pharmacodynamic signaling. The corresponding Hill function is then visualized in Fig 1.

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Fig 1. Hill function E (c).

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For a better conceptualization of the PK characteristics of the considered model, the state response to a single dose mg and liberation rate 1/h, is plotted in Fig 2. This state response documents that, after the dose administration, the amount of unliberated drug instantly jumps to mg, while the amount of liberated drug starts to rise continuously. The amount of absorbed drug in the bloodstream slightly lags behind , while the drug amount in the peripheral compartment lags about 5 hours behind the central compartment and peaks at 10 hours following the dose administration.

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Fig 2. State response to a single suboptimal dose.

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For the optimal dosing design, the desired therapeutic profile defined in Eq (22) was specified using a smooth, logistic-like function visualized in Fig 3, yielding a therapeutic effect that rises gradually to a clinically meaningful relative steady level (the mid-therapeutic range) with the onset time approximately 6 hours after the leading dose.

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Fig 3. Target therapeutic profile E_r(t).

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The logistic reference trajectory reflects qualitative features commonly desired in pharmacotherapy, namely a limited initial response during drug accumulation, a phase of increasing therapeutic effect as drug exposure approaches effective levels, and a subsequent stabilization of the effect within a desired therapeutic range. Such behavior corresponds to the general objective of dosing regimen design, which aims to achieve therapeutic effect within a reasonable time and maintain it at a stable level while avoiding excessive fluctuations [2,3].

Although such perfect behavior is idealized and cannot be achieved with real-world impulsive dosing, the logistic curve provides a physiologically plausible and clinically interpretable target profile against which optimal dosing and formulation parameters can be designed.

The dosing period is chosen as hours, the number of time points equals , and the number of doses is , which corresponds to the total optimization horizon hours.

The regularization matrix in Eq (24) was chosen as the block diagonal across all doses

(36)

This choice reflects the typical magnitudes of the optimized variables, i.e., the dose size and the liberation rate respectively, such that the regularization of both parameters is balanced.

The nominal parameters for the regularization were chosen as

(37)

The choice of the nominal dose size as zero is motivated by the need for minimizing the dose size (regularization toward the zero dose), whereas 1/h represents the liberation rate of the standard drug form.

Experiment results

For better conceptualization, the drug therapy will be evaluated first considering a standard fixed dose size mg, and the nominal liberation rate 1/h to demonstrate a suboptimal therapeutic response. The corresponding Fig 4 then shows a slow onset of the therapeutic effect, taking approximately 30 hours to achieve the desired value, and the presence of mild fluctuations between individual doses (). For comparison, by increasing the liberation rate to 1/h, the therapeutic response was faster, onset of the therapeutic effect to the desired level took approximately 20 hours, but it was achieved at the expense of more pronounced fluctuations () during the repeated dosing as documented in Fig 5. Both Figs demonstrate the obvious disadvantages of drug dosing with constant dose size and drug form, which constitute the inability to achieve fast onset of the therapy and to simultaneously suppress fluctuations of the therapeutic effect between the doses.

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Fig 4. Suboptimal therapeutic profile with fixed dose size D = 3.75 mg and liberation rate k_l = 0.3.

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Fig 5. Suboptimal therapeutic profile with fixed dose size D = 3.75 mg and fixed liberation rate k_l = 2.5.

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The next scenario considers optimizing the dose sizes by the means of the proposed methodology, but with the liberation rate fixed as , hence and the parameter vector gets . The purpose of this scenario is to demonstrate the advantages of variable-size optimal dosing over the conventional fixed-size dosing without the need to modify the drug form. The sequence of optimized doses was obtained as

(38)

It can be inferred that the first (leading) dose is significantly larger than the maintenance doses to ensure fast onset of the therapy, while the second dose turned out to be significantly smaller to prevent the subsequent overshoot and related toxicity. The corresponding therapeutic response can be seen in Fig 6, which demonstrates that optimal therapy with variable dose sizes achieves a fast therapeutic onset (within 7 hours) and provides satisfactory tracking of the desired effect level in the pseudo-steady state. However, the overshoot following the leading dose is too pronounced, which indicates a significant risk of toxicity in the blood circulation and is therefore considered undesired. Moreover, the magnitudes of the fluctuations during the remaining therapeutic windows are still too large, requiring a smoother response to achieve more consistent therapy.

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Fig 6. Suboptimal therapeutic profile with optimal variable dose size mg and fixed liberation rate .

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Unlike conventional dosing optimization approaches that treat formulation characteristics as fixed, the proposed framework explicitly includes liberation rate among the optimization variables. Consequently, the optimization simultaneously determines both how much drug should be administered and how rapidly it should become available for absorption.

Finally, by optimizing the full dosing regime, which includes the dose size D and the liberation rate k_l for each dosing event in terms of Eq (33), the following parameters were determined:

(39)

The corresponding optimal therapeutic profile is then plotted in Fig 7, while the full state response is plotted in Fig 8. The therapeutic profile shows a rapid onset of the effect towards the desired level within 6 hours, practically without an overshoot and the associated risk of toxicity, while the fluctuations during the subsequent therapeutic windows are more suppressed compared to Fig 6. This significant overall improvement is caused by the combined optimization of the dose size and the liberation rate for each dosing event, in such a way that, according to Eq (39), the leading (first) dose is much larger and has a faster liberation than the remaining doses. On the other hand, the maintenance doses are smaller and have slower liberation from the drug form. This dosing strategy leads to a steadier therapeutic response with minimal fluctuation closely resembling the pinnacle performance of continuously adjustable dosing or advanced controlled release drug forms. The state response in Fig 8 then confirms that the liberation following the first dose is faster than the liberation after the remaining doses (faster exponential decay of ), the state response shows that the drug amount in blood circulation still has an overshoot after the first dose, yet this temporal burst can be considered within safe (non-toxic) interval and is part of the fast-acting therapy. It is also important to emphasize that there is no accumulation of the unliberated drug in the stomach, which might cause digestive issues if the liberation rates are too low.

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Fig 7. Optimal therapeutic profile with variable dose size D mg and dose-dependent liberation rate k_l.

https://doi.org/10.1371/journal.pone.0354029.g007

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Fig 8. Optimal state response with variable dose size D mg and dose-dependent liberation rate k_l.

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The comparison between Figs 4–7 highlights the impact of progressively relaxing constraints on the dosing strategy. In the fixed-dose case (Figs 4–5), where identical doses are administered at each dosing event, the system exhibits a clear limitation: achieving faster therapeutic onset requires increasing the effective input intensity, which leads to larger fluctuations and overshoot in the pharmacodynamic response. As a result, the time required to reach the desired therapeutic effect remains on the order of 20–30 hours, depending on the chosen liberation rate, and is accompanied by significant steady-state oscillations.

Allowing variability in dose magnitudes (Fig 6) substantially improves transient performance. In this case, the desired therapeutic effect is reached within approximately 7 hours, representing a reduction in onset time by more than a factor of three compared to fixed-dose regimens. This improvement arises because non-uniform dosing enables an initial loading phase followed by smaller maintenance doses, thereby partially decoupling transient and steady-state behavior. However, fluctuations and overshoot remain relatively pronounced, indicating that dose optimization alone does not fully resolve the underlying dynamic trade-offs.

The fully optimized strategy (Fig 7), in which both dose magnitudes and liberation rates are optimized, provides the best overall performance. The target therapeutic effect is reached within approximately 6 hours, while fluctuations are reduced from approximately ±0.05 in the fixed-dose case to approximately ±0.012. This corresponds to both faster convergence and significantly improved tracking of the desired pharmacodynamic profile.

These results demonstrate that fixed-dose regimens represent a strongly constrained subset of admissible dosing strategies, which inherently limits achievable performance. Relaxing this constraint by allowing variable dose magnitudes introduces additional degrees of freedom that can be exploited to improve transient response, while further optimization of liberation rate enables simultaneous shaping of both transient and steady-state dynamics. Consequently, the proposed framework provides a systematic means of quantifying and exploiting these additional degrees of freedom, revealing performance improvements that are not attainable under conventional fixed-dose assumptions.

Because the resulting optimization problem is generally nonconvex, the Gauss–Newton algorithm provides only local convergence guarantees. To investigate the influence of initialization, the optimization was repeated from 100 randomly perturbed initial parameter vectors. Each initialization was generated by independently perturbing the nominal initial guess for every optimized dose magnitude and liberation rate by a uniformly distributed random factor within ±50% of its nominal value, while keeping the fixed pharmacokinetic parameters unchanged.

Fig 9 presents a histogram of the final objective function values obtained from 100 optimization runs initialized from randomly generated parameter vectors. The distribution indicates that despite the substantial variability of the starting points, all optimization runs converged to very similar objective function values and yielded highly similar optimal dosing regimens. This behavior indicates that, within the tested initialization region, the optimization landscape does not exhibit practically significant competing local minima and that the obtained solution is only weakly dependent on initialization. The small differences observed among the final objective function values are likely attributable to the stopping criterion of the Gauss–Newton algorithm, which is based on the norm of the gradient rather than on a prescribed residual tolerance. Consequently, individual runs may terminate at slightly different points within the same local neighborhood of the optimum, leading to minor variations in the final objective values. Although a formal guarantee of global optimality cannot be provided, these results suggest that the proposed sensitivity-based Gauss–Newton procedure exhibits robust convergence toward the same locally optimal solution for clinically relevant initial parameter guess.

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Fig 9. Histogram of the final objective function values obtained from multiple optimization runs initialized from randomly generated parameter vectors.

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To assess the robustness of the proposed optimal dosing framework beyond a single nominal parameter configuration, additional simulations were conducted under a set of different pharmacokinetic settings in terms of Monte Carlo analysis. Specifically, the absorption and elimination rate constants, as well as inter-compartmental transfer rates, were systematically varied to simulate a patient population with varying PK characteristics. The key non-optimizable patient-specific PK parameters, the rate constants , , , and , were independently perturbed using a uniform random distribution within ±30 of their nominal values.

In the first scenario, for each of realizations, the nonlinear least squares problem of the therapy design was re-solved using the Gauss–Newton optimization framework, resulting in a corresponding personalized optimal dosing policy .

The resulting distribution of optimized treatment parameters depicted in Fig 10. indicates that the proposed optimization algorithm remains stable under parameter perturbations, with bounded variability of parameters across all dosing intervals. The most significant variability was observed in the magnitude of the leading dose. As demonstrated in Fig 11., despite variability in the PK dynamics, the resulting pharmacodynamic response remains closely aligned with the desired reference trajectory . In particular, the ensemble mean response (Mean) closely follows the nominal solution from Fig 7., while the standard deviation STD remains relatively small throughout the dosing horizon, indicating moderate sensitivity to parameter variations. These results demonstrate that the proposed impulsive PK–PD optimization framework provides consistent therapeutic performance despite variations in pharmacokinetic parameters.

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Fig 10. Distribution of optimized treatment parameters obtained under Monte Carlo analysis of patient population with varying PK characteristics.

https://doi.org/10.1371/journal.pone.0354029.g010

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Fig 11. Ensemble mean pharmacodynamic response and its standard deviation obtained under Monte Carlo analysis of patient population with varying PK characteristics and reoptimization.

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In the second scenario, the nominal-patient optimal therapy defined by Eq. (39) was applied to each of the Monte Carlo realizations in order to evaluate the robustness of the dosing recommendation under pharmacokinetic parameter uncertainty.

As shown in Fig 12, the resulting pharmacodynamic response remains satisfactory despite the uncertainty in the PK dynamics. However, compared with the personalized optimization results presented in Fig 11, the variability of the response, quantified by the standard deviation STD, is increased. The standard deviation reaches approximately in terms of the relative therapeutic effect , corresponding to approximately variability with respect to the target therapeutic effect . Despite this increased variability, the pharmacodynamic response remains within an acceptable range, indicating a reasonable degree of robustness of the optimal dosing regimen with respect to pharmacokinetic parameter uncertainty. In particular, the observed deviations do not appear to induce clinically relevant underdosing or toxicity within the considered simulation setting.

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Fig 12. Ensemble mean pharmacodynamic response and its standard deviation obtained under Monte Carlo analysis of patient population with varying PK characteristics and nominal therapy.

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As an additional illustrative example, a scenario with impaired drug absorption was considered. This condition was modeled by reducing the absorption rate constant to ( 1/h), representing substantially slower transfer of the drug into the systemic circulation. From a pharmacological perspective, impaired absorption is expected to require larger administered doses to achieve the same therapeutic effect. Furthermore, faster drug liberation may be advantageous to compensate, at least partially, for the delayed absorption and to maintain the onset of action prescribed by the reference pharmacodynamic trajectory ().

By optimizing the full dosing regime, which includes the dose size and the liberation rate for each dosing event in terms of Eq (33), the following parameters were determined.

(40)

Comparison with the optimal solution presented in Eq (39) shows that the optimized doses are approximately three times larger, while the corresponding liberation rates are approximately 20% higher. These changes are consistent with the reduced absorption capacity of the patient and illustrate how the proposed optimization framework automatically adapts both the administered dose and the formulation-dependent liberation kinetics to maintain the desired therapeutic response.

The resulting optimal pharmacodynamic profile is shown in Fig 13. Despite the substantially impaired absorption, the optimized treatment strategy achieves a therapeutic response that remains closely aligned with the reference trajectory (), exhibiting behavior qualitatively similar to that observed in Fig 7.

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Fig 13. Optimal therapeutic profile with variable dose size D mg and dose-dependent liberation rate under impaired absorption ( 1/h) Model dependence, identifiability, and robustness considerations.

https://doi.org/10.1371/journal.pone.0354029.g013

The proposed framework is formulated as an open-loop nominal optimization problem and therefore its performance depends on the predictive accuracy of the underlying PK–PD model. Consequently, uncertainties in pharmacokinetic or pharmacodynamic parameters, structural model mismatch, inter-patient variability, adherence deviations, or unmodeled physiological disturbances may affect the resulting therapeutic response.

In particular, inaccuracies in elimination rates, absorption dynamics, receptor sensitivity, or concentration–effect relationships may alter both the transient onset and steady-state behavior of the optimized therapy. Since the present methodology performs optimization offline without feedback correction, such discrepancies may lead to deviations between the predicted and actual pharmacodynamic trajectories.

These limitations are intrinsic to deterministic open-loop optimization approaches relying on predicted system dynamics. Therefore, the present framework should primarily be interpreted as a nominal model-informed optimization methodology rather than a guaranteed robust control strategy.

Nevertheless, the present deterministic formulation constitutes an essential prerequisite for future robustness-oriented extensions. In control-theoretic and model-informed precision dosing frameworks, robust or adaptive optimization fundamentally relies on the existence of an underlying nominal model and corresponding parameter sensitivities.

Importantly, the analytical sensitivity equations derived in this work provide explicit quantitative information regarding how perturbations in PK–PD parameters influence the therapeutic trajectory. This creates a natural foundation for future incorporation of uncertainty-aware methodologies

In particular, repeated therapeutic measurements could be used to periodically update patient-specific PK–PD parameters and re-optimize the remaining dosing strategy online, effectively transforming the present open-loop formulation into an adaptive feedback optimization framework.

From a mathematical standpoint, the present methodology constitutes a structured nonlinear inverse problem. As with all inverse problems, practical limitations may arise due to parameter non-identifiability, parameter correlation, insufficient excitation of the system dynamics, measurement noise, or structural model mismatch.

In particular, if multiple parameter combinations produce similar therapeutic effect trajectories, the Jacobian matrix of the optimization problem may become ill-conditioned or locally rank-deficient. Such situations reduce numerical robustness of the Gauss–Newton iterations and increase sensitivity to perturbations in model parameters or measured data.

Several structural aspects of the proposed framework partially mitigate these difficulties. First, the optimization variables were intentionally restricted to clinically and technologically modifiable parameters, namely dose magnitude and liberation rate. This substantially reduces the dimensionality of the optimization problem and limits parameter correlation. Second, the repeated impulsive excitation generated by multiple dosing events improves distinguishability of transient and steady-state pharmacodynamic behavior. In particular, the liberation rate primarily influences the temporal shape and phase characteristics of the concentration trajectory, whereas the dose magnitude predominantly scales the amplitude of the response.

Practical implementation and constrained pharmaceutical optimization

In practical pharmaceutical applications, several additional constraints must typically be considered. These include:

1. • discrete tablet or capsule strengths,
2. • limited sets of available formulations,
3. • manufacturable release kinetics,
4. • excipient and technological limitations,
5. • and clinically acceptable administration schedules.

Consequently, the present methodology should be interpreted as a continuous relaxation of a more general constrained pharmaceutical co-design problem. Importantly, the proposed framework naturally enables systematic incorporation of such practical constraints.

Discrete dose strengths may be introduced through mixed-integer optimization formulations in which admissible dose values are selected from predefined therapeutic sets. Similarly, finite formulation choices may be represented through discrete candidate release profiles corresponding to available pharmaceutical technologies.

Manufacturability limitations may additionally be incorporated as inequality constraints imposed on liberation dynamics, release-rate bounds, smoothness of dissolution profiles, or achievable kinetic parameters associated with specific drug delivery technologies.

Under such conditions, the resulting optimization problem becomes a hybrid continuous-discrete optimization problem, potentially requiring mixed-integer nonlinear programming (MINLP), branch-and-bound techniques, evolutionary optimization, or hierarchical optimization methodologies.

Nevertheless, the analytical sensitivity framework derived in the present work remains highly valuable even in such constrained settings because it provides local gradient information for the continuous components of the optimization problem and substantially reduces the dimensionality of the clinically relevant decision space.