Relevance of Half-Life in Drug Design - Journal of Medicinal

Perspective Cite This: J. Med. Chem. XXXX, XXX, XXX-XXX

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Relevance of Half-Life in Drug Design Miniperspective Dennis A. Smith,† Kevin Beaumont,‡ Tristan S. Maurer,‡ and Li Di*,§ ‡

Medicine Design, Pfizer Inc., Cambridge, Massachusetts 02139, United States Medicine Design, Pfizer Inc., Groton, Connecticut 06340, United States

§

ABSTRACT: Drug half-life has important implications for dosing regimen and peakto-trough ratio at the steady state. A half-life of 12−48 h is generally ideal for once daily dosing of oral drugs. If the half-life is too short, it may require more frequent dosing in order to maintain desired exposures and avoid unnecessarily high peak concentrations. This may pose challenges to achieving optimal efficacy, safety, and patient compliance. If the half-life is too long, the time over which accumulation and subsequent elimination occur may be prolonged. This may pose problems with managing adverse effects and the design of efficient clinical trials. Half-life is a key parameter for optimization in research and development. Structural modification to affect clearance, and to a lesser extent volume of distribution, is the preferred means of modulating halflife. An effective approach to half-life optimization requires an understanding of the many pitfalls associated with its estimation and interpretation.

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INTRODUCTION

BASIC CONCEPTS, RELEVANCE TO DRUG DESIGN AND OPTIMAL SPACE The classical one compartment pharmacokinetic (PK) model provides a simple and generally applicable framework within which to consider the relevance of t1/2 to drug design (Figure 1a). In this model, a dose can be administered directly via iv bolus (Figure 1b and Figure 1c) or absorbed via a first order rate (ka) following oral dosing (Figure 1d and Figure 1e). Drug is eliminated via another first order rate constant (kel) which determines the t1/2 of the drug under conditions in which the rate of absorption is more rapid than the rate of elimination. Under this condition, kel can be estimated from the slope obtained after plotting the log10 of concentration versus time as indicated in Figures 1c and Figure 1e. The kel parameter determines the t1/2 of a drug according to eq 1.

Half-life (t1/2) is a useful parameter to put a rate of decline into an intuitive numerical context. The term was initially coined by Nobel Laureate Ernest Rutherford in the study of radioactive decay, and this area of application remains the most common in the popular press. From a drug discovery perspective, t1/2 is defined as the time required for the concentration of a drug (typically in blood or plasma) to reduce to half of its initial value when the concentrations of the drug are in simple exponential (log−linear) decline. Despite its relatively simple visualization and mathematical derivation, the role of t1/2 in drug discovery and development is surprisingly complex. In our recent review on volume of distribution,1 the importance of drug distribution in defining t1/2 was exemplified, as this is likely to be the major determinant of dosing frequency for many drugs. For ease of administration, orally delivered small molecule drugs are often required to be dosed on a once daily basis. Since the duration of pharmacodynamic (PD) effect will be determined, in part, by the duration of exposure, t1/2 is often a critical parameter in drug design. When considering a design effort to optimize t1/2, there are several key questions that need to be understood, namely:

t1/2 = ln(2)/kel

t1/2 is an important determinant in almost every aspect of a PK profile. Given the fixed relationship between t1/2 and kel, its relevance can be seen throughout the equations of Table 1 (keeping in mind that each instance of kel is exactly equivalent to ln(2)/t1/2). For example, dose requirements to achieve a steady-state maximum or minimum target exposure will be related to the t1/2 through kel. In addition, as a rule of thumb, the time to steady-state exposure will require 3−5 t1/2 of multiple dosing. t1/2 is also related to the extent of steady-state accumulation through its relationship to kel. Other determinants include the absorption rate and dosing frequency as depicted in the somewhat complex equation for R in Table 1.

• What is the optimal t1/2 to support once daily administration? • How can t1/2 be modulated? • What are the important considerations in measuring t1/2 during compound optimization? This manuscript will provide a review and perspective of t1/2 in the context of these generally relevant drug design questions. © XXXX American Chemical Society

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Received: June 30, 2017 Published: November 7, 2017 A

DOI: 10.1021/acs.jmedchem.7b00969 J. Med. Chem. XXXX, XXX, XXX−XXX

Journal of Medicinal Chemistry

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Figure 1. Basic characteristics of a one compartment PK model. Panel A illustrates the model structure shown with first order absorption and elimination rates. Panels b and c illustrate the characteristic exponential decline following iv bolus administrations. Panels d and e illustrate the absorptive phase followed by monoexponential decline that is typical following oral administration.

Table 1. Key Equations That Characterize the One-Compartment PK Model2,a parameter

equation

C(t)ss, steady-state concentration at time t

C(t )ss =

FpoDpo k a ⎛ e−kelt ⎜ Vss k a − kel ⎝ 1 − e−kel Tau

Cmax,ss, steady-state maximum concentration

Cmax,ss =

FpoDpo k a ⎛ e−kelt max,ss ⎜ Vss k a − kel ⎝ 1 − e−kel Tau

−

e−ka t max,ss ⎞ ⎟ 1 − e−ka Tau ⎠

Cmin,ss, steady-state minimum concentration

Cmin,ss =

FpoDpo k a Vss k a − kel

−

1 1 − e−ka Tau

Cavg,ss, steady-state average concentration

Cavg,ss =

FpoDpo CL tau

tmax,ss, steady-state postdosing time at which maximum concentration occurs

tmax,ss =

1 k a − kel

R, steady state accumulation at postdosing time t

R=

(

1 1 − e−kel Tau

−

e−kat

⎛ e−kelt ⎜ − ⎝ 1 − e−kel Tau

P /Tss =

Twash, the time it takes to “wash out” a specified fraction ( fel) of the drug concentration after sessation of dosing

⎟

)

⎛ ka(1 − e−kel Tau) ⎞ ⎟ ln⎜⎜ −k Tau ⎟ ⎝ kel(1 − e a ) ⎠ e−kat

⎞ ⎟

1 − e−ka Tau ⎠

(e−kelt − e−kat )

⎛ −kelt max,ss

P/Tss, steady-state peak-to-trough concentration ratio

⎞

1 − e−ka Tau ⎠

−k at max,ss ⎞

(ekaTau − 1)(ekelTau − 1)⎜⎜⎝ 1e− e−kelTau − 1e− e−kaTau ⎟⎟⎠ eka Tau − ekel Tau

(

ln 1 − fel

Twash = − ln(2) / t

)

1/2

a

All relationships correspond to a one compartment pharmacokinetic model. Fpo, Dpo, Vss, CL, and Tau represent oral bioavailability, oral dose, steady-state volume of distribution, clearance rate, and dosing interval, respectively. kel and ka represent the first order rates of elimination and absorption.

important aspect to consider in this regard is the relationship of t1/2 to steady-state peak-to-trough exposure ratio (P/Tss). t1/2 values that are short relative to a desired dosing interval (Tau) will potentially incur safety risks due to large swings in exposure around the target steady-state average exposure. As illustrated in Figure 2B, while the Cavg,ss remains constant, P/Tss increases

An important concept here is that accumulation will increase as t1/2 becomes a larger fraction of the target dose interval (Figure 2A). In contrast, t1/2 is not a determinant of steady-state average concentrations (Cavg,ss; see equations in Table 12). However, on a practical level, t1/2 remains an important consideration even when targeting a Cavg,ss. Perhaps the most B



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Figure 3. Semigeneralized relationship between average pharmacodynamics effect (expressed as a fraction of maximal) and half-life-to-dose interval ratio (t1/2/Tau). Relationship was simulated using the C(t)ss equation of Table 1 assuming a ka of 1.8/h. Simulations depict four dosing levels designed to deliver Cavg,ss values equivalent to arbitrarily selected EC20, EC50, EC80, and EC90 values. Actual average effect was calculated as the dose interval normalized AUC under an effect curve defined by (100 × C(t)ss)/(C(t)ss + EC50). Volume was varied in order to affect t1/2/Tau while maintaining a constant Cavg,ss.

other properties (e.g., intrinsic potency and safety) preclude the establishment of absolute values, the relationships described above support a common design rule of thumb that t1/2 should be no less than 25% of the intended dose interval in order to maintain the desired PD effect and minimize P/Tss. While increasing t1/2 is a common design goal, it is important to note that exceedingly long t1/2 can also have untoward implications. For example, the number of doses required to reach a specified fraction of target steady-state levels increases as t1/2 increases. As such, t1/2 values that are exceedingly long can pose issues to rapidly achieving target concentrations in clinical trials and in the therapeutic setting. Likewise, the time it takes to wash drug out (Twash) by a specified fraction ( fel) also increases as t1/2 increases (Table 1). For example, the time it takes to wash out (Twash) 50%, 75%, and 90% (fel = 0.5, 0.75, and 0.9) of a drug with a 48 h t1/2 is 48, 96, and 160 h, respectively. Here, t1/2 values that are exceedingly long can pose problems in managing adverse effects as they arise. For these reasons, it is generally desirable to target half-lives that are no more than 48 h for once daily dosing. It is also difficult to run crossover studies when the drug t1/2 is long. In some cases, however, longer terminal t1/2 may result from slow but extensive distribution into and out of particular tissues. The extreme example of prolonged t1/2 of this nature is seen with the oral bisphosphonates. These drugs avidly bind to their therapeutic target which is bone. After oral absorption, the apparent clearance from plasma is rapid by binding to bone quasi-reversibly and excretion by the kidney (20−80%). Alendronate, for example, has an initial volume of distribution excluding bone of 0.4 L/kg and a plasma clearance of 10 mL min−1 kg−1 indicative of active secretion in the kidney.3 After bone uptake, the return of the drug to the circulation is controlled by the resorption of the bone. This renders the terminal t1/2 of bisphosphonates very long (5−10 years). Drug is not detectable in plasma during this period, but the long t1/2 can be determined by monitoring the excretion of the drug in urine. In PK terms, the system has moderate clearance but the affinity for bone leads to a remarkably high volume of distribution (approximately 70 000 L/kg) and leading to long t1/2. Another similar example was observed during the development of the cholesteryl ester transfer protein

Figure 2. Semigeneralized relationships of the extent of steady-state accumulation (A) and peak-to-trough concentration ratio (B) (plotted as log P/Tss values) to half-life-to-dose interval ratio (t1/2/Tau). Relationships were simulated using the corresponding R(t) and P/Tss equations in Table 1 and assuming an absorption rate of 1.8/h. (A) Accumulation (as calculated as the ratio of AUC at steady state to AUC following a single dose) increases as the t1/2 becomes a smaller fraction of the target dose interval. (B) While the steady-state average concentration is constant, P/Tss increases exponentially as t1/2 becomes a smaller fraction of the target dose interval.

exponentially as t1/2 becomes a smaller fraction of the Tau. This figure is semigeneralized for a one compartment model in that it is universal under the assumption that ka is 1.8/h (providing for Tmax values between 1 and 2 h). As such, a molecule dosed once daily with a t1/2 value of 4 h (t1/2/Tau = 0.17), is expected to have a P/Tss ratio greater than 40 (log P/Tss > 1.6). Such large swings in drug exposure around a target Cavg,ss bring the potential for safety-related issues (due to unnecessarily high maximal concentrations). In addition, large swings around a target Cavg,ss can result in suboptimal efficacy. This arises from the fact that target binding and modulation are inherently nonlinear processes, resulting in a disconnect between average exposure and average PD effect as t1/2 decreases. This disconnect is again illustrated in a semigeneralized way (assuming a ka of 1.8/h and the simple sigmoidal Emax PD model) in Figure 3. In this case, the impact is greater when targeting higher ECxx values. For example, Figure 3 shows that a molecule dosed once daily with a t1/2 value of 4 h (t1/2/Tau = 0.17) is expected to achieve an average effect of 17%, 39%, 65%, and 78% despite maintaining Cavg,ss values equivalent to EC20, EC50, EC80, and EC90, respectively. The disconnect becomes larger as t1/2 continues to decrease to 2 h, with an average effect being approximately 50% at a Cavg,ss value equivalent to the EC90. As such, regardless of whether the target exposure is Cmin,ss, Cavg,ss, or Cmax,ss, t1/2 is an important aspect to consider in drug design as it will influence dose requirements and therapeutic index in the manner discussed above. Although C



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0.2, 1, and 4 mL min−1 kg−1, respectively (1%, 5%, and 20% of human hepatic blood flow, respectively).

(CETP) inhibitor anacetrapib; the terminal t1/2 in healthy volunteers and patients increased with increasing length of exposure.4 Initially thought to have a clinical t1/2 of around 60 h, long-term studies showed residual concentrations of anacetrapib remain significantly elevated after the cessation of therapy. Twelve weeks after stopping treatment of anacetrapib, plasma concentrations were approximately half the on-treatment levels. In a small subset of patients, anacetrapib was detectable in plasma as late as 4 years after the last dose. The lipophilicity of the compound prompted further preclinical studies which illustrated that white fat retention, rather than brown fat retention, was the probable source of the long t1/2.5 Not surprisingly simple PK reveals why long-term therapy exposes the long terminal t1/2. If 12 weeks is a broad estimate of the terminal t1/2 and it relates to redistribution from white fat, then steady state concentrations of anacetrapib will not be achieved in white fat until approximately 1 year of dosing (approximately 4-fold the terminal or white fat t1/2). Amiodarone is another drug that accumulates into white fat and exhibits a long terminal t1/2.6−9 The difficulty of measuring the long t1/2 of multiphasic compounds in short studies (single doses) is illustrated by the variation in t1/2 values quoted. Initially values of 4 and 17 h were published for the t1/2 following iv and oral administration. Subsequent publications have described the t1/2 as 35 h before a more realistic value of around 40 days. On long-term therapy, it is probably nearer 100 days. A study after chronic dosing in rabbits showed that amiodarone total concentrations were the highest in fat tissue followed by lung, liver, and muscle, with the lowest concentration in serum and only traces in the brain.10 Interestingly, the active metabolite desethylamiodarone had similar concentrations in liver, lung, and kidney to the parent drug but lower accumulation in fat tissue. The problems with t1/2 determination lead to similar huge variation in volume of distribution with values from 0.9 and 148 L/kg, with the latter probably the most accurate estimate.

Vss = Vp +

CL =

(3)

Qh fu p CL int Qh + fu p CL int

(4)

Optimization of CLint is often the primary focus in design efforts to improve t1/2. For oxidative metabolism, CLint is relatively easily measured in liver microsomes or hepatocytes in high throughput in vitro systems which facilitate an efficient development of structure−activity relationships (SARs). In addition, because it is also a determinant of Cavg,ss and hepatic extraction, optimization efforts focused on CLint are also likely to pay additional dividends with respect to dose optimization. While tempting when looking at Vss or CL in isolation, optimization of t1/2 through changes in plasma protein binding is not generally a productive endeavor. As can be seen in eqs 3 and 4, decreases in fup may improve clearance but may also reduce Vss. Likewise, increases in fup may improve Vss but will also tend to increase clearance. Consequently, modulation of plasma protein binding cannot modulate t1/2, unless clearance is reduced or Vss increased independent of the modulation of fraction unbound. It is a common misconception that reducing plasma protein binding can result in longer t1/2. Generally, in order to reduce plasma protein binding, lipophilicity is reduced. It is well-known that reducing lipophilicity is likely to reduce intrinsic metabolic clearance with an overall reduction of the clearance of a molecule. It is this reduction in metabolic clearance that drives any t1/2 improvement, not the reduction in plasma protein binding. The impact of plasma protein binding on drug disposition has been thoroughly reviewed previously.11,12

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MODIFIED RELEASE FORMULATIONS Another option for extending t1/2 is the use of modified release formulations (MR). Such formulations can effectively extend the PK t1/2 of molecules with suboptimal values by providing for sustained release along the intestinal tract. This has been successfully used in a number of marketed drugs.13−16 The principles of dose size and frequency of administration as well as the potential value of MR formulations were demonstrated early using the two calcium channel blockers nifedipine and amlodipine (Table 2).1,17−20 Amlodipine exhibits t1/2 value that

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OPTIMIZATION OF HALF-LIFE. VOLUME AND CLEARANCE MODULATION Keeping with the one-compartmental model, t1/2 can be appreciated in terms of its determinants (eq 2). t1/2 = ln(2)Vss/CL

∑ (fu p/fu t)Vt

(2)

From this relationship, it is clear that t1/2 will be proportional to volume and inversely proportional to clearance. In turn, eqs 3 and 4 provide conceptually useful information regarding the common physiological determinants of Vss and CL. In these equations, fup, fut, Vp, and Vt represent the unbound fraction in plasma, unbound fraction in tissue, volume of the plasma, and volume of the tissue, respectively. Further, in the depicted well stirred model of hepatic clearance, Qh, and CLint represent the hepatic blood flow and intrinsic clearance, respectively. As covered in our previous review, Vss is most sensitive to ionization state with acids typically displaying low volumes (high binding to albumin in plasma, low fup) and bases displaying higher volumes (high partitioning to acidic phospholipids in tissues, low fut). This has important implications for design, as the requisite clearance values in order to achieve a particular t1/2 will often depend on ionization state for this reason. For example, in order to achieve a t1/2 of 12 h, acids (Vss ≈ 0.2 L/kg), neutrals (Vss ≈ 1 L/kg), and bases (Vss ≈ 4 L/kg) will typically require clearances at least as low as

Table 2. Comparison of Nifedipine and the Long t1/2 Calcium Channel Blocker Amlodipinea

nifedipine amlodipine

IC50 (nM)

Cav,u (nM)

t1/2 (h)

CLint,u (mL min−1 kg−1)

daily dose (mg)

4 2

6 2

2 40

175 85

10 TID 5−10 QD

a Daily dose size reflects the potency (IC50) and CLint,u of the drugs. Dose frequency reflects the t1/2.

is amenable to once daily (QD) administration.1 The short t1/2 of nifedipine necessitated a 3 times daily (TID) dose regimen. Nifedipine was eventually developed as a slow release formulation (zero order) termed nifedipine gastrointestinal therapeutic system (GITS) to allow once a day administration with doses of 30 and 60 mg.21 A more recent example of the application of modified release to extend the t1/2 and reduce D



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defined as previously described in eqs 1 and 2. However, in certain circumstances, the rate of absorption can be slower than the rate of elimination, resulting in a t1/2 which is determined by the rate of absorption rather than the rate of elimination. This phenomenon, known as “flip-flop” kinetics, can occur when dealing with molecules that have particularly poor permeability or solubility. In these cases, the t1/2 following oral administration may exceed that observed following iv dosing due to absorption or dissolution rate-limited absorption. This phenomenon is not uncommon in preclinical PK studies (particularly in rodents) on molecules with suboptimal PK properties in the course of research and development. There are several pitfalls that arise in this situation. For example, as a compound progresses in development, formulations may improve the dissolution rate, and hence the t1/2 value will collapse to that defined by elimination. In addition, accumulation will often be much less than predicted from an oral t1/2 in this circumstance. This results from the fact that there is a discrete window of time over which ka is operational as determined by the physiological intestinal transit times and physiochemical drug properties. Further, this highlights the limitation of empirical compartmental PK models which do not explicitly account for such physiological limitations and may lead to erroneous expectations when these limitations are not properly considered in experimental design and interpretation. Finally, attention to oral t1/2 in this circumstance can lead to a misprioritization in favor of less soluble or poorly permeable molecules that will ultimately suffer from absorption challenges in the clinic where formulation options are more limited (e.g., limited absolute bioavailability with an associated high degree of inter and intrasubject variability). A much more appropriate use of “flip-flop” kinetics to extend t1/2 is the use of modifiedrelease formulations. Often PK data are extrapolated to infinity using the apparent terminal t1/2. This is an exercise in which the concentration in the last time point collected is divided by the apparent kel value (typically determined though regression of last few data points) in order to infer the remainder of the AUC not explicitly measured through infinite collection of samples. When the oral t1/2 exceeds the iv one, there is a dilemma. Extrapolating to infinity with the oral t1/2 is using potentially the ka and not the kel in preclinical species, which may not be the case in the human situation leading to much shorter t1/2. Here, it is worth pointing out that the t1/2 of metabolite can be equally confounded by formation-rate-limited disposition (analogous to “flip-flop” kinetics) when the rate of formation from parent is slower than the rate of metabolite elimination.25 Thus, a metabolite may, if administered discretely, show a shorter t1/2 than parent, but when the parent is administered, the decline of the metabolite will parallel that of the parent. Consequently, the t1/2 of a metabolite when derived from administration of a parent compound can never be shorter than that of the parent. Of course, some metabolites can possess longer t1/2 values than the parent, normally due to their being more stable than the parent to metabolic processes, or they exhibit Vss greater than the parent. Prime examples are the Ndealkylation reactions of tertiary amines where the generated metabolites exhibit lower intrinsic clearance by subsequent metabolism combined with potentially higher Vss values.

dose frequency is the extended release formulation of tofacitinib.22,23 Tofacitinib is JAK inhibitor with a human t1/2 of 3 h. It is presented as an immediate release tablet formulation with a dose of 5 mg twice daily. A modified release device using an extrudable core system (ECS) has been developed that drives similar AUC to 5 mg twice daily with an 11 mg once daily device. Although a thorough review of the practical considerations for solving t1/2 issues with this technology is beyond the scope of this manuscript, it is important to point out that there are limitations to what this technology can achieve. Generally, the compound must have a degree of aqueous solubility and relative rapid dissolution, since released drug will need to dissolve rapidly under all environments encountered in the human gastrointestinal (GI) tract. Membrane permeability also needs to be sufficient to allow for rapid absorption of dissolved compound across the GI tract wall. Finally, affinity for efflux transporters at the wall of the GI tract needs to be minimized. Generally, immediate release devices present very high drug concentrations to the upper GI tract which are capable of saturating any potential efflux mechanism. Modified release devices steadily present much lower concentration throughout the GI tract, which may be below those required to saturate efflux proteins in the GI tract. Aqueous solubility, permeability, and efflux transport are important considerations for MR, since the drug needs to be as well absorbed in the colon as it is in the small intestine. The human colon has less fluid, a lower surface area for absorption, and higher expression of efflux transporters (such as Pglycoprotein) than the small intestine. Finally, due to the need for excipients to drive MR, the drug loading (dose) for a MR device needs to be limited. Thus, potency and clearance properties of a candidate for MR need to be optimized to maintain dose requirements.

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PRODRUG APPROACH Finally, because a number of drugs generate long-lived active metabolites and therefore act as (accidental) prodrugs, the idea of adding a promoiety to a promising molecule that lacks duration due to a short t1/2 is regularly raised. This is not likely to be successful for the following reasons: (1) The design of optimal prodrugs from a preclinical perspective is difficult. In vitro assays need to be available in preclinical species and human to examine intestinal, hepatic, and extrahepatic hydrolysis of the prodrug moiety to the active drug. Preclinical species can modulate the prodrug to a significantly different extent to humans, making extrapolation and prediction fraught with difficulty. (2) If the short t1/2 of the active drug is due to high unbound intrinsic clearance or total clearance, the total daily dose of the prodrug is likely to be very high, even if there is extensive conversion of the promoiety to the active drug. (3) The rate of metabolism of the prodrug to the active drug must be slow, and this raises the likelihood of nonproductive clearance of the prodrug by metabolism via other routes or other mechanisms leading to the need for higher doses to compensate.24

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PITFALLS IN HALF-LIFE OPTIMIZATION. “FLIP-FLOP” KINETICS Most drugs are absorbed at a rate (ka) that exceeds the elimination rate (kel). In this circumstance, the observed t1/2 is

DOSE-DEPENDENT HALF-LIVES Dose-dependent PK usually occurs due to saturation of absorption or clearance processes as doses are increased. The E



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concentration−time course, particularly after iv injection, typically shows at least two or more discrete phases when viewed on a log10 Cp against time plot (Figure 5). This is due to

most common dose-dependent processes observed are saturation of clearance in preclinical toxicity studies, where doses are high. In many low dose PK studies, the unbound concentrations of drug are likely to be below the Km (e.g., substrate concentration that gives half maximal velocity of transport or metabolism) for the clearance process. Under these conditions, the parameters of clearance and t1/2 will be dose independent (i.e., will not change with increasing doses). However, if doses are used that drive to unbound concentrations that exceed the Km, then the PK will change from first order to zero order processes. If the dose is high enough for zero order kinetics to be observed, the plot of Cp versus time will be linear and a log10 Cp against time will be curvilinear. Once concentrations fall below the Km, the normal linear decline is observed for the log10 Cp against time curve. Therefore, clearance will decrease and t1/2 will increase with increasing dose (Figure 4). Consequently, it is important that

Figure 4. Plasma concentration versus time curve for a compound following escalating oral administration exhibiting dose-dependent (nonlinear) PK due to saturation of clearance with increasing dose.

Figure 5. Basic characteristics of a two-compartment PK model. Panel a illustrates the model structure shown with first-order rates of absorption, elimination, and distribution. Panel b illustrates the characteristic biexponential decline in plasma concentrations following iv bolus administration. β is the elimination rate constant.

PK studies which are extrapolated to the clinical situation are completed at doses that generate unbound plasma concentrations below the Km of the rate-determining clearing process. In rare instances, nonlinear pharmacokinetics may result from saturation of plasma protein binding. In these cases, concentrations exceeding the binding capacity of plasma proteins lead to higher free fractions which, in turn, will increase clearance and increase volume of distribution (all else being equal). The potential for this phenomenon should be considered in circumstances where plasma drug concentrations approach physiological levels of the major binding proteins in plasma (e.g., ∼15 μM and 600 μM for α1-acid glycoprotein and albumin, respectively). Valproic acid is an example of a drug associated with therapeutic exposures in this range (total plasma concentrations of ∼350−700 μM) and which also exhibits saturable protein binding and nonlinear clearance. In other, more complex situations, saturable tissue binding may occur. In these cases, a decrease in volume of distribution may be seen at concentrations that begin to saturate tissue binding.

the rate of entry into compartment 2 from the first compartment (k12) and from compartment 2 back to the first compartment (k21). Consequently, t1/2 can no longer be described as the time taken for the plasma concentrations to fall by half, as there are at least two distinct phases with different rates of decline. The initial decline is often viewed as a distribution phase followed by a terminal elimination rate (β). In this model, Vss, CL, β, and terminal t1/2 are defined by eqs 5, 6, 7, and 8 respectively. ⎛ k ⎞ Vss = V1⎜1 + 12 ⎟ k 21 ⎠ ⎝

(5)

CL = V1kel

(6)

β = 0.5[(k12 + k 21 + kel)

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MULTICOMPARTMENTAL PK PROFILES In the above discussion, a one compartmental model, which provides for a monoexponential decline in drug concentration, was assumed in order to discuss basic concepts related to the t1/2. This one compartment situation requires essentially instantaneous (or very rapid) distribution of a compound into tissues. In the case of most acidic drugs, the majority of total drug is confined to plasma and so a single compartment model is often observed. However, for lipophilic neutral and basic compounds, this is seldom the case. The decline of plasma

(k12 + k 21 + kel)2 − 4k 21kel ]

terminal t1/2 = ln(2)/β

(7) (8)

Although the relationships appear to be much more complicated, it is important to note that β is closely approximated by CL/Vss (i.e., another way to define kel in the one compartment model) so long as the distributional rates exceed that of the elimination rate. As such, under most circumstances, the basic concepts for modulating terminal t1/2 according to Vss and CL remain relevant with this more complicated profile. In addition, it is not uncommon for the F



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multiexponential decline to be lost or attenuated following oral administration due to the relative rates of absorption versus distribution. However, in a manner analogous to “flip-flop” kinetics, these rates may be difficult to interpret once the distribution rates become slower than the elimination rate. In some cases, a prolonged terminal half-life may be observed after concentrations have declined significantly providing for a portion of the concentration−time curve containing very little of the overall area under the curve (AUC). In this case, while the terminal elimination phase t1/2 may well approach a value that would be associated with once daily administration in humans, it would not lead to significant accumulation of drug exposure at steady state owing to the fact that it contributes very little to the overall AUC and clearance of the drug. Such a drug would still need to be administered more than once daily to achieve appropriate steady state exposure. To account for this complexity in a practical way, the concept of an “effective” or “operational” t1/2 has been developed that describes t1/2 in relation to the observed or predicted degree of accumulation and shape of the concentration−time profile upon steady-state dosing.26,27 This is exemplified by the NK2 antagonist UK224671 [(S)-(1)-4-(1-H2-[1-(cyclopropylmethyl)-3-(3,4-dichlorophenyl)-6-oxo-3-piperidyl]ethylazetidin-3-yl)-1piperazinesulfonamide].28 The PK of UK-224671 in human following iv administration is characterized by a multiple compartmental profile (at least three phases, Figure 6). The terminal elimination phase t1/2 is calculated at 28 h. However, this terminal phase describes approximately 20% of the AUC. Consequently, a once daily regimen for this compound would not accumulate to an appreciable steady state trough concentration. Paradoxically, this compound would be best suited to three times daily administration, despite an apparently once daily t1/2.

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ESTIMATION OF HALF-LIFE The normally quoted (terminal) t1/2 is commonly determined by estimating the final slope of the plasma concentration−time curve. This now brings error in data analysis. After an iv dose, time points will often be taken with three samplings in the first hour and then hourly up to 4 h followed by samplings at perhaps 6, 8, and 24 h. The sampling time of the later time points is mostly governed by practical reasons. The t1/2 calculation will probably use the final three or so time points that have measurable concentrations. In the estimation of t1/2, at least three points should be used. This is an inadequate number to gain a robust estimate of t1/2. Having any drug detectable at 24 h postdose usually significantly prolongs the apparent t1/2, which means assay sensitivity between compounds may be a major factor in selecting compounds rather than real PK differences. The assay differences may be compounded across species. For instance rodent studies usually have a lower number of samples of the different time points with a smaller volume of plasma samples than larger species like dog. Therefore, the larger species has the potential to generate a significantly longer t1/2 than rodents simply because of analytical sensitivity. The same analytical sensitivity issue will be seen in the AUC and area under the first moment curve (AUMC) so that clearance and volume will also be defined by the analytical sensitivity issue. Clearly, these are aspects of the PK study design that may complicate the estimation of the true t1/2 of the compound being tests and affect the extrapolation to human PK.

Figure 6. PK of UK-224,671 in human following single intravenous administration of 0.1 mg/kg: (a) overall PK profile; (b) isolated initial phase profile; (c) isolated terminal phase profile.

Often oral studies use higher doses than iv studies for safety reasons (oral exposure is usually less than that for the same iv dose due to oral bioavailability). This adds to the complexity of t1/2 determination since the oral study concentrations will be higher than iv dosing in the terminal phase, assuming the same analytical method is used. Consequently, the “true” terminal t1/2 of the molecule could be exposed with the chance of therefore a significantly longer apparent t1/2 after oral administration. The difficulty in interpretation will then be whether this is a fundamental parameter of the compound or driven by slow dissolution in the GI tract (as discussed previously). The problems of detection of drug in later samples or not sampling long enough also apply to the estimation of CL and Vss. Although model free approaches are used largely in drug discovery, there is still a need for determination of the final elimination phase to extrapolate AUC values to infinity. Underestimating the terminal t1/2 therefore results in an underestimation of Vss and an overestimation of CL. When this is exaggerated by the low iv dose compared to the oral dose in a crossover study, overestimates (including >100%) will be made for bioavailability. G



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POPULATION VARIABILITY OF HALF-LIFE Variability of the t1/2 across populations can be of major concern. Variability in t1/2 should be a consideration in any patient population for which a high degree of variability in CL is expected due to the known mechanisms of clearance (hepatic impairment, renal impairment, genotype, ethnicity, age, etc.). In other cases, variability can arise due to much more complicated mechanisms. For example, high variation in t1/2 is accentuated when the drug’s t1/2 is partially dependent on reversible metabolism (metabolic conversion back to parent). This is illustrated by benoxaprofen, a nonselective COX inhibitor (Figure 7). Most of the drugs in this class contain a carboxylic

Perspective

PERSPECTIVE The simplest PK definition of t1/2 as the time taken for plasma concentrations to fall by half only applies under one compartment PK following iv administration. Under other conditions, such as oral administration and multicompartmental PK, the effect of rate constants other than kel needs to be taken into consideration; however, Vss and CL remain important factors to consider in any approach to optimize t1/2. We introduced this perspective by posing several important questions with respect to t1/2 during a drug discovery project. • What t1/2 is optimal to support once daily administration? Deriving dose regimen from t1/2 should drive to minimize peak-to-trough ratio, which will limit the potential for Cmax driven side effects and lack of efficacy at Cmin. An acceptable peak-to-trough ratio is below 4, which leads to a t1/2 greater than 12 h (if the PK is one compartment). t1/2 of 24−48 h would further minimize peak-to-trough ratio and potentially allow “forgiveness” of a single missed dose in minimizing loss of steady state exposure. Compounds with t1/2 values greater than this can lead to potential issues in terms of time to achieve steady-state, washout in the event of adverse effects and longer drug development time. • How can t1/2 be modulated? t1/2 is determined by the plasma CL and Vss of a compound. To improve t1/2, medicinal chemistry strategy should mainly focus on reducing plasma CL (e.g., increase metabolic stability, reduce renal clearance, reduce hepatic uptake by active transporters and biliary clearance) and/or second increasing Vss.1 Generally, modulation of plasma protein binding in the absence of changing CL or Vss will not affect t1/2. This is a consequence of the relationship of t1/2 to CL and Vss (eq 2). Changing fraction unbound in plasma will often change CL and Vss in equal and opposite directions. Should a clinical agent not possess the appropriate t1/2 to suit its proposed dose regimen, it is possible to improve the t1/2 by modifying the delivery of the compound. Subject to appropriate properties of aqueous solubility sufficient to support dose requirements, high membrane permeability, and low efflux transport, a compound can be formulated as MR to enhance t1/2. This formulation will constantly release the compound throughout its passage through the GI tract. Effectively, the MR formulation gives zero order delivery of the compound and results in “flip-flop” PK, meaning t1/2 in plasma will reflect the rate of delivery.

Figure 7. Structures of benoxaprofen and acylglucuronidation metabolite. Oxidative metabolism is likely to be very low due the chlorine substituent and oxidazole ring leading to acylglucuronidation as the major clearance pathway.32

acid moiety. It is difficult to obtain a long t1/2 with such agents due to the low Vss. Benoxaprofen was developed because the plasma elimination t1/2 of the drug was in the range of 19−26 h,29 almost ideal for a once a day administration. However, the development studies were conducted only in younger volunteers and patients. At the time a once a day drug with a simple single dose size was commercially attractive against the many other drugs in the class. Very soon after launch, toxicity problems were noted affecting the skin and liver. With both skin toxicity and hepatotoxicity, the major proportion of effects was observed in elderly patients.30 When PK of these patients were studied, t1/2 extended out to 150 h which coincided with reduced creatinine clearance.31 Acyl glucuronidation was the major clearance pathway of benoxaprofen.32 Excretion of this metabolite could occur renally and hepatobiliary. Enterohepatic recirculation of this metabolite after hepatobiliary clearance and subsequent hydrolysis in the gut back to benoxaprofen were likely to be an important contributor to the long t1/2. GI stasis leading to higher recirculation and lowered kidney function lowering the renal clearance of the acyl glucuronide contributed to accumulation of the drug in the elderly. Today, studies in the elderly would have been conducted during drug development in addition to more detailed studies to understand the clearance and metabolism. These would have probably rendered the drug much less commercially attractive. The drug illustrates that thinking and basing decisions on a single average value for t1/2 are problematic. Drugs can exhibit quite different t1/2 in different patients and populations leading to the need for different dosing regimens.

• What are the important considerations for measuring t1/2 during compound optimization? Complexities related to the multiple rates (absorption, distribution, and elimination) can confound efforts to optimize t1/2 by obscuring the interpretation of preclinical and clinical data. A careful analysis of PK profiles under different dose regimens and formulations (particularly iv and oral administration) are required in order to put an observed t1/2 into a context useful to design. In addition, nonlinearities in absorption and elimination can equally confound the interpretation of t1/2. PK studies employing a range of doses and exposure levels are often required to detect these issues. Finally, population variability in t1/2 (particularly in the clinic) can confound the interpretation of t1/2. Here an understanding of the mechanisms of clearance and distribution specific to a molecule and how those mechanisms vary across potential H



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patient populations is important to ensure the appropriate interpretation of observed t1/2.

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Perspective

ACKNOWLEDGMENTS

The authors greatly appreciate the technical assistance from David Tess.

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AUTHOR INFORMATION

ABBREVIATIONS AUC, area under the curve; C0, initial drug concentration; Cavg,ss, average steady state drug concentration; Cavg,ss,u, average unbound steady state drug concentration; Cavg,ss, steady state average concentration; Cmin,ss, steady state minimum concentration; Cmax,ss, steady state maximum concentration; CL, clearance; CLint, intrinsic clearance; CLint,u, unbound intrinsic clearance; Cmax, maximum drug concentration; Cmin, minimum drug concentration; Cp, drug concentration in plasma; ECxx, effective concentration at certain “xx” percentage of maximal, e.g., EC50 is the concentration that causes 50% of maximum effect; fa, fraction absorbed; fel, fraction of drug being washed out; fg, fraction escaped from gut; fss, fraction of target steadystate level; fup, unbound fraction in plasma; fut, unbound fraction in tissue; GITS, gastrointestinal therapeutic system; iv, intravenous; ka, first-order absorption rate constant; kel, firstorder elimination rate constant; Km, substrate concentration that gives half maximal velocity of an enzymatic reaction; MR, modified release formulation; MRT, mean residence time; PBPK, physiologically based pharmacokinetic modeling; PD, pharmacodynamics; PK, pharmacokinetics; P/Tss, steady-state peak-to-trough concentration ratio; QD, once daily; Qh, hepatic plasma flow; t, time; t1/2, half-life; Tau, dosing interval; TID, three times daily; Tmax, time at maximum plasma drug concentration; Twash, the time it takes to wash drug out by a specified fraction; Vd, volume of distribution; Vss, steady state volume of distribution; Vp, volume of the plasma; Vt, volume of the tissue

Corresponding Author

*Phone: 860-715-6172. E-mail: li.di@pfizer.com. ORCID

Li Di: 0000-0001-6117-9022 Notes

The authors declare no competing financial interest. † Retired. Biographies Dennis A. Smith worked in the pharmaceutical industry for 32 years after gaining his Ph.D. from the University of Manchester, U.K. During this period he directly helped in the discovery and development of eight marketed NCEs. More recently his roles include advisory boards or expert panels with many drug research based organizations, both industrial and academic, including Medicines for Malaria Venture and Cancer Research UK. His research interests and publications span all aspects of drug discovery and development particularly where drug metabolism knowledge can impact on the design of more efficacious and safer drugs. Kevin Beaumont has worked extensively in the discovery and development drug metabolism field throughout his 30 years in the pharmaceutical industry. His major area of expertise is in the modulation of physicochemistry to affect drug disposition and prediction of human PK. He is author on over 40 peer reviewed publications. Overall, Kevin has worked on many drug discovery and development projects throughout his career. He has been responsible for the DMPK input to at least 30 FIH studies as well as 10 phase II compounds, including one marketed agent. Kevin now provides DMPK input to the Inflammation and Immunology research unit and Rare Disease research unit, based in Cambridge, MA.

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REFERENCES

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Tristan S. Maurer received his Pharm.D. from the University of Georgia in 1993 and his Ph.D. from the University of Buffalo, State University of New York in 1999. During his 18-year tenure with Pfizer, his work has focused on the development and application of quantitatively rigorous, biologically based methods to predict human pharmacokinetics and pharmacodynamics from preclinical data. He has coauthored over 60 manuscripts illustrating the utility of these methods to drug design and early clinical development. Currently, Dr. Maurer sits on the Medicine Design leadership team responsible for scientific and operational strategies spanning from idea to loss of exclusivity. He also leads a modeling and simulation group responsible for both computational chemistry and quantitative translational pharmacology across Pfizer’s small molecule portfolio. Li Di has about 20 years of experience in the pharmaceutical industry including Pfizer, Wyeth, and Syntex. She is currently a research fellow at Medicine Design Department, Pfizer Global Research and Development, Groton, CT. Her research interests include the areas of drug metabolism, absorption, transporters, pharmacokinetics, blood−brain barrier, and drug−drug interactions. She has over 120 publications including two books and presented more than 80 invited lectures. She is a recipient of the Thomas Alva Edison Patent Award, the New Jersey Association for Biomedical Research Outstanding Woman in Science Award, the Wyeth President’s Award, Peer Award for Excellence and Publication Award. I



Perspective

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Relevance of Half-Life in Drug Design - Journal of Medicinal

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