Half life of first order reaction whose rate constant is 0.25 per minute is 1 point

Elimination Rate Constant (k)

The elimination rate constant (usually a first-order rate constant) represents the fraction of xenobiotics that is eliminated from the body during a given period of time. For instance, when the value of the elimination rate constant of a xenobiotic is 0.25 per hour, this means that ∼25% of the amount remaining in the body is excreted each hour. The rate constant is calculated from the slope (−k/2.303) of the blood concentration and time curve (log–linear scale) as shown in Figure 2a (Figure 2b shows the same data on linear scale). Its value is affected by all processes (e.g., distribution, biotransformation, and excretion) that contribute to clear xenobiotics from the blood.

19.2.4 Half-Life (t½)

The time for the concentration of a drug in systemic circulation to reduce by half is termed half-life (t½). Drug clearance typically follows first order kinetics, so a plot of log drug concentration versus time provides the elimination rate constant (k) as the slope (Figure 19.9). t½ is calculated from k using the first order kinetics expression:

t½=0.693/k

t½ can also be calculated from Vd and CL using the expression

t½=0.693×Vd/CL.

Thus, PK half-life is determined by Vd and CL. Increasing CL decreases t½, because drug molecules are being removed from the blood at a higher rate. Increasing Vd increases t½, because tissue is a depot for drug, so a higher Vd increases the amount of drug in tissue that can diffuse back into the blood as drug molecules are removed from the blood.

One application of half-life is dosing interval. Pharmaceutical therapy usually involves regular dosing of a drug product to maintain the in vivo drug plasma concentration in a favorable range. Re-dosing typically is performed every 1-3 half-lives to maintain the concentration level, thus a human PK t½ of 8-24 h is conducive to once-per-day dosing (see Figure 19.10).

Drug discovery scientists should be careful not to confuse in vivo PK t½ with in vitro metabolic stability t½. They do not always correlate. One reason for this is shown by the above equation, in which Vd is a major determinant of in vivo PK t½.

Another related PK parameter is mean residence time. This is the time for elimination of 63.2% of the IV dose.

Elimination Half-Life (t1/2)

Elimination half-life is the time required to produce a 50% reduction in blood or plasma concentration. This value is estimated using the following equation:

[9]t1/2=0.693×VDCl

Since the first-order elimination rate constantske and β can be calculated by dividing VD by Cl, the half-life of a xenobiotic that follows a one- or two-compartment model can be calculated as follows: (1) one-compartment model – t1/2 = 0.693/ke and (2) two-compartment model – t1/2 = 0.693/β. These values should remain relatively consistent in xenobiotics following these models. Conversely, the half-lives of xenobiotics undergoing zero-order elimination, such as overdoses with aspirin and acetaminophen, cannot be calculated using eqn [9] because the time required to produce a 50% reduction in blood/plasma concentration is variable (i.e., elimination is not proportionally related to concentration).

1.08.2.1.4 Clearance

Total clearance (CL) is a proportionality factor that relates elimination rate (vel) to concentration:

(8)vel=CL⋅C

The units of clearance are that of flow, for example l min−1 or l h−1. Clearance can be conceptualized as the volume that is completely eliminated of the chemical per unit of time. Just as the volume of distribution, the clearance value is apparent and depends on the site of concentration measurements. In contrast to the elimination rate constant, clearance is a measure of elimination capacity that is independent of volume of distribution. This is easier to understand considering the intrinsic metabolic capacity of an enzyme (Vmax/Km) or glomerular filtration in the kidneys. The relationship between clearance and the elimination rate constant is obtained by combining eqns [4b] and [8]:

(9)kel=CLVd

CL can be calculated from the dose and the concentration–time integral, often called the area under the concentration–time curve or AUC:

(10)CL=D∫Cdt=DAUC

The AUC (Figure 3) may be either calculated from a fitted concentration–time curve (AUC = C0/kel in the one-compartment model) or directly from the observed data points by use of the trapezoidal rule (described in more detail, for example, by Rowland and Tozer (1968).

Duration of Pharmacodynamic Response

The duration of action may be defined as the time the drug concentration stays within the therapeutic range of a drug. Frequently, the duration of action (td) of a drug is influenced by the amount of drug in the body and the rate of drug elimination. The duration of action for a drug that obeys the one-compartment model after intravenous bolus administration may be calculated using Eq. (9), where A0 is the amount of drug administered; Amin is the minimum effective dose and kel is the first-order elimination rate constant of the drug. In this case, td is the time required for the initial amount of drug in the body to decline to the minimum effective amount. This expression demonstrates that there is a linear relationship between the duration of response and the logarithm of the amount of drug in the body. Thus, increasing the dose prolongs td but there is a risk of producing adverse effects if the therapeutic index of the drug is small. Furthermore, the equations show that the duration of action is inversely proportional to the first-order elimination rate constant of the drug. Therefore, td is expected to be prolonged in patients with renal or hepatic failure since impaired efficiency of drug elimination will lead to retention of drug in the body. Notwithstanding, the duration of action is also influenced by pharmacodynamic parameters of a drug. Consequently, some drugs such as β-blockers are usually administered once or twice daily, despite having short elimination half-lives, because the doses used are sufficient to produce a maximal effect for a significant part of the dosing interval:

(9)td=2.3kellogA0Amin

Duration of Pharmacodynamic Response

Frequently, the duration of action (td) of a drug is influenced by the amount of drug in the body and the rate of drug elimination. The duration of action may be defined as the time the drug concentration stays within the therapeutic range of a drug. The duration of action for a drug that obeys the one-compartment model after intravenous bolus administration may be calculated using eqn [7], where A0 is the amount of drug administered; Amin is the minimum effective dose and kel is the first-order elimination rate constant of the drug. In this case, td is the time required for the initial amount of drug in the body to decline to the minimum effective amount. This expression demonstrates that there is a linear relationship between the duration of response and the logarithm of the amount of drug in the body. Thus, increasing the dose prolongs td but there is a risk of producing adverse effects if the therapeutic index of the drug is small. Furthermore, the equations show that the duration of action is inversely proportional to the first-order elimination rate constant of the drug. Therefore, td is expected to be prolonged in patients with renal or hepatic failure since impaired efficiency of drug elimination will lead to retention of drug in the body. Notwithstanding, the duration of action is also influenced by pharmacodynamic parameters of a drug in question. Consequently, some drugs such as β-blockers are usually administered once or twice daily, despite having short elimination half-lives, because the doses used are sufficient to produce a maximal effect for a significant part of the dosing interval:

[7]td=2.3kel(logA0Amin)

Intravenous Administration

After administration, a drug is transferred into the body at a rate described by kabs (Fig. 1). However, for drugs administered intravenously, there is no absorption phase as the drug is immediately distributed throughout the body. Once in the body, the drug concentration decreases at a rate that is characteristic for it in that organism under a particular set of physiologic conditions. The rate of elimination for most drugs is proportional to its concentration in the body. These drugs are said to behave according to first-order processes. For a first-order drug, the rate of elimination is described as:

v = kel [A]

where v is the rate of drug elimination, kel is the elimination-rate constant, and [A] is the concentration of the drug in the body. The rate constant kel is determined by the physical properties of the drug and the mechanisms of elimination. For example, the kel for a drug that is excreted predominantly by glomerular filtration/tubular reabsorption is determined by its molecular size and its lipid solubility and ionization characteristics (see the records on Passive Diffusion of Drugs Across Membranes and Filtration of Drugs Across Membranes). The kel for a drug can differ among individuals and can even be different in the same individual if measured at different times. For example, induction of drug-metabolizing enzymes can lead to a significant increase in the kel for a drug.

When the time course for the disappearance of the drug is plotted (solid line, Fig. 2a), the rate of elimination (v) is rapid at first when the plasma concentration is high. As the drug is eliminated, the rate of elimination (which can be measured by the tangent to the curve at any given time) decreases. A straight line is obtained when this is plotted on a first-order (semilogarithmic) plot (Fig. 2b). The slope of the line from the semilogarithmic plot is equal to the kel.

The time required to eliminate half of the drug is its half-life. As can be observed from either plot in Fig. 2, the time required to eliminate 50% of the drug is independent of the time points selected. The relationship between the half-life and the elimination-rate constant is:

kel = 0.693/t1/2

Not all drugs are eliminated by simple first-order processes. Rather, some are eliminated by saturable mechanisms such as active transport, receptor-mediated endocytosis, or by way of drug-metabolizing enzymes. When these processes become saturated, the rate of elimination is no longer proportional to the drug concentration in the body. Under this circumstance, the drug is eliminated at a constant rate, which is referred to as a zero-order process (Fig. 3).

It is important to note that "zero order drugs" behave according to a first-order process at lower concentrations when the elimination mechanism is not saturated.

3.1 Equilibrium Sampling

As discussed earlier, EqP theory assumes that chemical activity and fugacity are the same in individual medium when the equilibrium of a chemical is reached among the lipids of biota, the OC of sediment, and pore water [35]. This relationship is similar for PSMs when the system is at equilibrium. After a passive sampler is introduced into the sediment for some length of time, a thermodynamic equilibrium is established for the target compounds among the sampler, sediment OC, and pore water. The sampler–medium partition coefficient (Ksampler–medium) is defined as the ratio of the concentration of HOC on the sampler (Csampler) and that in a given environmental medium (Cmedium, e.g., Cfree) at equilibrium (Eqn (5)). In addition, Ksampler–medium can be estimated from the uptake and elimination rate constants (Eqn (6)).

(5)

(6)

If 90% of equilibrium is assumed to represent the equilibrium state, then the time required to reach this state (t90) can be calculated using Eqn (7) [60–62].

(7)

Since equilibrium calibration is less impacted by the fluctuation of environmental conditions (e.g., temperature and hydrodynamics) and can generate more robust data than kinetically controlled sampling, it is preferable to determine Cfree by equilibrating the sampler with its environment [18]. Mayer et al. [43] reported that the analytical precision of PSMs at equilibrium was significantly better compared to that in the kinetic regime, with relative standard deviations increasing from 20% to 30% in the kinetic regime compared to <10% when the system was near equilibrium. Series measurements at different sampling times or parallel analyses using PSMs with different surface area/volume (A/V) ratios are generally conducted to confirm the equilibrium condition. Nevertheless, the time for the high octanol–water partition coefficient HOCs (log Kow > 6) to attain equilibrium might be too long to be useful in bioassays, especially in the case of field applications [63]. Therefore, kinetically controlled sampling has been developed for PSM applications without the need to reach equilibrium.

Piper nigrum

Piperine (PIP) is an alkaloid and is the key ingredient for the pungency of both long and black pepper. PIP has been reported to inhibit CYP3A4 and P-gp, which in turn increases the bioavailability of nevirapine, a medication used to treat and prevent HIV/AIDS. This significant interaction between the drug and the herb was predictable as nevirapine undergoes oxidative metabolism and glucuronidation using the common DMEs [89]. A significant increase in the level of carbamazepine, an anticonvulsant medication used in epilepsy, and an increase in the efficacy of midazolam is also noted when taken concomitantly with PIP due to inhibition of CYP3A4 [90,91,92]. When the healthy volunteers receiving diclofenac sodium (an antiinflammatory drug) were treated with PIP, then maximum plasma concentration and the half-life of the drug were enhanced along with a significant decrease in the elimination rate constant. This interaction might be attributed to the inhibition of CYP2C9 by PIP [93]. PIP significantly affects on the metabolism of glimepiride in STZ-induced (streptozotocin induced) diabetic rats due to inhibition of CYP2C9. The glucose-lowering effect of glimepiride is increased in the presence of PIP [94]. A significant PK interaction is also observed between PIP and chlorzoxazone (a muscle relaxant) due to inhibition of CYP2E1 by PIP [95]. The low oral bioavailability of domperidone (an antiemetic drug) was enhanced due to the inhibitory effect of black pepper on CYP3A1 (analogous to human CYP3A4) and P-gp across the intestine and liver of male Wistar rats [96]. It has been reported that PIP shows an inhibitory effect on the major DT (P-gp) in rats, which suggests that it may even modulate the P-gp–mediated drug efflux in the human beings. As per the reported experiments, PIP has an influential effect on the altered PK and increased bioavailability of fexofenadine (antihistamine drug and a substrate of P-gp) due to inhibition of the P-gp in humans [97]. Moreover, in rats, the intravenous PK of the same drug were not altered in the presence of PIP, but the gastrointestinal absorption was increased likely by the inhibition of P-gp–mediated cellular efflux during intestinal absorption [98].

PIP is considered as a modulator of a number of hepatic and intestinal pathways. These phytoconstituents might have the potential to inhibit the CYP2C enzyme system in the rats [96,99,100]. The concomitant administration of warfarin (an anticoagulant) and PIP resulted in a significant decrease in the concentration of the major metabolite of the drug, 7-hydroxywarfarin, and a statistically nonsignificant decrease in the total concentration of warfarin [100]. The co-administration of PIP with herbs has the potential to alter the PK of the latter without inhibiting or inducing any of the DMEs or DTs involved. For instance, the bioavailability of isoniazid (antibiotic for tuberculosis) is decreased in the presence of PIP, and the possible reason behind this could be delayed in the gastric emptying time [101]. In a similar manner, when ciprofloxacin (broad-spectrum fluoroquinolone antibiotic) was coincubated with P. nigrum, the permeability of the drug was significantly increased [102]. Another study suggested that PIP can be used along with phenytoin (an antiepileptic drug) as a bioavailability enhancer as the former might increase the plasma levels of the drug without causing any unwanted accumulation of the same in the body [92].

Half-life

The in vivo half-life (t1/2) of a compound is a major determinant of its success or failure as it moves down the path toward clinical study and eventual commercialization. Simply stated, the in vivo half-life of a compound is the time required for removal of 50% of the compound from the body. Dosing regimens are directly impacted by t1/2, as compound that have a short t1/2 are quickly removed from the body and require more frequent dosing. If the in vivo half-life is too low, it may not be possible for the compound to remain in the systemic circulation long enough for the compound to have an impact on a biological system. If, for example, a compound has a t1/2 of 15 minutes, then 1 hour after dosing, over 90% of the compound would be cleared from the body, and in 90 minutes less than 2% would remain in the systemic circulation. On the other hand, compounds with high in vivo half-lives will require less frequent dosing in order to maintain systemic availability. A compound whose t1/2 is 200 hours, such as the antimalarial drug Chloroquine,85 would require over 33 days in order to achieve 94% elimination from the systemic circulation. In general, most drug discovery and development programs focus on designing compounds capable of once or twice daily dosing, but as discussed earlier, the end goal of the program should be kept in mind in determining the required half-life. The requirements for a sleeping pill will be different from those of a pain medication or cancer treatment.

As previously mentioned, the in vivo half-life is dependent of on its clearance and volume of distribution. Understanding the relationship of these concepts can provide a great deal of insight into how to improve the characteristic of candidate compounds. In most cases the in vivo half-life of a given compound is assessed by measuring the plasma concentration of the compound after a single IV dose. Multiple concentration measurements are taken over a set time period, usually 12–24 hours. A plot of the log of the compound concentration over time can be used to determine the elimination rate constant (k) for the compound. The in vivo half-life is then calculated using Eq. (6.3) (Fig. 6.54). An alternate mathematical definition of the in vivo half-life shown in Eq. (6.4) describes the relationship between in vivo half-life, clearance, and volume of distribution.

As indicated in Eq. (6.4), the in vivo half-life is directly related to both CL and Vd. In vivo half-life is indirectly proportional to CL, but directly proportional to Vd. In other words, as CL increases, t1/2 falls and as Vd increases, t1/2 increases. Since CL is a driven by the rate of elimination (metabolism, excretion, etc.), it is easy to understand why in vivo half-life decrease as clearance increases. Higher CL values indicate that a compound is more rapidly removed from the body by any available mechanism, which logically shortens the time required to remove 50% of the compound from the systemic circulation. This phenomenon is readily apparent in the observed half-lives of the drugs Ethosuximide86 and Flucytosine87 (Fig. 6.55). The volume of distribution of both compounds is 49 L, but Ethosuximide’s t1/2 is 48 hours, while Flucytosine’s t1/2 is only 4.2 hours. This nearly 10-fold difference is a reflection of the nearly 10-fold difference in CL (0.7 vs 8.0 L/hour). Since clearance is a representation of the sum total of all mechanisms available for removal of the compound from the body, it stands to reason that efforts to slow down these mechanisms would lead to decreased clearance and, therefore, a longer t1/2. In other words, improving the metabolic stability of a compound or decreasing renal excretion can have a direct impact on t1/2 by decreasing clearance.

The impact of volume of distribution on t1/2 is not as readily apparent. As discussed earlier, volume of distribution is an imaginary volume used to describe how widely distributed a compound is through the body. Compounds that are readily distributed beyond the systemic circulation will have a large volume of distribution, while compounds that are more restricted to the blood will have a lower volume of distribution. The impact of volume of distribution on t1/2 is readily apparent in the comparison of Flucytosine and Digoxin88 (Fig. 6.55), as there is a 10-fold difference in t1/2, even though drug clearance is nearly identical. In this case the difference in half-life is mirrored by a 10-fold difference in volume of distribution, and Digoxin, which has a significantly larger volume of distribution, has a longer half-life. This phenomenon can be explained by considering the relative concentration of a compound in the blood versus tissue distribution. Compounds that are highly distributed into the tissues and organs (those that have a higher volume of distribution) are less concentrated in the blood. This means that less drug is available for processing by the liver and kidney with each cycling of the blood supply through these organs. Thus in a series of related compounds, changes in physical properties that alter volume of distribution, such as lipophilicity and transporter protein activity, will influence half-life. Changes that increase the volume of distribution will increase half-life, while changes that decrease the volume of distribution will decrease half-life.