Pharmacokinetics Of Oral Absorption PDF

**[PHARMACOKINETICS OF ORAL ABSORPTION]** **Learning Objectives**: The students will be able to: - Describe plasma concentration versus time profile obtained from the oral administration of a single dose of a drug - Discuss the pharmacokinetics of oral absorption of a drug following...

**[PHARMACOKINETICS OF ORAL ABSORPTION]** **Learning Objectives**: The students will be able to: - Describe plasma concentration versus time profile obtained from the oral administration of a single dose of a drug - Discuss the pharmacokinetics of oral absorption of a drug following a one-compartment model - Calculate the pharmacokinetic parameters after oral administration of a single dose - Describe the conditions that may lead to the flip-flop of absorption and elimination rate constants - Discuss how absorption and elimination rate constants influence the maximum plasma concentration and time to reach the maximum plasma concentration - Use the method of residual to estimate the following parameters in the compartmental analysis: - The absorption rate constant - The elimination rate constant - T~max~ - Vd - AUC - CL - Bioavailability **Write equations by hand to prepare for the test!** **Introduction:** Plasma concentrations of a drug after oral, intramuscular, or subcutaneous administration will show an additional phase when compared to those after IV bolus administration. The overall process may be described as either a first-order or zero-order input process. In general, however, the first-order absorption model is considered. **Figure.** Drug absorption and distribution (Do: dose, Ka: first-order absorption rate constant, K first-order elimination rate constant, C: plasma concentration, V: volume of distribution). ***Assumptions:*** -one-compartment system -first-order absorption -first-order elimination -complete bioavailability (F=1)  **Figure**. Plasma-level time curve after a single oral and IV dose **Note:** This model applies to the solution, rapidly dissolving dosage forms, suppositories, IR tablets, capsules, IM, and SQ injectables (everything that is not IV). The terminal phases after oral administration will parallel to that after IV administration. ***The equation for absorption***: \ [\$\$C\_{p} = \\frac{K\_{a}\\text{FD}\_{0}}{V\\left( K\_{a} - K \\right)}\\left( e\^{- Kt} - e\^{- K\_{a}t} \\right)\\text{\\ \\ }\$\$]{.math.display}\ \ [\$\$\\text{lnC}\_{p} = ln\\frac{K\_{a}\\text{FD}\_{0}}{V\\left( K\_{a} - K \\right)} - Kt\$\$]{.math.display}\ \ [\$\$\\log C\_{p} = log\\frac{K\_{a}\\text{FD}\_{0}}{V\\left( K\_{a} - K \\right)} - \\frac{\\text{Kt}}{2.3}\$\$]{.math.display}\ Where: Do: Dose Ka: first absorption rate constant K: first-order elimination rate constant F: fraction of dose absorbed (when 100% of the dose is absorbed, the value of F becomes 1) V: volume of distribution **Figure**. Plasma drug concentrations vs time after a single oral dose - Calculation of the maximum plasma concentration after oral dose, C~max~, and the time needed to reach maximum concentration, t~max~, is necessary, as the direct measurement may not be possible due to the inexact timing of blood sampling. - C~max~ is directly proportional to the dose (Do) and the fraction of dose absorbed (F). - The t~max~ is independent of dose and is dependent on the rate constants for absorption (Ka) and elimination (K) - At C~max~, the rate of drug absorption is equal to the rate of drug elimination. Hence the net change in concentration at C~max~ is zero. \ [\$\$t\_{\\max} = \\frac{\\ln K\_{a} - lnK}{K\_{a} - K} = \\frac{2.3\\log\\left( K\_{a}/K \\right)}{K\_{a} - K}\\text{\\ \\ }\$\$]{.math.display}\ **[Determination of Absorption Rate Constants from Oral Absorption Data]:** ***[Method of Residuals]***: \ [\$\$C\_{p} = \\frac{K\_{a}\\text{FD}\_{0}}{V\\left( K\_{a} - K \\right)}\\left( e\^{- Kt} - e\^{- K\_{a}t} \\right)\\text{\\ \\ }\$\$]{.math.display}\ Assumptions: - Ka \>\> K - The drug absorption is complete The above equation becomes: \ [\$\$C\_{p} = \\frac{K\_{a}\\text{FD}\_{0}}{V\\left( K\_{a} - K \\right)}e\^{- kt}\\ \$\$]{.math.display}\ \ [\$\$\\text{If\\ }\\frac{K\_{a}\\text{FD}\_{0}}{V\\left( K\_{a} - K \\right)} = A\$\$]{.math.display}\ \ [*C*~*p*~ = *Ae*^ − *Kt*^]{.math.display}\  1. Plot the drug concentrations vs time of a semi-log paper 2. Obtain the slope of the terminal phase by extrapolation 3. Take any points on the upper part of the extrapolated line (e.g., X~1~', X~2~', X~3~') and drop vertically to obtain corresponding points (e.g., X~1~, X~2~, X~3~). Then calculate the differences in values between Xs and X's. 4. Plot the differences at corresponding time points. A straight line will be obtained with a slope of *--K~a~/2.3* **[Example 1]**: A 100 mg drug was administered to a subject orally with 50 mL of water. The drug plasma concentrations were determined as follows: +-------------+-------------+-------------+-------------+-------------+ | Time (hr) | Cp (mcg/mL) | A*e^-Kt^* | A*e^-Kt^* - | A*e^-Kt^* - | | | | | Cp | Cp = | | | | | | Ae^-Kat^ | +=============+=============+=============+=============+=============+ | 0.5 | 5.36 | \ | 67.6 -- | \ | | ----- | ------- | [*C*~*p*~ = | 5.36 = 62.2 | [*C*~*p*~ = | | 1 | 9.95 |  70*e*^ − 0 | |  70*e*^ − 0 | | 2 | 17.18 |.0693 × 0.5 | 55.35 |.24*t*^]{.m | | 4 | 25.78 | ^ = 67.6]{. | | ath | | 8 | 29.78 | math | 43.7 |.display}\ | | 12 | 26.63 |.display}\ | | | | 18 | 19.4 | | 27.2 | | | 24 | 13.26 | \ | | | | 36 | 5.88 | [*C*~*p*~ = | 10.4 | | | 48 | 2.59 |  70*e*^ − 0 | | | | 72 | 0.49 |.0693 × 1^  | 3.9 | | | | | = 65.3]{.ma | | | | | | th | 0.7 | | | | |.display}\ | | | | | | | | | | | | \ | | | | | | [*C*~*p*~ = | | | | | |  70*e*^ − 0 | | | | | |.0693 × 2^  | | | | | | = 60.9]{.ma | | | | | | th | | | | | |.display}\ | | | | | | | | | | | | \ | | | | | | [*C*~*p*~ = | | | | | |  70*e*^ − 0 | | | | | |.0693 × 4^  | | | | | | = 53.0]{.ma | | | | | | th | | | | | |.display}\ | | | | | | | | | | | | \ | | | | | | [*C*~*p*~ = | | | | | |  70*e*^ − 0 | | | | | |.0693 × 8^  | | | | | | = 40.2]{.ma | | | | | | th | | | | | |.display}\ | | | | | | | | | | | | \ | | | | | | [*C*~*p*~ = | | | | | |  70*e*^ − 0 | | | | | |.0693 × 12^ | | | | | |  = 30.5]{.m | | | | | | ath | | | | | |.display}\ | | | | | | | | | | | | \ | | | | | | [*C*~*p*~ = | | | | | |  70*e*^ − 0 | | | | | |.0693 × 18^ | | | | | |  = 20.1]{.m | | | | | | ath | | | | | |.display}\ | | | | | | | | | | | | \ | | | | | | [*C*~*p*~ = | | | | | |  70*e*^ − 0 | | | | | |.0693*t*^]{ | | | | | |.math | | | | | |.display}\ | | | +-------------+-------------+-------------+-------------+-------------+ Plot on a semi-log paper (3-cycle); extrapolate the elimination phase line and get the y-intercept, which is A = 70. From that line find out the t ½ = 10; K = 0.0693, and Ka = 0.24. The equation is: [*C*~*p*~ = 70(*e*^ − 0.0693*t*^−*e*^ − 0.24*t*^)]{.math.inline} [\$AUC = Intercept(A)\\left( \\frac{1}{K} - \\frac{1}{K\_{a}} \\right)\$]{.math.inline} or Intercept(A) ([\$\\frac{Ka - K}{K\_{a}K}\$]{.math.inline}) \ [\$\$AUC = 70\\left( \\frac{1}{0.0693} - \\frac{1}{0.24} \\right) = 718.2\\ \\ \$\$]{.math.display}\ **Figure**. For example 1. ***[Flip-Flop of K~a~(absorption rate constant) and K(elimination rate constant)]*:** In the previous example, we have considered that Ka \>\> K (as such, e*^-Kat^* becomes almost zero leaving e*^-Kt^* as terminal phases). In certain cases (e.g. isoproterenol), however, after IV bolus administration, we see it the other way around, i.e., Ka \K (Which is usually true initially), \ [\$\$C\_{\\max} = \\frac{D\_{0}}{V}e\^{- Kt\_{\\max}}\\text{\\ \\ \\ }\$\$]{.math.display}\ ***Note: C~max~ is a function of [D~0~, Ka, V, and K]*** ***[AUC: ]*** \ [\$\$AUC = \\int\_{0}\^{\\infty}{C\_{p} = \\frac{K\_{a}D\_{0}}{V\\left( K - K\_{a} \\right)}\\left( \\frac{1}{K\_{a}} - \\frac{1}{K} \\right)}\\text{\\ \\ }\$\$]{.math.display}\ \ [\$\$AUC\\ = A\\left\\lbrack \\frac{1}{K} - \\frac{1}{K\_{a}} \\right\\rbrack\$\$]{.math.display}\ \ [\$\$AUC = \\frac{D\_{0}}{\\text{KV}} = \\frac{D\_{0}}{\\text{CL}}\\text{\\ \\ }\$\$]{.math.display}\ For two-compartment IV administration, [\$AUC = \\frac{C\_{0}}{K}\\text{\\ \\ }\$]{.math.inline} For two-compartment extravascular administration, [\$AUC = \\frac{\\text{FD}\_{0}}{\\text{CL}}\\text{\\ \\ }\$]{.math.inline} **Note**: 1. AUC is the measurement of the extent of bioavailability (either in one compartment, or, two compartments after IV, oral, or im administrations) of a drug. It reflects the total amount of the drug which reaches the systemic circulation. 2. Under linear conditions in which CL is constant, AUC is directly proportional to dose. 3. AUC is independent of the route of administration, once the drug reaches the circulation, the AUC value will be the same no matter after IV, oral, im, etc. 4. [AUC is equal to the ratio of intercept to slope]. [\$AUC = \\frac{C\_{0}}{K}\\text{\\ \\ }\$]{.math.inline}= [\$\\frac{D\_{0}}{\\text{CL}}\$]{.math.inline} ***[Half-life]*:** \ [\$\$t\_{\\frac{1}{2},elimination} = \\frac{0.693}{K}\$\$]{.math.display}\ \ [\$\$t\_{\\frac{1}{2},\\ \\ absorption} = \\frac{0.693}{K\_{a}}\$\$]{.math.display}\ ***[CL]***: \ [\$\$CL = \\frac{D\_{0}}{\\text{AUC}} = KV\\ \\ \$\$]{.math.display}\ **[Complications (lag time and bioavailability)]**: \(i) ***Lag time***: the time delay prior to the commencement of first-order drug absorption. This lag time may be due to physiologic factors such as gastric emptying time and intestinal motility. Suppose the absorption is delayed until t = t~lag~(lag time).  **Figure**. Lag time \ [\$\$C\_{p} = \\frac{K\_{a}D\_{0}}{V\\left( K - K\_{a} \\right)}\\left( e\^{- K\_{a}\\left( t - t\_{\\text{lag}} \\right)} - e\^{- K\\left( t - t\_{\\text{lag}} \\right)} \\right)\\text{\\ \\ }\$\$]{.math.display}\ ***(ii) Bioavailability***: Certain drugs are not absorbed 100% of the dose after oral administration. In this case, the fraction of the dose absorbed should be taken into consideration, instead of the dose administered. If F is the fraction absorbed, \ [\$\$C\_{p} = \\frac{FD\_{0}K\_{a}}{V\\left( K - K\_{a} \\right)}\\left( e\^{- k\_{a}t} - e\^{- Kt} \\right)\\text{\\ \\ }\$\$]{.math.display}\ \ [\$\$AUC = \\frac{FD\_{o}}{\\text{CL}}\$\$]{.math.display}\ \ [\$\$t\_{\\max} = \\frac{\\ln{K\_{a} - \\ln K}}{K\_{a} - K}\$\$]{.math.display}\ \ [\$\$C\_{\\max} = \\frac{FD\_{0}}{V}e\^{- Kt\_{\\max}}\$\$]{.math.display}\ ***Note: Except t~max~, other parameters decrease, when 100% of the dose is not absorbed.*** \ [\$\$Absolute\\ bioavailabiliy = F = \\frac{\\text{AUC}\_{\\text{oral}}/\\text{Dose}\_{\\text{oral}}}{\\text{AUC}\_{\\text{IV}}/\\text{Dose}\_{\\text{IV}}}\\text{\\ \\ }\$\$]{.math.display}\ \ [\$\$\\text{AUC}\_{\\text{oral}} = \\frac{FD\_{0}}{\\text{CL}};\\ \\ \\ \\text{AUC}\_{\\text{IV}} = \\frac{D\_{0}}{\\text{CL}}\\ \$\$]{.math.display}\ \ [\$\$Relative\\ Bioabailability = \\frac{\\text{AUC}\_{\\text{test}}/\\text{Dose}\_{\\text{test}}}{\\text{AUC}\_{\\text{reference}}/\\text{Dose}\_{\\text{reference}}}\\text{\\ \\ }\$\$]{.math.display}\ ***Note**: The basic assumption with these formulae is that the CL value is constant (linear PK)*. ***\*\*\*Two major issues in absorption kinetics: absorption rate and doses.*** 1. Ka value: when the Ka value decreases, T~max~ increases and C~max~ decreases, yet AUC is unchanged 2. Dose: when the dose increases, AUC and C~max~ increase proportional to the dose, yet, T~max~ is unchanged. 3. C~max~ is a function of Ka and dose 4. [The absorption process does not stop at T~max~. It just slows down after T~max~.] Definition of Linear Pharmacokinetics: 1. For IV: the disposition parameters are not changed at different doses One compartment: K, Cl, V~D~ Two compartments: K~12~, K~21~, K~el~, V~d~c, V~d~ss, CL Non-compartment: MRT, V~d~ss, CL 2. For absorption: the disposition parameters are not changed at different doses and for a particular formulation the absorption parameters, Ka and F, are not changed at different doses. (However, those absorption parameters may be different for different formulations.) **[Example 2]**: A 100 mg drug was administered to a subject orally with 50 mL of water. Blood samples were analyzed and the following parameters were obtained: A=70, Ka = 0.231, and K = 0.0693. Assuming complete absorption, calculate AUC, C~max~, t~max~, CL, V, absorption and elimination half-lives. \ [*V* = 2.04 *Liters*  ]{.math.display}\ \ [*t*~max~ = 7.4 hr ]{.math.display}\ \ [*t*~1/2, *elimination*~ = 10 hr ]{.math.display}\ \ [*t*~1/2, *absorption*~ = 3 hr ]{.math.display}\ \ [*AUC* = 707.3 *mcg*.*hr**hr*   ]{.math.inline} , CL = D/AUC [*C*~max~ = 29.35 *mcg**mL*  ]{.math.display}\ \ [\$\$\\frac{\\text{CL}}{F} = 0.69\\ L/hr\\ \\ \$\$]{.math.display}\ \ [\$\$\\frac{\\text{Vd}}{F} = 49.5\\ L\\ \\ \$\$]{.math.display}\ \ [*t*~1/2~ = 49.5 hrs. ]{.math.display}\ \ [*t*~max~ = 7.5 hrs]{.math.display}\ \ [*C*~max~ = 4.60 *mcg**mL* ]{.math.inline} , [\$C\_{\\max} = \\frac{\\text{FD}\_{0}}{V}e\^{- Kt\_{\\max}}\$]{.math.inline} where F = 0.80 [T~max~ = 7.5 hrs] \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \(k) What are the estimated plasma concentrations at 6 hours and 120 hours (5 days) post-dosing (250mg)? \ [*C*~*p*~ = 4.5 *mcg**mL*  ]{.math.display}\ \(l) What are the estimated plasma concentrations at 6 hrs and 120 hrs (5 days) after a 2-capsule dose (500 mg in total) assuming 80% bioavailability? \ [*C*~*p*~ = 9.04 *mcg**mL*]{.math.display}\ \(m) An available soft gelatin marketed product with identical strength claims that its Ka is greater than the above capsule product, yet with identical bioavailability. Indicate the direction (increase, decrease, remains the same) of the following parameters of this soft gelatin product when compared to the above capsule: T~max~: INCREASE/DECREASES/SAME C~max~: INCREASE/DECREASE/SAME F: INCREASE/DECREASE/SAME AUC: INCREASE/DECREASE/SAME K: INCREASE/DECREASE/SAME Vd: INCREASE/DECREASE/SAME CL: INCREASE/DECREASE/SAME \(n) Another marketed tablet with identical strength claims that its Ka is identical to the above capsule product, yet with less bioavailability (only 70%). Indicate the direction (increase, decrease, remains the same) of the following parameters of this tablet product when compared to the above capsule values: T~max~: INCREASE/DECREASE/SAME C~max~: INCREASE/DECREASE/SAME F: INCREASE/DECREASE/SAME AUC: INCREASE/DECREASE/SAME K: INCREASE/DECREASE/SAME Ka: INCREASE/DECREASE/SAME Vd: INCREASE/DECREASE/SAME CL: INCREASE/DECREASE/SAME **[Example 4]**: \(a) Plasma concentration of phenobarbital (mcg/mL) after an oral dose of a 150-mg tablet in a patient can be described by the following equation: 2.79(e^-0.1386t^ -- e^-10t^). Calculate its absorption and elimination half-lives. (unit for t is day) [so to get the answer in hrs, you need to multiply the half-life by 24] Absorption half-life = 1.7 hrs; Elimination half-life = 119.7 hrs. \(b) In another study on the same patient, plasma concentrations of Phenobarbital (mcg/mL) after 100-mg iv bolus dose can be described by the following equation: 2.04 x e^-0.1386t^ (unit of t is day). Calculate its absolute bioavailability. F = 89% \(c) Based upon the information obtained in (b), calculate the values of Vd and CL. Vd = 49.01 L; CL = 6.79 L/hr. **[Example 5: ]**  **Figure**. Use this figure for problems 5, 6, & 7 5\. The same doses (500 mg) of three different dosage forms of a lipid-soluble drug are administered orally. Product X is an aqueous solution, product Y is a suspension, and product Z is a tablet. If the clearance values of the three formulations are compared, which of the following is true? (Assume linear pharmacokinetics). CL= Dose/AUC \(A) X \> Y \> Z \(B) Y \> Z \> X \(C) Z \> Y \> X \(D) X = Y = Z **[Example 6:]** If the Ka (absorption rate constant) of the three dosage forms in example 7-5 are compared, which of the following will the correct: Ka increses if Cmax increase and Tmax decreses \(A) X \> Y \> Z \(B) Y \> Z \> X \(C) Z \> Y \> X \(D) X = Y = Z **[Example 7:]** If bioavailability (F) is compared for the three dosage forms, which of the following will be true?F increses if AUC increases \(A) X \> Y \> Z \(B) Y \> Z \> X \(C) Z \> Y \> X \(D) X = Y = Z **[Example 8]**: 500 mg of a lipid-soluble drug has been formulated as tablet (X) and capsule (Y). After oral administration to a healthy individual, the following plasma concentration-time profile was obtained. Assuming linear pharmacokinetics, compare the absorption rate constants (Ka) of the two formulations. Since tmax is same, Ka will also be same \(A) X \> Y \(B) Y \> X \(C) X = Y \(D) cannot be determined. **[Example 9]**: The plasma concentrations-time profile of a drug after oral administration has been plotted as follows (P). If the drug is given by iv bolus, which of the following profiles (X, Y, and Z) is impossible to occur? AUC of X and Y are lower than P(oral) so, Z is the only possible IV bolus.  \(A) X \(B) Y \(C) Z \(D) X and Y \(E) Y and Z