Pharmacokinetic and comparment model intoduction.pptx

What is mathematical model?
What is compartment model?
Classification of pharmacokinetic models.
Classification of compartment models.
One-compartment open model.
Two-compartment open model.

1. Pharmacokinetics & Compartment Model D r . Rameshwar Dass Guru Gobind Singh College of Pharmacy, Yamunanagar Dr. Rameshwar Dass

2. Pharmacokinetics Basic considerations, Pharmacokinetic models, Compartment modeling : One compartment model- IV bolus, IV infusion, Extra-vascular. Multi compartment model: Two compartment - model in brief, Non-linear pharmacokinetics : Cause of non-linearity, Michaelis – Menten equation, Estimation of k max and v max . Drug interactions : Introduction and the effect of Protein-binding interactions Tissue-binding interactions Cytochrome p450- based drug interactions, Drug interactions linked to transporters. Dr. Rameshwar Dass

3. Content What is mathematical model? What is compartment model? Classification of pharmacokinetic models. Classification of compartment models. One compartment open model. Two compartment open model. References. Dr. Rameshwar Dass

4. Mathematical Model Pharmacokinetic (PK) models are mathematical models used to describe and predict the time course of absorption, distribution, metabolism, and excretion (ADME) of a drug in the body. A mathematical description of biologic system and is used to express quantitative relationships. It is a collection of mathematical quantities, operations and relations together with their explanations and they must be realistic and practical. It is a hypothesis that employs mathematical terms to concisely describe quantitative relationships. Dr. Rameshwar Dass

5. Qualities of a Mathematical Model Validity : It should have practical applicability and should be valuable in describing events chosen accurately with high precision. Prediction ability : These models predict the qualitative and quantitative changes in the parameters that are rate constants and half lives of drugs. Computability: has the ability to solve a problem in an effective manner using mathematical logics. Consistency of results : Reproducibility is an important quality of a mathematical model. Dr. Rameshwar Dass

6. Applications of Pharmacokinetic Models These models help understand how a drug moves through the body, which is important for determining appropriate dosing regimens, assessing drug safety, and optimizing therapeutic effectiveness . Characterizing the behaviour of drugs in patients. Calculating the optimum dosage regimens for individual patients. Evaluating the bioequivalence between different formulations of same drug. Determining the influence of altered physiology or disease state on drug ADME. Explaining the drug interactions Dr. Rameshwar Dass

7. Types of Pharmacokinetic Models Pharmacokinetic models: Compartmental models Physiological models Non compartmental analysis Dr. Rameshwar Dass

8. Compartmental Models A group of tissues with similar blood flow and drug affinity. It is physiologic or anatomic region. Compartment is the traditional and most widely used approach to pharmacokinetic characterization of drug. These models simply interpolate the experimental data and allow on empirical formula to estimate drug concentration with time. Dr. Rameshwar Dass

9. Assumptions of Compartmental Models The body is represented as a sequence of compartments arranged in series or parallel to each other. The rate of drug movement between compartment is described by first order kinetics. Rate constants are used to represent rate of entry into and exit from compartment. A statistical analysis of plasma concentration with time data is another method used to find out numbers of compartments. Dr. Rameshwar Dass

10. Properties of Compartment Modelling It is a simple and flexible approach and is widely used. It gives a visual representation of various rate processes involved in drug disposition. It is useful in predicting drug concentration time profile in both normal and pathological conditions. It is useful in relating plasma drug levels in therapeutic and toxic effects in body. Its simplicity allows for easy tabulation of Vd,t1/2 etc. Dr. Rameshwar Dass

11. applications of Compartment Modelling 1. Drug Development and Optimization: Dose Optimization : Helps determine the appropriate dosing regimen by predicting drug concentration profiles in the body over time. Drug Formulation : Assists in evaluating how different formulations affect drug absorption and bioavailability. Bioequivalence Studies : Used to compare the pharmacokinetics of a generic drug with the reference (brand-name) drug. 2. Clinical Pharmacokinetics: Therapeutic Drug Monitoring (TDM) : Guides dosage adjustments in patients to achieve optimal drug concentrations, especially for drugs with narrow therapeutic windows. Adjusting for Special Populations : Helps tailor drug dosing in special populations such as the elderly, pediatric patients, or those with renal or hepatic impairments . Dr. Rameshwar Dass

12. applications of Compartment Modelling 3. Prediction of Drug Interactions: Drug-Drug Interactions : Compartment models can predict how the presence of another drug affects the pharmacokinetics of the primary drug, helping to prevent adverse effects. Food-Drug Interactions : Can be used to assess how food intake affects drug absorption and metabolism. 4 . Simulation and Scenario Analysis: Virtual Clinical Trials : Allows simulation of different dosing scenarios to predict outcomes without conducting physical trials, saving time and resources. Scenario Testing : Evaluates how changes in physiological or pathological conditions (e.g., organ dysfunction) impact drug kinetics. 5. Toxic kinetics and Safety Assessment: Toxicity Prediction : Predicts drug accumulation and potential toxic effects, especially for drugs with long half-lives or complex kinetics. Safety Margins : Helps establish safe dosing ranges and identify potential overdose risks . Dr. Rameshwar Dass

13. applications of Compartment Modelling 6. Pharmacokinetic-Pharmacodynamic (PK-PD) Modeling: Linking Drug Exposure to Effect : Compartment models are often used in conjunction with pharmacodynamic models to predict the time course of a drug's effect based on its concentration in the body. Efficacy and Safety Optimization : Enables the prediction of therapeutic and adverse effects, guiding dose adjustments. 7 . Regulatory Submissions: Support for Regulatory Approvals : Compartment models are often included in regulatory submissions to demonstrate a drug’s pharmacokinetic profile and justify dosing recommendations. Dr. Rameshwar Dass

14. Types of Compartment Models Based on arranged of compartment either parallel or series, models are classified as: Mammillary model Catenary model Open model Closed model Dr. Rameshwar Dass

15. Mammillary Model This is the most common compartment used in pharmacokinetics. The model consists of one or more peripheral compartments connected to a central compartment. The central compartment consists of plasma and highly perfused tissues in which drug distributes rapidly. No. of rate constants in a particular compartment model is given by R. For intravenous administration R=2n-1. For extravascular administration R=2n. where n= no. of compartments Dr. Rameshwar Dass

16. Mammillary Model Muscle Plasma Fat Input of Drug Excretion of Drug Dr. Rameshwar Dass

17.   Depiction of Various Models X = VC Excretion of Drug X = VC Excretion of Drug ka k E   Dr. Rameshwar Dass

18. Depiction of Various Models Two compartment open model i.v. injection Two compartment open model with first order absorption X1 = VC1 K 13 X2 = VC2 K 12 K 21 X1 = VC1 K 13 X2 = VC2 K 12 K 21 K a Dr. Rameshwar Dass

19. Catenary Model In this model compartments are joined to one another in a series like compartments. This model is directly linked to blood and this model is rarely used. X 1 X 2 X 3 Excretion of Drug Ka K 12 K 21 K 13 K 31 K 14 Dr. Rameshwar Dass

20. Advantages It gives a visual representation of various rate processes involve in drug disposition. It show how many rate constants are necessary to describe these processes. It enables the pharmacokineticist to write differential equations for each of the rate processes in order to describe drug-concentration changes in each compartment. It is useful in predicting drug concentration-time profile in both normal physiological and in pathological conditions. It is important in the development of dosage regimens. Dr. Rameshwar Dass

21. Disadvantages The compartments and parameters bear no relationship with the physiologic functions or the anatomic structure of the species; several assumptions have to be made to facilitate data interpretation. Extensive efforts are required in the development of an exact model that predicts and describes correctly the ADME of a certain drug. The model is based on curve fitting of plasma concentration with complex multi-exponential mathematical equations. The model may vary within a study population. The approach can be applied only to a specific drug under study. The drug behavior within the body may fit different compartmental models depending on the route of administration. Dr. Rameshwar Dass

22. Compartment Model Intravenous Bolus Mr Rameshwar Dass GGSCOP, Yamunanagar Dr. Rameshwar Dass

23. One Compartment Open Model It is the simplest model which depicts, the body as a single, kinetically homogeneous unit that has no barriers to the movement of drug. It applies only to those drugs that distribute rapidly though out the body. The drug concentration in plasma represents the drug conc. in all body. The term “open” indicates that the input and output are unidirectional and must be eliminated from the body. Dr. Rameshwar Dass

24. Flow Chart One compartment open models can be defined: Intravenous bolus administration Continuous intravenous infusion Extra vascular zero-order absorption Extra vascular first-order absorption. X= Vd.C p Drug (Absorption) Drug ( Elimination ) K a K E one-compartment open model showing input and output process Dr. Rameshwar Dass

25. Intravenous Bolus Administration Drug distributes rapidly in the body given by rapid I.V. injection (2 to 3 minutes) X= Vd.C Drug ( Elimination ) K E Elimination Rate Constant Change in amount of drug in the body = ( dX / dt ) where X = amount of drug in the body remained to be eliminated at time, t. Log C t Dr. Rameshwar Dass

26. Apparent Volume of Distribution ( V d ) Volume of distribution governs the plasma concentration of the drug after a given dose. The volume of distribution help in estimating the amount of drug in the body using the conc. of drug found in the sampling compartment. It is called the apparent volume of distribution because the value of the Vd does not have a true physiologic meaning in terms of an anatomic space Method-I : It is determined by administering it by rapid i.v . injection Method-II : V d can be determined by another way if the AUC and the first order elimination rate constant, k E is known Dr. Rameshwar Dass

27. Significance of V d The apparent V D is dependent on C p . for a given dose, if a very small C p may occur in the body means the drug in peripheral tissues and organs Drugs with a large apparent V D are more concentrated in extra-vascular tissues and less concentrated intra- vascularly If a drug is highly bound to plasma proteins or remains in the vascular region, then C p will be higher, resulting in a smaller apparent V D If V D is a very large number— ie , >100% of body weight—then it may be assumed that the drug is concentrated in certain tissue compartments. Thus, the apparent V D is a useful parameter in considering the relative amounts of drug in the vascular and in the extra-vascular tissues If the value for the apparent V D of a drug has the same value as a real anatomic volume. Then a drug is considered to be distributed in a true physiologic volume, then an investigation is needed to test this hypothesis. Dr. Rameshwar Dass

28. For example of V d In edematous conditions, the total body water and total extracellular water increases; this is reflected in a larger apparent V D value for a drug that is highly water soluble. Similarly, changes in total body weight and lean body mass (which normally occur with age) may also affect the apparent V D . Dr. Rameshwar Dass

29. Problems/Assignment An antibiotic has a volume of distribution of 10 L and a k of 0.2 hr – 1 . A steady-state plasma concentration of 10 g/ mL is desired. The infusion rate needed to maintain this concentration can be determined as follows. An infinitely long period of time is needed to reach steady-state drug levels. However, in practice it is quite acceptable to reach 99% C SS ( ie , 99% steady-state level). Using Equation 5.6, we know that the steady state level is? A patient was given an antibiotic ( t 1/2 = 6 hr) by constant IV infusion at a rate of 2 mg/hr. At the end of 2 days, the serum drug concentration was 10 mg/L. Calculate the total body clearance Cl T for this antibiotic. Dr. Rameshwar Dass

30. Compartment Model Intravenous Infusion Mr Rameshwar Dass GGSCOP, Yamunanagar Dr. Rameshwar Dass

31. Intravenous Infusion Intravenous (IV) drug solutions may be infused slowly through a vein into the plasma at a constant or zero-order rate. The main advantage Infusion allows precise control of plasma drug conc. to fit the individual needs of the patient. (narrow therapeutic window ( eg , heparin) IV infusion maintains constant MEC by eliminating wide fluctuations between the peak and trough Cp. Example: the IV infusion of drugs, such as antibiotics, electrolytes and nutrients. Css Elimination of drug from the plasma is a first-order process The infused drug follows zero-order input and first-order output Amount of drug in the body at any time Dr. Rameshwar Dass

32. Loading Dose A physician wants to administer an anesthetic agent at a rate of 2 mg/hr by IV infusion. The elimination rate constant is 0.1 hr – 1 , and the volume of distribution (one compartment) is 10 L. What loading dose should be recommended if the doctor wants the drug level to reach 2 g/ mL immediately? What is the concentration of a drug 6 hours after administration of a loading dose of 10 mg and simultaneous infusion at 2 mg/hr (the drug has a t 1/2 of 3 hr and a volume of distribution of 10 L)? A patient was infused for 6 hours with a drug ( k = 0.01 hr – 1 ; V D = 10 L) at a rate of 2 mg/hr. What is the concentration of the drug in the body 2 hours after cessation of the infusion Dr. Rameshwar Dass

33. Compartment Model Extra-vascular Delivery Mr Rameshwar Dass GGSCOP, Yamunanagar Dr. Rameshwar Dass

34. Extra-vascular Delivery The particularly oral dosing, are important and popular means of drug administration. Unlike IV administration, in which the drug is injected directly into the plasma Pharmacokinetic models after extravascular drug administration must consider systemic drug absorption from the site of administration Drug absorption from the gastrointestinal (GI) tract or from any other extravascular site is dependent on The physicochemical properties of the drug, The dosage form used, and The anatomy and physiology of the absorption site X=C. Vd Drug (Absorption) Drug ( Elimination ) K a K E Dr. Rameshwar Dass

35. Extra-vascular Delivery X=C. Vd Drug (Absorption) Drug ( Elimination ) K a K E Time (Hrs.) Plasma Conc.) Absorption phase Post Absorption phase Elimination phase At Absorption phase At peak drug concentration Post Absorption phase Elimination phase Dr. Rameshwar Dass

36. Zero-Order Absorption Model Zero-order absorption (controlled-release delivery system) Absorbed constant rate, k The rate of first-order elimination at any time is equal to D B k . Integration of this equation 1 with substitution of V D C p for D B produces Dr. Rameshwar Dass

37. First-Order Absorption Model First-order absorption (immediate-release delivery system) Absorbed constant rate, k The rate of first-order elimination at any time is equal to D B k . Disappearance of drug in GIT Integration of the differential equation 1 Elimination is described by a first-order rate process Time (Hrs.) Plasma Conc.(µg/ml) Dr. Rameshwar Dass

38. Frequently Asked Questions What is the difference between a rate and a rate constant? Why does k always have the unit 1/time ( eg , hr –1 ), regardless of what concentration unit is plotted? If a drug is distributed in the one-compartment model, does it mean that there is no drug in the tissue? How is clearance related to the volume of distribution and k ? If we use a physiologic model, are we dealing with actual volumes of blood and tissues? Why do we still use volumes of distribution that often are greater than the real physical volume? A 70-kg volunteer is given an intravenous dose of an antibiotic, and serum drug concentrations were determined at 2 hours and 5 hours after administration. The drug concentrations were 1.2 and 0.3 g/ mL , respectively. What is the biologic half-life for this drug, assuming first-order elimination kinetics? Dr. Rameshwar Dass

39. Multi-compartment Models Intravenous Bolus Administration Mr. Rameshwar Dass GGSCOP, Yamunanagar Dr. Rameshwar Dass

40. Pharmacokinetic Model Compartment models are classical pharmacokinetic models that simulate the kinetic processes of drug absorption, distribution and elimination with little physiologic detail This may be used to represent drug distribution and elimination in the body. Ideally, a model should mimic closely the physiologic processes in the body Dr. Rameshwar Dass

41. Grouping of Tissues According to Blood Supply General Grouping of Tissues Blood Supply Tissue Group Percent Body Weight Highly perfuse Heart, brain, hepatic-portal system, kidney, and endocrine glands 9   Skin and muscle 50   Adipose (fat) tissue and marrow 19 Slowly perfuse Bone, ligaments, tendons, cartilage, teeth, and hair 22 Dr. Rameshwar Dass

42. Two Compartment Model It assumes that all transfer rate processes for the passage of drug into or out of individual compartments are first-order processes The plasma level–time curve for a drug that follows a two-compartment model shows that the Cp declines biexponentially as the sum of two first-order processes—distribution and elimination The central compartment (blood, extracellular fluid, and highly perfused tissues) The tissue or peripheral compartment, contains tissues in which the drug equilibrates more slowly Dr. Rameshwar Dass

43. Two Compartment Model Central compartment X= Vd.C Peripheral Compartment Xt = Vd.Ct k12 k21 k10 Central compartment X= Vd.C Peripheral Compartment Xt = Vd.Ct k12 k21 k20 Central compartment X= Vd.C Peripheral Compartment Xt = Vd.Ct k12 k21 k10 k20 Dr. Rameshwar Dass

44. Intravenous Bolus The plasma level–time curve for a drug may be divided into two parts, A distribution phase and An elimination phase The rate of drug change in and out of the tissues: Plasma (Central) Tissue Time Plasma Conc. where X = dose given intravenously, t = time after administration of dose, and a and b are constants that depend solely on k 12 , k 21 , and k . Dr. Rameshwar Dass

45. Intravenous Bolus where X = dose given intravenously, t = time after administration of dose, and a and b are constants that depend solely on k 12 , k 21 , and k . The values for these micro-constants cannot be determined by direct measurement but can be estimated by a graphic method. Dr. Rameshwar Dass

46. Method of Residuals 100 mg of a drug was administered by rapid IV injection to a 70-kg, healthy adult male. Blood samples were taken periodically after the administration of drug, and the plasma fraction of each sample was assayed for drug. The following data were obtained: The rapid distribution phase is confirmed with the constant a being larger than the rate constant b . Therefore, at some later time the term Ae –at will approach zero Plasma Conc. Time a b Dr. Rameshwar Dass

47. Method of Residuals Time (hr) Plasma Concentration (g/ mL ) 0.25 43.00 0.5 32.00 1.0 20.00 1.5 14.00 2.0 11.00 4.0 6.50 8.0 2.80 12.0 1.20 16.0 0.52 Dr. Rameshwar Dass

48. Multi-compartment Models Intravenous infusion Mr. Rameshwar Dass GGSCOP, Yamunanagar Dr. Rameshwar Dass

49. Why Intravenous Infusion? When precise control of Cp require to fit the individual needs of the patient The drugs with a narrow therapeutic window (heparin), IV infusion maintains an effective constant Cp If the IV infusion of drugs, such as antibiotics co-administered with electrolytes and nutrients If the duration of drug therapy may be maintained or terminated as needed using IV infusion. Dr. Rameshwar Dass

50. Intravenous Infusion The distributions of theophylline and lidocaine in humans are described by the two-compartment open model It requires a distribution and equilibration of the drug before a Css reached no net change in the amount of drug in the tissue occurs at Css The time needed to reach a Css depends entirely on the distribution t1/2 of drug Dr. Rameshwar Dass

51. Intravenous Infusion where a and b are hybrid rate constants and R is the rate of infusion. At steady state ( ie , t = ∞), Equation reduces to Loading Dose Plus IV Infusion Drugs with long half-lives require a loading dose to more rapidly attain Css . It is clinically desirable to achieve rapid therapeutic drug levels. Initially it produce either slightly higher or lower than Css no loading dose LD= R / b (rapid infusion). Loading dose = R / b (slow infusion) C time Dr. Rameshwar Dass

52. V d at Steady State At ss , the rate of drug entry into the tissue compartment from the central compartment is equal to the rate of drug exit from the tissue compartment into the central compartment This volume is not useful in calculating the amount of drug in the body during pre-steady state It yields the amount of drug in the body at SS To determine the loading drug dose necessary to upload the body to a desired Cp Dr. Rameshwar Dass

53. Frequently Asked Questions 1. What is the main reason for giving a drug by slow IV infusion? 2. Why do we use a loading dose to rapidly achieve therapeutic concentration for a drug with a long elimination half-life, instead of increasing the rate of drug infusion or increasing the size of the infusion dose? 3. What are some of the complications involved with IV infusion? Dr. Rameshwar Dass

54. Frequently Asked Questions A female patient (35 years old, 65 kg) with normal renal function is to be given a drug by IV infusion. According to the literature, the elimination half-life of this drug is 7 hours and the apparent V D is 23.1% of body weight. The pharmacokinetics of this drug assumes a first-order process. The desired steady-state plasma level for this antibiotic is 10 g/ mL. Assuming no loading dose, how long after the start of the IV infusion would it take to reach 95% of the C SS? What is the proper loading dose for this antibiotic? What is the proper infusion rate for this drug? What is the total body clearance? If the patient suddenly develops partial renal failure, how long would it take for a new steady-state plasma level to be established (assume that 95% of the C SS is a reasonable approximation)? If the total body clearance declined 50% due to partial renal failure, what new infusion rate would you recommend to maintain the desired steady-state plasma level of 10 g/ mL ? Dr. Rameshwar Dass

55. Multi-compartment Models Extra-vascular Mr. Rameshwar Dass GGSCOP, Yamunanagar Dr. Rameshwar Dass

56. Significance of Absorption Rate Constants The overall rate of systemic drug absorption from an orally administered solid dosage form encompasses many individual rate processes, including dissolution of the drug, GI motility, blood flow, and transport of the drug across the capillary membranes and into the systemic circulation. The rate of drug absorption represents the net result of all these processes. The selection of a model with either first-order or zero-order absorption is generally empirical. The actual drug absorption process may be zero-order, first-order, or a combination of rate processes that is not easily quantitated . For many immediate-release dosage forms, the absorption process is first-order due to the physical nature of drug diffusion. For certain controlled-release drug products, the rate of drug absorption may be more appropriately described by a zero-order rate constant. Dr. Rameshwar Dass

57. Significance of Absorption Rate Constants k a is useful in designing a multiple-dosage regimen. k a and k allows for the prediction of peak and trough plasma drug conc following multiple dosing. In bioequivalence studies, drug products are given in chemically equivalent ( ie , pharmaceutical equivalents) doses, and the respective rates of systemic absorption may not differ markedly. Therefore, for these studies, t max , or time of peak drug concentration, can be very useful in comparing the respective rates of absorption of a drug from chemically equivalent drug products. Dr. Rameshwar Dass

58. Oral Dose of a Drug Follow Two-compartment Model kinetics: The amount of drug absorbed is calculated as the sum of Central compartment X= Vd.C Peripheral Compartment Xt = Vd.Ct k12 k21 k Drug (Absorption) K a Dr. Rameshwar Dass

59. Oral Dose of a Drug Follow Two-compartment A plot of the fraction of drug unabsorbed, (1 – Ab / Ab ∞ ), versus time – k a /2.3 = slope (the value for the absorption rate constant) Dr. Rameshwar Dass

60. Cumulative Relative Fraction Absorbed The fraction of drug absorbed at any time t ( eq ” 5) may be summed or cumulated for each time period for which a plasma drug sample was obtained Cumulative Relative Fraction Absorbed (CRFA): In the Wagner–Nelson equation, Ab / Ab ∞ or CRFA =1, (or 100%), even not be 100% systemically bioavailable. Ab % is based on the total amount of drug absorbed ( Ab ∞ ) rather than the dose X . Because the amount of the drug ultimately absorbed, Ab ∞ , is equal to k [AUC] ∞ , the numerator will always equal the denominator, whether the drug is 10, 20, or 100% bioavailable. % absorbed based on Ab / Ab ∞ is therefore different from the real % absorbed unless F = 1. However, for the calculation of k a , the method is acceptable. Dr. Rameshwar Dass

61. Cumulative Relative Fraction Absorbed To determine the real %absorbed, a modification of the Wagner–Nelson equation was suggested by . A reference drug product was administered and plasma drug concentrations were determined over time. CRFA was then estimated by dividing Ab / Ab ∞ ref , where Ab is the cumulative amount of drug absorbed and Ab ∞ ref is the cumulative final amount of drug absorbed from a reference. In this case, the denominator of Equation 7 is modified as follows: where k ref and [AUC] ∞ ref from the reference product. The terms in the numerator of Equation 7 refer to the product, as in Equation 6. Each fraction cumulated Ab and plotted against the time interval of sample was obtained . Dr. Rameshwar Dass