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Pharmacokinetic models

  2. 2. OVERVIEW • Basic considerations in pharmacokinetics • Compartment models • One compartment model • Assumptions • Intravenous bolus administration • Intravenous infusion • Extravascular administration (zero order and first order absorption model) • Multi-compartment model
  3. 3. BASIC CONSIDERATIONS IN PHARMACOKINETICS • Pharmacokinetic parameters • Pharmacodynamic parameters • Zero, first order & mixed order kinetic • Rates and orders of kinetics • Plasma drug conc. Time profiles • Compartmental models – physiological model • Applications of pharmacokinetics • Non compartment model
  4. 4. Common units in Pharmacokinetics Pharmacokinetic parameter Abbreviation Fundamental units Units example 1. Area under the curve AUC Concentration x time μg x hr/mL 2. Total body clearance ClT Volume x time Litres/time 3. Renal clearance ClR Volume x time Litres/time 4. Hepatic clearance ClH Volume x time Litres/time 5. Apparent volume of distribution VD Volume Litres 6. Vol. of distribution at steady state VSS Volume Litres 7. Peak plasma drug concentration CMAX Concentration mg/L 8. Plasma drug concentration CP Concentration mg/L 9. Steady-state drug concentration Css Concentration mg/L 10. Time for peak drug concentration TMAX Time Hr 11. Dose DO Mass mg 12. Loading dose DL Mass mg 13. Maintenance dose DM Mass mg 14. Amount of drug in the body DB Mass Mg 15. Rate of drug infusion R Mass/time mg/hr 16. First order rate constant for drug absorption Ka 1/time 1/hr 17. Zero order rate constant for drug absorption KO Mass/time mg/hr 18. First order rate constant for drug elimination K 1/time 1/hr 19. Elimination half-life t½ Time hr
  5. 5. Mixed Order Kinetics  Kinetics of a pharmacokinetic process changes from First order to Zero order with increasing dose or chronic medication.  Deviations from original Linear kinetic profile – Non Linear kinetics.  Dose dependent kinetics  Seen when P’kinetic process Carriers / Substrates Capacity Limited – get saturated at Higher drug Conc. Michaelis – Menten Kinetics  Describes velocity of Capacity limited, enzyme reactions and non linear pharmacokinetics
  6. 6. Some examples; Absorption (Vitamin C), Distribution (Naproxen), and Elimination MICHAELIS MENTON EQUATION -DC/DT = VMAX . C / KM + C KM = Michaelis constant VMAX = Theoretical maximum Rate of process (Riboflavin)
  7. 7. PLASMA DRUG CONCENTRATION – TIME PROFILE Effectiveness of Dosage Regimen Concentration of Drug in the Body Conc. at Site of action Conc. in whole Blood (Plasma, Serum), Saliva, Urine, CSF PK Parameters determine drug Conc.
  9. 9. PHARMACOKINETIC PARAMETERS Three important parameters useful in assessing the bioavailability of a drug from its formulation are: 1. Peak plasma concentration ( cmax ) the point at which, maximum concentration of drug in plasma. Units : μg/ml • Peak conc. Related to the intensity of pharmacological response, it should be above MEC but less than MSC. • The peak level depends on administered dose and rate of absorption and elimination.
  10. 10. 2. Time of peak concentration (tmax ) the time for the drug to reach peak concentration in plasma (after extra vascular administration). Units : hrs • Useful in estimating onset of action and rate of absorption. • Important in assessing the efficacy of single dose drugs used to treat acute conditions (pain, insomnia ).
  11. 11. 3. Area under curve (AUC) It represents the total integrated area under the plasma level-time profile and expresses the total amount of the drug that comes into systemic circulation after its administration. Units : μg/ml x hrs • Represents extent of absorption – evaluating the bioavailability of drug from its dosage form. • Important for drugs administered repetitively for treatment of chronic conditions (asthma or epilepsy).
  12. 12. PHARMACODYNAMIC PARAMETERS 1. Minimum effective concentration (MEC) Minimum concentration of drug in plasma/receptor site required to produce therapeutic effect. • Concentration below MEC – sub therapeutic level • Antibiotics - MEC 2. Maximum safe concentration (MSC) Concentration in plasma above which adverse or unwanted effects are precipitated. • Concentration above MSC – toxic level
  13. 13. 3. Onset time Time required to start producing pharmacological response. Time for plasma concentration to reach mec after administrating drug 4. Onset of action The beginning of pharmacologic response. It occurs when plasma drug concentration just exceeds the required mec. 5. Duration of action The time period for which the plasma concentration of drug remains above MEC level. 6. Intensity of action It is the minimum pharmacologic response produced by the peak plasma conc. Of drug. 7. Therapeutic range the drug conc. Between MEC and MSC
  14. 14. CONCEPT OF “HALF LIFE”  ½ Life = how much time it takes for blood levels of drug to decrease to half of what it was at equilibrium  There are really two kinds of ½ life…  “Distribution” ½ life = when plasma levels fall to half what they were at equilibrium due to distribution to/storage in body’s tissue reservoirs.  “Elimination” ½ life = when plasma levels fall to half what they were at equilibrium due to drug being metabolized and eliminated.  It is usually the elimination ½ life that is used to determine dosing schedules, to decide when it is safe to put patients on a new drug.
  16. 16. Pharmacokinetic Modelling Compartmen t Models Non-Compartment Models Physiologic Models Caternary Model One compt Mamillary Model Multi compt Two compt i v bolus Single oral Dose i v infusion Intermittent i v infusion Multiple doses i v bolus Oral drug AUC, MRT, MAT, Cl, VSS
  17. 17. PHARMACOKINETIC MODELS Means of expressing mathematically or quantitatively, time course of drug through out the body and compute meaningful pharmacokinetic parameters. Useful in : • Characterize the behavior of drug in patient. • Predicting conc. Of drug in various body fluids with dosage regimen. • Calculating optimum dosage regimen for individual patient. • Evaluating bioequivalence between different formulation. • Explaining drug interaction. Pharmacokinetic models are hypothetical structures that are used to describe the fate of a drug in a biological system following its administration. Model • Mathematical representation of the data. • It is just hypothetical
  18. 18. WHY MODEL THE DATA ? There are three main reasons due to which the data is subjected to modelling. 1. Descriptive: to describe the drug kinetics in a simple way. 2. Predictive: to predict the time course of the drug after multiple dosing based on single dose data, to predict the absorption profile of the drug from the iv data. 3. Explanatory: to explain unclear observations.
  19. 19. PHARMACOKINETIC MODELING IS USEFUL IN :- • Prediction of drug concentration in plasma/ tissue/ urine at any point of time. • Determination of optimum dosage regimen for each patient. • Estimation of the possible accumulation of drugs/ metabolites. • Quantitative assessment of the effect of disease on drug’s adme. • Correlation of drug concentration with pharmacological activity. • Evaluation of bioequivalence. • Understanding of d/i.
  20. 20. COMPARTMENTAL MODELS • A compartment is not a real physiological or anatomic region but an imaginary or hypothetical one consisting of tissue/ group of tissues with similar blood flow & affinity. • Our body is considered as composed of several compartments connected reversibly with each other.
  21. 21. ADVANTAGES • Gives visual representation of various rate processes involved in drug disposition. • Possible to derive equations describing drug concentration changes in each compartment. • One can estimate the amount of drug in any compartment of the system after drug is introduced into a given compartment. DISADVANTAGES • Drug given by IV route may behave according to single compartment model but the same drug given by oral route may show 2 compartment behaviour. • The type of compartment behaviour i.E. Type of compartment model may change with the route of administration.
  22. 22. TYPES OF COMPARTMENT 1. Central compartment Blood & highly perfused tissues such as heart, kidney, lungs, liver, etc. 2. Peripheral compartment Poorly per fused tissues such as fat, bone, etc. MODELS: “OPEN” and “CLOSED” models: • The term “open” itself mean that, the administered drug dose is removed from body by an excretory mechanism ( for most drugs, organs of excretion of drug is kidney) • If the drug is not removed from the body then model refers as “closed” model.
  23. 23. LOADING DOSE • A drug dose does not show therapeutic activity unless it reaches the desired steady state. • It takes about 4-5 half lives to attain it and therefore time taken will be too long if the drug has a long half-life. • Plateau can be reached immediately by administering a dose that gives the desired steady state instantaneously before the commencement of maintenance dose x0. • Such an initial or first dose intended to be therapeutic is called as priming dose or loading dose x0,l.
  24. 24. CALCULATION OF LOADING DOSE • After e.V. Administration, cmax is always smaller than that achieved after i.V. And hence loading dose is proportionally smaller. • For the drugs having a low therapeutic indices, the loading dose may be divided into smaller doses to be given at a various intervals before the first maintenance dose. • A simple equation for calculating loading dose is : xo,l = css,av vd F
  25. 25. CALCULATION…., • When vd is not known, loading dose may be calculated by the following equation : xo,l = 1___________ Xo (1 – e-ket) (1 – e-kat) • Given equation applies when ka >> ke and drug is distributed rapidly. • When drug is given i.V. Or when absorption is extremely rapid, the absorption phase is neglected and the above equation reduces to accumulation index:
  26. 26. ASSUMPTIONS 1. One compartment  The drug in the blood is in rapid equilibrium with drug in the extra-vascular tissues. This is not an exact representation however it is useful for a number of drugs to a reasonable approximation. 2. Rapid mixing  We also need to assume that the drug is mixed instantaneously in blood or plasma. 3. Linear model  We will assume that drug elimination follows first order kinetics.
  28. 28. MATHEMATICALLY • This behavior can be expressed mathematically as :
  29. 29. ONE COMPARTMENT MODEL One compartment model can be defined : • One com. Open model – i.V. Bolus. • One com. Open model - cont. Intravenous infusion. • One com. Open model - extra vas. Administration (zero-order absorption) • One com. Open model - extra vas. Administration (First-order absorption ) • INTRAVENOUS (IV) BOLUS ADMINISTRATION
  30. 30. RATE OF DRUG PRESENTATION TO BODY IS: • Dx =rate in (availability)–rate out( Eli) Dt • Since rate in or absorption is absent, equation becomes dx = - rate out dt • If rate out or elimination follows first order kinetic Dx/dt = -kex (eq.1) ELIMINATION PHASE: Elimination phase has three parameters: • Elimination rate constant • Elimination half life • Clearance
  31. 31. ELIMINATION RATE CONSTANT • Integration of equation (1) • In x = ln xo – ke t (eq.2) Xo = amt of drug injected at time t = zero i.E. Initial amount of drug injected X=xo e-ket ( eq.3) • Log x= log xo – ke t 2.303 (eq.4) • Since it is difficult to directly determine amount of drug in body x, we use relationship that exists between drug conc. In plasma C and X; thus • X = vd C (eq. 5) • So equation-8 becomes log c = log co – ke t 2.303 (eq.6)
  32. 32. KE = KE + KM +KB +KL +….. (Eq.7) (KE is overall elimination rate constant)
  33. 33. ELIMINATION HALF LIFE T1/2 = 0.693 KE (eq.8) • Elimination half life can be readily obtained from the graph of log c versus t • Half life is a secondary parameter that depends upon the primary parameters such as clearance and volume of distribution. • T1/2 = 0.693 V d Cl T (eq.9)
  34. 34. APPARENT VOLUME OF DISTRIBUTION • Defined as volume of fluid in which drug appears to be distributed. • Vd = amount of drug in the body = x Plasma drug concentration C (eq.10) Vd = xo/co =I.V.Bolus dose/co (eq.11) • Example: 30 mg i.V. Bolus, plasma conc.= 0.732 mcg/ml. • Vol. Of dist. = 30mg/0.732mcg/ml =30000mcg/0.732mcg/ml = 41 liter. • For drugs given as i.V.Bolus, Vd (area)=xo/KE.Auc …….12.A • For drugs admins. Extra. Vas. Vd (area)=f xo/ke.Auc ……..12.B
  35. 35. CLEARANCE Clearance = rate of elimination Plasma drug conc.. (Or) cl= dx /dt C ……., (eq.13) Thus, renal clearance = rate of elimination by kidney C Hepatic clearance = rate of elimination by liver C Other organ clearance = rate of elimination by organ C Total body clearance: Clt = clr + clh + clother ……, (eq.14)
  36. 36. • According to earlier definition cl = dx /dt C • Submitting eq.1 dx/dt = KE X , above eq. Becomes ,clt = KE X/ C .., (Eq 15) • By incorporating equation 1 and equation for vol. Of dist. ( Vd= X/C ) we can get clt =KE vd (eq.16) • Parallel equations can be written for renal and hepatic clearance. Clh =km vd (eq.17) Clr =ke vd (eq.18) • But, KE= 0.693/t1/2 • So, clt = 0.693 vd (eq.19) t1/2
  37. 37. • For non compartmental method which follows one compartmental kinetic is : • For drug given by i.V. Bolus clt = xo …..20.A Auc • For drug administered by e.V. Clt = f xo …..20.B Auc • For drug given by i.V. Bolus renal clearance = xu∞ …….(eq. 21) auc
  38. 38. ORGAN CLEARANCE • Rate of elimination by organ= rate of presentation to the organ – rate of exit from the organ. • Rate of elimination =q. Cin- Q.Cout (Rate of extraction) =Q (cin- cout) Clorgan=rate of extraction/cin =q(cin-cout)/cin =Q.Er …………….(eq 22) • Extraction ratio: ER= (cin- cout)/ cin • ER is an index of how efficiently the eliminating organ clear the blood flowing through it of drug.
  39. 39. According to ER, drugs can be classified as • Drugs with high ER (above 0.7) • Drugs with intermediate ER (between 0.7-0.3) • Drugs with low ER (below 0.3) • The fraction of drug that escapes removal by organ is expressed as F= 1- ER • Where f=systemic availability when the eliminating organ is liver.
  40. 40. HEPATIC CLEARANCE Clh = clt – clr  Can also be written down from eq 22  Clh= QH ERH  QH= hepatic blood flow. ERH = hepatic extraction ratio.  Hepatic clearance of drug can be divided into two groups : 1. Drugs with hepatic blood flow rate-limited clearance 2. Drugs with intrinsic capacity- limited clearance
  41. 41. HEPATIC BLOOD FLOW • F=1-erh = AUC oral AUC i.V
  42. 42. INTRINSIC CAPACITY CLEARANCE • Denoted as clint, it is defined as the inherent ability of an organ to irreversibly remove a drug in the absence of any flow limitation.
  43. 43. ONE COMPARTMENT OPEN MODEL: INTRAVENOUS INFUSION • Model can be represent as : ( i.v infusion) Drug Dx/dt =ro-kex …eq 23 X=ro/ke(1-e-ket) …eq 24 Since X =vdc C= ro/kevd(1-e-ket) …eq 25 = Ro/clt(1-e-ket) …eq 26 Blood & other Body tissues R0 Zero order Infusion rate KE
  44. 44. • At steady state. The rate of change of amount of drug in the body is zero ,eq 23 becomes Zero=ro-kexss …27 Kexss=ro …28 Css=ro/kevd …29 =Ro/clt i.E infusion rate ....30 Clearance Substituting eq. 30 in eq. 26 • C=css(1-e-ket) …31 Rearrangement yields: • [Css-c]=e-ket . ...32 Css Log CSS-C = -ket …33 Css 2.303
  45. 45. • If n is the no. Of half lives passed since the start of infusion(t/t1/2) • Eq. Can be written as • C=CSS [1-(1/2)n] …34
  47. 47. ONE COMPARTMENT OPEN MODEL EXTRA VASCULAR ADMINISTRATION • When drug administered by extra vascular route (e.G. Oral, i.M, rectal ), absorption is prerequisite for its therapeutic activity.
  48. 48. ONE COMPARTMENT MODEL: EXTRA VASCULAR ADMIN ( ZERO ORDER ABSORPTION) • This model is similar to that for constant rate infusion. Drug at site zero order elimination Blood & other Body tissues R0 Absorption o Rate of drug absorption as in case of CDDS , is constant and continues until the amount of drug at the absorption site (Ex. GIT) is depleted. o All equations for plasma drug conc. Profile for constant rate i.V. Infusion are also applicable to this model.
  49. 49. ONE COMPARTMENT MODEL: EXTRA VASCULAR ADMIN ( FIRST ORDER ABSORPTION) • Drug that enters the body by first order absorption process gets distributed in the body according to one compartment kinetic and is eliminated by first order process. • The model can be depicted as follows and final equation is as follows Blood & other Body tissues Drug at site Ka KE First order absorption elimination C=Ka F Xo/Vd (Ka-KE) [e -Ket-e-Kat] …41
  51. 51. • Ideally a true pharmacokinetic model should be the one with a rate constant for each tissue undergoing equilibrium. • Therefore best approach is to pool together tissues on the basis of similarity in their distribution characteristics. • The drug disposition occurs by first order. • Multi-compartment characteristics are best described by administration as i.v bolus and observing the manner in which the plasma concentration declines with time. The no. Of exponentials required to describe such a plasma level-time profile determines the no. Of kinetically homogeneous compartments into which a drug will distribute. The simplest and commonest is the two compartment model which classifies the body tissues in two categories : 1. Central compartment or compartment 1 2. Peripheral or tissue compartment or compartment 2.
  52. 52. TWO COMPARTMENT OPEN MODEL-IV BOLUS ADMINISTRATION: Elimination from central compartment Fig: 1 Central 2 peripheral • After the iv bolus of a drug the decline in the plasma conc. Is bi-exponential. • Two disposition processes- distribution and elimination. • These two processes are only evident when a semi log plot of C vs. T is made. • Initially, the conc. Of drug in the central compartment declines rapidly, due to the distribution of drug from the central compartment to the peripheral compartment. This is called distributive phase.
  53. 53. Extending the relationship X= vd C Dcc = K21 xp – K12 xc – KE xc Dt vp vc vc X= Amt. Of drug in the body at any time t remaining to be eliminated C=drug conc in plasma Vd =proportionality const app. Volume of distribution Xc and xp=amt of drug in C1 and C2 Vc and vp=apparent volumes of C1 and C2 = K12 xc – K21 xp Vc vp On integration equation gives conc of drug in central and peripheral compartments at any given time t Cp = xo [( K21 – a)e-at + (K12 – b)e-bt] Vc b – a a – b Xo = iv bolus dose
  54. 54. • The relation between hybrid and microconstants is given as : a + b = K12 + K21 + KE A b = K21 KE Cc = a e-at + be-bt Cc=distribution exponent + elimination exponent A and B are hybrid constants for two exponents and can be resolved by graph by method of residuals. A = X0 [K21 - A] = CO [K21 – A] VC B – A B – A B = X0 [K21 - B] = CO [K21 – B] VC A – B A – B CO = Plasma drug concentration immediately after i.v. Injection
  55. 55. • Method of residuals : the biexponential disposition curve obtained after i. V. Bolus of a drug that fits two compartment model can be resolved into its individual exponents by the method of residuals. C = a e-at + b e-bt From graph the initial decline due to distribution is more rapid than the terminal decline due to elimination i.E. The rate constant a >> b and hence the term e-at approaches zero much faster than e –bt C = B e-bt Log C = log B – bt/2.303 C = back extrapolated pl. Conc. • A semilog plot of C vs t yields the terminal linear phase of the curve having slope –b/2.303 and when back extrapolated to time zero, yields y-intercept log B. The t1/2 for the elimination phase can be obtained from equation • t1/2 = 0.693/b. • Residual conc values can be found as- Cr = C – C = ae-at Log cr = log A – at 2.303 A semilog plot cr vs t gives a straight line.
  56. 56. Ke = a b c A b + B a K12 = a b (b - a)2 C0 (A b + B a) K21 = A b + B a C0 • For two compartment model, KE is the rate constant for elimination of drug from the central compartment and b is the rate constant for elimination from the entire body. Overall elimination t1/2 can be calculated from b. Area under (auc) = a + b The curve a b App. Volume of central = X0 = X0 compartment C0 KE (AUC)
  57. 57. App. Volume of = VP = VC K12 Peripheral compartment K21 Apparent volume of distribution at steady state or equilibrium Vd,ss = VC +VP Vd,area = X0 B AUC Total systemic clearence= clt = b vd Renal clearence= clr = dxu = KE VC Dt The rate of excretion of unchanged drug in urine can be represented by : dxu = KEA e-at + KE B e-bt Dt The above equation can be resolved into individual exponents by the method of residuals.
  58. 58. TWO – COMPARTMENT OPEN MODEL- I.V. INFUSION The plasma or central compartment conc of a drug when administered as constant rate (0 order) i.V. Infusion is given as: C = R0 [1+(KE - b)e-at +(KE - a)e-bt] VCKE b – a a - b At steady state (i.E.At time infinity) the second and the third term in the bracket becomes zero and the equation reduces to: Css = R0 Vc ke Now VC KE = vd b Css = r0 = r0 Vdb clt The loading dose X0,L = css vc = R0 Ke 1 Central 2 Peripheral
  59. 59. TWO-COMPARTMENT OPEN MODEL-EXTRAVASCULAR ADMINISTRATION • First - order absorption : • For a drug that enters the body by a first-order absorption process and distributed according to two compartment model, the rate of change in drug conc in the central compartment is described by three exponents : • An absorption exponent, and the two usual exponents that describe drug disposition. The plasma conc at any time t is C = n e-kat + l e-at + m e-bt C = absorption + distribution + elimination Exponent exponent exponent • Besides the method of residuals, ka can also be found by loo-riegelman method for drug that follows two-compartment characteristics. • Despite its complexity, the method can be applied to drugs that distribute in any number of compartments.
  60. 60. CALCULATING Ka using Wagner-nelson method(Bioavailability parameters)
  61. 61. WAGNER-NELSONS METHOD THEORY: The working equations can be derived from the mass balance equation: Gives the following eqaution with time and mass balance • Above equation Integrating gives • To the equation amount absorbed VERSUS TIME
  62. 62. WAGNER-NELSONS METHOD • Taking this to infinity where cp equals 0 • Finally (Amax - A), the amount remaining to be absorbed can also be expressed as the amount remaining in the GI, xg • We can use this equation to look at the absorption process. If, and only if, absorption is a single first order process
  63. 63. WAGNER-NELSONS METHOD • Example data for the method of wagner-nelson kel (from IV data) = 0.2 hr- Time (hr) Plasma Concentratio n (mg/L) Column 3 ΔAUC Column 4 AUC Column 5 kel * AUC A/V [Col2 + Col5] (Amax - A)/V 0.0 0.0 0.0 0.0 0.0 0.0 4.9 1.0 1.2 0.6 0.6 0.12 1.32 3.58 2.0 1.8 1.5 2.1 0.42 2.22 2.68 3.0 2.1 1.95 4.05 0.81 2.91 1.99 4.0 2.2 2.15 6.2 1.24 3.44 1.46 5.0 2.2 2.2 8.4 1.68 3.88 1.02 6.0 2.0 2.1 10.5 2.1 4.1 0.8 8.0 1.7 3.7 14.2 2.84 4.54 0.36 10.0 1.3 3.0 17.2 3.44 4.74 0.16 12.0 1.0 2.3 19.5 3.9 4.9 - ∞ 0.0 5.0 24.5 4.9 4.9 -
  64. 64. WAGNER-NELSONS METHOD • The data (Amax-A)/V versus time can be plotted on semi-log and linear graph paper
  65. 65. WAGNER-NELSONS METHOD • Plotting (Amax-A)/V versus time produces a straight line on semi-log graph paper and a curved line on linear graph paper. This would support the assumption that absorption can be described as a single first process. The first-order absorption rate constant, ka, can be calculated to be 0.306 hr-1 from the slope of the line on the semi-log graph paper. ADVANTAGES: • The absorption and elimination processes can be quite similar and accurate determinations of ka can still be made. • The absorption process doesn't have to be first order. This method can be used to investigate the absorption process. DISADVANTAGES: • The major disadvantage of this method is that you need to know the elimination rate constant, from data collected following intravenous administration. • The required calculations are more complex.
  66. 66. RESIDUAL METHOD OR FEATHERING TECHNIQUE • Absowhen a drug is administered by extravascular route, absorption is a prerequisite for its therapeutic activity. • The absorption rate constant can be calculated by the method of residuals. • The technique is also known as feathering, peeling and stripping.
  67. 67. φ It is commonly used in pharmacokinetics to resolve a multiexponential curve into its individual components. φ For a drug that follows one-compartment kinetics and administered extravascularly, the concentration of drug in plasma is expressed by a biexponential equation. C= 퐾푎퐹푋0 [e-K 푉푑(퐾푎−퐾퐸) t – e-K E a t] (1) If KaFX0/Vd(Ka-KE) = A, a hybrid constant, then: C = A e-KEt – A e-Kat (2)
  68. 68. φ During the elimination phase, when absorption is almost over, Ka<<KE and the value of second exponential e-Kat approaches zero whereas the first exponential e-KEt retains some finite value. φ At this time, the equation (2) reduces to: 퐶 − = 퐴 푒 − 퐾퐸푡(3) φ In log form, the above equation is: Log C− = log A - 퐾퐸푡 2.303 (4)
  69. 69. Where , C− = back extrapolated plasma concentration values φ A plot of log C versus t yield a biexponential curve with a terminal linear phase having slope –KE/2.303 φ Back extrapolation of this straight line to time zero yields y-intercept equal to log A. 70
  70. 70. Plasma conc.-Time profile after oral administration of a single dose of a drug
  71. 71. φ Subtraction of true plasma concentration values i.e. equation (2) from the extrapolated plasma concentration values i.e. equation (3) yields a series of residual concentration value Cτ. (C− - C) = Cτ = A e-Kat(5) φ In log form , the equation is: log Cτ= log A - 퐾푎푡 2.303 (6)
  72. 72. φ A plot of log Cτ versus t yields a straight line with slope - Ka /2.303 and y-intercept log A. φ Thus, the method of residual enables resolution of the biexponential plasma level-time curve into its two exponential components. φ The technique works best when the difference between Ka and KE is large (Ka/KE ≥ 3).
  74. 74. THREE COMPARTMENT MODEL • Gibaldi & feldman described a three compartment open model to explain the influence of route of administration .I.E. Intravenous vs. Oral, on the area under the plasma concentration vs. Time curve. • Portman utilized a three compartment model which included metabolism & excretion of hydroxy nalidixic acid.
  76. 76. Three compartment model consist of the following compartments . Central compartment. Tissue compartment. Deep tissue compartment.  In this compartment model drug distributes most rapidly in to first or central compartment.  Less rapidly in to second or tissue compartment . Very slowly to the third or deep tissue compartment. The third compartment is poor in tissue such as bone & fat. • Each compartment independently connected to the central compartment. • Notari reported the tri exponential equation c=a e-t+ b e-βt+ c e-γt • A,B,C are the y-intercept of extrapolated lines. • Α,β,γ are the rate constants
  77. 77. RAPID I.V BOLUS ADMINISTRATIONS • When the drug is administered by i.V the drug will rapidly distributed in c.C ,less rapidly in to t.C. Very slowly in to deep tissue compartment. Plasma profile • When the drug is administered by i.V the plasma conc. Will increased in c.C this is first order release. • The conc. Of drug in c.C. Exhibits an initial distribution this is very rapid. • Drug in central compartment exhibits an initial distribution this is very rapid .
  78. 78. Pharmacokinetic parameters Bioloigical half-life :: • It is defined as the time taken for the amount of drug in the body as well as plasma to decline by one half or 50% its initial value. • Concentration of drug in plasma as a function of time is c=a e - t+ b e -β t+ c e -γ t • In this equation α>β>γ some time after the distributive phase (i.e. When time become large) the two right hand side terms values are equal to zero. • The eq.. Is converted in to c=a e-αt Taking the natural logarithm on both sides the rate constant of this straight line is ‘α’ and biological half life is t1/2 =0.693/α
  79. 79. VOLUME OF CENTRAL COMPARTMENT • At time=0 C=A e –α t+ B e –β t+ C e –γ t This equation becomes CO = A+B+C -----1 CO =conc. Of plasma immediately after the i.V administration • When administered the dose is not distributed in tissue compartment. • Therefore the drug is present in c.C only . • If D is dose administered then CO = D /V C---------2 Vc=volume of drug in c.C Combining the 1&2 eq.. We get Vc = d/co (c o----- conc. Of drug in plasma) ELIMINATION RATE CONSTANT:  Drug that follows three compartment kinetics and administered by i.V injection the decline in the plasma drug conc. Is due to elimination of drug from the three compartments. Ke=(a+b+c) α β γ/a β γ +b α γ+ cα β
  80. 80. PHYSIOLOGICALLY BASED PHARMACOKINETIC MODELS • Blood flow rate limited or perfusion rate limited model. • Drawn on the basis of anatomic and physiologic data.(More realistic) • Organs or tissues having no perfusion are excluded. • Drug movement to a particular region is much more rapid than its rate of delivery to that region by blood - perfusion rate limited model. • Thus, applicable to highly membrane permeable drugs, i.e. Low molecular weight, poorly ionized and highly lipophilic drugs. • For highly polar, ionized and charged drugs, the model is referred to as membrane permeation rate limited. 81
  81. 81. 82
  82. 82. TISSUE DOSIMETRY • Measure of the level of some reactive metabolites reaching the target tissue provide a better dose parameter for risk assessment purpose than administered doses. • The effects of growth and ageing(since the fat increasing proportional to the body weight, as animal grows), topical adsorption( in inhalation studies), pregnancy and lactation(for example changes in body weight, total body water, plasma proteins, body fat and cardiac output will alter the distribution of many drugs and their metabolites.)And competitive multiple metabolites are illustrated in PBPK modelling. 83
  84. 84. THANK YOU!