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# Compartment Modelling

This PPT gives a detailed description about Compartment modelling and its types

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### Compartment Modelling

1. 1. COMPARTMENT MODELLING Presented by T. Joseph Kumar Reddy 11AB1R0097 UNDER THE GUIDANCE OF Mrs.PALLAVI.K M.pharm Assistant professor Department of pharmaceutics VIGNAN PHARMACY COLLEGE Vadlamudi, Guntur-522213. 1
2. 2. Contents  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. 2
3. 3. MATHEMATICAL MODEL  A model is a mathematical description of biologic system and used to express quantitative relationships.  Mathematical models are a collection of mathematical quantities,operations and relations together with their definitions and they must be realistic and practical.  A model is a hypothesis that employs mathematical terms to concisely describe quantitative relationships. 3
4. 4. 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 these parameters that are rate constants and half lives of drugs.  Computability:  Consistency of results: Reproducibility is an important quality of a mathematical model. 4
5. 5. APPLICATIONS OF PHARMACOKINETIC MODELS  Characterizing the behavior 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. 5
6. 6. TYPES OF PHARMACOKINETIC MODELS  There are three different types of pharmacokinetic models they are: Pharmacokinetic models Compartmental models Physiological models Non compartmental analysis 6
7. 7. COMPARTMENTAL MODELS  A compartment is a group of tissues with similar blood flow and drug afinity. A compartment 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 emperical formula to estimate drug concentration with time. 7
8. 8. ASSUMPTIONS OF COMPARTMENTAL MODELS  The body is represented as a series of compartment 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 time data is another method used to find out no.of compartments.
9. 9. APPLICATIONS OF COMPARTMENT MODELING  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. 9
10. 10. TYPES OF COMPARTMENT MODELS Based on whether the compartment is arranged in parallel or series the compartmental models are classified into four types they are: 1. Mammillary model. 2. Catenary model. 3. Open model. 4. Closed model. 10
11. 11. 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.  The no.of rate constants which will appear 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. 11
12. 12. DEPICTION OF VARIOUS MODELS:  Model 1: one compartment open model i.v injection 1 푙표푔푐 = 푙표푔푐0 − 푘퐸푡 2.303 .  Model 2: one compartment open model with first order absorption. Ka K 1 퐶 = 퐾푎퐹푋0/푉푑(퐾푎 − 퐾푒)[푒−푘푒푡 −푒−퐾푎푡] 12
13. 13.  Two compartment open model i.v injection. K12 K21 K13 푐 = 푅표/푉푐퐾퐸(1 + (퐾퐸 − 훽 2 훽−훼)푒−α푡+(KE-α/α─β)푒−훽푡). o Two compartment open model with first order absorption. Ka K12 K21 퐾13 1 1 2 13
14. 14. 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. Ka K12 K13 1 2 3 K10 K21 K31 14
15. 15. ONE COMPARTMENT OPEN MODEL 15
16. 16. ONE COMPARTMENT OPEN MODEL  The time course of drug concentration determined after the administration can be satisfactorily explained by assuming the body as single well mixed compartment with first order disposition process.  The body is constituted as a single kinetically homogenous unit that has no barriers to movement of drug.  Elimination is a first order process with first order rate constant.  Drugs move dynamically in and out of the compartment then rate of input is greater than rate of output. 16
17. 17.  The term OPEN indicates that input and output are unidirectional and that the drug can be eliminated from body. metabolism Blood and other body tissues Ka Ka KE drug Excretion output  One compartment open model is generally used to describe plasma levels following administration of a single dose. 17
18. 18. CLASSIFICATION OF ONE COMPARTMENT OPEN MODEL(INSTANTANEOUS DISTRIBUTION DATA)  Depending upon rate of input several one compartment models are defined:  One compartment open model- intravenous bolus administration.  One compartment open model- continuous intravenous infusion.  One compartment open model- extravascular zero order absorption.  One compartment open model- extravascular first order absorption. 18
19. 19. INTRAVENOUS BOLUS ADMINISTRATION  When a drug that distributes rapidly in the body is given in the form of a rapid intravenous injection, it takes about one or three minutes for complete circulation. The model can be diagrammatically depicted as K1 Blood and other body tissues  It can be mathematically represented as 19
20. 20.  The rate of drug presentation in body is expressed as 푑푥 = availability- Elimination 푑푡  In bolus injection absorption of drug is absent so availability is zero then the equation is depicted as 푑푥 푑푡 = -KEX  After applying integrations to the above it can be written as 푥 = 푥표푒−푘푡  Transforming the above equation 푙표푔푥 = 푙표푔푥0 − 퐾퐸푡 2.303 20
21. 21. ESTIMATION OF PHARMACOKINETIC PARAMETERS BY I.V BOLUS  From i.v bolus the elimination phase can be characterized by three parameters they are:  Elimination rate constant.  Elimination half life.  Clearance.  Elimination Rate Constant It can be expressed as 푙표푔푐 = 푙표푔푐0 − 퐾퐸푡 2.303 21
22. 22.  A graph is plotted by taking log concentration versus time it gives a straight line and slope gives the elimination rate constant i.e. −퐾퐸 2.303 = slope 22
23. 23.  The elimination rate constant is an additive property and they are said to be overall elimination constants 퐾퐸 = 퐾푒 + 퐾푚 + 퐾푏 + 퐾1------------  The units of elimination rate constant are 푚푖푛−1.  Elimination half life can be also be called as biological half life. It can be defined as the time taken for the amount of drug in body to decline by one half or 50% of its initial volume. It can be depicted as t1 2 . t1 2 0.693 퐾 = 23
24. 24.  Clearance:  It is the most important parameter in drug clinical drug applications and is useful in evaluating the mechanism by which a drug is eliminated.  Clearance is a parameter that relates plasma drug concentration with rate of drug elimination from below equation 푐푙푒푎푟푎푛푐푒 = 푟푎푡푒 표푓 푒푙푖푚푖푛푎푡푖표푛 푝푙푎푠푚푎 푑푟푢푔 푐표푛푐푒푛푡푟푎푡푖표푛 푐푙푒푎푟푎푛푐푒 = 푑푥/푑푡 푐  The clearance is expressed as ml/min or lit/hr. 24
25. 25. INTRAVENOUS INFUSION  Rapid i.v injection is unsuitable when the drug has the potency to precipitate toxicity.  Advantages  Ease of control of rate of infusion to fit individual patient needs.  Prevents fluctuating maxima and minima plasma levels. It is especially when a drug has narrow therapeutic index.  In critically ill patients the drugs that administered by infusions such as electrolytes and nutrients. 25
26. 26.  The model can be diagrammatically depicted as Drug Ka KE Elimination zero order absorption Blood and other body tissues  In an infusion the rate of change in amount of drug in a body is the difference between zero orderrate of infusion R0 and first order elimination –KEX. 푑푥 푑푡 = R0 − KEX Applying proper integration the above can be written as 26
27. 27. 푐 = 푅0 푐푙푇 (1-푒−퐾퐸푡)  The concentration of drug in plasma approaches a constant value called as steady state plateau or infusion equilibrium. 27
28. 28.  Transforming the above equation in concentration terms 퐶푠푠 = 푅표 퐾퐸 푉푑 = 푅푂 퐶푙푇 or 푖푛푓푢푠푖표푛 푟푎푡푒 푐푙푒푎푟푎푛푐푒 substituting in above equation 퐶 = 퐶푠푠(1 − 푒−퐾퐸푡)  Transforming into log form the equation becomes 퐶푠푠−퐶 퐶푠푠 log( )= −퐾퐸푡 2.303 퐶푠푠−퐶 퐶푠푠 log( ) 28
29. 29.  The time to reach steady state concentration is dependent upon elimination half life and infusion rate.  If n is the no.of half lives passed since start of infusion can be written as 퐶 = 퐶푠푠(1 − (1 2)n )  For therapeutic response more than 90% of steady state concentration in blood desired which reached 3.3 half lives.  It takes 6.6 half lives for concentration to reach 99% of steady state.  Shorter the half life sooner is the steady state concentration reached. 29
30. 30. EXTRAVASCULAR ADMINISTATION  When a drug is administered by extravascular absorption is an essential for its therapeutic activity and efficacy.  The rate of absorption can be mathematically depicted by two processes they are: 1. Zero order absorption process. 2. First order absorption process. 30
31. 31. ZERO ORDER ABSORPTION MODEL  It is similar to that of constant rate of infusion. Ro KE Elimination Drug at e.v.site zero order absorption  The rate of drug absorption incase of several controlled drug delivery systems. Blood and other body tissues 31
32. 32. FIRST ORDER ABSORPTION MODEL  When ever the drug enters into body it follows first order absorption process according to one compartment kinetics.  The model can be represented as Ka KE Elimination Drug at e.v.site first order absorption The differential form of above equation can be 푑푥 푑푡 =퐾푎푋푎 − 퐾푒푋 Blood and other body tissues 32
33. 33.  Integrating the above equation 푋 = 퐾푎퐹푋0 퐾푎−퐾퐸 [푒−퐾퐸푡-푒−퐾푎푡] Transforming into c terms as X= VdC 푐 = 퐾푎퐹푋0 퐾푎−퐾퐸 푉푑 [푒−퐾퐸푡-푒−퐾푎푡] F=Fraction of drug absorbed systematically after e.v.administration  Here from the i.v infusion studies the tmax and cmax cam be calculated tmax= 2.303log( 퐾푎 퐾푒 ) 퐾푎−퐾푒 EV absorption study can 퐹푋0 푉푑 Cmax = 푒−퐾푒푡푚푎푥 33
34. 34. DETERMINATION OF ABSORPTION RATE  It is the most important pharmacokinetic parameter when a drug follows first order absorption. It can be calculated by two methods in one compartment open model they are: Absorption rate constant determination Method of residuals Wagner nelson method 34
35. 35. METHOD OF RESIDUALS  This method also called as feathering, peeling and stripping. For a drug that follows one compartment and it is a biexponential equation.  퐶 = 퐾푎퐹푋0 푉푑(퐾푎−퐾퐸) [푒−퐾퐸푡 − 푒−퐾푎푡]  퐶 = 퐴푒−퐾퐸푡 − A푒−퐾푎푡  퐶 = 퐴푒−퐾퐸푡 푙표푔퐶 = 푙표푔퐴 − −퐾퐸푡 2.303 35
36. 36. 36 Graphical representation of method of residuals
37. 37.  Subtraction of true values to extrapolated concentration that is residual concentration. 퐶푟 = 퐴푒−퐾푎푡 푙표푔퐶푟=푙표푔퐴 − −퐾푎푡 2.303  If KE/Ka≥3 the terminal slope eliminates Ka and not KE. The slope of residual lines gives KE and not Ka.  This is also known as flip-flop kineticsas the slopes of two lines exchanged their meanings. 37
38. 38. Extrapolated concentration (CE) Original concentration ( C0) 5.8mg/ml 2.5mg/ml 3.3mg/ml Residual concentration= CE-C0  Lag time: Residual concentration (CE-C0)  It is defined as the time difference between drug administration and start of absorption. It is denoted by t0.  This method is best suited for drugs which are rapidly and completely absorbed and follow one compartment kinetics even when given i.v. 38
39. 39. WAGNER NELSON METHOD  One of the best alternatives to curve fitting method in the estimation of Ka is Wagner Nelson method.  This method involves the determination of Kafrom percent unabsorbed time plots and does not require the assumptions of zero or first order absorption.  According to this the amount of drug in body is depicted as 푋퐴 = 푋 + 푋퐸 39
40. 40. 푡=푉푑퐶 + 푉푑퐾[퐴푈퐶]0  푋퐴 푡 (1 ) ∞=푉푑퐶∞+푉푑퐾[퐴푈퐶]0  푋퐴 ∞ (2 )  (1) (2) = 푡 푋퐴 ∞ = 푋퐴 푉푑퐶+푉푑퐾[퐴푈퐶]0 푡 푉푑퐾[퐴푈퐶]0 ∞ = 푡 푋퐴 ∞ = 푋퐴 퐶+퐾[퐴푈퐶]0 푡 퐾[퐴푈퐶]0 ∞ If the fraction of total amount of drug absorbed = 1 40
41. 41.  Amount remaining to be absorbed = 1- 푡 푋퐴 ∞ . 푋퐴  Amount remaining to be absorbed = 1- 퐶+퐾[퐴푈퐶]0 푡 퐾[퐴푈퐶]0 ∞  % amount remaining to be absorbed= (1- 퐶+퐾[퐴푈퐶]0 푡 퐾[퐴푈퐶]0 ∞ ) 100  A graph is plotted by taking log%ARA on Y-axis and time on X-axis a straight line is obtained from that slope Ka is determined. Log%ARA 41
42. 42. URINARY EXCRETION STUDIES 42
43. 43.  It is the best method to describe and determine the excretion and elimination rate constants.  Advantages  The method is non-invasive and therefore better subject compliance is assured.  This method is more convenient since it involves the collection of urine samples in comparison to drawing of blood periodically.  When coupled with plasma level time data it can also be use to estimate renal clearance of unchanged drug according to the formula 푐푙푅 = 푡표푡푎푙 푎푚표푢푛푡 표푓 푑푟푢푔 푒푥푐푟푒푡푒푑 푢푛푐ℎ푎푛푔푒푑 푎푟푒푎 푢푛푑푒푟 푝푙푎푠푚푎 푙푒푣푒푙 푡푖푚푒 푐푢푟푣푒 43
44. 44.  Estimation of excretion rate constant: Excretion rate constant determination Excretion rate method Sigma minus method 44
45. 45. EXCRETION RATE METHOD  The appearance of drug in urine can be mathematically depicted as 푑푥푢 푑푡 = KEX 푑푥푢 푑푡 = kEX0푒−푘푡 (X=X0푒−푘푡) 푙표푔 푑푥푢 푑푡 = 푙표푔kEX0- 푘푡 2.303  In this both the elimination and excretion rates are determined.  The main disadvantage is that the rate of excretion is not constant and the values are scattered. 45
46. 46. SIGMA MINUS METHOD  The appearance of drug in body can be represented as 푑푥푢 푑푡 =퐾퐸푋 푑푥푢 푑푡 = kEX0푒−푘푡 integrate on both sides 푥푢 = 푥푢∞ 1 − 푒−푘푡 Applying log on both sides 푙표푔 푥푢 ∞ − 푥푢 =푙표푔푥푢 ∞ − 푘푡 2.303  The main disadvantage is the total urine collected has to be carried out until no unchanged drug can be detected in urine up to 7 half lives which may be tedious for drugs having long t1/2. 46
47. 47. TWO COMPARTMENT OPEN MODEL 47
48. 48.  It is common in all multi compartment models.  It consists of central compartment i.e., compartment 1 comprising of blood and highly perfused tissues and peripheral compartment i.e., compartment 2 comprising of poorly perfused tissues such as skin and adipose tissues.  INTRAVENOUS BOLUS: K12 1 2 K21 KE The model can be mathematically represented as 48
49. 49. 푑푐푐 푑푡 = K21Cp − K12Cc − KECc 퐶푐 = 퐴푒−훼푡- 퐵푒−훽푡 α and β are hybrid constants then α +β=K12+K21+KE α β =K21KE Intravenous infusion R0 K12 1 2 K21 KE 푐 = 푅표/푉푐퐾퐸(1 + (퐾퐸 − 훽 훽−훼)푒−α푡+(KE-α/α─β)푒−훽푡). 49
50. 50. EXTRAVASCULAR ADMINISTRATION  In two compartment open model for extravascular administration it can be depicted as Ka K12 K21 KE Central compartment Peripheral compartment 퐶 = 푁푒−퐾푎푡 + L푒−훼푡 + M푒−훽푡 The absorption rate is calculated by Loo-Regeilmann method. 50
51. 51. 51
52. 52.  A graph is plotted by taking log %ARA on y-axis and time on x-axis a straight line is obtained and slope of that gives absorption rate i.e., slope= −퐾푎 2.303 . 52
53. 53. CONCLUSION  Compartment model provides a framework for the study of the dynamic flow of chemicals (nutrients, hormones, drugs, radio-isotopes, etc.) between different organs which are assumed as compartments in the human body.  Multi-compartment models have applications in many fields including pharmacokinetics, epidemiology, biomedicine, systems theory, complexity theory, engineering, physics, information science and social science. 53
54. 54. REFERENCES  Comprehensive pharmacy review by Leon shargel pg.no.113-129.  Bio pharmaceutics and pharmacokinetics by D,M.Brahmankar and Sunil.B.Jaiswal, Vallabh prakash publications pg.no.245- 299.  The Pearson book to GPAT by Ashok.H.Akabari and Anil Kumar Baser pg.no.1.114-1.120.  Bio pharmaceutics and pharmacokinetics by venkateswarlu second edition pg.no.176-182. 54
55. 55. ACKNOWLEDGEMENT I would like to thank my project guide Mrs.PALLAVI.K for encouraging me with positive spirit. I wish to express my gratitude to our principal Dr.P.Srinivasa Babu and the seminar committee. 55
56. 56. THANK YOU 56

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