Compartment Model. Zero and First Order

A compartment model is a simplified mathematical framework used to describe how substances (like drugs, chemicals, or even populations) move between different "compartments" of a system. Each compartment represents a homogeneous, well-mixed space where the substance is evenly distributed, and transfer between compartments is defined by rate constants.
It is widely used in pharmacokinetics, toxicology, biology, and systems modeling.
Types of Compartment Models in Pharmacokinetics
One-Compartment Model
The drug distributes instantly and uniformly throughout the body.
Body = a single compartment.
Elimination occurs directly from this compartment.
Example: IV bolus injection with simple exponential decay.
Two-Compartment Model
Body is divided into:
Central compartment (plasma + highly perfused tissues like heart, liver, kidney).

Peripheral compartment (less perfused tissues like muscle, fat).

Drug moves between compartments and elimination happens from the central one.
Captures distribution + elimination phases.
Multi-Compartment Models
More than two compartments (complex distribution).
Used for drugs with multiple tissue binding sites or slow redistribution.
Zero-Order Kinetics
Definition: A constant amount of drug is eliminated per unit time, regardless of concentration.
Characteristics:
Rate of elimination = constant.
Linear decline in plasma concentration vs time.
Saturation of enzymes or carriers (capacity-limited process).
First-Order Kinetics-
Definition: A constant fraction (percentage) of the drug is eliminated per unit time.
Characteristics:
Rate of elimination ∝ plasma concentration.
Exponential decline in plasma concentration vs time.
Most drugs follow first-order kinetics at therapeutic doses.
Graphical Representation-
Zero-order: Straight-line decline.
First-order: Curved exponential decline.
Zero-Order Kinetics-
Examples: Phenytoin, Ethanol, High-dose Aspirin.
First-Order Kinetics-
Examples: Most drugs (e.g., penicillin, paracetamol).
Applications-
Pharmacokinetics: predicting plasma drug levels after dosing.
Environmental modeling: movement of pollutants in soil/water compartments.
Physiology: modeling glucose/insulin regulation, oxygen transport, etc.
Epidemiology: spread of diseases (e.g., SIR model as a compartmental epidemiological model).
Key Parameters-
k₁₂, k₂₁ = transfer rate constants between compartments.
k₁₀ = elimination rate constant.
Vd (Volume of distribution) = hypothetical volume to describe drug concentration in plasma.
t½ (Half-life) = time required for drug concentration to reduce by half.

1. Compartment Modelling Presented by: Miss. Shraddha Prakash Bandgar M. Pharm (Assistant Professor) Department of Pharmaceutics

2. Content I.v. Infusion Extravascular Administration - Zero Order Absorption Model - First Order Absorption Model 2

3. One Compartment Open Model – I.V. Infusion Rapid I.V. injection is unsuitable when the drug has potential to precipitate toxicity. In such a situation, the drug (for ex- Several antibiotics, theophylline, procainamide, etc.) is administered at a constant rate (zero-order) by i . v. infusion . In contrast to the short duration of i.v. infusion of an i.v. bolus (few seconds), the duration of constant r ate infusion is usually much longer than the half life of the d r ug. 3

4. Advantages of Zero-order infusion of drugs include- Ease of control of rate of infusion to fit individual patient needs. P revents fluctuating maxima and minima (Peak and valley) plasma level. This is desired especially when the d r ug has a narrow therapeutic index. Other drugs, electrolytes and nutrients can be conveniently administered simultaneously by the same infusion line in critically ill patients. 4

5. The model can be represented as follows: At any time during infusion, the rate of change in the amount of drug in the body, dX /dt is the difference between the Zero-order rate of drug infusion R and a first order rate of elimination, - K E X: Integration and rearrangement of above equation: - (9.33) - (9.34) 5

6. Since X = V d C , the eq n (9.34) can be transformed into concentration terms as follows: At the start of constant rate infusion, the amount of drug in the body is zero and hence, there is no elimination. As time passes, the amount of drug in the body rises gradually (elimination rate less than the rate of infusion) until a point after which the rate of elimination equals the rate of infusion i.e. Concentration of drug in plasma approaches a constant value called as steady state , plateau or infusion equilibrium . Fig : Plasma concentration- Time profile for a drug given by constant rate i.v. Infusion (the two curves indicate different infusion rate R and 2R for the same drug.) - (9.35) 6

7. At steady state the rate of change in amount of drug in the body is zero, hence the equation becomes: Transforming to Concentration terms and rearranging the equation: Where, Xss and Css are amt. of drug in the body and Conc n . of drug in plasma at steady state resp. The value of K E Can be obtained from the slope of Straight line obtained after, a semilogarithmic plot. (log C vs t) of the plasma conc n -time data gathered from the time when infusion is stopped. - (9. 36) Or - (9. 37) 7

8. Alternatively, K E can be calculated from the data collected during infusion to steady state as follows: Substituting R / Cl T = C SS from equation (9.37) in equation (9.35) we get, Rearrangement yields: Transforming into log form, the equation becomes: - (9. 38) - (9. 39) - (9. 40) 8

9. A semilog plot of log (C SS –C) / C SS vs t results in Straight line with slope -K E /2.303 Fig : Semilog plot to compute K E from infusion data upto steady state. 9

10. Infusion + Loading Dose It takes a very long time for the drugs having longer half-lives before the plateau Conc n is rea c hed (eg. Phenobarbital, 5 days). Thus, initally such drugs have sub therapeutic conc n This can be overcome by administering an i.v. loading dose large enough to yield the desired steady-state immediately upon injection prior to starting the infusion. Fig : I.V. infusion with loading dose As the amount of bolus dose remaining in the body falls, there is a complementary rise resulting from the infusion. 10

11. Recalling once again the relationship X= V d C, the equation for computing the loading dose X O,L can be given: Substitution of C SS = R / K E V d from equation (9.37) in above equation yields another expression for loading dose in terms of infusion rate: The equation describing the plasma conc n – time profile following simultaneous i.v. loading dose ( i.v. bolus) and constant rate i.v. infusion is the sum of two equation describing each process (i.e. modified equation 9.5 and 9.35) : If we substitute C SS V d for X O,L (from eq n 9.42) and C SS K E V d for R (From eq n 9.37) in above equation and Simplify, it reduces to C = C SS indicating that the conc n of drug in plasma retains constant (Steady) throughout the infusion time. - (9. 42) - (9. 43) - (9. 44) 11

12. Assessment of Pharmacokinetic Parameter The 1 st Order elimination rate const. and elimination half- life can be computed from a semilog plot of post-infusion Con c n - time data. Eq n (9.40) can also be used for the same purpose. Apparent vo lume of distribution and total systemic clearance can be estimated from steady state conc n . and infusion rate ( eq n 9.37) . These two parameters can also be computed from the total area under the curve t ill the end of infusion. Where, T – Infusion Time The above equation is a general expression which can be applied to several pharmacokinetics models. - (9. 45) 12

13. One Compartment Open Model- Extravascular Administration When a d r ug is administered by extravascular route ( eg. oral, rectal, etc.) absorption is a prerequi s ite for its therapeutic activity . The rate of absorption may be described mathematically as a zero order or first-order process. 13

14. One Compartment Open Model- Extravascular Administration The differences between zero -order and first-order kinetics are: Fig : Distinction between zero- order and first- order absorption processes. Fig. a) is a regular plot and Fig. b) is a semilog plot of amount of drug remaining to be absorbed (ARA) vs time (t). 14

15. First order absorption process: On Semilog Plot – Give straight line On Regular Plot – Give Curvilinear Zero order absorption process: On Semilog Plot – Give Curvilinear On Regular Plot – Give Straight line After E.V. administration, the rate of change in the amount of drug in the body dX/ dt is the difference between the rate of input ( absorption) dX ev / dt and rate of output (elimination) dX E / dt. For a drug that follows one compartment kinetics, the plasma concentration- time profile is characterized by absorption phase, post absorption phase and elimination phase. dX/dt = Rate of absorption – Rate of elimination - ( 9.46 a) 15

16. Fig: The absorption and elimination phases of the plasma concentration- time profile obtained after Extravascular Administration of a single dose of drug. 1) During the absorption phase , the rate of absorption is greater than the rate of elimination. - ( 9.46 b ) 16

17. 2) At peak plasma concentration , the rate of absorption equals the rate of elimination and the change in amount of drug in the body is zero . 3) During the post absorption phase , rate of elimination is greater than the absorption rate. - (9. 46 c) - (9. 46 d) 17

18. Zero-Order Absorption Model- E.V. Administration This model is similar to that for constant rate infusion The rate of drug absorption, as in the case of several controlled drug delivery systems, is constant and continues until the amount of drug at the absorption site (e.g. GIT) is depleted . All equation that explain the plasma concentration- time profile for constant rate i.v. Infusion rate also applicable to this model. 18

19. First-Order Absorption Model- E.V. Administration For a drug that enters the body by a first- order absorption process, get distributed in the body according to one compartment kinetics and is eliminated by a first- order process, the model can be depicted as follows: The differential form of equation (9.46 a) is: Where, K a – 1 st order absorption rate constant X a – Amount of drug at absorption site remaining to be absorbed i.e. (ARA) K E X – Amount of drug to be eliminated - ( 9. 47) 19

20. Integration of equation (9.47) yields: Transforming into concentration terms, the equation becomes: Where, F – Fraction of drug absorbed systematically after e.v. administration - (9. 48) - ( 9. 49) 20

21. Reference Brahmankar DM and Jaiswal SB. Biopharmaceutics and Pharmacokinetics . 2014 Nov 12;(3 rd edition): 2 70 – 81. WWW. Wikipedia.Com 21

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