Download concept of pharmacokinetics

DEPARTMENT OF PHARMACEUTICS
introduction
oMany
drugs in a single intravenous bolus dose demonstrate a plasma
level time that does not decline as a single exponential (first
order)process.
oThe
plasma level time curve for a drug that follows a two-compartment
model shows that the plasma drug concentration declines biexponentially
as the sum of two first order processes-distribution and elimination.
o
In this model, the drug distributes into two compartments, the central
compartment and the tissue/peripheral compartment

The central compartment represents the blood,ecf and highly perfused
tissues.

The drug distributes rapidly and uniformly in the central compartment.

Tissue/peripheral compartment contains tissues in which drug equilibrates
more slowly.

Drug transfer between two compartments is assumed to takes place by first
order process
Blood supply
Tissue group
% body weight
Highly perfused
Heatr,brain,liver
9
Kidney,endocrine glands 50
Skin muscle adipose
tissue & marrow
Slowly perfused
19
Bone,ligaments,tendons,t 22
eeth&hair


Intravenous bolus administration:The model can be depicted as shown below with elimination from the central
compartment.
After the i.v. bolus of a drug that follows two compartment kinetics,the decline in
plasma conc.,is biexponential indicating the presence of two disposition
processes viz.,Distribution and Elimination.
Initially,the concentration of drug in central compartment declines rapidly,this is
due to distribution of drug from the central compartment to peripheral
compartment.
The phase during which this occurs is therefore called as the ‘distributive phase’.

After sometime, a pseudo-distribution equilibrium is achieved between
the two compartments following which the subsequent loss of drug from
the central compartment is slow and mainly due to elimination.

This is second slower rate processes is called as the ‘post-distributive
/elimination phase’.

Let k12 and k21 be the first order distribution rate constants depicting
drug transfer between the central and the peripheral compartments and
let subcript ‘c’ and ‘p’ define central and peripheral compartment
respectively.

The rate of change in drug concentration in central compartment is given

Extending the relationship,X=Vd.C to the above equation, we have

dCc/dt= K21Xp/Vp-K12Xc/Vc-KeXc/Vceq.2

Where Xc and Xp are the amounts of drug in the central and peripheral
compartments .,and Vc and Vp are the apparent volume of central and
peripheral compartment respectively.

The rate of change chage in drug concentration in the peripheral compartment
is given by;

dCp/dt=K12Cc-K21Cp

dCp/dt=K12Xc/Vc-K21Xp/Vp---------------eq.4


Integrating of equations 2&4 yields equations that describe the concentration
of drug in the central and peripheral compartments at any given time ‘t’:-

Cc=Xo/Vc[(k21-α/β-α)e-αt +(k21-β/α-β)e-βt.

Cp=Xo/Vp[(k12/β-α)e-αt +(k12/α-β)e-βt.

Where Xo=i.v.bolus dose,α andβ are hybrid first –order constants for the rapid
distribution phase and slow elimination phase respectively.,which depend
entirely upon the first order rate constants K12,K21 and ke.

The constants k12 and k21 that depict reversible transfer of drug between
compartments are called as “microconstants (or) transfer constants.

The mathematical relationships between hybrid and microconstants are given
as :-

α+β= K12+K21+Ke. Eq.7

αβ=K21.Ke.eq.8

eq.5 can be written in simplified form as:-

Cc= Ae-αt+Be-βt.

Cc= Distribution exponent+Elimination exponent.

Where A and B are also hybrid constants for two exponents and can be
resolved graphically by the method of residuals.

A=Xo/Vc[k21-α/β-α].
=Co[k21-α/β-α].eq.10
B=Xo/Vc[k21-β/α-β].
=Co[k21-β/α-β]. Eq.11

Where Co= plasma drugconcentration immediately after i.v. injection.






Method of residuals:- The biexp onential 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.it is also called as feathering
/peeling/stripping
.rewriting the eq.9:Cc=Ae-αt+Be-βt.

As apparent from the biexponential curve given in figure ,the initial
decline due to disrtibution is more rapid than the terminal decline due to
elimination i.e., the rate constant α>>β and hence the term e-αt
approaches zero much faster than does e-βt.

The eq.9 reduced to.

C=Be-βt.eq.12

In log form, the above equation becomes:-

Log C=logB-βt/2.303…eq.13
(9.98)

substration of extrapolated plasma concentration values of the
elimination phase eq.12 from the corresponding the plasma concentration
values eq.9.yields a series of residual concentration values Cr.

Cr=C-C =Ae-αt.

In log form ,the equation becomes:-

Log Cr= logA-αt/2.303.

A semi log plot of Cr values ‘T’ yields a staight line with slope –
α/2.303.and y-intercept log ‘A’.
Assessment of pharmacokinetic parameters—Iv bolus administration;

All the parameters of eq.9,can be resolved by the method of residuals.

Other parameters of the model viz.,K12,K21 and Ke etc can now be derived by
proper substitution of these values.

Co=A+B.eq.16

Ke=αβCo/Aβ+Bα.eq.17

α+β=k12+k21+ke.eq.18

K12= (α+β)-k21-ke.

k12=(α+β)-k21-(αβCo/Aβ+Bα).Eq.19.

αβ=k21ke.eq.20

It must be noted that for two compartment, Ke is the elimination rate constant
of drug from central compartment and β is elimination rate constant from the
entire body.

Overall elimination t1/2 should therefore be calculated from β.

Area under the plasma concentration-time curve can be obtained by the
following equation;

AUC=A/α+B/β.EQ.22

The apparent volume of central compartment Vc is given as;

Vd=Xo/Co=Xo/ke.AUC.

Apparent volume of peripheral compartment can be obtained from eq,

Vp=Vck12/k21.eq.23

Similarly,Vc=Vpk21/k12.eq.24.

The apparent volume of disrtibution at steaty state or equilibrium can be defind
as.;

Vd,ss=Vc+Vp.eq.25.

It is also given as ;-Vd,area=Xo/β.AUC.eq.26.

Total systemic clearance isgiven as.:-

clT=β.Vd..eq.27.

The pharmacokinetic parameters can also be calculated by using urinary
excretion data:-

Dxu/dt=ke.vc..eq28.

An eq,identical to eq .9 can bederived for rate of excretion of unchanged
drug in urine.:-

Dxu/dt=keAe-αt+keB-Βt.eq.29,

Renal clearance is given as;

clR=KeVc..eq.30.
Intravenous infusion:
The plasma or central compartment concentration of a drug that fits two –
compartment model when administered as constant rate (zero order)
i.v.infusion.,is given by equation;

C=Ro/VcKe[1+(ke-β/β-α)e-αt+ (ke-α/α-β)e-βt..

At steady state (i.e., at time infinity),the 2nd and 3rd terms in the bracket
becomes zero and the eq. reduces to…

Css=Ro/Vc.ke.

Now Vc.Ke=Vd.β

Css=Ro/Vd.β=Ro/clT

The above eq .is rearranging gives loading dose Xo.L

Css =Ro/vc.Ke

Xo.,L=Css.Vc=Ro/ke
the model can be depicted as follows:

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
concentration in the central compartment is described by 3 exponentsabsorption exponent,the two usual exponents that describe drug disposition.

The plasma concentration at any time ‘t’ is givenby equation.,

C=Ne-ka.t+Le-αt+Me-βt.

C= Absorption exponent+ Distribution exponent+ Elimination exponent.

By the metWhere ka,α,β have usual meanings;L,M,N. are co-efficients.

hod of residuals above exponents are resolved,assuming

ka>α>β.

This method is in contrast to the wagner-nelson method for
determination of ka of drug with one compartment model.

The loo-riegelman method requires plasma conc.-time data both after oral
and i.v.administred drug to the same subject at different times in order to
obtain all the kinetic constants.



Bio-Pharmaceutics and Pharmacokinetics A
Treatise by D.M BRAHMANKAR; SUNIL B.
JAISWAL.
Leon Shargel; et.al ; applied biopharmaceutics.
Biopharmaceutics and pharmacokinetics by
Venkateswarulu
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