Nonlinear Pharmacokinetics-1

NON-LINEAR PHARMACOKINETICS:: 

NON-LINEAR PHARMACOKINETICS: PRESENTED BY: Sudharani

MICHAELIS MENTEN EQUATION:: 

MICHAELIS MENTEN EQUATION: It is named for Leonor Michaelis and Maud Menten. Dose dependent kinetics are most commonly expressed by Michaelis-Menten kinetics, according to which the rate of a process is given by Where, Dc/dt is elimination rate Vmax is the maximum rate of the process km is the Michaelis-Menten constant C is the Concentration of drug in plasma.

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Here,elimination rate is change with change in drug concentration. (1) When the drug concentration is larger then to kM( Cp  kM), saturation of enzyme occurs and the value of kM negligible. which is the expression for a zero order process. Thus at higher concentrations we have a zero order process.

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(2) A saturable process can also exhibit linear elimination, when drug concentration are much less then enzyme conc. Here, Cp Km so, Cp is negligible and the elimination rate becomes first order.

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( 3) If drug concentration is same as Km,then Cp=Km. dCp = Vmax dt 2 so it is evident that the rate of process becomes equal to one half its maximum rate. It is also called mixed order elimination rate

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Zero-order at high doses Mixed order at intermediate doses First-order rate at low dose

Estimation of Km and Vmax: 

Estimation of K m and V max The parameters of capacity-limited process like metabolism, renal tubular secretion and biliary excretion can be easily defined by assuming one-compartment kinetics for the drug and that elimination involves only a single capacity-limited process. The parameters Km and Vmax can be assessed from the plasma conc.- time data collected after i.v.bolus administration of a drug with nonlinear elimination characteristics.

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Rewriting michaelis-menten equation, conversion to log base 10 yields, Log C= Log Co + (Co-C) – Vmax.t ( 1) 2.303 Km 2.303 Km

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A semilog plot of c verses t yields a curve with a terminal linear portion having slope –Vmax/2.303. Km and when back extrapolated to time zero gives - intercept log

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The equation that describes this line is Log C= log - Vmax.t (2) 2.303.Km Equation1& 2 simplify, Log (C0-C) = log 2.303 Km Log Co Here Km thus can be obtained from the above equation. Vmax can be computed by substituting the value of Km in the slope value.

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Vmax and Km is determining the rate of change of plasma drug conc.at different times & using the reciprocal of michaelis-menten equation, Where, Cmid=the plasma drug conc. At middle point of sampling interval.

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The linewever-burk plot of 1/dc/dt &1/Cm yield straight line with slope=Km/Vmax & intercept=1/Vmax. The disadvantages of lineweaver-burk plot is that the points are clustered.

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Another mothod in which points are uniformly spread are Hanes-woolf plot C/v= 1.C + Km Vmax Vmax And Woolf –Augustinnson-Hofstee plot. v= Vmax – Km.v/c Here , v=(1/dc/dt)

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A plot of C/v verses C yield a straight line 1/Vmax as a slope & Km/ Vmax as intercept.

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A plot of v verses v/C yield slope of Km&intercept of Vmax. Slope of Km

Estimation of Km & Vmax from steady state concentration:: 

Estimation of Km & Vmax from steady state concentration: When a drug is administered as constant rate i.v.infusion or in a multiple dose regimen, the steady-state conc.Css is given in terms of dosing rate DR as: DR = Css.Cl T (A) DR= Ro when the drug is administered as a zero-order i.v infusion & it is equal to Fxo/T when administered as multiple oral dosage regimen. F is a function of bioavailable Xo= oral dose T= dosing interval

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At a steady state ,the dosing rate equals to rate of decline in plasma drug conc. & if the decline is due to a single capacity limited process then, DR= Vmax.Css Km+ Css

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A plot of Css Vs DR yields a typical hockey-stick shaped curve as,

Three ways for estimating Km & Vmax:: 

Three ways for estimating Km & Vmax: Lineweaver-Burk Plot: (2) Direct Linear Plot: (3) The Third graphical method:

(1) Lineweaver-Burk Plot:: 

(1) Lineweaver-Burk Plot: Taking reciprocal of, DR= Vmax.Css Km+ Css 1 = Km + 1 DR Vmax.Css Vmax A plot of 1/DR Vs 1/Css yields a straight line with slope Km/Vmax & y-intercept 1/Vmax

(2) Direct Linear Plot:: 

(2) Direct Linear Plot: A pair of Css is Css1 & Css2obtained with two different dosing rates DR1 & DR2 are plotted. The points Css1 & DR1 are joined to form a line & a second line is obtained similarly by joining Css2 & DR2. The points where these two lines intersect each other is extrapolated on DR axis to obtain Vmax & on X-axis to get Km.

(3)The third graphical Method:: 

(3)The third graphical Method: DR= Vmax.Css Km+ Css DR.Km +DR.Css =Vmax.Css DR.KM + DR = Vmax Css DR = Vmax – DR.Km Css A ploy of DR Vs DR/Css yield a straight line with slope -Km & y-intercept Vmax.

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Km & Vmax can also be calculated numerically by setting up simultaneous equations as shown below: DR1 = Vmax.Css1 Km + Css1 DR2 = Vmax.Css2 Km + Css2 Combine both equation gives, Km = DR1 –DR2 DR1 – DR2 Css1 Css2

REFERENCE;: 

REFERENCE; BIOPHARMACEUTICS AND PHARMACOKINETICS BY BRAHMANKAR AND SUNIL B. JAISWAL. BIOPHARMACEUTICS AND PHARMACOKINETICS BY MADAN. BIOPHARMACEUTICS AND PHARMACOKINETICS BY SHARGEL. INTERNET SOURCE.

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