1. Nonlinear Pharmacokinetics
Quantitative Pharmacokinetics
Nonlinear Pharmacokinetics
Dr. Chalet Tan

2. Learning Objectives profiles of nonlinear kinetics
sources and effects of dose dependency on ADME
Michaelis-Menten equation and parameters (Vmax, Km) for capacity-limited metabolism

3. Case Study
An epileptic patient who has not responded to phenytoin after 2 weeks on 300 mg/day is observed to have a plasma concentration of 4 mg/ml. Twenty days after the daily dose is subsequently increased to 500 mg/day, the patient develops severe toxicities. The plasma concentrations of phenotoin is now 36 mg/L.
4 mg/mL is the CPSS, but not enough to cause therapeutic effect. Has a narrow therapeutic window.
Phenytoin must not be linear! There is a 9-fold increase in new CPSS when you only increased the dose by about 1.67x. Doesn’t line-up.

4. Review of Linear Pharmacokinetics
ADME all obey first-order kinetics
Pharmacokinetic parameters, e.g. elimination half-life (t1/2), the elimination rate constant (k), the apparent volume of distribution (V) and the clearance (CL) remain constant.
Plasma drug concentration at a given time and AUC are directly proportional to the dose.
Concentrations of drug in plasma and tissues are below protein binding saturation , i.e. fu and fuT remain constant.
They’ve all been linear so far because we have made that assumption. ADME all follow first-order. So there is a rate constant for all of those 4.
All of the parameters t1/2, k, CL, and V al remain constant.

5. Review of Linear Pharmacokinetics
time
i.v. bolus
100 mg
1 mg
10 mg
Log C
100 mM
10 mM
1 mM
i.v. bolus
1 mg
time
Log C
1 h
normalized
by dose
1mg IV bolus, linear relationship between LogC and time.
1 h
Drug plasma concentrations are proportional to the dose.
Drug plasma concentration-time profiles are superimposable when normalized to the dose.

6. Review of Linear Pharmacokinetics
p. o.
1 mg
Log C
time
tmax
0.1 mM
p. o.
Log C
2.5 mM
normalized
by dose
0.5 mM
25 mg
1 mg
5 mg
0.1 mM
Convinced it’s the same thing. Superimposable profile is linear kinetics.
tmax
time
Drug plasma concentrations are proportional to the dose.
tmax remains unchanged.
Drug plasma concentration-time profiles are superimposable when normalized to the dose.

7. Review of Linear Pharmacokinetics
Cp or AUC
VD or CL or t1/2
When you increase dose, you expect a linear increase in plasma concentration and AUC. The VD, CL, or t1/2 remain unchanged.
Linear = Dose-independent pharmacokinetics.
Dose
Dose

8. Nonlinear Pharmacokinetics
time
i.v. bolus
100 mg
1 mg
10 mg
Log C
1 h
1 mM
2 mM
8 mM
i.v. bolus
100 mg
1 mg
10 mg
time
Log C
20 mM
1 mM
800 mM
normalized
by dose
Same drug at 3 different doses. When you increase the dose, the sloe changes, so you know K has changed. Less steep slope, drug is eliminated slower.
Rate constant of those processes, went up. They are no longer following 1st order kinetics.
1 h
Drug plasma concentrations are not proportional to the dose.
Drug plasma concentration-time profiles are not superimposable when normalized to the dose.

9. Nonlinear Pharmacokinetics
p. o.
100 mg
10 mg
Log C
time
1 mg
1 mM
0.5 mM
0.1 mM
p. o.
10 mM
normalized
by dose
Log C
5 mM
1 mM
100 mg
10 mg
1 mg
When increase dose by 10x, absorption can not occur to 10x higher extent.
Increases 100x, only increases 10x. Increase 10x, only increase 5x.
Normalize the dose – divide all concentrations on 100mg curve by 100x. 100mg ends up being the lowest.
time
Drug plasma concentrations are not proportional to the dose.
tmax may or may not change.
Drug plasma concentrations are not superimposable when normalized to the dose.

10. Nonlinear Pharmacokinetics
F, V, CL or t1/2
Cp or AUC
linear
linear
Non-linear kinetics only occur when the dose becomes high enough. Is linear to the last point on last graph, then curves up or down, depending on whether it gets more or less effective.
Same with the parameters: linear to a point, then curve up or downward.
Dose-dependent kinetics.
Dose
Dose

11. Common Sources for Nonlinear Pharmacokinetics
Transport of drug into gut wall can be saturated. Drug can be comparitively insoluble. Distribution can be an issue. Metabolism can occur in GI/Liver and cause altered first pass effect.

12. Linear vs. Nonlinear Pharmacokinetics
(dose-dependent)
(dose-independent)
at least one of the ADME processes is saturable.
≥1 PK parameters are dose-dependent.
AUC is disproportional to the dose.
Concentration vs. time profile is not superimposable for different doses.
ADME all obey first-order kinetics.
PK parameters (CL, V, F, Ka, and t1/2) are constant.
AUC is directly proportional to the dose.
Concentration vs. time profile is superimposable for all doses.
Important slide. Summarizes first 10 slides.

13. Most Common Sources for Nonlinear Pharmacokinetics
Capacity-limited oral absorption (F)
Capacity-limited metabolism (CLH )
Saturable protein binding (CLH, CLR, V )
Capacity-limited excretion (CLR )
Capacity-limited excretion, affects tubular secretion/reabsorption, which in turn affects renal clearance.

14. Capacity-Limited Oral Absorption (F)
limited dissolution/solubility as the oral dose increases
saturable transport across the intestinal mucosa as the oral dose increases
saturable first-pass metabolism in the intestinal epithelium (gut wall) and/or liver as the oral dose increases
Reasons that cause capacity-limited oral absorption.
Limited dissolution, true of very lipophilic drugs.

15. limited dissolution/solubility in the GI tract
normalized to the dose
- Griseofulvin is poorly water-soluble (10 mg/L).
- Less proportion of the drug is being dissolved and absorbed with the higher dose.
F decreases as the dose increases.
tmax remains the same.
4T is lower because it’s normalized by dose. Not proportionally high. Very lipophilic.
Tmax remains the same because…

16. Saturable transport across the intestinal epithelium
e. g.
Saturable transport across the intestinal epithelium
375 mg
750 mg
1500 mg
3000 mg
- Amoxicillin is actively transported by peptide transporter in the small intestine.
- The active transport becomes saturated as the dose increases.
F decreases as the dose increases.
tmax remains the same.
Very polar hydrophilic, mediated by active transport.
At 8x dose, it’s almost half of F. Because active transport is saturated, and rate of absorption is being.
You normalize to show which is better per 375 tab (for example).

17. Saturable first-pass metabolism
e. g.
Saturable first-pass metabolism
- Nicardipine is metabolized by CYP3A4 in the intestinal epithelium and hepatocytes.
- First-pass metabolism is saturated as the dose increases.
F increases as the dose increases.
CCB.
Would expect concentration to double when dose doubles, and F remains the same.
First-pass metabolism is being SATURATED, which means less can be metabolized on the first-pass.

18. Saturable first-pass metabolism
e. g.
Saturable first-pass metabolism

19. Saturable Drug-Plasma Protein binding (CL,V)
Drug-plasma protein binding is saturable
The saturation drug concentrations for binding with plasma albumin and a1-acid glycoprotein are ~ 600 mM and 15 mM, respectively.
May increase CLH and/or CLR
May increase V
May be difficult to identify due to effect on both V and CL
When drug conc. Reaches plasma concentration, binding between drug/protein will be saturated.

20. saturable plasma protein binding
- AUC and Cp of trandolaprilat do not increase proportionally with D; Cp does not accumulate with multiple doses.
As the dose increases, binding to ACE (angiotensin-converting enzyme) in plasma is saturated.
Trandolaprilat is elminated by glomerular filtration
CLR= fu GFR
As fu increases with higher Cp, CLR increases.
2 g/day
Top left: if linear would follow orange line. In this dosing level, drug is having non-linear kinetics. But why?
Bottom right: protein binding assay. When PPB increases from fold increase in fu.
Top right: ose increase from 0.5 to 4. 8x incresae in dose, but terminal concentration, it’s almost identical. Very little difference.
Bottom left: multiple dosing. Expect an accumulation effect. But there is no increase in plasma concentration.
ACE inhibitor – TIGHT binding to ACE enzyme. Trandolaprilat, eliminated by renal excretion (Glomerular filtration).
So plasma conc. Increases but elimination also gets much higher due to increase in free fraction.

21. Capacity-Limited Excretion (CLR)
Active secretion and active reabsorption are saturable processes.
Saturated tubular secretion decreases CLR
Saturated tubular reabsorption increases CLR
CLR = fu GFR + (CLsecretion – CLreabsoption)
All saturable carrier-mediated processes. Tubular secretion is saturated, renal clearance will decrease. When Tubular reabsorption is saturated, renal clearance will INCREASE.
ROTS will reach a maximum, and stop going up, but CP will continue rising, and make CLsecretion and CLreabsorption go down.
ROTS will make renal clearance decrease.
RORA will make renal clearance increase.
FIRST BREAK

22. capacity-limited renal excretion
e. g.
When concentration is about lower than 7 mg/L, it could be linear. Since clearance is linear. But once it gets above 7, the clearance rises, which makes it non-linear.
capacity-limited renal excretion
CLinulin = GFR
p.o mg
When Cp above 10 mg/L starts to saturate renal reabs of Vit C.
i.v g
Secretion is the process that drug goes from bloodstream, and into the lumen. When this is saturated, drug is less likely to be eliminated, so renal clearance will decrease.
Reabsorption is process where drug goes back into the bloodstream. (passive if lipophilic). When this is saturated, more drug is eliminated.
Active reabsorption of vitamin C is a saturable process. Severely decreased renal clearance of vitamin C when increase in dose.
You would expect renal clearance to stay the same if you didn’t think about non-linear kinetics.
Clearance of inulin is GFR.
Inulin is there to give a sense/comparison of CLR or VC and CLR of inulin. So ratio is telling you that CLR can keep increasing, but can’t be higher than GFR. Y-axis is the RATIO of VC CLR and CLInulin. Will be about 100 ml/min cuz GFR is 120 ml/min. Can’t go over 1 because it’s a ratio.
Renal Clearance will reach 120 ml/min eventually, as it becomes saturated. Can’t exceed.
- Vitamin C is reabsorbed from urine by active transporter.
Tubular reabsorption becomes saturated as Cp increases, i.e. as Cp increases, CLreabsorption (= Ratereabsorption /Cp) decreases.
ClR (=fu GFR –CLreabsorption) approaches GFR (fu=1) as Cp increases.

23. Capacity-Limited Metabolism (CLH ,F)
Enzymatic reactions are saturable.
Saturated hepatic metabolism decreases CLH.
Saturated first-pass metabolism increases F.
When hepatic metabolism gets saturated, drug will stay in body longer, increasing F.

24. capacity-limited metabolism
e. g.
capacity-limited metabolism
- Phenytoin is eliminated by hepatic metabolism only.
- As the dosing rate increases, Cp increases disproportionally.
As the dosing rate increases, hepatic metabolism is saturated and CL decreases.
As the dosing rate increases, it takes longer time to reach steady state.
Plasma drug Conc. Increases dramatically, and disproportionately. When drug has linear kinetics, you can use this equation to calculate. When you increase the dose, expect an increase in proportion of steady state concentration. BUT when Hepatic metabolism is saturated, clearance will decrease, plasma conc. goes UP disproportionately.
Half-life increases due to saturation, so it takes a lot longer to reach steady state with more drugs.
Arrows denote when Steady State is reached. As it gets more and more saturated, clearance becomes less and less, and higher the steady state. Half-life is longer, which shows you that it can’t be linear.

25. Michaelis-Menten Kinetics Applied to Metabolism
n: rate of metabolism
Vmax : maximum rate of metabolism
Km : Michaelis constant, disassociation constant of ES
[S]: drug concentration

26. Rate of Metabolism is NOT ALWAYS proportional to drug concentrations
Michaelis-Menten kinetics
Rate of Metabolism is NOT ALWAYS proportional to drug concentrations n
Zero order
- When [S] = Km , n=1/2 nmax
Km is the drug concentration at which half of the active sites on enzymes are occupied.
Non linear
Vmax and Km are constants.
- When [S] <<< Km ,
First order Km
[S]
- When [S] >>> Km ,

27. Michaelis-Menten Kinetics Applied to CLM
Rate of elimination = CL x Cp
Rate of metabolic elimination =
Metabolic clearance can be dependent on CP. For linear, CP is MUCH lower than Km, so remains a constant. Cp is close to Km and sometimes higher in non-linear so drug could be saturating metabolic enzymes, so clearance will change. Clearance will decrease when Cp increases.

28. Michaelis-Menten Kinetics Applied to Metabolism
zero-order
nmax
When Cp << Km , linear PK
non-linear
When CP is way higher than Km, clearance is such a low level compared to the drug amount in the body, elimination is almost not occurring.
first-order Km
[Drug]
When Cp >> Km ,

29. Linear vs. Saturable Metabolism
nonlinear
linear
Clearance is independent of Cp CL CL Cp Cp

30. Michaelis-Menten Kinetics Applied to Metabolism
Rate of metabolic elimination =
At the steady-state following multiple dosing ,
If you know Vmax/F and Km, then you can design a dosing rate that will produce the plasma drug concentration that you have/are looking for.
D/tau is the dosing rate. Tau = time.

31. Linear vs. Saturable Metabolism
nonlinear
CSS
CSS
D/t
D/t

32. Most Common Sources for Nonlinear Pharmacokinetics
Capacity-limited oral absorption (F)
Capacity-limited metabolism (CLH )
Saturable protein binding (CLH, CLR, V )
Capacity-limited excretion (CLR ) 32 32

33. Case Study
At a daily intake of 75 mg of ascorbic acid (vitamin C), the steady-state plasma concentration is 9 mg/L, whereas at a daily dose of 10,000 mg, the steady-state concentration is about 19 mg/L in a healthy volunteer. The renal clearance of ascorbic acid is less than 0.5 ml/min at the plasma concentration of 9 mg/ml, whereas the renal clearance is 21 ml/min at 19 mg/L.
Renal clearance is increasing because saturation of renal reabsorption.
Saturated intestinal absorption decreases the oral bioavailability also. It is carrier-mediated.
Vitamin is absorbed by passive facilitated diffusion in the small intestine, and undergoes tubular reabsorption in the kidney.

34. Maintenance Dose Selection for Phenytoin
Phenytoin is eliminated by hepatic metabolism (CYP2C9) only.
Variability in Vmax and Km values in patients causes a wide range in the effective doses needed to achieve therapeutic levels.
Phenytoin has a low extraction ratio drug. Enzyme has a very limited capacity to metabolize phenytoin.
Very narrow therapeutic range. Different color indicates a different patient. Hits therapeutic window at very different doses. Have to individualize dose based on their own Vmax.
Limited therapeutic range and easily saturable. Cyt P450 enzyme, so gene expression varies a lot between people. Different people’s enzymes have different Vmax and Km, so different dosing rate per person!
therapeutic range= mg/ml = mg/L

35. Michaelis-Menten Kinetics Applied to Metabolism
Rate of metabolic elimination =
At the steady-state following multiple dosing ,

36. How to Obtain Vmax/F and Km
y= m x - b
Slope = m
= y2-y1
x2-x1 * x y -b *
Css / Dose rate
Css
Slope = Vmax /F
-Km
Simple way to get Vmax. Vmax/F (treat as one term)
Y-intercept is Km, Slope is (Vmax/F)
X is Dose/Time
Y is Css.

37. The Direct Linear Plot Vmax/F Km dosing rate 2 dosing rate 1 -C2 -C1
Need Dosing rate and [Css]. On minus side of x-axis, both points. Where the lines intercept you get Km and (Vmax/F)
More straight-forward. Km
-C2
-C1
Biochem J, 139: (1974)

38. Maintenance Dose Selection for Phenytoin
A patient has been taking phenytoin (PHE) 150 mg b.i.d for 4 months. His plasma levels of PHE averaged 5 mg/L on this dose. Adjustment in dose to 250 mg b.i.d eventually led to a new plateau level of 20 mg/L. Assuming true steady state, strict patient compliance and that the measured plasma concentrations represent average levels over the dosing interval.
a) use a graphical method to estimate the patient's operative Vmax/F and Km values;
b) estimate a daily dose which should provide a steady-state plasma level of 12 mg/L.

39. * * Rate of metabolic elimination = Css Slope=Vmax /F Vmax/F Km -C1
Drug-Protein Binding
Nonlinear Pharmacokinetics
Rate of metabolic elimination =
Clearance Concepts *
Css *
Slope=Vmax /F
This is equation sheet for Dr. Tan’s part.
End of exam 3 materials!
Vmax/F Km
-C1
-C2
dosing rate 1
dosing rate 2
Css / Dose rate
-Km

40. Very first pass can eliminate 70% of drug
Very first pass can eliminate 70% of drug. So that’s why oral bioavailability is inversely proportional to protein binding in high excretion ratio drugs.
Protein binding may or may not affect hepatic clearance and renal clearance.
Clearance by definition is rate of elimination over plasma concentration. But real meaning is the flow rate concept. Volume of plasma being completely removed of drug per unit of time. Clearance is a hypothetical volume. CLT = Dose/AUC = Ro/Css. At steady state, rate of dosing is equal to rate of elimination. So when you divide by CSS, get CLT.
Clearance is additive. If you know CLT and CLR, you can calculate CLH. The fe (fraction excreted) is how you calculate hepatic clearance. Can’t direclty measure hepatic clearance, it is inferred from CLR.

41. Hepatic flow rate = fraction of flow rate x the flow itself (volume of blood being removed per drug per unit of time in liver).
For low extraction ratio drug, high bioavailability. They have low intrinsic capacity, hepatocytes don’t have enough metabolic activity or biliary excretion capacity. Sometimes it can limited by protein-binding or permeability. Regardless, low ER drug, can not remove it very efficiently. Clearance is restricted by Clint and fu.

42. Capacity-limited metabolism, happens to low extraction ratio drug
Capacity-limited metabolism, happens to low extraction ratio drug. Capacity not only is low, but is easily saturable. It’s being saturated, because Cp is approaching michaelis constant, or is going above it. Binding site to drug is being saturated, so clearance (metabolic) is being saturated, so when you increase Cp, the clearance continues to decrease to a point, when you are completely saturated, and clearance becomes ZERO.

43. Renal clearance is net result of all 3 processes
Renal clearance is net result of all 3 processes. Doesn’t mean Secretion and Reabsorption occur to every drug, BUT GFR occurs to every drug, as well as fu.

44. Graphs:
How do you tell there is non-linear kinetics going on? You make the decision by looking at whether the concentration or AUC is proportionally increased as dose increased.
VD CL or t1/2 remain unchanged in linear. In Non-linear at least one of F, V, CL, or t1/2 will be different.