Introduction to Pharmacokinetics��Novartis-Academia Hackathon

Andrew Stein, PhD

Associate Director Pharmacometrics

Cambridge, MA August 2019

Pharmacometrics

Pharmacometrics

Overview

Pharmacometrics

Acknowledgements�(Based on material from)

• Rowland, M., Tozer, T. N. Clinical pharmacokinetics and pharmacodynamics: concepts and applications 
• Peter Bonate, Astellas
• Richard Brundage, U Minnesota
• Leon Aarons, U Manchester
• Jean-Louis Steimer, Novartis
• Martin Fink, Novartis
• Nick Holford, U Auckland http://holford.fmhs.auckland.ac.nz/Teaching/pharmacometrics/advanced.php

3

Pharmacometrics

Motivation – the importance of finding the right dose of a drug

• Tylenol helps to reduce pain and fever and is safe at daily doses ≤3,000 mg/day
• >5,000 mg/day can cause liver damage
• >10,000 mg/day can be lethal.
• You can learn about safety and efficacy from clinical trials
• Making best use of this data requires models to integrate the data

4

(paracetamol)

Pharmacometrics

The “PKPD” pathway of drug effect

5

Oral Dose

Elimination

from body

Pharmacokinetics (PK):

How body affects drug

Pharmacodynamics (PD):

How drug affects body

Should children and adults receive the same dose?

What dose is needed to shrink a tumor without causing severe neutropenia

Drug Concentration

Measurement

Pharmacometrics

Measuring PKPD

6

PK (Pharmacokinetics) example

Measurement of drug concentration from circulation

PD (Pharmacodynamics) example

Change in tumor size, as measured by X-Ray

Dose Regimen

• amount given
• frequency
• method (oral, patch, etc.)

Pharmacometrics

Understanding PKPD can help in picking the optimal dose regimen

7

From Rowland and Tozer

Drug doesn’t work

Drug is too toxic

Pharmacometrics

Types of Drugs

8

Small Molecule

Antibody

Cell Based

Therapies

Humira

Keytruda

Aspirin

Tylenol

Kymriah

Stem Cell Transplant

~1 nm

500 Da

~10 nm

150,000 Da

100,000 nm

~10¹⁴ Da

Pharmacometrics

Introduction to Pharmacokinetics

9

Pharmacometrics

10

Why Do We Measure Blood or Plasma Concentrations When the Site of Action is Someplace Else?

Pharmacometrics

Key process of pharmacokinetics�(ADME)

• Absorption
• Distribution
• Metabolism
• Excretion

11

Pharmacometrics

Volume of Distribution

12

Pharmacometrics

Key Lessons

• For intravenous dose
• Initial Concentration = Dose/Volume
• Volume is a “theoretical concept” and does not necessarily reflect blood or body volume

13

Pharmacometrics

Definition of dose and concentration

14

Dose

Measurement: Mass

Units: mg

Concentration (C)

Measurement: Mass/Volume

Units: mg/ml

Key Formula

Right after IV dosing (time = 0)

C(0) = Dose/V → V = Dose/C₀

Volume (V)

“Theoretical Volume” needed to contain administered drug at the measured concentration

Pharmacometrics

Volume of distribution example�C₀ = Dose/V or Dose=C₀·V

15

Drug Concentration

in Beaker

From Pete Bonate

Drug Concentration

with Activated Charcoal

(absorbs drug)

Dose = 1000 mg

C₀ = 0.2 mg/mL

V = 5000 mL

​

Dose = 1000 mg

C₀ = 20 mg/mL

V = 50 mL

​

Pharmacometrics

Volumes for different drugs

16

Erythropoietin

Chloroquine

Morphine

Erythromycin

Doxorubicin

10,000

1000

100

10

Volume (L)

~ Dose/C(0)

Drug

Blood Volume = 5

Total Body Water = 42

Total Body = 70

Pembrolizumab (antibody)

3

Pharmacometrics

How can chloroquine have a volume of 10,000L, the size of an X?

• A drug may appear to have a larger volume if
• It is lipophilic (sticks to fat, instead of water) and distributes to body and is not observed in blood.
• Accumulates in red blood cells and not present in the plasma sample
• Key lesson: Volume (and all PK parameters) are “apparent” and cannot always be easily related to physiological properties

17

https://www.watertankfactory.com.au/water-tanks/r10000-litre-rainwater-tank/

10,000 L

Pharmacometrics

Key Lessons

• Volume relates amount in body to concentration measured in circulation.
• Volumes are not necessarily physiologically meaningful and can be as large as 10,000 L.

18

Pharmacometrics

Pharmacokinetics after single intravenous bolus dose

Part 1 – focus on elimination (ignore distribution)

19

Pharmacometrics

Examples of intravenous drugs

• Chemotherapies (doxorubicin)
• Pain medications (morphine)
• Increase blood pressure (epinepherine)

20

Pharmacometrics

Drug concentration over time

21

Dose = 10 mg

C₀ = 0.04 mg/L

V = Dose/C₀

= 250 L

Pharmacometrics

Drug concentration over time�Log scale

22

Dose = 10 mg

​

C₀ = 0.04 mg/L

​

V = 250 L

k

k = 0.23/h

= elimination rate

units = 1/time

Pharmacometrics

Half-life definition – how long for half of drug to be eliminated

23

k

How long does it take for half the drug to be eliminated?

At what time does:

Pharmacometrics

Half-life example

24

k = 0.23/h

Every 3 hours,

concentration declines by half

0.04 (100%)

0.02 (50%)

0.01 (25%)

0.005 (12%)

0.0025 (6%)

0.00125 (3%)

​

​

​

Pharmacometrics

The shorter the terminal half-life, the more frequently the drug is dosed

25

Dosing interval also depends on duration of effect

Pharmacometrics

SEE QUIZ QUESTIONS

26

Pharmacometrics

Compartmental Model

27

A

Rate of change of amount in body

Dose

k

Pharmacometrics

Mathematical introduction to Clearance [L/h]

28

Clearance is a flow rate of plasma that is completely eliminated of drug

Pharmacometrics

Mechanical Analogy

29

From Leon Aarons

Vessel

Volume (V)

Concentration (C)

Metabolizer (Liver)

Filter (Kidney)

Pump (Heart)

Filter

Pharmacometrics

Mechanical Analogy

30

From Leon Aarons

Vessel

Dose

Volume (V)

If filter eliminates all drug that passes through, then CL is the flow rate

Concentration (C)

Perfect

Filter

Pharmacometrics

Clearance is the most important PK parameter

31

Pharmacometrics

Clearance is helpful in understanding the average concentration at steady state

• After sufficient time under the same dosing regimen, one reaches “steady state”, where
• Rate In = Rate Out = CL·C
• Let’s say:
• You want to maintain concentration ~ 10 mg/L
• You know that CL = 3 L/h
• Then:
• Rate Out = (3 L/h)·(10 mg/L) = 30 mg/h
• The maintenance dose (Rate In) is 30 mg/h.

Pharmacometrics

33

Total drug in = Total drug out

If clearance does not depend on concentration (linear)

Formula doesn’t depend on specific model for C(t), Volume or half-life. It only requires linear clearance

Pharmacometrics

Major Sites of Elimination

34

Kidneys

From Pete Bonate

Liver

Lysosome

Mainly metabolism

Mainly excretion

(urine)

Proteolysis

(in cells)

Small Molecules

Biologics

All cells

Pharmacometrics

Proteins in general are eliminated through lysosomal degradation

mAb PMX Overview | Business Use Only

35

This process can occur in many tissues. Endothelial cells is one major location

AS

Uptake via fluid phase or receptor

In lysosome, proteins are degraded

Cell Membrane

Pharmacometrics

Function of Kidneys

36

• “Cleans the blood”
• Filter the blood and remove endogenous metabolic waste, such as urea, uric acid, and creatinine
• Remove foreign substances

​

• Regulate body water content, hormones, mineral composition, and acidity

​

• Can also metabolize drugs (glucuronidation)

​

From Pete Bonate

Pharmacometrics

Phase 1 Metabolism: add small polar groups. Hydroxylation

37

From Pete Bonate

Pharmacometrics

Phase 2 Metabolism: add larger polar molecules

38

From Pete Bonate

Pharmacometrics

Metabolic Pathways Can Be Complex and Consist of Both Phase 1 and Phase 2 Pathways

39

From Pete Bonate

Pharmacometrics

Cytochrome P450 (CYP)�metabolizes drug

• Catalyze most Phase I biotransformations
• Found in every species and every tissue
• In particular in the liver, kidney, GI tract, and brain

​

​

40

From Pete Bonate

Pharmacometrics

Nonlinear pharmacokinetics occurs at “critical concentration” = C_crit

41

Burmester, G. R. et al. Mavrilimumab, a human monoclonal antibody targeting gm-csf receptor-α, in subjects with rheumatoid arthritis: a randomised, double-blind, placebo- controlled, phase I first-in-human study. Annals of the rheumatic diseases 70, 1542–1549 (2011).

Pharmacometrics

If route of elimination saturates, elimination is reduced

42

At small concentrations,

At large concentrations,

Faster elimination at lower concentrations

Pharmacometrics

“Derivation” for C_crit

43

For large doses (and concentrations), C ≫ K_m

Define C_crit to be where the linear (CL·C) and nonlinear (V_m) components contribute equally to total elimination

Stein, Andrew M., and Lambertus A. Peletier. "Predicting the Onset of Nonlinear Pharmacokinetics." CPT: pharmacometrics & systems pharmacology 7.10 (2018): 670-677.

Pharmacometrics

Nonlinear pharmacokinetics occurs at “critical concentration” = C_crit

44

AM Stein and L Peletier, to appear in CPT:PSP (2018)

Presented at ACoP in 2017

Pharmacometrics is a relatively young field.

​

There are opportunities to analyze simple models and contribute to understanding

Pharmacometrics

SEE QUIZ QUESTIONS

45

Pharmacometrics

Key Lessons

• Clearance tells you the average concentration at steady state (given an input dosing rate)
• Rate In = Rate Out = CL · C
• Small molecules are eliminated mainly by the liver and the kidneys.
• Large molecules (proteins) are eliminated mainly by endocytosis and proteolysis.

46

Pharmacometrics

Continuous Dosing: Infusion

47

Pharmacometrics

Continuous infusion

48

A

Dose Rate (R₀, mg/h)

k

Pharmacometrics

Continuous infusion

49

Pharmacometrics

Continuous infusion

50

C_SS

Pharmacometrics

Continuous infusion

51

C_SS

Pharmacometrics

At what time do you reach steady state?

52

C_SS

Pharmacometrics

Key Lesson: Sensitivity

• If you double the infusion rate, you double the steady state drug concentration
• If a patient has reduced clearance (e.g. due to liver impairment for a drug metabolized in liver) that reduces CL, then the steady state drug concentration is reduced.

53

Pharmacometrics

Repeated Dosing - Bolus

54

Pharmacometrics

Repeated dosing

55

Similar to infusion dose

Pharmacometrics

Recall formula for single bolus dose and amend for dose at another time

56

C₁ is the response of a single dose at time 0

Pharmacometrics

Each dose contributes one “bolus dose” term

57

Concentration

Time (τ)

During first dose (time 0-τ)

During second dose (τ-2τ)

During third dose (2τ-3τ)

0 1 2 3

Pharmacometrics

The general formula for N doses

58

Pharmacometrics

This formula is a geometric series

59

As N 🡪 ∞

Pharmacometrics

Formulas for Cmax,ss and Cmin,ss

60

What about average concentration C_avg,ss?

Recall that k and C₀ depends on both V and CL

Pharmacometrics

Formula for Cavg,ss (for linear PK)

61

Total drug in = Total drug out

If clearance doesn’t depend on concentration (linear)

Formula doesn’t depend on specific model for C(t)

It only requires linear clearance

Pharmacometrics

Repeated dosing similar to infusion

62

Infusion Rate ~ Dose Rate

Pharmacometrics

Cool and useful result:�AUC_0-∞ = AUCτ

63

Pharmacometrics

AUC_0-∞ = AUCτ �Graphical Demonstration

64

Dose N

Dose N-1

Dose N-2

Dose N-3

All earlier

doses

Pharmacometrics

AUC_0-∞ = AUCτ�Mathematical Demonstration

65

Pharmacometrics

Key Insights

66

Pharmacometrics

SEE QUIZ QUESTIONS

67

Pharmacometrics

Distribution

68

Pharmacometrics

Distribution

69

From Leon Aarons

Systemic Circulation

Other

Tissues

Distribution

Pharmacometrics

Factors that can effect distribution

• Lipophilicity (able to dissolve and move through fats)
• Transport proteins can move drugs around
• Blood proteins (e.g. albumin) will bind to low molecular weight drugs and prevent them from moving into tissue

70

From Leon Aarons

Pharmacometrics

Profile of a single dose (Linear Space)

71

Dose = 10 mg

C₀ = 0.055 mg/L

Pharmacometrics

Profile of a single dose (Log Space)

72

Dose = 10 mg

C₀ = 0.55 mg/L

​

V₁ = Dose/C₀

= 18 L

​

α = initial decline

= 12/h

​

β = terminal slope

= 0.5/h

​

t_1/2β = 1.4 h

α

β

Pharmacometrics

Compartmental Model formulation

73

A₁

Rate of change of amount in each compartment

Dose

k₁₀

A₂

k₁₂

k₂₁

Pharmacometrics

Finding the analytical solution

74

M

α and β are the eigenvalues of M

A and B come from the eigenvectors of M and the initial dose

Details in www.pfim.biostat.fr/PFIM_PKPD_library.pdf

Pharmacometrics

Compartment model and prolife each have 4 parameters

75

A+B

α

β

B

A₁

Dose (V₁)

k₁₀

A₂

k₁₂

k₂₁

Pharmacometrics

Elimination from the peripheral compartment not identifiable.

76

A+B

α

β

B

A₁

Dose (V₁)

k₁₀

A₂

k₁₂

k₂₁

k₂₀

4 parameters

of information

5 parameters

In model is too many

Pharmacometrics

Alternative parameterization for �2 compartment model

77

A₁

Dose (V₁)

k₁₀

A₂

k₁₂

k₂₁

CL = k₁₀·V₁

Q = k₁₂·V₁

V₂ = V₁·(k₁₂/k₂₁)

{V₁, k₁₀, k₁₂, k₂₁}

A₁

Dose (V₁)

CL

A₂

Q

V₂

{V₁, CL, Q, V₂}

Pharmacometrics

78

Total drug in = Total drug out

If clearance doesn’t depend on concentration (linear)

Pharmacometrics

Be careful in interpreting parameters from the model

79

A₁

Dose (V₁)

CL

A₂

Q

V₂

Pharmacometrics

Really, there are many more peripheral compartments, but the 2 compartment approximation often works

80

(wikipedia for PBPK)

Pharmacometrics

Key Lessons

• Distribution leads to two (or more) compartment kinetics
• Multiple “half-lives” are observable in the data.
• Even for two models, AUC_0-∞, AUC_τ and C_avg,ss are still driven by CL, when CL is independent of concentration
• This result holds for any number of compartments
• The volume becomes a function of time and the behavior of this function at time zero (V₁) or as time goes to infinity (V_z, V_ss) are often what is reported

81

Pharmacometrics

Absorption and bioavailability

82

Pharmacometrics

Routes of Administration (examples)

83

Inhalation (lung)

Nasal (nose)

Oral (mouth)

Intravenous (vein)

Intramuscular (muscle)

Subcutaneous (under skin)

Transdermal (skin patch)

Pharmacometrics

Oral dose of drug is absorbed through the gastrointestinal tract

84

Wikipedia

Pharmacometrics

Bioavailability (some drug is lost)

85

Oral

dose

small

molecule

Subcutaneous

dose

biologic

Oral

Absorption

Subcutaneous

Absorption

Intestinal

First Pass Elimination

Inject Site

First Pass Elimination

Hepatic

First Pass Elimination

Lymphatic

First Pass Elimination

AJM

(

systemic

circulation

Pharmacometrics

Oral dose vs intravenous dose

86

β

Intravenous Bolus

Oral Dose

Pharmacometrics

Oral Dose

87

Cumulative drug

absorbed over time

Pharmacometrics

Compartment Model with Absorption

88

A₁

Dose

k₁₀

A₂

k₁₂

k₂₁

A_depot

F, k_a

absorption

elimination

distribution

k_a = absorption rate

F = bioavailability [0-1]

fraction of drug

absorbed

Pharmacometrics

Application – extended release prolongs exposure and can improve safety (reduce peak-trough ratio)

89

k_a = 0.03/h

​

k_a = 0.3/h

Pharmacometrics

More complex absorption models can be developed

90

Abuhelwa, Ahmad Y., et al. "Food, gastrointestinal pH, and models of oral drug absorption." European Journal of Pharmaceutics and Biopharmaceutics 112 (2017): 234-248.

Pharmacometrics

Question

• How would you expect changing the rate of absorption(ka) to affect the average drug concentration at steady state?

​

​

​

• How does changing the bioavailability affect the average drug concentration

91

Pharmacometrics

Updating average concentration formula

92

Total drug in = Total drug out

If clearance is linear

Formula doesn’t depend on absorption, but it does depend on F

Pharmacometrics

Key Lessons

• When drugs are given by alternative routes to intravenous (e.g. oral, subcutaneous, etc.), they must be absorbed before entering blood stream.
• Some of the drug may be lost before entering the blood.

93

Pharmacometrics

Allometric Scaling

Effect of body size on PK parameters

Pharmacometrics

Often, we want to predict PK in humans from animal data

Moore, Brioni R., et al. "Pharmacokinetics, pharmacodynamics, and allometric scaling of chloroquine in a murine malaria model." Antimicrobial agents and chemotherapy 55.8 (2011): 3899-3907. Pictures from wikipedia

Slope of line ~ ¾ in log-space

Pharmacometrics

Many biological processes scale with body weight

• The weight of lifeforms spans 21 orders of magnitude:
• Smallest microbe ~ 10^-13 grams
• Largest whales ~ 10⁸ grams
• Many fundamental biological processes scale with weight (WT) in a surprisingly simple way.
• Volume ~ WT
• Metabolic Rate ~ WT^3/4
• Lifespan ~ WT^1/4
• Heart rate ~ WT^-1/4
• Length of aorta ~ WT^1/4
• Height of tree ~ WT^1/4

​

96

West, Geoffrey B., and James H. Brown. "The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization." Journal of experimental biology 208.9 (2005): 1575-1592.

Total number of heart beats before death is independent of weight:

(Heart rate)·(Lifespan) = WT⁰

​

Mammals get about 1 billion heart beats in their lifetime (from hamster to elephant)

Pharmacometrics

Metabolic rate ~ WT^3/4

97

slope = 3/4

slope = 3/4

Pharmacometrics

The reason for metabolism ~ WT^3/4 �is to maximize surface area (SA)

• Evolution maximizes surface area (SA), in order to maximize nutrient absorption and metabolism.
• For standard geometries (cube, sphere), surface area relates to length (L), volume (V), and weight (WT) by:
• SA ~ L²
• V ~ L³
• SA ~ V^2/3 ~ WT^2/3 (because volume is proportional to weight)
• Because evolution pushes life to maximize surface area and a little more area can be obtained using the “fractal dimension” The maximum possible exponent is ¾.
• Therefore, SA ~ WT^3/4 > WT^2/3

98

Pharmacometrics

Blood vessels follow fractal patterns

99

https://dejavouz.wordpress.com/reflections/tree2/

https://blogs.uoregon.edu/artofnature/2013/12/03/fractal-of-the-week-blood-vessels/

What is the optimal fractal dimensions to maximize surface area?

Pharmacometrics

A fractal is a self similar as you zoom in to smaller scales

100

Pharmacometrics

An example of building a fractal

101

Koch Curve

Length

1

fractal dimension =

log(4)/log(3) = 1.26

Fractal

Pharmacometrics

Fractal geometry can increase the dimensionality of the space

102

0°

https://www.researchgate.net/figure/Von-Koch-curve-with-bend-angle-from-0-to-90-degrees_fig1_309391846

0°

90°

60°

30°

D = 1

D = 1.26

D = 2

The largest a fractal dimension can be is 1 + spatial dimension

The surface area can go from 2 🡪 3 dimensions

angle of triangle

Pharmacometrics

Life has found a 4th dimension using fractal geometry! This is why CL ~ WT^3/4

• For “normal” shapes:
• SA ~ L²
• V ~ L³
• SA ~ V^2/3 ~ WT^2/3

103

1. West, Geoffrey B., James H. Brown, and Brian J. Enquist. "The fourth dimension of life: fractal geometry and allometric scaling of organisms." Science 284.5420 (1999): 1677-1679.

• For blood vessels
• SA ~ L³
• V ~ L⁴
• SA ~ V^3/4 ~ WT^3/4

Pharmacometrics

Key Lessons

• Generally
• Clearance scales with CL ~ WT^3/4
• Volume scales with V ~ WT
• This is a rule of thumb and does not always apply. Important caveats:
• Differing mechanisms of drug metabolism across animals
• Difference between “lean body weight” and total body weight
• Differences in metabolism with age, especially under 1 year old.
• In practice, estimating the exponent for a particular drug can be challenging due to the limited range in weights that are tested. So often, the ¾ exponent is assumed.

104

Pharmacometrics

Where PK measurements and models are used

105

Pharmacometrics

Bioequivalence Postulate

• If the concentration-time profiles for two drugs or dosing conditions are the same then efficacy and toxicity should be the same
• Basis for generic drugs
• Basis for formulation changes
• Basis for examination of drug interactions, food effects, etc.

106

From Pete Bonate

Pharmacometrics

107

Mean clozapine plasma concentrations after 7 days of 100 mg BID for 2 formulations in 18 schizophrenic subjects

Tassaneeyakul W et al. Steady-state bioequivalence of clozapine tablet in schizophrenic patients. J Pharm Pharmaceutic Sci 8, 47, 2005

From Pete Bonate

Pharmacometrics

For drugs that are cleared by kidney, kidney function will affect exposure

108

~ Measure of kidney function

Roland and Tozer, Figure 13-14

Pharmacometrics

Some factors that may impact drug concentrations

109

Pharmacometrics

Backups

110

Pharmacometrics

Acetaminophen Overdose

• Acetaminophen is Phase 2 conjugated by glutathione
• Large doses deplete the glutathione and sulfate pools
• Toxic metabolite then binds to cellular macromolecules
• Clinical course:
• Nausea, vomiting (Day 1)
• Hepatic enzymes ↑, right upper quadrant sensitive to touch (Day 2 to 3)
• Hepatic necrosis, possibly renal failure, and cardiac abnormalities (Days 3 to 5)

​

111

From Pete Bonate

Pharmacometrics

112

• Smallest dose recorded for hepatotoxicity in adults is 5.4 g
• 10 g is lethal
• Therapy is targeted at repleneshing sulfate pools

From Pete Bonate

Pharmacometrics

Biliary Elimination

• Bile is secreted from the liver to the gallbladder and to the the intestine
• Bile aids absorption of fats
• breaks down large fat globules into smaller ones
• Drugs may be excreted by biliary route

​

113

From Pete Bonate and wikipedia

Liver

Gall-

bladder

Bile

Duct

Intestine

Pharmacometrics

In the two compartment model, “volume” is a function of time

• Recall that volume relates amount in body to concentration measured in circulation.
• For 1 compartment model:
• C(t) = A(t)/V
• There is only one volume.
• For a two compartment model with central CL, there are:
• Two volume parameters: V₁ and V₂.
• There is also a volume function: C(t) = A(t)/V(t)
• At time 0, C(0) = A(0)/V₁
• After a single dose, at long times, C(t) = A(t)/V_z where V_z = CL/β
• At steady state after multiple doses:
• C(t_ss) = A(t_ss)/V_ss where: V_ss = V₁·[1 + k₁₂/k₂₁]

114

Pharmacometrics

Sketch of derivation of V_z and V_ss

• For the derivation, we start with the definition of V(t)

​

​

​

• To derive V_z and V_ss, we then:
• Use the analytic expression for C(t) for the either single dose (V_z) or multiple dose (V_ss)
• Take the limit as t🡪∞

115

Pharmacometrics

Definition of a dimension

• Let N = number of shapes (sticks, squares or cubes) needed to cover a big shape.
• Now reduce the length of each side of shape by a factor ε (e.g. 1/3)
• Then the dimension (D) is such that: N ~ ε^-D

116

ε = 1

ε = 1

ε = 1/2

ε = 1/3

Pharmacometrics

For normal shapes, surface area relates to weight by 2/3 power

• Cube

​

SA = 6L²

V = L³

SA = c₁·V^2/3 ~ WT^2/3

c₁ is a constant

117

• Sphere

​

SA = 4𝜋R²

V = (4/3)𝜋R³

SA = c₂·V^2/3 ~ WT^2/3

c₂ is a constant

R

L

• As V increases, SA increases too, but the SA:V ratio decreases.
• This 2/3 ratio holds at the macroscale, but on the microscale, it can be possible for SA to have > 2 dimensions.

Pharmacometrics