# Presentation: 661904

## COMPARTMENTAL ANALYSISOF DRUG DISTRIBUTION

**Juan J.L. Lertora, M.D., Ph.D****.**
**Director**
**Clinical Pharmacology Program**
**Office of Clinical Research Training **
**and Medical Education**
**National Institutes of Health**
**Clinical Center**

## DRUG DISTRIBUTION 

**The post-absorptive transfer of drug from one location in the body to another.**** **

-   **Compartmental Models **
- (ordinary differential equations)
**  ****Distributed Models **
- (partial differential equations)

## Pharmacokinetic Models Using Ordinary Differential Equations*

**    **

- * **From Atkinson AJ Jr, et al. Trends Pharmacol Sci 1991;12:96-101.**

## Mathematical vs. Physical Models*

**   ****MATHEMATICAL MODEL:**
**    ****Functions or differential equations are employed without regard to the physical characteristics of the system.**
**   ****PHYSICAL MODEL:**
**    ****Implies certain mechanisms or entities that have physiological, biochemical or physical significance.**

- * **Berman M: The formulation and testing of models.**
**    ****Ann NY Acad Sci 1963;108:182-94**

## Goals of Drug Distribution Lecture

**   ****Significance**** of Drug Distribution Volumes**
**   ****Physiological Basis**** of Multi-Compartment ****	****Pharmacokinetic Models**
**   ****Clinical Implications**** of Drug Distribution ****	****Kinetics**
**	**

## DIGOXIN DISTRIBUTION VOLUME

## Body Fluid SpacesCatenary 3-Compartment Model 

- cell membranes

## Volume of Distribution and Physiological Fluid Spaces

** ****Intravascular Space:**
**       ****None**
** ****Extracellular Fluid Space:**
**     ****Inulin**
**       ****Proteins ****and other**** Macromolecules**
**       ****Neuromuscular Blocking Drugs ****(N****+****)**
**       ****Aminoglycoside Antibiotics ****(initially)**
**	**

##                   Volume of Distribution and               	        Physiological Fluid Spaces 

**     ****Total Body Water**
**         ****Urea**
**         ****Ethyl alcohol**
**         ****Antipyrine ****(some protein binding)**
**         ****Caffeine**

## Factors Affecting Volume of Distribution Estimates 

**	****Binding to Plasma Proteins**
**	****	**** ****Thyroxine**
**	****	****Theophylline**
**	****Tissue Binding**** ****(partitioning)**
**	**** ****	****Lipophilic Compounds**
**           ****	****Digoxin**** ****(Na****+**** -  K****+**** ATPase****)****	**
**	**

## Effect of Plasma Protein Binding on Drug Distribution

**ECF***** ***

**ICF**

-   **Elimination**

**Cell Membranes**

**BINDING**
**PROTEINS**

## Effect of Plasma Protein Binding on Apparent Volume of Distribution*

*** Atkinson AJ Jr, et al. Trends Pharmacol Sci 1991;12:96-101.**

**f****u**** is the “free fraction”, the fraction of drug in plasma that is not bound to plasma proteins.**

## Impact of Protein Binding on Thyroxine Distribution Volume*

*** From Larsen PR, Atkinson AJ Jr, et al. J Clin Invest 1970;49:1266-79.**

***f******u****** ****** *****=***** *****0.03%**

**V****d**** = V****ECF**

## Impact of Protein Binding on Theophylline Distribution Volume*

*** From Atkinson AJ Jr, et al. Trends Pharmacol Sci 1991;12:96-101.**

***f******u******  *****=***** *****60%**

**V****d**** = V****ECF**** + *****f******u*****V****ICF**

## Basis for Increased Theophylline Volume of Distribution in Pregnancy*

*** From Frederiksen MC, et al. Clin Pharmacol Ther 1986;40;321-8.**

## Effect of Plasma Protein and Tissue Binding  on the Volume of Distribution of Most Drugs*

*** Atkinson AJ Jr, et al. Trends Pharmacol Sci 1991;12:96-101.**

**Ф is the ratio of tissue/plasma drug concentration.**

## LIPID SOLUBILITY(Doct) and 

## Apparent Volume of Distribution for Digoxin

**Φ**** includes**** binding to Na****+****-K****+ ****ATPase**.

## Tissue vs. Plasma Digoxin Levels

**	**

**HOURS**

## GOALS OF DRUG DISTRIBUTION LECTURE 

**   ****Significance of drug distribution volumes**
**   ****Physiologic basis of multi-compartment ****	****pharmacokinetic models**
**   ****Clinical implications of drug distribution ****	****kinetics**
**	**

## First Multicompartmental Analysis of Drug Distribution*

*** From** **Teorell T.  Arch Intern Pharmacodyn 1937;57:205-25****.**

## Analysis of Experimental Data

**    **

** ****How many compartments?**

**  *****Number of exponential phases*****        in plasma level vs. time curve determines the *****number of***** *****compartments*****.**

## TECHNIQUE OF CURVE PEELING

**    **

## COMPARTMENTAL ANALYSIS

**    **

**Data Equation:**
- C  =  A ́e -αt  +  B ́e -βt

**Model Equation:**
- dX1/dt = -(k01 + k21)X1 + k12X2

## TWO-COMPARTMENT MODEL

***V******d(ss)******  =  V******1******  +  V******2***

## 3 DISTRIBUTION VOLUMES

## TWO-COMPARTMENT MODEL

***CL******E******  =  k******01******V******1******  ***

## TWO-COMPARTMENT MODEL

***CL******I******  =  k******21 ******V******1******  =  k******12****** V******2***

## INTERCOMPARTMENTAL CLEARANCE*

**          ****Volume-Independent Parameter Characterizing the Rate of Drug Transfer     ****	**** Between Compartments of a Kinetic****      ****	****	****	****    ****Model**
**	****	****	**

*** From** **Saperstein et al.  Am J Physiol 1955;181:330-6.**

## Is Central Compartment Intravascular Space?

**   ****Usually ****not**** identified as such ****unless**** drug is ****	****given ****rapidly IV****.**
**   ****NEED TO CONSIDER:**
**    ****- ****If distribution is ****limited to ECF****, compare the ****	****central compartment volume with ****plasma**** ****	****volume****.**
**    ****- ****If distribution volume ****exceeds ECF**** compare central**
**      ****	****compartment with ****blood**** volume.***
**     *****(account for RBC/Plasma partition if [plasma] measured)**

**        **

## Analysis of Procainamide and NAPA Central Compartment Volumes*

**    **

- * **From Stec GP, Atkinson AJ Jr. J Pharmacokinet Biopharm 1981;9:167-80.**

| DRUG | VC
(L) | RBC/P | INTRAVASCULAR SPACE
(L)
PREDICTED     OBSERVED |  |
| --- | --- | --- | --- | --- |
| PA | 6.7 | 1.52 | 5.6 | 5.5 |
| NAPA | 7.5 | 1.62 | 5.6 | 6.0 |

## If Central Compartment Volume is Based on Plasma Concentration Measurements

**RBC/P  =  red cell/plasma partition ratio**
**Hct  =  hematocrit**

## Analysis of Inulin Kinetics with a 2-Compartment Model*

**Analysis of ****Inulin**** Kinetics with a ****2-Compartment Model***

*** Gaudino M. Proc Soc Exper Biol Med 1949;70:672-4.**

## 3-Compartment Model of Inulin Kinetics

***V******S***

***CL******F***

***V******F***

***CL******S****** ***

***V******C****** ***

**Dose**

-  ***CL******E***

**EXTRACELLULAR FLUID**

## Basis for Kinetic Heterogeneity of Interstitial Fluid Space

## ENDOTHELIAL FENESTRAE IN HEPATIC SINUSOIDS 

**   **

## INTERENDOTHELIAL CELL JUNCTION IN CONTINUOUS CAPILLARY

**   **

## UREA-15N2 KINETICS INA NORMAL SUBJECT

## Multicompartment Model ofInulin and Urea Kinetics*

*** From Atkinson AJ Jr, et al. Trends Pharmacol Sci 1991;12:96-101.**

**UREA**

## ROLE OF TRANSCAPILLARY EXCHANGE

**The ****central**** compartment for both ****urea**** and ****inulin**** is the ****intravascular ****space.**

**Therefore,**** transcapillary exchange ****is the**** rate-limiting ****step**** ****in the distribution of urea and inulin**** ****to the**** peripheral ****compartments**** ****of**** ****the mammillary**** 3-compartment model. **

## RENKIN EQUATION*

*** From Renkin EM.** **Am J Physiol 1953;183:125-36**.

**Q = capillary blood flow**
**P = capillary permeability coefficient-surface**
**      ****area product (sometimes denoted P****•S).**

## SIMULTANEOUS ANALYSIS OF INULIN AND UREA-15N2 KINETICS

## 3-COMPARTMENT MODEL

***V******S***

***CL******F ****** =  Q******F****** (1 – e ******P******F/Q******F******)***

***V******F***

***CL******S****** =  Q******S****** (1 – e ******P******S/Q******S******)***

## For Each Peripheral Compartment

**U = urea;  I = inulin**
**D = free water diffusion coefficient**

## SIMULTANEOUS ANALYSIS OF INULIN AND UREA-15N2 KINETICS

**How does Q****F ****+ Q****S**** compare with C.O.?**

## CARDIAC OUTPUT AND COMPARTMENTAL BLOOD FLOWS*

*** From Odeh YK, et al. Clin Pharmacol Ther 1993;53;419-25.**

|  | QF | QS | QF + QS |  |
| --- | --- | --- | --- | --- |
|  | L/min | L/min | L/min | % CO |
| MEAN† | 3.87 | 1.52 | 5.39 | 99 |

**†**** ****MEAN OF 5 SUBJECTS**

## TRANSCAPILLARY EXCHANGEMechanisms

_**TRANSFER OF SMALL MOLECULES (M.W. < 6,000 Da)**_**:**
** ****Transfer proportional to D**
**         ****- Polar, uncharged (urea, inulin) **

**         ****- Facilitated diffusion (theophylline)**

** ****Transfer rate > predicted from D**
***         ******- *****Lipid soluble compounds (anesthetic gases)**

** ****Transfer rate < predicted from D**
***         ******- *****Highly charged (quaternary compounds)**
**         ****- Interact with pores (procainamide)  **

## THEOPHYLLINE KINETICS Referenced to Inulin and Urea

** **

## THEOPHYLLINE  Intercompartmental Clearance and Compartmental Blood Flows*

*** From Belknap SM, et al. J Pharmacol Exp Ther 1987;243:963-9.**

## Urea and Theophylline  Diffusion Coefficients*

*** From Belknap SM, et al. J Pharmacol Exp Ther 1987;243;963-9.**

|  | MOLECULAR WEIGHT

(DALTONS) | CORRECTED STOKES-EINSTEIN
RADIUS
(Å) | Dm @ 37o C

(x 10-5 cm2/sec) |
| --- | --- | --- | --- |
| UREA | 60 | 2.2 | 1.836 |
| THEOPHYLLINE | 180 | 3.4 | 1.098 |

## PRESUMED CARRIER-MEDIATED TRANSCAPILLARY EXCHANGE

## GOALS OF DRUG DISTRIBUTION LECTURE 

**   ****Significance of drug distribution volumes**
**   ****Physiologic basis of multi-compartment ****	****pharmacokinetic models**
**     **
**   ****Clinical implications of drug distribution **
**	****kinetics**
**     **
**	**

## SIGNIFICANCE OF DRUG DISTRIBUTION RATE

**  ****	****1.****   ****Affects toxicity of IV injected drugs**
**	****	****Theophylline, lidocaine**
**   ****	****2.   Delays onset of drug action****	****	****	****	****Insulin, digoxin**** **
**          ****3.****   ****Terminates action after IV bolus dose**
**	****	****Thiopental, lidocaine**
**	****	****	**

## PK Model of THEOPHYLLINE Distribution

**CO = *****Q******F *****+ *****Q******S***

## DIGOXIN is NOT the First Drug Given to Patients with Acute Pulmonary Edema

## PK-PD Study of INSULIN Enhancement of Skeletal Muscle Glucose Uptake*

*** From Sherwin RS, et al. J Clin Invest 1974;53:1481-92.**

## DISTRIBUTION TERMINATES EFFECT BOLUS LIDOCAINE DOSE*

**DISTRIBUTION TERMINATES EFFECT**** ****BOLUS LIDOCAINE DOSE***

*** From Atkinson AJ Jr. In: Melmon KL, ed. Drug Therapeutics: Concepts for Physicians, 1981:17-33.**

**THERAPEUTIC RANGE**

## CONSEQUENCES OF VERYSLOW DRUG DISTRIBUTION 

** ****“****Flip-Flop” Kinetics**
** ****Effective Half-Life**
** ****Pseudo Dose Dependency**
**	****	****	**

## GENTAMICIN Elimination Phase Preceeds Distribution Phase*

*** From Schentag JJ, et al. JAMA 1977;238:327-9.**

## GENTAMICIN ELIMINATION  Nephrotoxic vs. Non-Toxic Patient*

*** From Coburn WA, et al. J Pharmacokinet Biopharm 1978;6:179-86.**

**NON-TOXIC**

**NEPHROTOXIC**

## CONSEQUENCES OF VERYSLOW DRUG DISTRIBUTION 

** “****Flip-Flop” Kinetics**
** ****Effective Half-Life**
** ****Pseudo Dose Dependency**
**	****	****	**

## TOLRESTAT Cumulation with Repeated Dosing*

***From Boxenbaum H, Battle M: J Clin Pharmacol 1995;35:763-6.**

## CUMULATION FACTOR

## TOLRESTAT CUMULATION

**Predicted C.F. from T****1⁄2**** = 31.6 hr:****	****  4.32**
**Observed C.F.:****	****	****	****	****  1.29**

## EFFECTIVE HALF- LIFE*

*** From Boxenbaum H, Battle M. J Clin Pharmacol 1995;35:763-66.**

## EFFECTIVE HALF-LIFE OF TOLRESTAT*

*** From Boxenbaum H, Battle M. J Clin Pharmacol 1995;35:763-66.**

## CONSEQUENCES OF VERYSLOW DRUG DISTRIBUTION 

** “****Flip-Flop” Kinetics**
** ****Effective Half-Life**
** ****Pseudo Dose Dependency**
**	****	****	**

##  AREA UNDER THE CURVE Measure of Dose Proportionality

** ****AREA UNDER THE CURVE ****Measure of Dose Proportionality**

**CL = ABSORBED DOSE/AUC**

## HYPOTHETICAL Phase I Trial Results

|  | DOSE 1 | DOSE 2 | INCREASE |
| --- | --- | --- | --- |
| DOSE
(mg) | 25 | 100 | 4 x ↑ |
| AUC
(μg·hr/mL) | 1.32 | 17.91 | 13.6 x ↑ |

## Dependency of PK Estimates on Identified Terminal Phase

## DISTRIBUTION VOLUME  Representative Macromolecules

## CLOTTING FACTOR PHARMACOKINETICS*

**  “****The ****V****d(ss)****..... always ****exceeds**** the actual ****plasma****              ****	****volume****, implying that ****no drug****, not even large ****	****molecular complexes as F-VIII, is ****entirely**** ****	****confined to the plasma space****.”**
**   **
**   “****A too ****short blood sampling**** protocol gives ****flawed**** ****	****results**** not only for terminal T1/2 but also for ****	****the model independent parameters.”**
**   **

*** Berntorp E, Björkman S. Haemophilia 2003;9:353-9.**

## Topics for Further Study (See end of Chapter 3)

**  ****Graphical Analysis of PK Data**
**  ****Calculate Parameters of Compartmental ****	****Models**
**  ****Comparison of Three Different Distribution ****	****Volumes**