Human Body Drug Simulation

Nathan LilesBenjamin Munda

Presentation Outline

Objective

Background

Model Overview

Organ and Body

Theory

Case Studies

Objective

Pharmacokinetics: Seeks to determine fate of substances administered externally to a living organism

Our focus, Oral Administration

a) most common, practical

b) model can be expanded

Our project focuses on the first objective above: creating an overall PBPK model that is accurate, physiologically correct, and innovative

1) Overall PBPK ADME model

a) How to divide the body and describe movement?

b) How to make mathematical model consistent with physiology/anatomy of the human body

2) Specific Component of a PBPK model

a) How does a drug’s structure and charge affect its movement in capillaries?

b) How does an enzyme within the liver interact with a drug? Are any intermediates created?

Background: Potential Applications

On Pharmaceuticals every year:

a) Companies R&D ≈ $70 billion

b) Consumers spend≈ $200 billion A Mathematical Body Model

Could:

a) For Companies:Accelerate R&D

b) For consumers:Help doctors optimize

dosesLower costs of prescription

drugs

Background: What Has Been Done

Previous Work:

Our Work

≈1,100 Simultaneous ODEs

≈200 constants≈100,000 steps

1) Began with current models of key organs

2) 23 Tissues included, avoided well-mixed assumption

3) Incorporated the tissues into a circulatory system with mass transfer on the capillary level

1) Compartment models

2) Well-mixed regions

3) Few parameters <15

4) Few equations < 25

Overview of Our Proposed Model

Path of drug

1) Enters stomach

2) Moves through small intestine

3) Enters the blood, first pass through the liver

4) Liver to the heart

5) Heart to the entire body

6) Interacts with the body

a) Reacts at intended site

b) Eliminated in liver, kidney

7) Returns to the Heart

1 2 34

5

6a

6b

6b 7

Absorption: Stomach

Stomach information:

1) “Churning” creates a well mixed volume

2) Exit flow rate of mass depends on the mass inside the stomach

3) Little absorption of the drug into the bloodstream occurs in the stomach

Governing Equations:

1) For liquids 2) For solids

Flow rate:

1) Dose (mg) enters stomach, which has some mass inside

2) The drug exits with a semi-constant concentration and a flow rate that varies with:

a) L or S?b) Massc)

Liberation

Concentration:

Stomach Results

0 0.5 1 1.5 2 2.5 30

100000

200000

300000

400000

500000

600000

Mass in Stomach vs. Time after consuming a 500 g drink

Time (hours)

Mass (

mg

)

Absorbtion: Small IntestineSmall Intestine Information:

1) Main site of drug absorption

2) ≈7 meters long with an average diameter of 2.5-3 cm

3) Modeled as a PFR

Governing Equation

Assumptions a) Radial variations are

unimportant

b) Diffusive flux term is negligible

Final Form:

c) Superficial velocity is a variable

Absorbtion: Small Intestine

A LaPlace transform was performed in the

z-dimension. The equation became a linear

homogenous ODE in the time dimension.

This integral cannot be solved analytically, thus

the inverse Laplace transform cannot be solved.

Numerical MethodsMethod of Lines

1) Discretize Space Variable Z2) System of Equations to Solve

Numerical Methods4th Order Runge-Kutta

Method of lines means we have many ordinary differential equations=4RK

Used to integrate a function:

Described by a 1st Order ODE:

Given initial values for y estimates next y in time by:

Where values for k (slope estimates) come from:

Small Intestine Results

0 2 4 6 8 10 12 14 160

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

Concentration vs. Time for Different Lengths

Entrance Mid LengthEnd

Time (hours)

Con

cen

trati

on

(m

g/m

l)

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

Concentration vs Dis-tance for Different Times

Early Middle End

Length (m)

Con

cen

trati

on

(m

g/m

l)

Absorbtion: Small Intestine

Amount absorbed through the gut wall for each time interval

Distribution

Estimations used in distribution:

1) Time from SI to liver negligible

2) Time from Inferior vena cava to heart negligible.

3) 8 seconds to get from the right heart to the left heart

4) 3 Seconds to get to the extremities.

5) 3 seconds back to the heart

1 2

3

45

Distribution: Blood Concentration

Governing Equations

1)

2)

1) Blood concentration is the ‘heart’ of our model

2) Heart chosen as site to track

Heart Information:

1) Well mixed tank

2) Receives influx from entire body and outputs to the entire body

Distribution: Capillaries

Governing Equations

Capillary Information:

1) Literature ≈ 40 billion capillaries

Calculated (A, d, l) ≈ 20 billion capillaries

2) Number of capillaries in an organ estimated by percent of total body blood flow

Example: Brain and kidney are small, but receive a large (≈30-35%) amount of blood, requires dense capillaries

1)

2)

From the Heart

Return to the Heart

Organ Capillary

Organ Tissue

≈1 billion capillaries

Capillary Results

0 2 4 6 8 10 12 14 16 18 200

0.001

0.002

0.003

0.004

0.005

0.006

Concentration vs Distance for Different Times

Early Middle End

Length (10^-4 m)

Con

cen

trati

on

(m

g/m

l)

Metabolism: The Liver

Governing Equations

Macrostructure

Microstructure

Liver Information1. Modeled as PFR

Simultaneous with tissue

2. Blood Mixing produces convection

3. Movement slow enough for Dispersion to matter

4. Mass transfer between vascular and tissue compartments

1)

2)

Hepatic Portal Vein

Return to the Heart

Sinusoid Volume

Liver Tissue Volume

`

Metabolic Elimination

Hepatic Artery

Liver Results

0 2 4 6 8 10 12 14 160

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Concentration vs Time for Different Lengths

Entrance Mid Length

Time (Hours)

Con

cen

trati

on

(m

g/m

l)

0 5 10 15 20 25 300

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

Concentration vs Dis-tance for Different Times

Early Mid End

Length (cm)

Con

cen

trati

on

(m

g/m

l)

Liver Results

0 2 4 6 8 10 12 140

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

Tissue Concentration vs. Time for Different

Lengths

Entrance Midlength

Time (hours)

Con

cen

trati

on

(m

g/m

l)

0 5 10 15 20 25 300

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

Tissue Concentration vs Distance for Different

Times

Early Mid

Length (cm)

Con

cen

trati

on

(m

g/m

l)

Excretion: The Kidney

Governing Equations

Kidney Information:

1) Blood enters kidney vascular system

2) Some flow rate transferred to bladder by GFR

3) Rest passes through capillaries where it can interact with tissue

4) From tissue moves to bladder, where excreted

1)

2)

3)

From the Heart

Return to the Heart

Kidney Capillary

Kidney Tissue

`

The Bladder

GFR

Kidney Results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

Urine Concentration vs. Time for Different Lengths

Time (Hours)

Co

nce

ntr

ati

on

(m

g/m

l)

Case Studies: LimitationsModel Requires ≈200

Physiological Parameters:1) Drug Differences2) Human Differences3) Literature Limitations

Our Strategy:1) Values have data/theory behind them2) Human differences don’t matter3) Drug parameters optimized to reproduce data

Case Study: Atenolol

Important Information

1) Acts to treat hypertension

2) Acts in the brain

3) 11.1% of the dose was absorbed into the brain

4) The compartmental model does not predict the double peak

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.000.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04 Blood Concentration of

Atenolol vs Time

Experimental Value Model PredictionCompartmental Prediction

Time (hours)

Co

nce

ntr

ati

on

(m

g/m

l)

Case Study: Imatinib Mesylate Important

Information

1) Common anti-cancer drug

2) Acts within tissue where tumors located

3) 18.9% was absorbed into the esophagus and stomach

4) The compartmental model and our model predict similar results for the blood concentration, however, the compartmental model would be unable to predict tissue concentrations.

0 5 10 15 200

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

0.00045

0.0005

Concentration vs. Time for Imatinib

Experimental ModelCompartmental

Times (hours)

Co

nce

ntr

ati

on

(m

g/m

l)

Conclusion:

What did we accomplish?

1) A mathematical model that can accurately describe the way a drug moves through the body

2) Integrated all organs at the capillary level - A novel approach

3) Includes spatial variations in all body tissues

What should be done in the future?

4) Develop a method to determine the model parameters

5) Account for differences between people6) Compare more extensively to simpler models