PBPK of Diazepam, IV and Oral Dosing

A PBPK Model of Diazepam. Including open source software simulation code. The simulator models both IV and oral dosing.

Disclaimer: This article has not been peer reviewed or published. Provided for informational purposes only.

Contents 1 Abstract 2 Introduction 3 Methods 3.1 The Physiological Compartment Model 3.1.1 Figure 1.0 – Physiological Compartment Model 3.2 The Physiological Parameters 3.2.1 Volume Data 3.2.2 Blood Flow Data 3.3 Substance Specific Parameters 3.3.1 Blood to Plasma Ratio 3.3.2 Figure 2.0 – Detailed Substance SubCompartment Model 3.3.3 Fraction Unbound, plasma (fup) 3.3.4 Partition Coefficients 3.3.4.1 Partition Coefficients: Plasma vs. Blood 3.3.4.2 Partition Coefficients: Animal to Human Scaling 3.3.4.3 Partition Coefficients: In the literature 3.3.5 Clearance 3.4 Mass Balance Equations 3.4.1 Figure 3.0: Mass Balance Diagram, Liver compartment 3.5 Oral Dosing Model 3.5.1 Figure 4.0: Oral Dosing CAT Model 3.5.2 Oral Mass Balance Equations 3.6 The simulation code 3.6.1 Open Source PBPK Simulation Code 3.7 Experimental Data 3.7.1 Igari1983 3.7.2 Ochs1985 3.7.3 Klotz1976 3.7.4 Klotz1975 3.7.5 Klotz1975 Oral 3.7.6 Friedman1992 Oral 4 Results 4.1 Ochs1985 4.2 Klotz1975 4.3 Klotz1976 4.4 Igari1983 4.5 Klotz1975 Oral 4.6 Friedman1992 Oral 5 Discussion 6 Appendix 6.1 Derivation of Tissue and Blood Clearance formulas 7 References 7.1 PubMed 7.2 Non-PubMed 8 See Also 9 Wikipedia Links 10 Resources

Abstract

A Physiologically Based Pharmacokinetic (PBPK) model of Diazepam in the human was developed, including Oral dosing using the digestive tract compartmental absorption and transit (CAT) model developed by Yu and Amidon Yu1999a.

On the whole, the simulator output matched the experimental data reasonably well for both IV and Oral dosing.

Introduction

Physiologically Based Pharmacokinetics (PBPK), is a computer modeling technique that attempts to simulate how a substance/drug will behave within the body. PBPK differes from traditional compartmental Pharmacokinetics in that instead of using “virtual” compartments, PBPK uses physiologically based compartments, such as brain, kidney and liver, etc.

In principal, a PBPK model permits prediction of drug concentration in any tissue at any time and may provide considerable inside into drug dynamics. Igari1983.

The first Physiologically Based Pharmacokinetic (PBPK) model of Diazepam was developed by Igari, et al, (Igari1983) in the 1983 article, “Prediction of Diazepam Disposition in the Rat and Man by a Physiologically Based Pharmacokinetic Model”.

This current article attempts to extend the work of Igari etal by:

• Adding a model for Oral dosing
• Using “updated” pharmacological and substance specific parameters from a number of different literature sources.
• Comparing the simulation vs. a number of different sources of experimental results (Oral and IV).

Methods

Developing a PBPK simulation model, requires several steps:

• Determine the Physiological Compartment Model
• Determine the Physiological Parameters (substance independent)
  • Volume
  • Blood Flow
• Determine the Substance Dependent Parameters
  • Blood to Plasma Ratio
  • Partition Coefficients
    • FractionUnbound
  • Clearance Data
• Formulate the Mass Balance Equations

Once the necessary parameters are known, (Compartment Volume, Compartment Blood Flow, Partition Coefficients, Blood to Plasma Ratio and Liver Clearance), the Mass Balance Equations can be calculated and the Model is ready for simulation.

The Physiological Compartment Model

The Physiological Compartment Model used in the simulation follows from Igari1983. (and also see Langdon2007)

It depicts the body as being composed of 12 compartments (including a catch all compartment named “rest”) See Figure 1.0 below.

Figure 1.0 – Physiological Compartment Model

Notes:

• Q stands for Blood Flow, (not Quanity.)
• Compartments not modeled separately are “lumped” into a “catch-all” compartment named “Rest”.
• The blood vessels are “lumped” into two large compartments, one for arteries and one for veins.
• For each Tissue/Organ Compartment the associated blood capillaries and interstitial fluid are “lumped” together with the tissue/organ.
• Regarding the heart: For the flow from the lungs through the heart to the arteries, the heart is not included. Note: the value Q_heart is only the flow to the cardiac arteries (The arteries that supply blood to the heart to keep it alive).

A number of assumptions are made in the model development:

• intercompartmental transport occurs via blood flow.
• instantaneous equilibrium between tissue and blood within the tissue.
• drug concentration in the effluent blood is in equilibrium with that in the tissue.
• elimination of diazepam is by liver metabolism, not kidney excretion.

The Physiological Parameters

There are two physiological parameters (substance independent) needed:

• volume of each compartment
• blood flow into and out of each compartment

The data for the physiological Parameters is available from a number of literature sources. There is a fairly wide range in the numbers, basically due to the wide range in human body types. The parameters used in this simulation are from a composite of sources.

Volume Data

For volume, most sources derive their numbers from Brown1997 with some variations:

• Volume Data: Brown1997, Levitt2006, Langdon2007, Levitt2005

Volume Physiological Data:

Compartment	Volume
	Final (in %)	Final (kg)	Langdon2007 (kg)	Levitt2005 (kg)
venousBlood	5.57	3.9	3.9	–
arterialBlood	2.43	1.7	1.7	–
rest	13.47	9.424	13.83	9.524
adiposeTissue	25.00	17.5	12.5	17.5
skin	3.71	2.6	2.6	2.6
muscle	41.43	29.0	30.0	29.0
gastrointestinalTract	2.14	1.5	1.16	1.5
kidneys	0.44	0.31	0.31	0.31
liver	2.57	1.8	1.8	1.8
heart	0.47	0.33	0.33	0.33
brain	2.00	1.4	1.4	1.4
lungs	0.77	0.536	0.47	0.536
Total	100%	70kg	70kg	70kg

Note: The column labeled “final” are the values used in the simulation. Also, the data assumes a density of 1gm/ml, so a 70kg human has a volume of 70,000 ml.

Blood Flow Data

The Physiological Parameter for Blood Flow showed more variation across sources. Below are the numbers for five sources, with the final being a composite of the five.

• Blood Flow Data: Luttringer2003 Levitt2006, Levitt2005, Langdon2007, DeBuck2007, Kawai1994

Blood Flow Physiological Data

Compartment	Blood Flow (% of CO)
	Final	Luttringer	Levitt	Langdon2007	DeBuck	Kawai
venousBlood	–	–	–	–	–	–
arterialBlood	–	–	–	–	–	–
rest	8	13	2	16	15	8
adiposeTissue	10	5	13	5	10	6
skin	5	5	5	5	4	6
muscle	17	17	11	17	10	14
gastrointestinalTract	19	19	20	15	17	23
kidneys	19	19	22	19	17	21
liver	6 (out: 25)	6 (out: 25)	8 (out: 28)	7 (out: 22)	8 (out: 25)	6 (out: 29)
heart	4	4	5	4	4	3
brain	12	12	14	12	15	13
lungs	–	–	–	–	–	–
CO (ml/min)	6	6.5	5.6	6	6.08	5.2

Note: The column labeled “final” are the values used in the simulation.

Substance Specific Parameters

These simulation modeling parameters are specific to the substance Diazepam. They include:

• Blood to Plasma Ratio
• Fraction Unbound, (fup)
• Partition Coefficients
• Clearance

The “Fraction Unbound” is not actually used by the simulation model, but may be used in calculating
the partition coefficients.

Blood to Plasma Ratio

Once a substance enters the body it spreads out into the bloodstream (The substance may stay in it’s “free” form in the plasma, or it may enter into the red blood cells (RBC), or bind to plasma proteins) From the bloodstream it may enter into interstitial fluid (ISF) and into Tissues. Below is a detailed model from Kawai1998 (also see Kawai1994), showing the various areas (subcompartments) where the substance may reside.

Figure 2.0 – Detailed Substance SubCompartment Model

Where, Ar = Amount of Substance bound to and within Red blood cells Apf = Amount of Substance unbound (free) in the blood plasma Apb = Amount of Substance in the blood plasma bound to plasma proteins Aif = Amount of Substance unbound (free) in the interstitial fluid Aib = Amount of Substance in the interstitial fluid bound to proteins At = Amount of Substance in the tissue. Ad = Amount of Substance within “deep pools” in the tissue Kawai1998

In developing a PBPK model, one must be aware of what is being simulated vs. what is being measured. Case in point, the concentration of the substance in the blood. In general, what is being measured is the concentration of the substance in the plasma (See Figure 2.0 above). However, what is being simulated is the total blood concentration (Substance in the plasma plus red blood cells).

The Blood to Plasma Ratio is used to convert the simulation result (Concentration of Diazepam in the total blood), to what is being measured (concentration of Diazepam in the plasma).

Where,

where,

Also note:

Typical values:

The Blood to Plasma Ratio will vary for each substance, depending on its propensity to bind to plasma proteins and red blood cells. The Blood to Plasma Ratio for Diazepam is available from a number of literature sources: Klotz1976, Gueorguieva2006, Igari1983, Jones2004

Blood to Plasma Ratio (Diazepam):

Source	blood2PlasmaRatio	Notes
Final	0.57	Based on Klotz1976 and Jones2004
Klotz1976	0.58	also Klotz1975
Gueorguieva2006	0.65	from Greenblatt1980
Igari1983	1.037	assumed same as rat
Jones2004	0.56	From: P/B ratio: 1.79:1 when hematocrit was 45%

Note: The row labeled “final” is the value used in the simulation.

Fraction Unbound, plasma (fup)

The “Fraction Unbound, plasma” (fup), is the unbound fraction of the substance in the plasma. From figure 2.0 above:

and also,

Where,

Fup is not actually used by the simulation model, but may be used in calculating the partition coefficients.

Fup will vary for each substance, depending on its propensity to bind to plasma proteins. Values of fraction Unbound, plasma in the literature: Klotz1976, Ochs1985, Klotz1975, Gueorguieva2006, Ochs1981

Fraction Unbound, plasma (Diazepam):

Source	Fraction Unbound in Plasma (fup)	Notes
Final	0.015	-
Klotz1976	0.032	also Igari1983
Klotz1975	0.026	-
Ochs1985	0.0142	-
Ochs1981	0.0134	-
Gueorguieva2006	0.015	from Greenblatt1980, fup:0.009 to 0.027

Note: The row labeled “final” is the value used in the simulation.

Partition Coefficients

In the same way that the blood to plasma ratio, describes the relationship between total blood concentration vs. blood plasma concentration, the Partition Coefficients describe the relationship between the concentration of a substance in a tissue/organ vs the concentration of the substance in the blood plasma:

Where,

and,

and,

The Partition Coefficient formula makes a number of assumptions and approximations:

• When measuring substance concentration within a tissue (for example the muscle, skin or brain compartments) “everything” is measured, including the blood capillaries and interstitial fluid Bjorkman2002.
• That in calculating C_tissue it is valid to add all the various amounts and volumes (in the blood capillaries, interstitial fluid, deep pools, etc.) together to obtain an “average” tissue concentration.
• The Partition Coefficient is a constant.
• Typically, there is a different Partition Coefficient for each tissue compartment (ie brain, liver, muscle, etc.)

Partition Coefficients: Plasma vs. Blood

An important note is that in the Simulation it is Kp_blood that is used, not Kp_plasma .Kp_blood is related to Kp_plasma by the formula:

Where,

One must be careful to note which partition coefficient is being used in the literature, Sometimes it is Kp_plasma, sometimes it is Kp_blood and sometime it is Kp_{plasmaUnbound}.

Kp_{plasmaUnbound} is defined as:

and Kp_{plasmaUnbound} is related to Kp_plasma:

Also, sometimes the values in the literature describe partition coefficients for a species other than human.

Partition Coefficients: Animal to Human Scaling

A critical element in human PBPK modeling is the uncertainty in the values of the Partition Coefficients. In general, the tissue partition has a complex dependence on proteins and lipid binding and can vary significantly from tissue to tissue.
For legal, ethical and moral reasons, the partition coefficients cannot be directly measured in humans. Therefore it may be necessary to derive human partition coefficients based on extrapolations from animal measurements.

A conventional way to scale partition coefficients between animal and man is to measure the Kp_{plasmaUnbound} (partition coefficient, unbound in plasma) in the animal. It is then assumed that the animal Kp_{plasmaUnbound} and the human Kp_{plasmaUnbound} are identical.

Then given fup and the bloodToPlasmaRatio, the human Kp_blood can be derived. This method has been applied successfully to a number of drugs.

(Note: this section is adapted from Levitt2005, Igari1983 and Bjorkman2001)

Partition Coefficients: In the literature

Values for Partition Coefficients from the literature: Langdon2007, Rogers2006, Igari1983

Partition Coefficients, Blood (Kp_blood):

Compartment	Partition Coefficient (Kpblood)
	Final	Rogers2006	Igari1983	Langdon2007 priors	Langdon2007 Winbugs
rest	0.07	–	–	2.20	0.07
adiposeTissue	4.40	4.40	2.50	3.34	3.37
skin	0.80	0.80	0.65	0.46	0.38
muscle	0.38	0.38	0.26	0.56	0.19
gastrointestinalTract	0.69	0.69	0.37	0.75	0.76
kidneys	0.77	0.77	0.46	0.73	0.71
liver	0.84	0.84	0.95	1.35	0.66
heart	0.90	0.90	0.44	0.86	0.82
brain	0.35	0.35	0.19	0.32	0.31
lungs	0.59	0.59	0.63	0.71	0.70

Note: The column labeled “final” are the values used in the simulation.

Clearance

Clearance (CL) is defined as, the volume of plasma (or blood) cleared of a substance per unit time (units: volume/time, such as: L/hr, or ml/min) Birkett2002 p.1,2

In the literature, clearance may be plasma clearance CL_plasma or blood clearance CL_blood.

For many substances (such as Diazepam) the major source of elimination is metabolism by the liver. In which case what is wanted for the simulation is not CL_plasma or CL_blood, but
CL_liver.

These are all related by the following: (See appendix for derivation of these formulas)

and

where,

Note: In the literature, Clearance can refer to the whole body or a particular organ, such as the liver. The following values are Total Plasma Clearance (whole body).

Literature values of Clearance for Diazepam: Klotz1976, Klotz1975, Ochs1985, Igari1983, Ochs1981

Total Plasma Clearance (Diazepam):

Source	Clearance, Total Plasma (ml/min)	Notes
Final	28.0	-
Klotz1976	26.0	Also, CLblood=47.3 ml/min, Liver blood flow = 1500ml/min, ExtractRatio: 47.3/1500 = 0.032
Klotz1975	28.0	20-32ml/min
Ochs1985	30.8	0.44ml/min/kg
Ochs1981	29.4	0.42 man, 0.63 woman, ml/min/kg
Igari1983	28.7	CLint=0.598ml/min/gLiver, CL=0.598 * 1500gLiver * 0.032 = 28.7ml/min, (note:fup=0.032)

Note: The row labeled “final” is the value used in the simulation.

Mass Balance Equations

The heart of the PBPK simulation are the Mass Balance Equations. A mass balance equation describes the time rate of change of a substance within a compartment. When taken together, the mass balance equations for all compartments describe how the substance flows within the body.

The point of the mass balance equations is that the amount (aka mass) of substance entering a compartment should equal the amount of substance leaving the compartment plus the amount retained within the compartment:

amount in = amount out + amount metabolized (or excreted) + amount retained in compartment

Below is a mass balance diagram for the liver compartment.

Figure 3.0: Mass Balance Diagram, Liver compartment

The mass balance equation for the liver compartment would be:

Where,

and

for a generic compartment without metabolism the mass balance equation reduces to:

for the lungs compartment:

for the ArterialBlood compartment:

for the VenousBlood compartment:

Where,

Mass balance equations are ordinary differential equations (ODEs) and can be solved with the usual differential equation solving algorithms (Eulers, Runge Kutta). In addition to simply writing the mass balance equations in Fortran or C, specialized simulation software is available to simplify the task.

Oral Dosing Model

An extension to the original PBPK model was made for Oral Dosing. This required adding additional compartments for the digestive tract: stomach, small intestine and colon. The model is based on the work by Yu and Amidon in the 1999 article, “A compartmental absorption and transit model for estimating oral drug absorption” Yu1999a

In their compartmental absorption and transit “CAT” model the digestive tract is broken into 9 compartments:

• 1 stomach compartment
• 7 small intestine compartments
• 1 colon compartment (final reservoir of the digestive tract path)

The CAT model makes the following assumptions:

• Absorption of a substance from the digestive tract to the gastrointestinal compartment is through the small intestine, not stomach or colon.
• Transport across the small intestinal membrane is passive.
• Dissolution (substance in “pill” form to substance in dissolved “liquid” form) is instantaneous.
• A substance/drug moving through the small intestine can be viewed as a process flowing through a series of segments, each described by a single compartment.
• Compartment flow is described by linear transfer kinetics.

The following is a diagram of the CAT model used in the simulation, (note: using 5 SI compartments instead of 7)

Figure 4.0: Oral Dosing CAT Model

Where,
Kge: gastric emptying rate constant,
Kt: intestinal transit time rate constant,
Ka: absorption rate constant

Oral Mass Balance Equations

The mass balance equations for the CAT model are:

Where,

The three Rate Constants: Kge, Kt, and Ka are determined as follows:

Where,

In the simulation, N is set to 5 (five small intestine compartments) Tge, Tsi and F are available from the literature:

Literature: Yu1999, Willmann2004, Yee1997, Usansky2005, Rinaki2004

Parameter	Literature Source
	Final	Yu1999	Willmann2004	Yee1997	Usansky2005	Rinaki2004
Tge	30 min	–	30 min (10-60min)	–	–	–
Tsi	199 min	199 min (40-360min)	4hr (2-6hr)	–	–	–
F (Diazepam)	98%	–	–	100%	100%	100%

Note: The column labeled “final” are the values used in the simulation.

The simulation code

PBPK modeling simulators can be written directly in Fortran or C.

There are a number of popular software packages that are also available:

• ACSL
• Matlab
• Mathematica
• Nonmem

The simulator for modeling Diazepam was written in a high-level descriptive PBPK language. A low-level program (written in C++) parses the high level description and automatically generates:

• the mass balance equations and ODE solver (Eulers, Runge Kutta, etc.)
• and outputs C code ready for compilation and simulation.

Open Source PBPK Simulation Code

• Oral Diazepam Model High level descriptive language
• pbpk2cpp CPP program that converts the High level descriptive language to C
• psim.cpp psim.cpp is the output of pbpk2cpp (ready to be compiled and simulated)

Note: The simulator works in “amounts”, not “concentrations”. It converts to concentrations when necessary, such as when printing results, by dividing amount by volume.

Experimental Data

The experimental data is from a variety of sources in the literature. The output of the simulator is compared with the experimental data.

Igari1983

• Intravenous Injection
• amount: 0.1mg/kg = 7.0e6 ng Diazepam
• 2 subjects, age 20-35, taken from Klotz1975 and Klotz1976
• Disclaimer: data below is visually “estimated” from graph.

Time (hours)	Diazepam, Cplasma (ng/ml)
0.5	250
0.75	230
2	150
4	110
6	100
7	80
8	95
12	92
13	50
24	70
25	45
36	45
37	30
48	32
49	28
62	20
71.8	15

Ochs1985

• Rapid Intravenous Injection
• amount: 5mg = 5.0e6 ng
• Vd (L/kg): 1.22
• Elim T1/2 (hr): 33.5
• Total AUC (ug/ml*hr): 4.73
• Total Clear: ml/min/kg: 0.44
• Fup: 1.42%
• Disclaimer: data below is visually “estimated” from graph.

Time (hours)	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)
0	300	0
0.25	270	–
0.5	170	–
0.75	125	3.5
1	100	–
1.5	90	7
2	75	–
4	65	7
6	61	–
8	59	10
10	40	9
12	55	14
24 (1day)	38	19
48 (2days)	20	20
72 (3days)	11	20
96 (4days)	4	17
120 (5days)	–	9
144 (6days)	–	8
168 (7days)	–	7

Klotz1976

• Rapid Intravenous Injection
• amount: 0.1mg/kg = 7.0e6 ng
• Vd area (L/kg): 0.95
• Vdss (L/kg): 0.89
• V1 (Liters): 23.6
• V1 (L/kg): 0.32
• T1/2: 24.5 hr
• Disclaimer: data below is visually “estimated” from graph.

Time (hours)	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)
0	200	0
1	130	5
2	90	6
4	75	8
6	65	7
12	52	9
28	42	19
40	28	17
50	20	14
60	16	11
70	12	10

Klotz1975

• Rapid Intravenous Injection
• amount: 0.1mg/kg = 7.0e6 ng
• 20 year old
• T1/2: 21.6 hr
• Disclaimer: data below is visually “estimated” from graph.

Time (hours)	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)
0	310	0
0.5	250	6
1	205	7
1.5	180	–
2	125	9
4	105	10
6	100	12
8	95	14
12	80	16
24	60	18
36	43	19
48	30	18
70	12	17

Klotz1975 Oral

• Oral
• amount: 10mg = 10.0e6 ng
• peak level: 221-440 ng/ml
• peak time: 1 hour
• rapid decline between: 6-9 hours
• Bioavailability: 75%
• initial rapid absorption, then slower absorption
• T1/2: 21.2 hr
• Disclaimer: data below is visually “estimated” from graph.

Time (hours)	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)
0	105	0
0.5	200	4
1	250	9
2	210	15
4	180	21
8	150	22
12	105	25
224	75	30
336	44	40
450	32	30
772	20	12

Friedman1992 Oral

• Oral
• amount: 2mg = 2.0e6 ng
• amount: 5mg = 5.0e6 ng
• amount: 10mg = 10.0e6 ng
• Disclaimer: data below is visually “estimated” from graph.

Time (hours)	2 mg		5 mg		10 mg
	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)	Diazepam, Cplasma (ng/ml)	Desmethyldiazepam, Cplasma (ng/ml)
.25 (15min)	12	2	12	–	3	0
.5 (30min)	60	2.5	119	–	80	2
.75 (45min)	64	3.5	123	3.5	170	5
1	62	4	120	4	235	8
1.5	45	4.5	115	5	260	11
2	35	4.3	90	5.5	220	13
2.5	28	4.1	85	6.5	185	15
3	23	4.0	75	6	155	14
4	19	4.5	65	7	120	15.5
6	20	5.0	70	9	130	21
8	19	5.5	55	10	110	23
12	18	6.0	50	12	95	25
Cmax	75	–	172	–	317	–
Tmax (hr)	.89	–	1.00	–	1.32	–
AUC 12hr	330	65	779	112	1530	215

Results

In general, the Simulator results match the experimental data reasonably well. Especially considering that the modeling parameters are a composite from many different sources.

Note: in the human, experimental data is only available for the blood compartments. The simulator output for other compartments is provided as a reference.

Ochs1985

Below: 5mg IV. Experimental data points from Ochs1985, should match Venous blood concentrations…

Klotz1975

Below: Simulated: 7mg IV. Experimental data points from Klotz1975, should match Venous blood concentrations…
(Simulation result: Thl: 25.5 hr)

Klotz1976

The Experimental data points are from Klotz1976, This data set is interesting. The experimental data from the article said it was for a 7mg IV. However, when simulated with a 7mg IV the simulator does not match very well with the experimental data. When re-simulated using a 5mg IV the data matched the experimental data.

Below: Original Simulation: 7mg IV. Experimental data points should match Venous blood concentrations… (which it doesn’t)

Below: Re-simulated using: 5mg IV. Experimental data points should match Venous blood concentrations… (The 5mg simulation does match the experimental data)

Unfortunately, it’s not exactly clear where the problem lies.

Igari1983

Below: 7mg IV. Experimental data points from Igari1983, should match Venous blood concentrations…

Klotz1975 Oral

This is the first Simulation using the Yu and Amidon CAT Oral model. Considering the opportunity for wide physiological variations in the digestive tract, the simulator results matched the experimental data extremely well.

Below: Simulated: 10mg Oral. Experimental data points from Klotz1975, should match Venous blood concentrations…

Friedman1992 Oral

The Friedman data was for three different oral dosing levels: 10 mg, 5 mg, 2 mg. The 10mg simulation matched the experimental data fairly well using the default Tge of 30 minutes.

Looking at the experimental data for all 3 dosing levels it is noticed that peak concentration happens earlier the lower the dosage. This implies that lower dosages enter the bloodstream faster than higher dosages. The CAT model used in the simulation does not have parameters to account for this effect.

In order to approximate this phenomenon the value Tge was manually adjusted as follows:

• 5mg dose: Tge = 20 minutes
• 2mg dose: Tge = 10 minutes

This allowed a better fit of the experimental data.

Even with this fix the 2mg simulation still didn’t fit the experimental data very well. The peak wasn’t high enough and the experimental data dropped off much faster from the peak than did the simulation.

Also, it should be noted that at all three dosage levels experimental data has a slight “bump” at 6 hours. It isn’t clear what accounts for this phenomenon.

Below: Simulated: 10mg Oral. Experimental data points from Friedman1992, should match Venous blood concentrations… (With Tge gastric emptying = 30 minutes)

Below: Simulated: 5mg Oral. Experimental data points from Friedman1992, should match Venous blood concentrations… (With Tge gastric emptying = 20 minutes)

Below: Simulated: 2mg Oral. Experimental data points from Friedman1992, should match Venous blood concentrations…
(With Tge gastric emptying = 10 minutes) (conclusion: not a good match)

Discussion

A Physiologically Based Pharmacokinetic (PBPK) model of Diazepam in the human was developed. This model included an Oral dosing extension to the original Igari1983 model.

On the whole, the simulator output matched the experimental data reasonably well.

Still to be resolved include:

• An extension to the Oral CAT model to account for the shift in peak concentration vs. oral dosage
• The Klotz1976 data: 7mg or 5mg IV ?
• The 2mg oral dosage in the Friedman1992 experimental data.
• The “bump” at 6 hours in the Friedman1992 experimental data.

Appendix

Derivation of Tissue and Blood Clearance formulas

The EliminationRate is defined as the Amount of substance eliminated per unit time (mg/hr)

likewise

and

Therefore,

so that,

And also,

so that,

References

PubMed

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Non-PubMed

• Birkett2002 – Pharmacokinetics Made Easy, 2nd ed

See Also

• Test Simulation of original Igari1983 Model
• Simulating Physics with Computers

Wikipedia Links

• Pharmacokinetics
• Pharmacodynamics
• Multi-compartment Model
• Rate Equations
  • Numerical ODEs
  • Euler Method
  • Runge-Kutta Method
• PBPK modeling
• gnuplot wikipedia
  • TarBall file format, at wikipedia (tar -xzf file_to_unpack.tar.gz)

Resources

Article Reviews (PubMed)

• Nestorov2003 – Whole body pharmacokinetic models., Nestorov, Clin Pharmacokinet. 2003;42(10):883-908.

Books

• Birkett2002 – Pharmacokinetics Made Easy, 2nd ed
• Ritchel – Handbook of Basic Pharmacokinetics, 6th ed

Web Sites

• PBPK
  • Link to Pharmacokinetic and Pharmacodynamic Resources
  • Link to PBPK.org
• Misc
  • Link to gnuplot homepage