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Pharmacokinetic model

What is pharmacokinetic model? Pharmacokinetic model is a quantitative description of drug absorption, distribution, metabolism and elimination from the body. Why do we need it? To compare conventional and our therapy. To calculate required drug production at target site. How was our pharmacokinetic model constructed? The model is physiologically based. Organs and tissues are presented as compartments. Processes in the body are described with ordinary differential equations (ODEs). The model is customized for chosen drug and its type of administration. For which drugs (diseases) did we test our model? interferon (hepatitis C) anakinra (ischemic heart disease) 3D animation 3D animation serves as a real-time visual representation of how drug concentrations change throughout the course of the therapy in various compartments. The animation was implemented in C++ programming language and OpenGL was used for 3D rendering. Results Our therapy does not produce concentration fluctuations - absence of concentration peaks and lows. With local drug production, drug concentration rises to a desired level within a day (for both tested drugs). For each drug, we obtained an estimation of the degree of drug localization. What did the results tell us? Results show us the drug is more localized; the degree of localization depends on characteristics of tissue (e.g. perfusion) where microencapsulated cells would be applied. Additionally, absence of high concentration peaks suggests that our therapy would have less systemic side effects than conventional methods. Since the drug would be constantly produced, drug concentration would not drop enough for the drug to be therapeutically ineffective. According to these findings, steady drug levels over time therefore imply an optimized treatment. With known drug production rate of microencapsulated cells, it is possible to adjust the dosage to reach the desired drug concentration in the target tissue. Therapeutic production was estimated according to activity assays with mammalian cells. After obtaining the results of pharmacokinetic simulation, we were able to calculate the number of microencapsulated cells needed for the therapy.

Introduction

The major downsides of standard interferon treatment are substantial side effects that to a large degree are the consequence of very high drug concentrations, occurring shortly after drug administration. If high concentration peaks could be avoided and lower levels maintained steadily over time, this would result in reduced side effects without compromising therapeutic effectiveness. This idea is currently being tested in clinical trials using the interferon infusion pump (COPE-HCV: Phase 2, randomized, open-label, active-control, dose-ranging study of interferon alfa-2b given via continuous sub-Q infusion; trial by Medronic Inc.).

One of our goals is to show that therapy with drug-producing cells is more beneficial than standard treatments based on drug injections, which are in use today. We predict that if the drug is constantly produced inside the body, it could reach a steady concentration at almost any desired level. We believe localization of therapeutic cells would also decrease the proportionate drug concentrations in non-target tissues, thus further reducing the side effects.

We tested this hypothesis and compared standard therapies with our proposed treatment.
Because of complex physiological mechanisms and an extensive set of biological parameters that cannot be accurately measured for either ethical or technical reasons, we developed a model that covers the most crucial aspects and processes. At the same time, computer simulation provides simpler and faster option than in vivo research.

3D visualization

We implemented a real-time 3D visualization of the pharmacokinetic simulation for hepatitis C and ischemia. This way, we were able to visually represent how drug concentrations change throughout the course of the therapy in various compartments, such as liver. In our 3D visualizations, drug concentration levels are represented as red color intensities - the compartments with high concentrations are closer to saturated red color, while the compartments with low concentrations are closer to white color. Red color intensity of a compartment was calculated as a ratio of the compartment's drug concentration at a given time to the maximal reached concentration of any compartment during the course of the therapy, or therapies compared.

For implementation, we used C++ programming language and OpenGL for 3D rendering. The 3D anatomical models, obtained from BodyParts3D database, were optimized for real-time rendering using MeshLab and rendered as vertex arrays in OpenGL. Compartment concentration values as a function of time were obtained as vectors from the pharmacokinetic simulation and integrated into the application.

A physiologically based pharmacokinetic model

A pharmacokinetic model is a quantitative description of drug absorption, distribution, metabolism and elimination from the body. A model is defined by a system of ordinary differential equations to represent essential drug kinetics.
We took into account the necessary biological parameters and based the simulated processes on actual physiological mechanisms to construct a physiologically plausible model.

Selecting the optimal physiological model

A physiologically based model is composed of multiple compartments which represent organs of the body. Parts were chosen in accordance with drug and tissue specifics, so that the relevant organs are represented as separate compartments, while other tissues were merged on the basis of common characteristics (Levitt et al., 2006).

Determing compartments

We used the following characteristics as criteria for splitting and merging organs:

• Liver is the target organ of therapy.
• Interferon alpha is widely distributed into body tissues – highest concentrations occur in kidney, liver and lung.
• Interferon is a water-soluble molecule; it is only poorly distributed in adipose tissue.
• Interferon alpha does not cross the blood-brain barrier.
• Skin and muscle tissues do not seem to have much higher concentrations of the drug in comparison to adipose tissue.
• Skin, musle and adipose tissue have similar, slow blood perfusion.
• Gut, spleen and heart are all rapidly perfused tissues.
• Interferon is mainly eliminated via renal catabolism, while hepatic metabolism accounts only for a minor pathway of elimination.

We decided to define separate compartments for the liver, kidney and lungs. All other rapidly perfused tissues are grouped together as one part. Since venous blood enters the lungs and arterial blood flows into all other organs, we separately simulate venous and arterial blood. Because interferon does not cross the blood-brain barrier it is not necessary for the brain to be modeled separately. Skin, muscle, fat and other slowly perfused tissues are merged together into one compartment.

We constructed three final models - two for standard interferon treatments and a third for a prospective therapy with drug-producing microencapsulated cells. The fundamental design is the same in all models; they are only modified for specific entry points of the drug and the corresponding absorption or production processes. On the diagram, blocks representing different administrations are shown in distinct colors: blue for the intravenous bolus, green for the subcutaneous injection and red for the interferon production by microencapsulated cells that are implanted into liver (Figure 1).

Figure 1. Multi-compartment physiologically based pharmacokinetic model, customized for different Interferon therapies. Main difference is the route of administration - for each administration, there is different compartment from which drug enters the body. Blue blocks are used in modeling for intravenous administration, green blocks are specific for subcutaneous administration and red blocks represent continuous production of the drug, in this case in the liver compartment. Therapies of course don't differ only in drug entry points, but also in absorption or production processes, percent of drug that enters the blood stream and so on.

Parameters

Types of parameters used:

1.) Species specific parameters

Q_i – blood flows to tissues
V_i – organ volumes

	Blood flows
Cardiac output	Qc	Qc = 5.58 L/min = 335L/h
Compartment	Parameter name	Percent cardiac output
lung	Qlung	100% Qc
venous blood	Qven	100% Qc
arterial blood	Qart	100% Qc
liver	Qliver	25% Qc
kidney	Qkidney	19% Qc
rapidly perfused tissue	Qrpt	18% Qc
slowly perfused tissue	Qspt	38% Qc

Sum of blood flows through liver, kidney, rapidly and slowly perfused tissue must always equal total cardiac output.

	Tissue Volumes
Body Weight	BW	BW = 70 kg => *BV = 70 L
Compartment	Parameter name	Percent body volume
lung	Vlung	0.8% BV
venous blood	Vven	5.57% BV
arterial blood	Vart	2.43% BV
liver	Vliver	2.60% BV
kidney	Vkidney	0.44% BV
rapidly perfused tissue	Vrpt	5.16% BV
slowly perfused tissue	Vspt	83% BV

* average body density = 1 kg /L

2.) Individual specific parameters

BW - body weight
Q_c - cardiac output
varying percents of tissue volumes (e.g. percent body fat)

The variability of parameters

Parameter values can range significantly between individuals, depending on factors such as age, sex, renal function, activity level and diet.
For instance, cardiac output can vary significantly even in one individual, depending on the current activity (sleeping, sitting, running etc.). There can be substantial differences in, for example, the percent of body fat (accounted for in slowly perfused tissue) comparing individuals with an otherwise similar profile (same age, sex, etc.).
Values used in our model present an average adult male, weighing 70 kg, with mean cardiac output and normal renal function.

3.) Chemical specific parameters

t_1/2 - drug half life
k_el - elimination rate (kidney)
P_blood:tissue - partition coefficient

Interferon is expected to be found only in the plasma and not in red blood cells; therefore we can conclude that the amount of interferon found in plasma is equal to the amount in blood. Since blood is comprised of four parts plasma and three parts red blood cells, we can calculate the blood to plasma ratio to be:

The equation has the same form for each kind of tissue:

Uncertainty of partition coefficients

A critical element in human physiologically based pharmacokinetic modeling is the uncertainty of values of the partition coefficients. Partition coefficients are an important aspect of pharmacokinetic modeling, because they denote how the drug distributes throughout body tissues. The value of each coefficient has a complex dependence on solubility, permeability, pH, binding affinity of the drug to the receptor receptor and many other factors. The difference in a few amino acids between subtypes of interferon alpha can impact values of coefficients quite noticeably. Even so, there are deviations of evaluated drug distribution of the same subtype of interferon alpha, depending on the detection method (radioactivity, ELISA).

For legal and ethical reasons these values cannot be directly measured in humans. We had to rely on various studies of interferon tissue distribution in rodents to calculate partition coefficients (Johns et al, 1990). It is generally assumed that animal and human partition coefficients are similar for the same kind of tissue.

Chemical and ROA* specific parameters

* ROA - route of administration

D – dose
k₀ – absorption rate constant (zero-order process)
k_a – absorption rate constant (first-order process)
F – bioavailability (percent of absorbed dose)
k_prod – production rate constant

General mass-balance equations

The equation below describes the change in concentration over time in non-eliminating tissues. The equation has the same form for both rapidly and slowly perfused tissue. Each compartment is then described with its distinctive values for blood flow, concentration and partition coefficient.

We can use the same form for the liver compartment, since the metabolism of interferon is negligible.

Although the lungs also represent a non-eliminating tissue in this model, the equation is slightly different, since venous and not arterial blood flows into the tissue.

The kidneys represent a site of elimination, so we had to include this process as well.

In the equation describing the change of concentration in venous blood, there is a sum of blood flows which flows in from multiple compartments. These include liver, kidney, rapidly and slowly perfused tissue.

Blood flow from the lungs is accounted for in the equation for the arterial blood compartment.

Modeling pharmacokinetic processes

1.) Absorption

Absorption processes depend on the route of administration:

a.) Intravenous administration

The drug is injected directly into the blood stream, so there is no special absorption process. The dose is bolus and enters the system completely.

b.) Subcutaneous administration

In subcutaneous administration, more than 80% of the initial dose enters systemic circulation. This percentage is defined as bioavailability and can reach up to 95% for interferon alpha given subcutaneously.
Time to peak Tmax
The absorption half-life is approximately 2.3 h

Relying on the available studies, we specified the absorption as a two phase process:

1.) At the beginning, when the drug concentration at the injection site is highest, only a limited amount of drug can be absorbed into the blood - the rate of absorption is maximal. This is described by a zero order process from t=0 to t=t_k.
Zero order process equation:

2.) At t=t_k the concentration drops to a level where rate of absorption is proportional to local concentration. Absorption follows first order kinetics from t= t_k until the bioavailable drug is completely absorbed.
First order process equation:

Constant k_a was taken from literature and the value of k₀ was calculated using parameters from the same source. Fz denotes the portion of the drug that is absorbed by zero order process.

2.) Distribution

Estimating partition coefficients

Distribution process is usually modeled indirectly with the use of partition coefficients, which are drug and tissue specific. We derived partition coefficients according to the tissue distribution data in available literature (Thitinian et al., 2009; Johns et al., 1990). However, these values do differ from study to study (e.g. that kidney concentration is 1-9x times that of plasma (Bohoslawec et al., 1986), but they overall agree that after subcutaneous or intravenous administration, interferon concentration is highest in the kidney, very high in the liver and lungs and to some extent in other rapidly perfused tissues, but lower in the adipose and other slowly perfused tissues.

Compartment	Parameter name	Value
lung	Pblood:lung	2.5
liver	Pblood:liver	2.8
kidney	Pblood:kidney	8.5
rapidly perfused tissue	Pblood:rpt	2.2
slowly perfused tissue	Pblood:spt	0.7

3.) Elimination

In the literature the terminal half-life for interferon alpha is from 3-8 h, with a mean around 5 h. Interferon alpha is detectable in the plasma for 4-8 h after the rapid intravenous injection and for 16-30h after subcutaneous administration.
The elimination constant for the kidneys was determined by curve fitting to data from the obtained studies.

Simulation data for different therapies

Interferon alpha subcutaneous therapy modeling

We simulated common type of interferon administration:

• 3 times per week, subcutaneous administration
• 3 MU = 11100 ng, dose
• F = 93%, bioavailability
• Ko, Ka - zero and first order absorption
• t_1/2 = 5h, half life

We constructed the model according to data from the literature (Roferon A, Hoffman-La Roche; Chatelut, 1999; Thitinian et al., 2009; Marcellin et al., 2003). Results are shown in graph below (Figure 2).

Figure 2. Standard subcutaneous therapy: high peaks right after administration, followed by rapid concentration decrease.

Calculating desired drug production

Our model assumes no cell divisions and therefore a constant therapeutic cell count in the body. In this case, cell production of a drug is mathematically equivalent to a continuous infusion of a drug into the tissue, where cells are located. To calculate the desired concentration level in the target tissue, we calculated AUC (area under the curve) of liver concentrations for subcutaneous administration over a period of one week. This allowed us to estimate the average concentration in the liver which corresponds to the therapeutic level we wish to reach and maintain with cellular drug production. The combination of kidney elimination and constant production leads to stable drug levels in tissues. After about four half-lives, drug concentrations rise to final levels and remain steady as long as the production rate stays the same (Figure 3).

Figure 3. Drug distribution troughout tissues with Switch-IT therapy.

Results and conclusions

If we compare result of different treatments it clearly shows that local therapy with drug producing cells does not result in concentration fluctuations, which are present in common therapies.
Concentrations never reach levels as high as in subcutaneous or intravenous administration. On the other hand, concentrations do not drop to a point, which would prevent the drug from becoming therapeutically inefficient.

When the drug is produced at the target site, concentration levels between target tissues and other organs are inclined in favor of the target tissue - although kidneys show a substantial uptake of the drug in subcutaneous administration, the relative difference between kidney and liver concentrations are smaller with localized therapy (Figure 4).

Figure 4. Comparison of subcutaneous and Switch-IT therapy. We can see the new therapy results in stable concentration levels, without peaks and lows.

From model to implementation

When we calculate the desired steady concentration in the target tissue, it was possible to estimate how many cells would need to be implanted. From the required production rate, which is thought to be linearly proportional to the cell count, we can calculate the total number of cells needed. Considering the average number of cells per microcapsule and mean capsule size, we can calculate the amount of microcapsules we would need to inject in order to attain the wanted therapeutic effect.

Anakinra therapy modeling

For modeling the local application of anakinra-producing microencapsulated cells in the ischaemic muscle tissue we adopted slightly different approach. Liver is perfused by sinusoid capillaries which allow for passage of macromolecules to the tissue, therefore even protein therapeutics can be rapidly equilibrated with blood and distributed around the organism. Muscle tissue, however, is perfused with continuous capillaries, which prevents fast passage of macromolecules from the tissue to the blood (Figure 5). Protein molecules pass the continuous capillaries only through slow trans-pinocytosis. Consequently this increases the local concentration of therapeutics permitting the achievement of high local concentration with relatively little loss to the system and elimination by kidneys.

Figure 5. Different capillary types in muscle tissue (continuous capillary, left) and in liver (sinusoid capillary, right). Image from: (Mescher, 2009)

Therefore we took this feature of the tissue into account when calculating partition coefficients between tissue and blood compartment. Myoglobin has similar molecular weight as anakinra; therefore we assumed its partition coefficient to be the same(Hall, 2010). We calculated the therapeutic concentration of anakinra to be 25 ng/mL, which we derived from the anakinra IC_50 value (Dahlén et al., 2008).

Compartmental model and simulation parameters

According to a study of anakinra distribution and elimination after subcutaneous injection, we calculated relevant compartments for physiological model. At first, we modeled standard regime for anakinra therapy: 100 mg per day subcutaneously with the following parameters:

• bioavailability: F = 95%
• half life: T_1/2 = 5h
• time to maximum concentration: T_max = 4h
• maximum concentration: C_max = 780 ng/mL
• average concentration: C_avg = 350 ng/mL

Figure 6. Multi-compartment model for anakinra. For local therapy model, we added one additional compartment (shown in purple color) inside the heart compartment. This represents the affected area, where microencapsulated cells would be administered to. In conventional therapy model, the whole heart, both infarcted and undamaged tissue, are represented as one, since standard treatments can not target selected area specifically.

We calculated following partition coefficients according to a study on drug distribution in rodents.

tissue	percent of dose
kidney	11 %
liver	9 %
gut	6 %
lung	2 %
muscle	43 %
plasma	26 %

Figure 7. Dose distribution plot, fitted to the literature data, which was used to determine the partition coefficients for the listed tissues.

Partition coefficients of tissues are proportionate to drug concentrations (for non-eliminating tissue). From drug amount in each tissue and tissue volumes, we could find calculate concentration ratios and then estimate coefficient values. Parameters are further tuned to fit measured data as much as possible.

Compartment	Parameter name	Value
lung	Pblood:lung	0.4
liver	Pblood:liver	0.8
kidney	Pblood:kidney	9
rapidly perfused tissue	Pblood:rpt	2.2
slowly perfused tissue	Pblood:spt	0.23
muscle	Pblood:muscle	0.25
heart	Pblood:heart	0.2
target tissue	Pblood:target	33

Figure 8. Concentration distribution throughout tissues after subcutaneous administration. In this kind of administration, affected myocard cannot be targeted specifically; drug concentration is equal in both infarcted and undamaged heart tissue. With this kind of administration, the drug is also distributed throughout the whole body. Drug concentration is even higher in plasma or kidneys than concentration in heart.

Figure 9. Concentration distribution throughout tissues in local therapy. Here, the drug would be constantly produced in microencapsulated cells, which would be implanted into the damaged part of heart. Because heart muscle tissue is differently perfused than liver, very small amount of drug enters systemic circulation and localization of drug is even more evident.

We achieved significantly better results with our proposed therapy comparing with conventional treatment.

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