A two-compartment open model with first-order absorption and elimination process was used to describe Levo-

floxacin plasma concentration time profile. The model is described by the following system of ordinary differential equations. Equation 1:. Equation (2):. Equation (3):. With X_a, X₁, X₂ are the milligram amounts of drug in the gut, the central (plasma) and the peripheral compartments, respectively. K_a (hr⁻¹) and K₀ (mg/h) are the absorption rate constant and the intravenous (IV) infusion rate of Levofloxacin, respectively. K_el (h⁻¹) is the elimination rate constant from the central compartment. K₁₂, K₂₁ are the between-compartment transfer rate constants (all in hr⁻¹).

The other pharmacokinetic parameters are the volume of distribution (V). Equation (4):. Equation (5):. With C₁, C₂ are Levofloxacin concentration in μg/mL in the central compartment and the peripheral compartments, respectively. V₁, V₂ represent volume of distribution in the central and peripheral compartments in mL, respectively. F is the fraction of dose absorbed. In extravascular models, the fraction of dose absorbed cannot be estimated separately. Therefore, V₁/F was estimated together in PK modeling. Dpo is the administered dose orally. Pharmacokinetic parameter values (Table 1), obtained from literature [18], were used in Monte Carlo simulation of drug concentration versus time profiles for Levofloxacin after single or multiple oral and IV infusion dosing administration.

Recall the ordinary differential Equations (2) and (3) above:

where: is the unit function

and T is the duration time of infusion.

Using the Laplace transform, we have the following

Table 1. Summary Levofloxacin pharmacokinetics used in Monte Carlo Simulation Derivations of drug concentration in plasma and tissue compartments after IV infusion single and multiple doses using Laplace transforms.

equation in the s domain:

where: are the initial conditions of.

At t = 0:

Linear equation and solution are as follows:

where:

and α, β are defined by:

, then:

Moreover,

where:

Now, by the inverse Laplace transform, we have the solution for in the time domain:

Then,

For a simulation of a 7 days treatment duration and 1 dose was administered a day, was replaced by in the above differential equations (Equations (6) and (7)). And as a result, was replaced by

in its representation in the s domain. Therefore, we had a similar solution for as follows:

And, see Equations (8) and (9).

Descriptive statistics of all levofloxacin PK parameters in healthy subjects and patients (mean, standard deviation, coefficient of variation) were obtained from literature. Pharmacokinetic simulation was then used to reveal the effect of input variables. A direct comparison of the experimental data (mean ± sd) obtained from simulation was involved. Comparison between the experimental and observed data from various studies reported in literature was conducted using a student t-test. Mean and standard deviations of PK parameters were determined to assess how well the model described the clinical data. The best PK model and log-normal distribution (mean and variance) of all the transfer rates were then used as input in the Monte Carlo simulation process to generate plasma and tissue concentrations.

A 2-compartment open model without a lag time was used to generate each concentration-time profile, where drug distribution rates (K₁₂, K₂₁) or drug elimination rate (K_el) were random variables associated with their distribution information. Their distributions were considered lognormal distributions. The mean concentration-time profile was generated by using mean values of PK parameters.

Investigation of varying the effect of elimination and distribution rates was performed using various values of those parameters. Monte Carlo simulation of 5000 drug concentration profiles was performed by importing elimination/distribution rates randomly into pharmacokinetic model and according gain blocks as in Figure 1. Random values were chosen from their associated log-normal distributions (mean and standard deviation) of reference values.

Random function (rand) was used to randomly select pharmacokinetic parameter values which were associated to their distribution. The bounds for each PK parameter were set in accordance with the pharmacokinetic parameter values obtained. Simulink (Matlab, Natick, Massachusetts, USA) was used to simulate signals and determine how these concentrations vary over time using a system of 2 differential equations to describe plasma concentrations in compartment 1 and tendon concentration were in compartment 2.

The block Plasma (X₁) and Tissue(X₂) were two integrators. They took the integration of the inputs dX1/dt, dX2/dt and returned the outputs X₁ and X₂. The block Pulse Generator 1 was to reset the integrator plasma compartment each 24 time steps (corresponding to 24 hours per day). The block Pulse Generator was to describe multiple doses: One dose oral or infusion calculated as amount over the time of infusion or duration of absorption (1 time step) dose per 24 hours (24 time steps). Other blocks including gain blocks (K₁₂, K₂₁, K_el, 1/V₁, 1/V₂) and sum blocks were to implement the left hand side of each of two differential equations. Concentration time profiles in two compartments were obtained in two scope blocks by running this simulation.

Plasma and well-perfused organs such as lung, skin, and

Figure 1. Simulink for intravenous infusion 1 hour q24 for 7 days.

spongiosa etc. were grouped in compartment 1. Tissues characterized by poor blood flow such as tendon tissue (including Achilles tendon) along with adipose and cartilage were grouped in compartment 2. Tendon was anticipated to have very low redistribution rate compared to other sites, and was grouped in compartment 2 where the distribution rate is comparable to those of cortical bone and adipose tissue [9,11,13].

Figure 2 shows that generated drug concentration time curves were approximately super-imposable in plasma and in tissue after oral administration and IV infusion. This is in agreement with clinical data when approximately 100% drug absorption occurs in the oral route. The drug peak concentration produced with IV infusion was higher than that of oral administration. This initial simulation of the average drug concentration ratio between plasma and tissue showed no obvious change with time 4 hr post dosing onward, and that equilibrium between plasma and tissue drug concentrations were actually achieved 4 hr after dosing and the tissue concentrations declined proportionally to the plasma concentrations.

In addition, simulated data also show that after a single dose, drug concentration values in tissue achieved at Cmax of 3.08 μg/mL and 3.28 μg/mL, and Tmax of 3.25 hrs and 3.11 hr, and AUC0-24 of 42.05 μg∙hr/mL and 42.91 μg∙hr/mL for 500 mg oral dose and IV infusion, respectively.

Figure 2. Simulated mean plasma and tissue levofloxacin concentration-time profiles following the oral and IV infusion administrations of a single 500 mg dose. Key […□]: drug concentrations in plasma; […+]: drug concentrations in tissue after a single intravenous infusion; [−∆]: drug concentrations in plasma, [−o]: drug concentrations in tissue after a single oral dosing.