# A Three-Compartment Pharmacokinetic Model to Predict the Interstitial Concentration of Talaporfin Sodium in the Myocardium for Photodynamic Therapy: A Method Combining Measured Fluorescence and Analysis of the Compartmental Origin of the Fluorescence

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurement of Plasma Concentration and Myocardial Fluorescence of Talaporfin Sodium in Canines

_{plasma}) of blood samples was obtained using the visible absorption spectrum measured using a microvolume spectrophotometer (Titertek-Berthold, Pforzheim, Germany) immediately before, and 5, 10, 15, 20, 30, 40, 50, 60, 70, 75, and 85 min after injection. The blood sample volume was 1 mL. Because we wanted to know the interstitial myocardial concentration of talaporfin sodium, its fluorescence in the canine myocardium was measured using a fluorescence-sensing probe, as described previously [16]. The fluorescence-sensing probe pad shown in Figure 1 was attached to the open-chested beagle canine heart while avoiding the major coronary arteries.

_{myo}) was measured immediately before, as well as 10 and 40 min after injection. The fluorescence represents the relative value of concentration; therefore, the measured fluorescence was translated to the absolute myocardial concentration (C’

_{myo}) using a conversion constant (R

_{myo}) as described in Equation (1).

#### 2.2. Determination of the Components of Talaparfin Sodium Myocardial Fluorescence Based on Histological Examinations

_{1}) in this cross-section, assuming a hematocrit level of 0.45 [17]. The number of cells in the cross-section was counted by binarized HE-stained cell nucleus imaging using ImageJ 1.51 (National Institute of Health, Bethesda, MD, USA). The total cell area in the cross-section was obtained by multiplying a cell unit area, assumed to be a sphere of 18.5 μm [18,19], by the number of cells. The cell area ratio (R

_{3}) in the cross-section was obtained by dividing the total cell area by the area of the cross-section. To determine the interstitial area ratio (R

_{2}) of the cross-section, fat and collagen fibers, 11.7 vol% [20] and 3.9 vol% [21], respectively, were extracted from the entirety.

#### 2.3. Determination of the Compartment Volumes in the Three-Compartment Model

_{1}) was obtained physiologically. The blood volume in a canine constitutes 7.7% of the total body weight [22]. The plasma compartment volume was 55% of that, as described in Section 2.2. The interstitial compartment volume (V

_{2}) and the cell compartment volume (V

_{3}) were obtained functionally. Glucose and talaporfin sodium are both water-soluble and close in molecular weight; therefore, we assumed that the pharmacokinetics for both would be similar, but of course it is not identical. The tissue compartment volume using a two-compartment model of glucose for canine was 1221 mL [23], after correcting for the canine weight of 9.3 kg. The corrected volume was considered to be the sum of the interstitial space and cell compartment volume in the constructed three-compartment model. These values were obtained using the measured interstitial and cell area ratios described in Section 2.2.

#### 2.4. Three-Compartment Mathematical Modeling of Pharmacokinetics

_{i}and C

_{i}indicate each compartment volume and concentration, respectively (i = 1, 2, and 3). The excretion rate constant from the plasma compartment and the rate constants between each compartment are written as k

_{10}, k

_{12}, k

_{21}, k

_{23}, k

_{32}, as shown in Figure 3.

_{1}, R

_{2}, R

_{3}, indicate the volume ratios of the plasma, interstitial space, and cell for the measured myocardial fluorescence, respectively. Our idea is that the fluorescence in the myocardium consists of the linear addition of fluorescence generated from three compartments using the volume ratios. Also, the fluorescence represents the relative value of the concentration. Therefore, we can describe the concentration in the myocardium as the linear addition of the concentration of the three compartments using the volume ratios.

_{1}, V

_{2}, V

_{3}, R

_{1}, R

_{2}, and R

_{3}are found based on the histological examinations described in Section 2.2 and 2.3. The initial concentration of the plasma compartment [C

_{1}(0)] was obtained by dividing the initial talaporfin sodium administration of X μg by V

_{1}, and the initial concentration of the interstitial compartment [C

_{2}(0)] and the cell compartment [C

_{3}(0)] were set to 0 μg/mL. A conversion constant (R

_{myo}: C′

_{myo}= R

_{myo}· M

_{myo}, see Section 2.1) to obtain the absolute talaporfin sodium concentration from the measured myocardial fluorescence was determined to match the initial values of the myocardial measured concentration data [C′

_{myo}(0)] and the myocardial concentration [C

_{myo}(0)] calculated from Equation (5). The rate constants, k

_{10}, k

_{12}, k

_{21}, k

_{23}, k

_{32}, were optimized to minimize fval, the sum of the squared errors between the measured plasma concentration (M

_{pla}) and calculated plasma concentration (C

_{1}), and the measured myocardial concentration (C′

_{myo}) and calculated myocardial concentration (C

_{myo}) divided by the number of data points using the conjugate gradient method with the solver “fmincon” in MATLAB R2016a (Mathworks, Natick, MA, USA).

## 3. Results

#### 3.1. Determination of Volume Ratios (R_{1}, R_{2}, and R_{3}) and Compartment Volumes (V_{1}, V_{2}, and V_{3})

_{1}, R

_{2}, and R

_{3}were obtained by the method described in Section 2.2. The blood vessel and plasma area ratios in the myocardium cross-section were 12.3 ± 2.1% and 6.77%, respectively. The cell area ratio in the myocardium cross-section was 61.7 ± 7.7%. The interstitial area ratio was calculated as 15.9%. The volume ratios of plasma (R

_{1}), interstitial space (R

_{2}), and cell (R

_{3}) to the myocardial fluorescence were 0.08, 0.189, and 0.731, respectively, with the sum fixed at 1. Finally, we obtained the following Equation (6).

_{1}, V

_{2}, and V

_{3}were obtained by the method described in Section 2.3. The plasma compartment volume was calculated physiologically to be 394 mL (V

_{1}). The interstitial compartment volume of 251 mL (V

_{2}) and the cell compartment volume of 970 mL (V

_{3}) were obtained functionally. A conversion constant (R

_{myo}) was set to be 189 μg/(mL·counts) described in Section 2.4 and Appendix A. Finally, we obtained the following Equation (7).

#### 3.2. Construction of the Three-Compartment Model Using the Measured Plasma Concentration and Myocardial Fluorescence

## 4. Discussion

#### 4.1. Methodology of the Constructed Three-Compartment Model for Talaporfin Sodium Using the Myocardial Fluorescence Time History and Volume Ratios Measured from Histological Examinations

_{myo}) was inserted into the model using the measured volume ratios (R

_{1}, R

_{2}, and R

_{3}) based on the histological examinations. A few studies have presented three-compartment models using fluorescence data [11,28]. These involved a pharmacokinetic model for chloro-aluminum sulfonated phthalocyanine in an implanted hamster cheek pouch carcinoma tumor model [28]. However, only the fluorescence in the tumor and normal tissue compartments was measured. In contrast, we measured the myocardial fluorescence considering the origin of the fluorescence from plasma, interstitial space, and cell compartments. Another three-compartment pharmacokinetic model used radioisotope to estimate the interstitial concentration [11]. In contrast, we measured myocardial florescence, which is an easier and safer approach compared with the reported compartment model method using radioisotope. We believe our methodology is the first to use measured changes in talaporfin sodium plasma concentration and myocardial fluorescence that considers the origin of the fluorescence from the compartments with the measured volume ratio based on histological examinations.

#### 4.2. Application of the Three-Compartment Model

#### 4.3. Limitations

_{2}and V

_{3}). These values for glucose are not perfectly applicable to those of talaporfin sodium. The pharmacokinetic properties, such as absorption, distribution, metabolism, and excretion process, could essentially be different between glucose and talaporfin sodium. Based on our model construction policy regarding the optimized parameters, we wanted to describe the functional volumes of each compartment for talaporfin sodium under the assumption that the pharmacokinetics for talaporfin sodium and glucose would be similar because both materials are water-soluble and close in molecular weight for canine model. Thirdly, we were only able to provide a few data points for the measured myocardial fluorescence because it was hard to stably measure the fluorescence from the beating heart of the open-chested animal. Finally, we only produced one series of interstitial fluorescence dynamics in one canine. We did not use multiple animals or data sets. The use of only one animal reduces the significance and reproducibility of the methods proposed.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A: How to Obtain a Conversion Constant (R_{myo})

_{myo}) was determined using the myocardial measured fluorescence data (M

_{myo}) and C’

_{myo}. We assumed that the initial value of the calculated myocardial concentration [C

_{myo}(0)], which was obtained from Equation (6) using the determined C

_{1}(0): 59.01 μg/mL and C

_{2}(0), and C

_{3}(0): 0 μg/mL described in Section 2.4, was equal to the initial value of the myocardial measured concentration data [C′

_{myo}(0)]. R

_{myo}was set to be 189 μg/(mL·counts) using M

_{myo}(0) = 0.025 counts and C

_{myo}(0) = 4.72 μg/mL.

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**Figure 1.**Fluorescence-sensing probe pad installation on an open-chested beagle heart. (

**a**) Image of the fluorescence-sensing probe pad installation; (

**b**) structure of the fluorescence-sensing probe pad.

**Figure 2.**Origin of the measured talaporfin sodium concentration changes in the myocardium with volume ratios, R

_{1}, R

_{2}, and R

_{3}, which indicate the volume ratios of the plasma, interstitial space, and cell in the measured myocardial fluorescence, respectively.

**Figure 4.**Diagram of the proposed optimization procedure. (* N

_{pla}and N

_{myo}represent the number of data points for the plasma concentration and myocardial fluorescence, respectively. ** fval

_{pre}represents the previously calculated value of fval).

**Figure 5.**Estimated talaporfin sodium concentration time history using the three-compartment model. (Red line: estimated concentration in the plasma compartment; orange line: estimated concentration in the interstitial space compartment; blue line: estimated concentration in the cell compartment; black line: estimated concentration in the myocardium; the red plots: measured plasma concentration [24]; and black plots: measured myocardial concentration.).

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**MDPI and ACS Style**

Uno, Y.; Ogawa, E.; Aiyoshi, E.; Arai, T. A Three-Compartment Pharmacokinetic Model to Predict the Interstitial Concentration of Talaporfin Sodium in the Myocardium for Photodynamic Therapy: A Method Combining Measured Fluorescence and Analysis of the Compartmental Origin of the Fluorescence. *Bioengineering* **2019**, *6*, 1.
https://doi.org/10.3390/bioengineering6010001

**AMA Style**

Uno Y, Ogawa E, Aiyoshi E, Arai T. A Three-Compartment Pharmacokinetic Model to Predict the Interstitial Concentration of Talaporfin Sodium in the Myocardium for Photodynamic Therapy: A Method Combining Measured Fluorescence and Analysis of the Compartmental Origin of the Fluorescence. *Bioengineering*. 2019; 6(1):1.
https://doi.org/10.3390/bioengineering6010001

**Chicago/Turabian Style**

Uno, Yuko, Emiyu Ogawa, Eitaro Aiyoshi, and Tsunenori Arai. 2019. "A Three-Compartment Pharmacokinetic Model to Predict the Interstitial Concentration of Talaporfin Sodium in the Myocardium for Photodynamic Therapy: A Method Combining Measured Fluorescence and Analysis of the Compartmental Origin of the Fluorescence" *Bioengineering* 6, no. 1: 1.
https://doi.org/10.3390/bioengineering6010001