### Distribution

#### Volume of distribution

The volume of distribution (Vd or Vu,ss - Unbound volume of distribution at steady state or Vss - (Bound) Volume of distribution at steady state) is a pharmacokinetic parameter that describes the apparent space in the body available to contain a drug. It is not a physical volume but rather a mathematical concept used to characterize the distribution of a drug between the bloodstream and the rest of the body tissues.
The formula for calculating the volume of distribution is:

Vd = Amount of drug in the body / Concentration of drug in the plasma[ 1 ]

Vd helps to estimate how extensively a drug distributes into tissues after administration. A high Vd suggests that the drug is extensively distributed into tissues, while a low volume of distribution indicates that the drug remains primarily in the bloodstream.

Vd can be influenced by various factors, including the drug's physicochemical properties, tissue binding, and lipid solubility. It is an important parameter in pharmacokinetics as it provides insights into the distribution characteristics of a drug within the body, which, in turn, can impact its therapeutic effectiveness and potential for side effects.

#### Tissue-to-plasma ratios

The tissue-to-plasma ratio (Kpu - Unbound tissue-to-plasma ratio, Kp - (Bound) tissue-to-plasma ratio), also known as the partition coefficient, is a pharmacokinetic parameter that quantifies the distribution of a drug between tissues and plasma. It represents the ratio of drug concentration in a specific tissue to its concentration in the plasma at a given point in time.

This ratio provides insights into the extent of a drug's distribution into tissues relative to its concentration in the bloodstream. A Kp greater than 1 indicates that the drug has a higher concentration in the tissue compared to the plasma, suggesting good tissue penetration. Conversely, a ratio less than 1 implies that the drug is more concentrated in the plasma than in the tissue.

Kp is a valuable parameter for understanding the pharmacokinetics of a drug, especially regarding its distribution into specific target tissues or organs. Different tissues may have varying affinities for a drug, and the Kp helps assess the drug's propensity to accumulate in certain areas of the body.

It's important to note that the Kp is influenced by factors such as the drug's physicochemical properties, lipid solubility, tissue binding, and the presence of transporters. Additionally, this ratio is often measured at specific time points after drug administration to capture the dynamic distribution process.

#### Vd Calculator

There are various Vd methods in the literature to compute the TP ratios and currently six methods are currently available on Teoreler:

### Validation of methods

The above methods from literature have been validated using provided equations and datasets from respective articles. The comparison between observed and predicted Vd (L/kg) for various drugs is presented below, highlighting the effectiveness of different methods to predict Kp and steady state volume of distribution.

Poulin and Theil, Rodgers and Rowland, PK-Sim® and Schmitt 2020 have reported predicted Vu,ss and/or Kpu using their method in their manuscript or software. The volume fractions from Poulin and Theil 2001 were used for the Vu,ss calculations during verification and volumes of organs/tissues obtained from Rodgers and Rowland 2007 were used for the calculation of the remaining Vd methods.The following figure shows the validations of the respective methods:

The available Vd methods were compared against a collective dataset of observed Vu,ss and Kpu from Poulin and Theil1 and Rodgers & Rowland2-4. A comparison of the performance of each method using (AAFE - Absolute average fold error, TSS - Total sum of squares) is shown below in rats and humans. A further detailed description about each of the methods is available from the respective method section.

AAFE - Absolute average fold error, TSS - Total sum of squares

#### Interactive comparison

The following plots show a visual comparison of the calculated Vu,ss and Kpuusing various methods against observed data for various drugs. The observed Vu,ss and Kpu values were obtained from Poulin and Theil1 and Rodgers & Rowland2-4 manuscripts and compared against predictions using various methods as shown in the scatter plots below. The data can be filtered using available Vd methods, property of the drug, log P and tissue.

#### Poulin and Theil

Poulin and Theil are one of the earliest to develop a method to describe the TP ratios and Vdbased on drug physicochemical properties that had a significant improvement over previously reported methods and works for a wide variety of drugs. In 2001, they have validated their method for 148 drugs in total (both human and rat). This method estimates the Kpu and Vd well for Neutral compounds and also for compounds with a higher LogP (> 3).
The following plots show the prediction outcome using Teoreler in both humans and rats against predicted values reported by Poulin and Theil, 2001.

Further comparison was done against Vu,ssobserved data collated from Poulin and Theil 2001 1 and Rodgers and Rowland 2007 4and the plots below demonstrate acceptable Absolute average fold error (AAFE) values.

#### Berezhkovskiy

Leonid M. Berezhkovskiy7 modified Poulin and Theil's equation to eliminate the assumption that tissue partition between tissue lipids, water and extracellular water space is mainly governed by passive distribution. Since Berezhkovskiy method has not reported any tissue compositions, Poulin and Theil 2001 data was used. Also predicted Kpu and Vu,ss values have not been reported, therefore, the validation of this method was done by comparing observed Vu,ss and experimental Kpu from Poulin and Theil1 and Rodgers and Rowland2-4 manuscripts.

AAFE - Absolute average fold error, TSS - Total sum of squares

#### Rodgers and Rowland

Rodgers, Leahy and Rowland published an improvised method in 2005 for moderate-to-strong basic compounds and further elaborated the method for remaining compounds in a manuscript published in 2006.2-4 This Vd method works well for most of the compounds with different properties and having a log P < 3.

Validation was done against available Kpu of various organs and tissue of various drugs. Below is the comparison of the values predicted using Teoreler against predicted values from Rodgers and Rowland.

Comparison between published observed data for various drugs against Teoreler predicted values using Rodgers and Rowland method2-4 in humans and rats are shown below:

AAFE - Absolute average fold error, TSS - Total sum of squares

#### Schmitt W method

Walter Schmitt8 in 2008 published this method that is universally applicable across species, estimates Kpu by analyzing tissue composition in water, lipids, phospholipids, and proteins, along with compound-specific parameters like lipophilicity, phospholipid membrane binding, pKa, and unbound plasma fraction. By considering compound charge and its fraction at physiological pH, the method accommodates neutral, acidic, basic, or multiply charged substances. This comprehensive approach greatly expands applicability compared to prior methods.
The tissue composition table provided in this manuscript has been modified using values provided from PK-Sim® since the prediction performance of this composition table was better. Schmitt method has not reported any predicted Kpu and Vu,ss values, therefore, validation of this method was done by comparing observed Vu,ss and Kpu from Poulin and Theil1 and Rodgers and Rowland2-4 manuscripts.

AAFE - Absolute average fold error, TSS - Total sum of squares

#### PK-Sim®

PK-Sim, an open source software, part of Open Systems Pharmacology Suite9 adapated a Kpu method proposed by Willmann et al. 2005 10 that computes using tissue composition of water, lipids, and proteins.
This method does not report any predicted Kpu or Vss values, therefore, Kpu values predicted using PK-Sim application for a limited number of different compound types have been compared against Teoreler predicted values in rats and humans.

The method was also compared against observed Vu,ss and Kpu values from Poulin and Theil1 and Rodgers and Rowland2-4 manuscripts

AAFE - Absolute average fold error, TSS - Total sum of squares

#### Schmitt et al. 2020

Schmitt et al. 2020 further elaborated Rogers and Rowland method based on lysosomal entrapment of drug. Earlier work by Schmitt et al. in 2019 provided evidence of lysosomal trapping of basic lipophilic drugs potentially impacting the overall drug distribution.6 Schmitt et al. 2020 have expanded Rogers and Rowland method to include lysosomal trapping for basic drugs and is identical for other compounds.

Schmitt's method was validated against predicted Kpu values obtained from the manuscript in rats as shown below:

The method shows a slight improvement in the R2 value from the original Rodgers and Rowland method (0.956 vs. 0.946). This method was also compared against observed Vu,ss and Kpu values from Poulin and Theil1 and Rodgers and Rowland2-4 manuscripts as shown below.

##### PBPK models

For PBPK modelling, a 'Distribution scalar' option is provided for the users, found in the 'Other Inputs' tab that allows users to input a scalar value. This scalar modifies all predicted Kp by the entered value. The default factor is 1, indicating no scaling. The Vd in the simulated output table is presented in L/kg.

### References

1. Poulin P, Theil FP. Prediction of pharmacokinetics prior to in vivo studies. 1. Mechanism-based prediction of volume of distribution. J Pharm Sci. 2002 Jan;91(1):129-56. https://doi.org/10.1002/jps.10005
2. Rodgers T, Rowland M. Physiologically based pharmacokinetic modelling 2: Predicting the tissue distribution of acids, very weak bases, Neutrals and Zwitterions. Journal of Pharmaceutical Sciences. 2006;95(6):1238-57. https://doi.org/10.1002/jps.20502
3. Rodgers T, Leahy D, Rowland M. Physiologically Based Pharmacokinetic Modeling 1: Predicting the Tissue Distribution of Moderate-to-Strong Bases. Journal of Pharmaceutical Sciences. 2005;94(6):1259-76. https://doi.org/10.1002/jps.20322
4. Rodgers T, Rowland M. Mechanistic Approaches to Volume of Distribution Predictions: Understanding the Processes. Pharmaceutical Research. 2007;24(5):918-33. https://doi.org/10.1007/s11095-006-9210-3.
5. Schmitt MV, Reichel A, Liu X, Fricker G, Lienau P. Extension of the mechanistic tissue distribution model of Rodgers & Rowland by systematic incorporation of lysosomal trapping: impact on Kpu and volume of distribution predictions in the rat. Drug Metabolism and Disposition. 2020:DMD-AR-2020-000161. https://doi.org/10.1124/dmd.120.000161.
6. Schmitt MV, Lienau P, Fricker G, Reichel A. Quantitation of Lysosomal Trapping of Basic Lipophilic Compounds Using In Vitro Assays and In Silico Predictions Based on the Determination of the Full pH Profile of the Endo-/Lysosomal System in Rat Hepatocytes. Drug Metabolism and Disposition. 2019;47(1):49-57. https://doi.org/10.1124/dmd.118.084541.
7. Berezhkovskiy LM. Volume of distribution at steady state for a linear pharmacokinetic system with peripheral elimination. J Pharm Sci. 2004;93(6):1628-40. https://doi.org/10.1002/jps.20073.
8. Schmitt W. General approach for the calculation of tissue to plasma partition coefficients. Toxicol In Vitro. 2008;22(2):457-67. https://doi.org/10.1016/j.tiv.2007.09.010.
9. Open Systems Pharmacology Suite. http://www.open-systemspharmacology.org/
10. Willmann S, Lippert J, Schmitt W. From physicochemistry to absorption and distribution: predictive mechanistic modelling and computational tools. Expert Opin Drug Metab Toxicol. 2005;1(1):159-68. https://doi.org/10.1517/17425255.1.1.159.