### Introduction

### Pharmacokinetic principles for understanding volume kinetics

### Kinetics of drugs

*e*refers to the fraction of the initial amount X (0) of the drug remaining in the body after the drug was removed by first-order kinetics during time t, and has a value between 0 and 1. Additionally, the disposition of fluids in the body can be explained by first-order kinetics.

^{-kt}### Mammillary compartmental model

*V*,

_{1}*k*,

_{10}*k*,

_{12}*k*). However, they can be expressed in terms of volume and clearance domain (

_{21}*V*volume of distribution in the central compartment;

_{1}*V*volume of distribution in the peripheral compartment; Cl, metabolic clearance;

_{2}*Q*, inter-compartmental clearance). The relationship between the two domains can be described as follows:

### Population analysis

^{2}, and it is a biologically natural phenomenon. In the population analysis, the part that can be explained by the patient’s characteristics (covariate) among the random inter-individual variability of the pharmacokinetic parameter is mathematically linked to the fixed-effect parameter. This is the essence of population analysis. Hence, the control stream of NONMEM is written as follows, using V

_{1}(volume of distribution in the central compartment) as an example.

_{1}(typical value, a typical human or population average value with zero variation between random individuals), and in this case, THETA(1). ETA(1) is the random effect parameter between individuals in

*V*. In other words, THETA(1) is connected by an exponential function with ETA(1). Additionally, they can be connected by multiplication or addition, and the one that best describes the data will be selected. Suppose that part of ETA(1) is described by the patient’s weight; in this case, the NONMEM control stream can be changed as follows:

_{1}### Calculation of plasma volume expansion caused by fluid administration using hemoglobin

*V*) at any time t after the start of intravenous infusion can be expressed as follows, based on the hematocrit (Hct).

_{p}*V*(t) is the blood volume at any time t. Assuming that the volume of erythrocytes is constant during the observation time, then the volume of erythrocytes at t = 0 before intravenous infusion is equivalent to the volume of erythrocytes at any time t after intravenous infusion.

_{b}_{p}, then A

_{p}at any time t, i.e., A

_{p}(t), during an intravenous infusion can be described as follows:

*D*is dimensionless and is referred to as plasma dilution. Substituting Equation (10) into Equation 11 yields

_{p}*V*(t) and

_{p}*D*(t), and this is equivalent to an increase in plasma volume due to the fluid administered. Substituting Equation (6) into Equation 12 yields

_{p}### Structure model for volume kinetics

### One-volume model

*k*) was not estimated and was set as a fixed value. In general,

_{b}*k*is 0.5–1.5 ml/min in adults [14]. Robert G. Hahn often set it to 0.8 or 1.0 ml/min for modeling [10,11]. If the data are fitted to the model, then the estimated value of renal clearance (

_{b}*k*) will exceed 100 ml/min; therefore, even if

_{r}*k*is fixed to a specific value in the range of 0.5–1.5 ml/min, the actual estimated value will not be affected significantly.

_{b}### Two-volume model

*k*and

_{12}*k*; however, in volume kinetics, it is estimated as

_{21}*k*. This can be understood by recalling that the observed value in volume kinetics is the plasma dilution measured based on the hemoglobin concentration. Because hemoglobin exists only in blood vessels, it cannot move to the peripheral compartment, and it may be impossible to estimate the rate at which fluid propagates from the peripheral compartment to the central compartment or is cleared through the lymphatic system. In addition, owing to the nature of crystalloids, the rate of transfer between the central and peripheral compartments will not differ significantly. Moreover, it is beneficial to reduce the number of parameters to be estimated to ensure model stability (parsimonism). In some studies,

_{t}*k*and

_{12}*k*were estimated separately using the two-volume model [6-9]. The objective function value of a model can be reduced significantly by distinguishing the two parameters; however, this often causes model instability. Considering these characteristics of volume kinetics, it is almost impossible to explain the plasma dilution-time data of crystalloids using a three-volume model. The NONMEM control streams for one- and two-volume models are presented in the Appendix 1.

_{21}### Calculation of renal clearance using urine volume

*k*). Because renal clearance is proportional to the plasma volume expansion, the following differential equation can be established:

_{r}_{TOT}) is known during the entire observation time T, then

*k*and

_{r}*k*can be calculated as follows:

_{b}*k*∙

_{r}*AUC*(

*D*) refers to the urine volume produced by intravenous infusion, and

_{p}*k*∙

_{b}*T*the urine volume induced by basal elimination. However, in reality, renal clearance is controlled by a complex mechanism that involves various hormones such as antidiuretic hormone, renin-angiotensin, atriopeptin, aldosterone, and angiotensin II [14]. Moreover, specific cell receptors are involved, including baroreceptors and osmoreceptors that regulate salt and water homeostasis [14]. Renal clearance can be obtained via two methods. The first is an estimation method from the structural model of volume kinetics, and the second is a calculation method based on the area under the time–dilution curve for the central compartment divided by the observed total urinary output [15]. The more accurate method is yet to be determined. However, it is reasonable to estimate

*k*from the same dataset with other volume kinetic parameters.

_{r}### Covariates describing inter-individual variabilities of volume kinetic parameters

### Clinical application of volume kinetics

### Limitations of fluid kinetics

*k*) removed from the peripheral compartment without passing through the central compartment. In this regard, a previous study that explained the disposition of Ringer’s lactate solution based on a three-volume model is difficult to understand [23].

_{20}