Volume Kinetic Models forPerioperative Fluid Therapy

DEGREE PROJECT, IN MEDICAL ENGINEERING , FIRST LEVEL STOCKHOLM, SWEDEN 2015

Volume Kinetic Models for

Perioperative Fluid Therapy

PEHR WESSMARK, VIKTOR WINTHER

This project was performed in collaboration with Karolinska Institutet.

Supervisors: Christer Svensen, Director for Doctoral Education and Peter Rodhe,

Karolinska Institutet, Department of Clinical Science and Education,

Södersjukhuset, Stockholm, Sweden.

Volume Kinetic Models for Perioperative Fluid Therapy

Volymkinetiska modeller för perioperativ vätsketerapi

P e h r W e s s m a r k V i k t o r W i n t h e r

Degree project in Medical Engineering

First level, 15 hp

Supervisor at KTH: Anna Bjällmark

Examinator: Lars-Gösta Hellström

School of Technology and Health

KTH Royal Institute of Technology

SE-141 86 Flemingsberg, Sweden

Abstract

Intravenous fluid infusion during surgeries is based on clinical practice guidelines. Many factors impact the fluid distribution in the body, mainly the effect of anesthetic gases and surgical stress. Volume kinetics is a method to simulate the distribution and elimination of infusion fluids by considering the dilution of plasma over time. In this work, two volume kinetic models for fluid therapy are described – the single and two-fluid space model.

The goal was to estimate five volume kinetic parameters for implementation in a population kinetic model. The method was based on data from an experiment at the University of Texas Medical Branch where the purpose was to examine the effect of the anesthetic gas isoflurane on fluid distribution after a controlled bleeding.

Sammanfattning

Intravenös vätsketillförsel under operationer grundar sig på klinisk praxis och beprövad erfarenhet. Vätskedistributionen i kroppen påverkas av många faktorer, framförallt av anestesigaser och kirurgiska interventioner. Volymkinetik är en metod för att simulera infusionsvätskors distribution och eliminering genom att använda beräkningar av plasmadilutionen över tid. I detta arbete beskrivs två volymkinetiska modeller för vätsketerapi – en- och tvåvolymsmodellen.

Målet var att estimera fem volymkinetiska parametrar för implementering i en populationskinetisk modell. Metoden baserades på mätdata från ett experiment vid University of Texas Medical Branch där syftet var att undersöka effekten av anestesigasen isofluran på vätskedistributionen efter en kontrollerad blödning.

Contents

List of Tables. . . . List of Figures . . . . List of Acronyms. . . .

1 Introduction. . . .1

1.1 Purpose and Goal . . . 1

1.2 Delimitations . . . 1

2 Background. . . .2

2.1 Pharmacokinetics . . . 2

2.1.1 Volume Kinetics . . . 2

2.2 Previous Work . . . 3

3 Materials and Methods . . . .4

3.1 Data from Previous Work . . . 4

3.1.1 Inclusion and Exclusion Criteria . . . 4

3.1.2 Protocol . . . 5

3.2 Kinetic Modelling of Experimental Data . . . 5

4 Results. . . .8

5 Discussion. . . .12

6 Conclusion. . . .14

References. . . .15

Appendix

A Decimal Fractional Plasma Dilution. . . .

B Curve Fitting Using Linear Polynomials. . . . B.1 Linear Interpolation and Extrapolation. . . .

C Volume Kinetic Models. . . . C.1 Single-Fluid Space Model. . . . C.2 Two-Fluid Space Model. . . .

List of Tables

Table 1 List of randomized experimental sessions at UTMB. . . 4 Table 2 Single-fluid space model. Mean fractional plasma dilution during fluid

infusion. . . 9 Table 3 Single-fluid space model. Mean fractional plasma dilution without fluid

infusion. . . 9 Table 4 Two-fluid space model. Mean fractional plasma dilution during fluid

infusion. . . 9 Table 5 Two-fluid space model. Mean fractional plasma dilution without fluid

infusion. . . 9 Table 6 Estimated mean of the single-fluid space model parameters. . . 11 Table 7 Estimated mean of the two-fluid space model parameters . . . 11

List of Figures

Figure 1 Fluid distribution from a central to a peripheral compartment . . . 2 Figure 2 Experimental protocol at UTMB . . . 5 Figure 3 Compilation of experimental data from UTMB in MATLAB . . . 6 Figure 4 Decimal fractional plasma dilution from changes in Hgb concentration

over time after 20 minutes of 25 ml·kg-1 fluid infusion. . . 8 Figure 5 Decimal fractional plasma dilution from changes in Hgb concentration

over time without fluid infusion . . . 8 Figure 6 Plasma dilution during fluid infusion including values from individual

experiments . . . 10 Figure 7 Plasma dilution without fluid infusion including values from individual

List of Acronyms

ABF Anesthetized, Blood withdrawal, Fluid infusion ABN Anesthetized, Blood withdrawal, No fluid infusion CBF Conscious, Blood withdrawal, Fluid infusion CBN Conscious, Blood withdrawal, No fluid infusion HCT Hematocrit

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1 Introduction

Body fluids are essential for proper organ functioning and are important for digestion, temperature regulation and elimination of waste products. Intravenous (IV) administration of fluids facilitates oxygenation of tissues and organs, which is vital when a patient is losing blood during surgery. Administered fluids are replenishing lost intravascular volume. However, since the capillary walls are semipermeable membranes some of the infused fluid will be lost to the interstitial space. From a functional point of view, this can be expressed by kinetic models as fluid being allocated in a central and peripheral fluid space.

Fluid is infused to counteract fluid losses from the body. Recommendations for how to infuse fluids are varying due to limited evidence regarding clinically perioperative outcomes. The outcome depends largely on the underlying pathology and amount of infused fluid. Proper fluid therapy can also reduce postoperative nausea [1]. Furthermore, hemorrhaging can force changes in surgical sequences. Therefore, there is a need to develop a robust kinetic model that can predict the amount of fluid to be administered to patients under anesthesia and help plan fluid therapy during surgeries with intraoperative bleeding.

Modelling physiological processes mathematically is a difficult task, yet it is critical to understanding the behavior of IV fluid distribution. When modelling a system, the model is kept as simple as possible and the essential features are imitated to approximate a model which simplifies the calculations needed to describe the system. The equations constructed by the model must first be solved in order for the model to give accurate accumulation (input–output) values.

This work describes an approach to population kinetic modelling using estimated volume kinetic parameters. The volume kinetic models are based on the fluid crystalloid 0.9% sodium chloride (normal saline) in order to accurately determine the distribution and retention of IV fluids.

1.1 Purpose and Goal

The goal is to evaluate the dilution of plasma over time from a set of experimental data and to compare the result to accumulation values from volume kinetic models. The reliability of the estimated model parameters will be evaluated when applied to subjects under anesthesia with or without IV administration of fluid. The parameters will later on be implemented in a population kinetic model.

The purpose is to be able to predict how IV fluids are distributed at any given time in the body. The population kinetic model will be used to plan fluid therapy during surgeries and the result will be of both academic and clinical interest.

1.2 Delimitations

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2 Background

2.1 Pharmacokinetics

Pharmacokinetics is one of the sciences underlying clinical anesthesia. It is the study of drug disposition [2] (absorption, distribution, metabolism and excretion) from compartments in the body. In pharmacokinetics, blood is sampled repeatedly in order to determine drug concentration in the body. It gives a mathematical basis for studying clearance and drug concentration in plasma. The concept of clearance is important in many kinetic models and can be defined as the rate of elimination of a drug divided by the plasma concentration. Pharmacokinetic simulation of distribution and elimination of IV fluids is made possible with volume kinetics [3].

2.1.1 Volume Kinetics

Volume kineticsis is used to analyse the effect of IV fluids on theoretical fluid compartments. Contrary to traditional pharmacokinetics, the walls of the fluid compartments are elastic and can change in size. Parameters for fluid movement are generated from first order differential equations and fluids are analyzed by the use of hemoglobin and hematocrit values (Appendix A). The main component of both infusion fluids and blood plasma is water. The concentration of an infusion fluid is therefore expressed as dilution of Hgb with respect to time.

In the basic volume kinetic model, the two-fluid space model, the fluid is thought to expand the space around the infusion site [4]. The infused fluid is supposed to expand a central body fluid space Vc to a larger volume vc. vc is the functional representation of the intravascular target volume Vc including the replenished volume after fluid infusion. Capillary fluid exchange will occur between the intravascular and interstitial compartments. This is modelled as fluid being distributed from vc to a peripheral compartment Vp which in turn can be expanded to vp (Figure 1). Fluid is eliminated from the central fluid space by baseline diuresis, evaporation and by a dilution dependent mechanism. The rate of volume equilibrium between the central and peripheral compartment is proportional to the relative change in volume _(vc−V_c) /V_c

by a constant of proportionality. The plasma and Hgb is part of Vc and the change in volume is directly correlated to plasma dilution. It follows that Hgb concentration can be used to calculate the dilution of plasma in the blood (Appendix A).

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2.2 Previous Work

The concept of volume kinetics was introduced some 20 years ago by Hahn, et al. [3, 5-7]. Several papers outlined the basis for modelling distribution and elimination of IV fluids by analyzing the volume effects of fluids.

In 2007 a paper by Norberg, et al. [8] was published in the medical journal Anesthesiology, describing a population volume kinetic approach to intravenous fluid therapy and supplementation in surgical patients. Concluding observations were made about arterial pressure, cardiac output, total protein concentration and plasma dilution using volume kinetic models. As a follow-up to the paper, Li and Svensen at the University of Texas Medical Branch (UTMB) [9], did several experiments aiming at kinetic evaluation of fluid distribution in both awake and anesthetized subjects. A controlled hemorrhage was added.

The work was further developed by Peter Rodhe in his dissertation at Karolinska Institutet [10]. The thesis included several volume kinetic models describing the dynamics of fluid distribution in relation to bleeding and anesthesia.

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3 Materials and Methods

The experimental data used in this section was collected by Li et al. at UTMB. Measurements were taken from twelve human subjects at four separate occasions in random order over a period of several months [9]. These occasions were denoted as ABF (Anesthetized, Blood withdrawal, Fluid infusion), ABN (Anesthetized, Blood withdrawal, No fluid infusion), CBF (Conscious, Blood withdrawal, Fluid infusion) and CBN (Conscious, Blood withdrawal, No fluid infusion) (summarized in Table 1). Hemodynamic data including information about systemic vascular resistance and stroke volume were also determined during each session. Relevant parameters were exported from the data sets and imported into MATLAB (Matlab 8.4.0.150421, The MathWorks, Inc., Natick, Massachusetts, USA, 2014b). The experimental data and details concerning the original research plan were supplied by Svensen.

Table 1: List of randomized experimental sessions at UTMB.

Session Conscious Anesthetized Fluid infusion Blood withdrawal

CBN Yes No No Yes

ABN No Yes No Yes

CBF Yes No Yes Yes

ABF No Yes Yes Yes

3.1 Data from Previous Work

Twelve volunteers consisting of five men and seven women aged 19-34 years of age of varied ethnicity were recruited to participate in the study at UTMB after written consent. The study was approved by the Institutional Review Board at UTMB at Galveston, Texas (IRB No. 04-379). The subjects underwent 48 experiments divided into four separate experiments separated by at least 10 days, each with or without fluid infusion of 0.9% sodium chloride. Blood withdrawal was performed according to the requirements set by the UTMB Blood Bank and reinfused after the experiment was completed.

3.1.1 Inclusion and Exclusion Criteria

Allthough rarely seen, edema or heart failure can occur after large volume IV fluid infusions. Therefore, the individuals had to undergo medical screening and a series of physical examinations to rule out any cardiac diseases and to ensure that each subject was healthy. In order to find suitable candidates for the study at UTMB, a set of inclusion and exclusion criteria for the original research plan were established [9]: Inclusion criteria:

 The subjects had to be 19–34 years of age  The subject had to be in good health Exclusion criteria:

 Pregnancy

 Sulfite or iodide allergies  Any form of fungal infection  Optic nerve damage (glaucoma)

 Impaired blood clotting or any other bleeding disorders

 Decreased amount of erythrocytes (male <0.14 g·ml-1, female <0.12 g·ml-1)  Peptic ulcers

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 The subject participates in a weight loss program  More than 20 pack–years cigarette smoking  Any cardiovascular disease

 Any eating disorder or obesity (BMI <21 or BMI >27)  Indications of hypertension

3.1.2 Protocol

The experiment at UTMB was conducted in four sessions. Healthy subjects underwent two sessions during conscious state and two sessions in which the subjects were anesthetized by administration of isoflurane. Hgb concentration (g·ml-1) and hematocrit values (%) were obtained by arterial blood sampling every five minutes for the first 60 minutes and then in 10-minute increments for another 120 minutes (Figure 2). An arterial radial catheter was used together with an IV cannula for blood sampling. Total plasma protein and albumin concentrations were sampled at 0, 20, 60, 120 and 180 minutes. During the blood collection and infusion phase, the subjects underwent a mild hemorrhage of 7 ml·kg-1 body weight and IV fluid infusion of 25 ml·kg-1 body weight of crystalloid 0.9% saline respectively. Each individual was scheduled to participate in all four sessions. Variables relevant to data analysis and hemodynamic parameters were monitored during the experiment.

Figure 2: Experimental protocol at UTMB. Blood sampling intervals during 180 minutes. Infusion time (A–B), 5 minute sample intervals (A–C) and 10 minute sample intervals (C–D).

3.2 Kinetic Modelling of Experimental Data

In this work the import of data was managed by the built-in MATLAB function

xlsread which is customized for sheets in Microsoft Excel workbooks [11]. Svensen had

gathered the data from UTMB in a worksheet that included time points, subject identification number (1-12), type of experimental session (CBN, ABN, CBF and ABF), heart rate, systolic and diastolic volume, Hgb and HCT values, albumin and total protein concentrations. The HCT data was used as mean baseline values and the arithmetic mean of the Hgb concentrations were compiled into a structure array according to each session (Figure 3).

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Figure 3: Compilation of experimental data from UTMB in MATLAB.

The infusion rate was set to 1.6 ml·min-1 body weight. The duration of the fluid infusion was set to 20 minutes. A row vector holding time intervals was matched to the blood sampling measurements. Initial values for the central and peripheral compartment were 3000 ml and 6000 ml respectively. Baseline diuresis was set to 1.5 ml·min-1, evaporation and dilution dependent mechanism was 30 ml·min-1 and the net rate of volume equilibration was initially set to 100 ml·min-1, all according to Rodhe´s original MATLAB algorithm [10]. A hemorrhage was added to the model and the baseline plasma dilution was shifted to fit the dilution–time graph. Baseline values for plasma dilution were calculated using the expression

Baseline = (7 ml ⋅ kg−1)⋅(Body weight)⋅(1 − HCT)

(1) The MATLAB function ode45 [12] was used to solve the following differential equations (2-4) generated by the experimental data and initial parameter values:

dv(t) dt = ki− kb− kr v(t)−V V (2) where, ki denotes the infusion rate (ml·min-1) of the incoming fluid, kb is the baseline diuresis and evaporation, kr governs the dilution dependent mechanism, v is the expanded central body fluid space (ml) and V is the baseline volume (ml). This equation describes the single-fluid space model.

A system of two differential equations describes the two-fluid space expansion of the central and peripheral compartments,

dv₁(t) dt = ki− kb− kr v₁(t)−V₁ V₁ − kt v₁(t)−V₁ V₁ − v₂(t)−V₂ V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ (3) dv₂(t) dt = kt v₁(t)−V₁ V₁ − v₂(t)−V₂ V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ (4)

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The solutions to (2–4) are presented in Appendix C. The solutions to the differential equations were stored in a 25 by 3 matrix, holding values for compartment volume changes and fluid elimination over time. The elimination of fluid from the central compartment was computed using a MATLAB function developed by Rodhe. The function described the dynamics of volume equilibrium between the central and peripheral fluid compartments.

Using the regression analysis method of least squares, a curve was fitted to the experimental measurements of the plasma dilution. As a result, deviations between predicted (model) and experimental Hgb data could be determined. The mean values including the standard deviations of the model and experimental data was calculated for each time point for every session. The results were compiled into separate dilution–time graphs and the corresponding estimated model parameters V1, V2, kr, kt and kb were established.

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4 Results

Dilution-time graphs for the single and two-fluid space model presented as mean values after 20 minutes of 25 ml·kg-1 fluid infusion (Figure 4) and without fluid infusion (Figure 5).

Figure 4: Decimal fractional plasma dilution from changes in Hgb concentration over time after 20 minutes of 25 ml·kg-1 fluid infusion. ABF (a) and CBF (b).

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Key dilution points at 0, 20 and 180 minutes, as shown in Figure 2, are given in complementary tables for each session, presented as mean plus or minus the standard deviation for both kinetic models. The mean fractional plasma dilution for the single-fluid space model is presented in Tables 2 and 3. Corresponding values for the two-fluid space model are given in Tables 4 and 5.

Single-fluid space model:

Table 2: Single-fluid space model. Mean fractional plasma dilution during fluid infusion.

Time (min) ABF ABF* CBF CBF*

0 0 0 0 0

20 0.5454 ± 0.0056 0.6964 ± 0.0184 0.4874 ± 0.0064 0.6682 ± 0.0123 180 0.2479 ± 0.0028 0.1809 ± 0.0017 0.2363 ± 0.0034 0.1726 ± 0.0011

Table 3: Single-fluid space model. Mean fractional plasma dilution without fluid infusion.

Time (min) ABN ABN* CBN CBN*

0 0 0 0 0

20 0.0592 ± 0.0003 0.0090 ± 0.0001 0.0488 ± 0.0002 0.0087 ± 0.0001 180 0.0906 ± 0.0021 0.0416 ± 0.0002 0.0780 ± 0.0014 0.0400 ± 0.0001

* Values obtained from the curve fit model.

The relative differences between the estimated plasma dilution and the measured values for the ABF, CBF, ABN and CBN sessions were 30.55%, 48.64%, 44.99% and 59.98%.

Two-fluid space model:

Table 4: Two-fluid space model. Mean fractional plasma dilution during fluid infusion.

Time (min) ABF ABF* CBF CBF*

0 0 0 0 0

20 0.5454 ± 0.0056 0.5699 ± 0.0122 0.4874 ± 0.0064 0.5471 ± 0.0081 180 0.2479 ± 0.0028 0.1783 ± 0.0014 0.2363 ± 0.0034 0.1708 ± 0.0009

Table 5: Two-fluid space model. Mean fractional plasma dilution without fluid infusion.

Time (min) ABN ABN* CBN CBN*

0 0 0 0 0

20 0.0592 ± 0.0003 0.0411 ± 0.0001 0.0488 ± 0.0002 0.0403 ± 0.0001 180 0.0906 ± 0.0021 0.0566 ± 0.0002 0.0780 ± 0.0014 0.0552 ± 0.0001

* Values obtained from the curve fit model.

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Additional line graphs showing the plasma dilution of individual experiments and optimal curve fit for each session. Individual experiments are shown in grey. The cross-trend outlier at 150 minutes in Figure 5b has been replaced by a linearly interpolated value (Figure 7b).

Figure 6: Plasma dilution during fluid infusion including values from individual experiments. ABF (a) and CBF (b).

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Final estimates of kinetic parameters for the single and two-fluid space model are given in Tables 6 and 7.

Table 6: Estimated mean of the single-fluid space model parameters.

Session V1 (ml) V2 (ml) kb (ml·min-1) kr (ml·min-1) kt (ml·min-1)

CBN 3184 – 110.8 2929 –

ABN 2718 – 6.2 460.8 –

CBF 5493 – 0.0001 40.46 –

ABF 4163 – 0.0002 40.08 –

Table 7: Estimated mean of the two-fluid space model parameters.

Session V1 (ml) V2 (ml) kb (ml·min-1) kr (ml·min-1) kt (ml·min-1)

CBN 2172 4452 1.375 46.64 141.4

ABN 1974 5235 1.860 50.48 120.4

CBF 2890 8280 0.136 15.24 224.1

ABF 2934 5344 1.087 19.43 168.9

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5 Discussion

As seen in Figures 4 and 5, there is a distinct difference between the dilution-time curves. This is because in Figure 4, 25 ml·kg-1 bodyweight crystalloid fluid was infused for 20 minutes. The single and two-fluid space model curve will increase during the infusion and begin to decrease after 20 minutes. There is also a transcapillary refill effect. However, this effect is likely reduced after fluid has been administered. Without the infusion of fluid, the physiological response will be to decrease blood flow and increase the total peripheral resistance. This is reflected in the plasma dilution curves as a gradual increase until a steady state is reached. This is indeed the case with the two-fluid space model as seen in Figure 5.

Bleeding can have adverse effects on determining plasma dilution by volume kinetic analysis. This is because the amount of Hgb is part of the central compartment in the kinetic models as shown in Figure 1. Hemorrhaging decreases cardiac output and increases total peripheral resistance. One of the physiological responses to hemorrhaging is a decrease in blood flow by vasoconstriction and also a transcapillary refill from the interstitial tissues. To support the lost fluid, IV fluid is administered and the total blood volume (Figure 1, Appendix A) is quickly restored. As a result, the plasma dilution will increase. The relative difference in total peripheral resistance will be greater for sessions with fluid infusion. This is because the fluid acts to decrease total peripheral resistance by diluting the blood, lowering the blood viscosity. A physical interpretation based on idealized conditions is presented in Appendix D.

The relative difference between the measured and estimated plasma dilution ranges from 3.38% to 18.36% with the two-fluid space model, and from 30.55% to 48.64% with the single-fluid space model. It is fair to say that the two-fluid space model produce more accurate accumulation values than the single-fluid space model. The presence of a cross-trend outlier at 150 minutes (Figure 5b) could affect the accuracy of the least squares approximation. Removing the outlier would change the precision, making the two-fluid space model more accurate (see Figure 7). From Figures 6 and 7 it can also be noted that the individual variability is larger when no fluid is infused. The cross-trend outlier is an indication of measurement error. This is a reasonable assumption due to the fact that the same method and algorithm for calculating plasma dilution was used for every session.

The model parameters V1, V2, kr, kt and kb were initially set according to values used in Rodhe´s dissertation [10]. The infusion rate ki was based on tests and infusion studies made by Svensen and the infusion time was set to 20 minutes on the same basis. The estimated mean value of the central compartment was 3.9 l for the single-fluid space model and 2.5 l for the two-single-fluid space model. 3.9 l coincides quite well with the expected plasma volume of about 3.5 l [4]. The estimated mean value of the peripheral compartment V2 was 5.83 l, which is less than the amount of interstitial fluid in the body [4]. The two-fluid space model is supposed to account for both the plasma and interstitial volume.

The estimated parameters kb, kr and kt for the two-fluid space model were deemed statistically valid and useful for implementation in a population kinetic model. According to the literature, kr was expected to lie in the interval 50-150 ml·min-1 and

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An F-test can be used to accurately determine which of the two models to implement in a population kinetic model. The F-test is a statistical test which compares variances from a set of data in a population to determine if they are equal [13]. The null hypothesis is that the variances of the volume kinetic models are the same. However, from looking at Figures 4 through 7 it is obvious that the two-fluid space model more accurately predicts the plasma dilution for each time point.

According to the sampling plan from UTMB, blood was supposed to be sampled 25 times during each experimental session. In some cases however, Hgb or HCT data could not be obtained from the subject. On some occasions, the subject did not take part in the session at all. This is more or less inevitable in human subject research when the study runs over a longer period of time. In order to avoid any significant effect on the curve fit model, the missing values (NaN entry in MATLAB) were estimated by linear interpolation. This was only possible for single pieces of data gaps with valid adjacent measurements. If the missing value was in the first or last measurement, linear extrapolation had to be used. The reasoning is explained in more detail in Appendix B. The measurements were omitted if the individual did not take part in an experimental session. The corresponding generated values of the kinetic curve fit model also had to be ignored. This resulted in 10 data points for the CBN session and 11 data points for the CBF and ABN sessions which is demonstrated in the individual experiments shown in Figures 6 and 7. These inconsistencies however did not have a significant impact on the end result.

The mathematical evaluation of the kinetic models were greatly simplified by using the numerical language of MATLAB. The programming environment was proven effective when visualizing data and incorporating Rodhe´s volume kinetic models. The aim was to use the estimated model parameters in population kinetic analysis. Conventional pharmacokinetic models fail to take account of physiological variation within populations, but provides an effective and safe way to administer fluids. With a population-based approach to modelling dilution of plasma, the population is modelled as one unit. Population-based pharmacokinetics can be used as a means of explaining variability between individuals and across populations [14]. In further research, the program NONMEM (NONMEM 7.0, ICON Development Solutions,

Hanover Maryland, USA, 2013) will be used to analyze the kinetic models presented in this work. The program develops techniques for pharmacokinetic data analysis and nonlinear mixed effects models. Mixed effects are divided into either fixed or random effects. Fixed effects are known properties of individuals that causes the individual to differ from the average. Random effects cannot be determined in advance.

Ethical concerns can be raised about the UTMB experimental protocol and human subject research in general. For instance, there is always a risk of infection as a result of invasive blood sampling. The blood was sampled by radial artery cannulation. This is a common procedure and considered safe with a rate of complications below one percent [15, 16]. The blood volume drawn from each subject corresponded to a little more than a mild controlled hemorrhage of 7 ml·kg-1 body weight [9]. The subjects were told to contact the staff at UTMB if they felt anything unusual during the experiment. Although anesthetic procedures are considered safe, a medical assessment had to be made before any anesthetic agent could be administered. Accordingly, the subjects underwent a medical screening prior to the participation in the experiments as described in chapter 3.1.1. Another ethical concern could be the number of subjects participating in the experiment. How many individuals need to be involved in order to get statistically significant results? If the sample size is too small, the results will not be reliable and the validity of the experiment may be questioned. The sample size should be appropriate in regards to both size and quality.

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6 Conclusion

Based on literature values, the estimated two-fluid space model parameters are suited for implementation in a population kinetic model. The two-fluid space model was kept simple for the purpose of minimizing the number of parameters that needed to be estimated. Naturally, the model has to be developed further in order to give more accurate accumulation values.

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References

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2 Caldwell J, Gardnet I, Swales N: An introduction to drug disposition: the basic principles of absorption, distribution, metabolism, and excretion. Toxicologic Pathology 23(2) Mar-Apr 1995. 102-114.

3. Hahn RG: Volume Kinetics for Infusion Fluids. Anesthesiology 113(2) Aug 2010. 470–481.

4. Svensén C, Drobin D, Edsberg L, Ståhle L, Hahn RG: Volymkinetik – ny metod styra intravenös vätsketillförsel. Läkartidningen 96(16) 1999. 1969-1970. 5. Svensén C, Hahn RG: Volume kinetics of Ringer Solution, dextran 70, and

hypertonic saline in male volunteers. Anesthesiology 87(2) Aug 1997. 204-212. 6. Ståhle L, Nilsson A, Hahn RG: Modelling the volume of expandable body fluid

spaces during i.v. fluid therapy. British Journal of Anaesthesia (Oxford Journals) 78(2) Feb 1997. 138-143.

7. Hahn RG, Drobin D: Urinary excretion as an input variable in volume kinetic analysis of Ringer´s solution. British Journal of Anaesthesia (Oxford Journals) 80(2) Feb 1998. 183-188.

8. Norberg Å, Hahn RG, Huson L, Olsson J, Prough D, et al.: Population Volume Kinetics Predicts Retention of 0.9% Saline Infused in Awake and Isoflurane-anesthetized Volunteers. Anesthesiology 107 2007. 24–32.

9. Svensen C, Prough S, Li H, McQuitty C, Esenaliev RO, et al.: The Effect of Isoflurane on Transcapillary Fluid Redistribution after Hemorrhaging in Humans. Research Plan, The University of Texas Medical Branch. The Department of Clinical Science and Education, Stockholm South General Hospital.

10. Rodhe P: Mathematical Modelling of Clinical Applications in Fluid Therapy. Ph.D Thesis (2010), Karolinska Insitutet, The Department of Clinical Science and Education, Stockholm South General Hospital.

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14. Wright P. M. C.: Population based pharmacokinetic analysis: why do we need it; what is it; and what has it told us about anaesthetics? British Journal of Anaesthesia 80(4) 1998. 488-492.

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emedicine.medscape.com/article/1999586-overview

Appendix

A Decimal Fractional Plasma Dilution

Water is the main component of plasma and intravenous fluids which is why the concentration of intravenous fluids _{(v −V ) /V} are expressed in terms of dilution of hemoglobin as a function of time1,

v(t)−V V = Hgb − Hgb(t) Hgb(t) 1 − HCT = Hgb Hgb(t)−1 1 − HCT (1) where the hematocrit or erythrocyte volume fraction (Figure 1) is the volume percentage of red blood cells (RBC) in the blood defined by

HCT =

RBC volume Total blood volume

(2)

Figure 1: Blood sample depicting plasma, red and white blood cells (WBC) in proportion to the total blood volume.

B Curve Fitting Using Linear Polynomials

B.1 Linear Interpolation and Extrapolation

Method of linear interpolation of a data set2.

Figure 2: Linear interpolation of an unknown value y at x.

Consider a linear function _y(x), a polynomial of degree one of one variable,

y(x) = mx +b

(3) where m is the slope of the line and b is the y-intercept. Then,

m = m₀ = m₁ ⇒y −y0 x − x₀ = y₁−y₀ x₁− x₀ (4) y = y₀+y1−y0 x₁− x₀

(

)

Linear extrapolation by means of modifying equation (4):

y₁ = y₀+x1− x0 x − x₀

(

)

(

)

(

)

2 _{Various Unit Operations and More: How to do Linear Interpolation. (2002, March 6}th_).

C Volume Kinetic Models

C.1 Single-Fluid Space Model

Figure 3: Diagrammatic representation of the single-fluid space model3.

The infusion starts at t=0 and it is assumed that the peripheral fluid space v=V.

dv(t) dt = ki−kb−kr v(t)−V V ⇒v(t)−V V = k_i−k_b k_r + Qe−krt /V V v(0) =V : v(t)−V V = k_i−k_b(1 −e−k_rt /V₎ k_r

where _{(v −V ) /V} is the relative volume change4. The solution becomes

v(t)−V V = v(t)−V V + k_b k_r ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟e −k_rt−T V ₋kb k_r (8)

3 _{Hahn RG: Volume Kinetics for Infusion Fluids. Anesthesiology 113(2) Aug 2010. 470–481.}

With permission from the author.

4 _{Ståhle L, Nilsson A, Hahn RG: Modelling the volume of expandable body fluid spaces during}

C.2 Two-Fluid Space Model

Figure 4: Diagrammatic representation of the two-fluid space model.

dv₁(t) dt = ki−kb−kr v₁(t)−V₁ V₁ −kt v₁(t)−V₁ V₁ − v₂(t)−V₂ V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ dv₂(t) dt = kt v₁(t)−V₁ V₁ − v₂(t)−V₂ V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ ⇒ v_1p = A =(ki−kb+ kt)V1 k_r , v2p = B = (k_i−k_b+ k_t)V₂ k_r v₁ = EeXt_{+ Fe}Yt_{+ A, v} 2 = Ge Xt _{+ He}Yt _{+ B} X =1 2− k_r+ k_t V₁ + k_t V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟+ k_r+ k_t V₁ − k_t V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ 2 + 4kt 2 V₁V₂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Y = 1 2 − k_r+ k_t V₁ + k_t V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟− k_r+ k_t V₁ − k_t V₂ ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟⎟ ⎟⎟ 2 + 4kt 2 V₁V₂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⇒ G = Ekt V₁X +ktV1 V₂ , H = Fkt V₁Y +ktV1 V₂ The general solution is given by

v₁(t) = EeXt _{+ Fe}Yt₊(ki−kb+ kr)V1 k_r (9) v₂(t) = ktEe Xt V₁X +ktV1 V₂ + ktFe Yt V₁Y +ktV1 V₂ +(ki−kb+ kt)V2 k_r (10)

Two initial conditions are considered at time t=0: the fluid spaces v1=V1 and v2=V2.

V1= E + F + A

V₂ = Ekt V₁X +ktV1 V₂ + Fkt V₁Y +ktV1 V₂ + B (12)

X, Y, Q1, Q2 and Q3 can be estimated from the dilution. Furthermore,

D Vasoconstriction and the Hagen–Poiseuille Law

Assuming that the pressure gradient ΔP remains constant, the systemic vascular resistance or the total peripheral resistance (TPR) can be expressed as

TPR =Pa− Pv Q = ΔP Q ⇒ Q = ΔP TPR (15) and by the Hagen–Poiseuille equation, the laminar flow Q in a blood vessel is

Q = πr4ΔP

8ηl

(16)

Figure 5: The effect of vasoconstriction on blood flow.

where l is the length of the vessel, r is the vessel radius and η is the viscosity of blood. It follows that a small change Δr in vessel radius _r0 corresponds to a dramatic decrease in flow or cardiac output (Figure 5):

Q₀ = πr0 4_ΔP 8ηl , Q = π(r₀− Δr)4_ΔP 8ηl where

Q0 denotes the non-constricted flow. The relative change in fluid flow _{Q /Q}0 is

Q Q₀ = π(r₀− Δr)4_ΔP 8ηl πr₀4_ΔP 8ηl = 1 − Δr / r

(

)

The total peripheral resistance can also be rewritten in the form

TRITA - STH 2015:043