A Variable-Volume Kinetic Model to Estimate

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tests (3-5) within each of several (2-6) HD treatments. ... piece of information, cannot inferred from RBV – indeed, patients with differing ABVs can exhibit similar.
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A Variable-Volume Kinetic Model to Estimate Absolute Blood

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Volume in Dialysis Patients Using Dialysate Dilution Protocol

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Abstract:

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Long- and short-term adverse outcomes in hemodialysis (HD) have been associated with

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intradialytic hypotension, a common HD complication and significant cause of morbidity. It has been

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suggested that knowledge of absolute blood volume (ABV) could be used to significantly improve

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treatment outcomes. Different dilution-based protocols have been proposed for estimating ABV, all relying

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on the classic mono-exponential back-extrapolation algorithm (BEXP). In this paper, we introduce a

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dialysate dilution protocol and an estimation algorithm based on a variable volume, two-compartment,

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intravascular blood water content kinetic model (VVKM). We compare these two algorithms in a study

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including 3 arteriovenous (AV) and 3 central-venous (CV) access patients, and multiple bolus injection

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tests (3-5) within each of several (2-6) HD treatments. Investigation of the distribution of differences

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between the two methods showed a negligible systematic difference between the mean values of ABVs

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estimated from the BEXP and VVKM algorithms, however, the VVKM estimates were 53% and 42% more

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precise for the CV and AV patients, respectively.

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1. Introduction:

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Volume management plays an important role in renal replacement therapies. Removing too much

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fluid by ultrafiltration triggers intradialytic hypotension, a significant cause of long- and short-term adverse

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outcomes, while removing too little fluid causes edema, left ventricular hypertrophy and heart failure

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Knowing a patient’s ABV at the start of ultrafiltration would better allow clinicians to return patients to

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their dry weight and significantly improve such outcomes

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1-3

1-3

.

. Isotope dilution, the gold standard for

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measuring ABV, is invasive, expensive, time consuming and impractical for routine clinical application 4.

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A practical technique for estimating ABV is needed.

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Current HD machine technology provides sensors such as the Crit-LineTM and the blood volume

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monitor (BVM) that measure a patient’s hematocrit (Crit-LineTM) and blood water content (BWC). From

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these measurements and assumptions for a single compartment, one can compute changes in a patient's

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intravascular blood volume - referred to as relative blood volume (RBV). However, ABV, the crucial

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piece of information, cannot inferred from RBV – indeed, patients with differing ABVs can exhibit similar

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RBVs 5.

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Recently, a technique that uses blood water content measurements to make estimates of ABV was

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introduced in 6. In this technique, a bolus injection of ultra-pure dialysate was administered and the back-

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extrapolation algorithm (BEXP) algorithm used to estimate the initial blood water concentration at the time

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of injection. This estimate together with the size of the bolus injection was then enough to estimate ABV

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at the time of injection.

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pharmacokinetic approach 7 which assumes that the indicator dynamics can be sufficiently represented by

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a single-compartmental model with constant coefficients.

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distribution of an indicator is not uniform within the bloodstream due to blood flow

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have considered models consisting of more than one compartment. to better reflect such distribution 10-14.

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Multi-compartmental modelling has been studied, including fixed-volume

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parallel and series compartment configurations 12. Applications of such models include the distribution of

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indicators in solute kinetics

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distribution in blood

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models such as

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impossible, estimation problem.

10-16

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Fitting an exponential function to a measured indicator is a standard

10-12

, hemodialysis

, and urea kinetics

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However, studies have shown that the

10-13

, and researchers

variable-volume

, β2-microglobulin kinetics

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8, 9

14

14, 15

, and

, indocyanine green

. However, application of high-order (>2) compartmental

involve an increasing number of unknown parameters resulting in a difficult, if not

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In this paper, we present a new, physiologically motivated, variable-volume, two-compartment

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model as the basis for estimating ABV corresponding to the technique in 6. Absolute blood volume

2

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estimates derived from this model are compared with estimates from the classic mono-exponential back-

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extrapolation algorithm.

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2. Materials & Methods

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A Fresenius 4008H-HDF machine equipped with a BVM and dedicated data acquisition software 17

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(Fresenius Medical Care, Bad Homburg, Germany)

provided hemodiafiltration (HDF) therapy and

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measurement of hematocrit and blood water content, the latter of which was used to calculate RBV changes.

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Dialysate was delivered at a flow of either 500 or 800 ml/min, and at 36 degrees C. Blood flows, dialysate

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[Na+] and HDF infusion volumes and the pre or post-dilution configuration were set as prescribed in 6.

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Indicator dilutions were administered using the bolus function in the HDF machine. This function

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delivers ultrapure dialysate in multiples of 30 ml at a constant infusion rate of approximately 150 ml/ min

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during the HDF session. This bolus volume was delivered with an accuracy of better than ±1.5% 6.

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Patients

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The study included 3 arterio-venous (AV9 and 3 central-venous (CV) access patients, and multiple

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(3-5) indicator dilution experiments within each of several (2-6) HD treatments. Patients consented to

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participate as approved by the Ethics Committee of the Medical University of Graz, Austria. Table 1

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summarizes the patient and treatment data.

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Modeling

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Following

10-16

, we modeled the intravascular circulatory system by two compartments loosely

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termed central and peripheral, reflecting vessels with high and low blood flow rates, respectively (Figure

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1). The water mass and fluid volume constituted the state for each compartment.

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The following assumptions are used:

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Ultrafiltration removes fluid from the central compartment at the prescribed rate qufr .

3

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The indicator fluid is injected into the central compartment at a rate of qind . Instantaneous mixing

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is assumed within each compartment. Following the injection, the indicator fluid is assumed to

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arrive at the measurement site with a fixed time delay after circulating throughout the body.

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Since an accurate model of the inter-compartment flow is beyond the scope of this work, we assume q1  t 

that, over the time period of interest (20 minutes), both

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the blood flow from the central to peripheral compartments, q2 is the blood flow from peripheral

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to central compartments and V1 and V2 are the fluid volumes for the central and peripheral

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compartments respectively.

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V1 (t )

and

q2  t 

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V2 (t )

are constants. Here q1 is

We assume that the fluid exchange between the interstitial and intravascular spaces, referred to as

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refilling/filtration, occurs between the interstitial and peripheral compartments. For simplicity,

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we take this nonlinear exchange q f as an affine function of the central volume as in q f  q f 0  V1 .

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Here we assume that q f depends only on the central volume since the interstitial volume is much

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larger than the volume of fluid removed by ultrafiltration within the simulation time period. The

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coefficient  models the sensitivity of q f to the lymphatic flow rate and the nonlinear Starling

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mechanism describing microvascular refilling/filtration flow into the peripheral compartment 18,

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.



We assume that the fluid removed from intravascular space by ultrafiltration and filtration have

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the same density and water content as the diluted indicator (i.e. ultra-pure dialysate) 20. The water

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content and density of dilution are 0.991 kg/kg and 1.0 kg/L, respectively.

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Under these assumptions, we can write mass balance equations for the indicator fluid (water) and blood

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in each compartment in our model:

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Central compartment: 4

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Indicator mass balance: dmw,1 (t )

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dt

q1 (t ) q (t ) mw,1 (t )  2 mw, 2 (t )  Wind ind qind (t )  Wufr ufr qufr (t ) , V1 (t ) V2 (t )

(1)

Blood mass balance: d  1V1    1 q1 (t )   2 q2 (t )  ind qind (t )  ufr qufr (t ) dt

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Peripheral compartment:

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Indicator mass balance: dmw, 2 (t )

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

dt



(2)

q1 (t ) q (t ) mw,1 (t )  2 mw, 2 (t )  W f  f q f (t ) . V1 (t ) V2 (t )

(3)

Blood mass balance: d  2V2   1 q1 (t )  2 q2 (t )   f q f (t ) dt

106 where

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i  1/ 27404  4.933 104 T  0.26378Wi  1.812 104 T Wi

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the water content. Blood and water mass define the state for each compartment. Subscript i  1, 2 denotes

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central compartment and peripheral compartment, respectively, and subscripts ufr and ind denote

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ultrafiltration and indicator dilution, respectively. For example, Wind denotes the water content of the

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indicator injection and Wufr is the water content of fluid removed by UFR. In the above equations,

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Wind ind qind (t ) and W f  f q f (t ) equal the rate of water mass added by indicator dilution and refilling/filtration,

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respectively, and Wufr ufr qufr (t ) is the rate of water mass removed by ultrafiltration.

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mw,1 (t ) V1 (t )

Vi

denotes fluid volume,

i is fluid density given by

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mw,i

denotes water mass,

(4)

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, T  360 C is temperature, and Wi  mw,i iVi is

mw, 2 (t ) V2 (t )

q2 (t ) and

q1 (t ) denote the convective inflow between compartments. Other terms can be interpreted in a similar

manner.

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The output is the measured water content defined as water mass over blood mass. In this study, we

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measure water content of blood Wm in the arterial line of extracorporeal circulation. Subscript m refers to 5

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the measurement. In AV patients, arterial blood from the fistula/graft enters the extracorporeal circulation

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with high flow rate before passing the arterial measuring site. We therefore assumed that Wm measures the

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central compartment’s water content. For CV patients, venous blood from the superior vena cava, a mix of

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blood from both compartments, enters the extracorporeal circulation before passing the measuring site.

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Therefore, Wm comprises a mix of water contents from each compartment.

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Parameter Estimation, Observability, and Identifiability

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The feasibility of obtaining reasonable estimates depends on several factors including model

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structure and model complexity relative to what is measured. A dynamic system is said to be observable if

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the initial states can be determined from system’s measured outputs

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condition for parameter identification, but is not a sufficient condition for identifiability 23-25. Our analysis

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based on linearization (see Appendix) shows that the model described by Eqs. (1)-(4) is unobservable when

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the output is an equal mix of water content of both compartments.

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. Observability is a necessary

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In the Appendix, we show that parameters of our two-compartment model for CV patients are not

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identifiable because the measurement Wm is an unknown function of W1  mw,1 1V1 and W2  mw,2 2V2 . To

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overcome this limitation, we assume that the states of central compartment and peripheral compartment are

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equal to each other (i.e. mW ,1  mW ,2 and 1V1  2V2 ) . This assumption transforms the unobservable two-

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compartment model into an observable, single-compartment model.

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A list of model parameters to be estimated is given in

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Table 2. The parameter estimation is conducted using the nonlinear least squares with the “trust-

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region-reflective” algorithm 26 in MATLAB, in which the parameters are identified to minimize the root-

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mean-square error (RMSE) between the water content measurements Wm and the water content estimates

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Westimates obtained by our algorithm

6

N

RMSE 

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 W k 1

estimates

 Wm 

2

.

N

Parameter estimation is conducted 5 minutes prior to and 10 minutes after indicator injection time, by taking 15 minute samples of Wm .

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Figure 2 summarizes estimation results for an AV patient. The left panel shows the variation of

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BWC in each compartment throughout the indicator dilution protocol and the right panel shows the

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variation in flow rates between compartments within the dilution protocol 6. The spike in the measured

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variation of BWC occurring at t  74 min is due to automatic transmembrane pressure tests (TMP) from

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the Fresenius on-line HD/HDF machines. These spikes are repeated every 15-minute and each spike affects

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measurements for 3 minutes. Since dilution starts immediately after these TMP tests, a 5-min period prior

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to dilution is required to ensure that spike-free data were collected. Since the two-compartment model

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equilibrated after an injection in about 10 minutes, we found a 15 min sampling period was a good

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compromise between practicality and the model’s approximation of the actual nonlinear and time-varying

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phenomena.

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Finally, the estimate of ABV at any time of interest V(t) is derived from the sum of the two

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estimated compartments (central and peripheral V  t0   V1  t0   V2  t0  ) at time of start of dilution t0 and

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measured relative blood volume ( RBVt , vol/vol) at injection time and at time of interest 6: RBV  t0 

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RBV  t 



V  t0  V t 

.

Note that at the start of HD treatment RBV (0)  1 .

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In the next section, we discuss and compare ABV estimates from our model with ABVs estimates

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obtained using the classic back-extrapolation algorithm. In obtaining ABV estimates using BEXP we

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followed 6. We found this estimation is very sensitive to the period of time used for back-extrapolation. For

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consistency with the results in [6], in all cases we used the time period of 4-10 minutes after the injection

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to estimate ABV.

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Statistical Analysis

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We assessed the differences between the algorithms with an approach motivated by Bland and

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Altman???, namely, using a two-sided statistical tolerance interval (TI) (confidence level 95%)

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27

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analysis of variance (ANOVA) 28 is used to compute and compare the intratreatment variability of estimates

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in the two algorithms.The ANOVA provides a more sophisticated comparison between the variability of

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the BEXP and VVKM algorithms. Patients were chosen as the main factor while treatments (within

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patients) were taken as the nested factor. Normally distributed results are reported using mean (SD),

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otherwise, median [first quartile-third quartile]. Shapiro–Wilk test is used to test normality.

for the population of differenceshaving a normal distribution with unknown variability. Nested one-way

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3. Results

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A total of 85 bolus dilution tests (60 to 210 mL) of ultrapure dialysate were performed over 21 HD

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treatments in 6 patients using multiple indicator dilutions within each treatment. The descriptive statistics

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of the estimation results are given in Table 3. Figure 3 shows measured water content and the estimation

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using the VVKM algorithm for AV patient AF300 and CV patient ST011. Good agreement was observed

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between measured and estimated BWC with RMSNE less than 0.02 kg/kg (2%) and 0.03 kg/kg (3%) for

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AV and CV patients respectively. The largest RMSNE values were at the fourth indicator dilution of KH110

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(RMSNE=0.03 kg/kg) for AV patients and at the second indicator dilution of FR170 (RMSNE=0.05 kg/kg)

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for CV patients (Figure 4).

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A normal probability plot (not shown) and a Shapiro-Wilk test for AV patients indicated that the

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differences between the ABV estimates of the BEXP and VVKM algorithms were normally distributed

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with mean of 0.02L, standard deviation (SD) of 0.52L, and with a 95% tolerance interval from −1.27L to

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1.32L. For CV patients, the differences are also normally distributed with mean of − 0.09L, SD of 0.42L,

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and with 95%TI factor from −1.10L to 0.91L. Thus, the systematic difference between the two algorithms

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is negligible. 8

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Since a patient could have different blood volumes on different treatment days, it is appropriate to

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compare the results of the two algorithms at each treatment day (intratreatment variability). Figure 5

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provides such a comparison. Within each treatment, three to five indicator dilutions were administered. The

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ABV estimates at the start of HD treatment obtained from dilutions within the same treatment are

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summarized as mean+/-SD. Results show that our algorithm has much better reproducibility by virtue of

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lower SDs in all 11 dilution tests in CV patients, and in 8 out of 9 instances in AV patients.

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The results of nested one-way analysis of variance are presented in Table 4. Intratreatment SDs

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for the BEXP estimates were 0.51L and 0.47L for CV and AV patients respectively; and the corresponding

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SDs for VVKM estimates were 0.24L and 0.27L for CV and AV patients, indicating significant reductions

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in variability by 53% and 42% respectively. The AV and CV intratreatment coefficients of variation were

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0.080 and 0.128 for BEXP, and 0.046 and 0.062 for VVKM.

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4.

Discussion

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This study shows that ABV during an HD treatment can be successfully estimated using an

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indicator dilution protocol and a new physiologically-motivated compartmental model. The dilution

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protocol delivers boluses of ultra-pure dialysate using the bolus function of a modern HDF dialysis

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machine. When compared to other solutions such as normal saline, this ultra-pure dialysate has the

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advantage of being readily available at the proper temperature and osmotic concentration. Though not

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shown here, the VVKM algorithm can be extended for hemoglobin and hematocrit measurements available

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within almost all HD machines.

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In this study, our central compartment was assumed to model the high-blood flow in organs

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including the heart, central veins and arteries, lungs, brain, and GI tract 10. This compartment is where the

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dilution indicator mixes with blood at a high rate. The estimated inter-compartmental flow rate is lower

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than typical cardiac output (1.64 +/- 0.39 L/min) that suggests that the positioning of the central

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compartment in our model differs from other models where the systemic blood circulation alone is

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considered as central compartment. It appears that without additional measurements beyond blood water 9

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content at a single site, it not be possible to evaluate the accuracy of estimated individual parameters except

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for ABV. The potential of a faster sampling frequency to allow estimation of these parameters is a topic

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for future research.

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The fidelity of our parameter estimation scheme was studies by analyzing the sensitivity of the

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model’s output to changes in the model’s parameter. To this end, we used forward sensitivity analysis (FSA)

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to compare sensitivities at each sampled point in time 29, 30. It is convenient to multiply forward sensitivity

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function by the model parameter to define the un-normalized forward sensitivity function in cases where

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magnitude order of parameters differs considerably. The un-normalized sensitivity is equal to 1 if 1%

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fractional change in the model parameter changes the output by 1%. The un-normalized forward sensitivity

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function of output Wm with respect to a model parameter is given by29-31 Si  pi

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 Wm  t , p  , pi

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where p is the vector of model parameters pi

Roughly speaking, it is said that convergence of an

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identification algorithm increases with larger sensitivity value

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sensitivity analysis for AV patient AF300. The plot is divided into three regions: region I captures dynamics

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prior to dilution, region II captures dynamics immediately after dilution which is dominated by dilution

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mixing between compartments, and region III captures post-mixing dynamics referred to as the elimination

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phase, starting 4 minutes after dilution. Figure 6 shows that the model output Wm has much lower sensitivity

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to model parameters in region I compared to the other regions. The output is dominated by refilling/filtration

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prior to dilution in region I, by time delay and compartmental volumes during mixing in region II, and by

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central compartment volume and refilling/filtration during the elimination phase (region III). The output is

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most sensitive to the central compartment’s volume in region II with sensitivity dropping significantly in

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region III. We observe that the shape of curves becomes similar to each other moving from region II to

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region III. Since the BEXP algorithm is limited to modelling only the elimination phase (region III), it is

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less likely to uniquely identify parameters.

10

30

. Figure 6 shows an example of this

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Figure 6 shows that our model has a higher sensitivity for estimates of central compartment volume

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V1  t0  compared to peripheral compartment volume V2  t0  , and lower sensitivities for other model

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parameters such as blood exchange between compartments. Sensitivity analysis would suggest higher

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variabilities in the estimates of these parameters which is consistent with actual estimation results. These

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low sensitivities are consistent with the results in

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from expected value. Separate sensitivity analysis using a modified model which has ABV as a state (work

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not shown here for brevity) showed good ABV  t0  sensitivity. Indeed, for example, for our AV patients,

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intratreatment SDs of estimates for V1  t0  and V2  t0  are 0.23L and 0.32L, while the SD of

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ABV  t0   V1  t0   V2 t0  is only 0.27L.

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where some of the estimated parameters are different

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It is worth nothing that sensitivity function is discrete in time and can indirectly indicate how to

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select sample points in time to enhance information extraction from the measurement since more

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information can be extracted from a sample point with high sensitivity. Note that absolute blood volume

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estimates in this study, similar to 6, include the added extracorporeal circulation volume estimated to be

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around 300 ±10 mL 6. This volume needs to subtracted from our estimates to obtain actual absolute blood

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volume.

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Study limitations include small patients sample size, lack of validation against gold-standard

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methods, and the assumption that the so-call F-cell ratio 32 is fixed. The change in F-cell ratio during HD

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may not be as large as previously assumed 32.

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In conclusion, the dilution protocol and the new VVKM-based estimation algorithm offer a

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noninvasive, inexpensive, safe, and practical approach for ABV estimation in routine HD settings. The

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estimation of ABV estimates is significantly more precise when compared with estimates derived from the

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classical BEXP algorithm. This ABV information can be the basis for hypothesis generating studies aimed

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at achieving better fluid balance management resulting in improved HD outcomes.

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Appendix 11

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Analysis of observability in nonlinear systems requires detailed theoretical considerations which 33

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are beyond the scope of this work

. However, we can discuss this important issue which affects

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identifiability by using a linearized version of the model. A general linear (time-invariant) two-

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compartment model can be described in state-space form as x  Ax  Bu

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y  Cx

,

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where x , u , y are respectively the state, input and output vectors. A, B and C denote the state matrix,

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input matrix, and output matrix, respectively. The rows of the state matrix of a two-compartment model are

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always symmetric with a negative sign as:  a11 A  a11

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a12  . a12 

The linear two compartment model is said to be observable if and only if

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C  rank    2 CA .  c11 c12  C  c21 c22 

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Application to the linearized model with y1 being the output, result in the following condition for

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observability:

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c11 c12   det    0  c11  c12  a11  c11  c12  a12  c11  c12 

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When c11  c12 , we have a situation where the output is an equal mixture of the states of both compartments

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and the system is not observable. In such cases, since the states are unobservable, the model parameters

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become unidentifiable

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occurs in nonlinear model.

23-25

. Our numerical simulations suggested that a similar loss of observability also

12

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Cobelli C, Romanin-Jacur G: Controllability, observability and structural identifiability of multi input and multi output biological compartmental systems. IEEE Transactions on Biomedical Engineering 2: 93-100, 1976. Anguelova M (ed): Observability and identifiability of nonlinear systems with applications in biology. Chalmers University of Technology, 2007. Karlsson J, Anguelova M, Jirstrand M: An efficient method for structural identifiability analysis of large dynamic systems. IFAC Proceedings Volumes 45: 941-946, 2012. MathWorks: Solve nonlinear least-squares (nonlinear data-fitting) problems. 2016: 2016 (abstr) ISO-16269-6: Statistical interpretation of data -- Part 6: Determination of statistical tolerance intervals. 2014 (abstr) Tabachnick BG, Fidell LS: Using multivariate statistics. Petzold L, Li S, Cao Y, Serban R: Sensitivity analysis of differential-algebraic equations and partial differential equations. Computers & chemical engineering 30: 1553-1559, 2006. ZivariPiran H: Efficient simulation, accurate sensitivity analysis and reliable parameter estimation for delay differential equations. pp., 2009. Eslami M (ed): Theory of sensitivity in dynamic systems: an introduction. Springer Science & Business Media, 2013. Schneditz D, Ribitsch W, Schilcher G, Uhlmann M, Chait Y, Stadlbauer V: Concordance of absolute and relative plasma volume changes and stability of Fcells in routine hemodialysis. Hemodialysis International 20: 120-128, 2016. Hermann R, Krener AJ: Nonlinear controllability and observability. IEEE Transactions on automatic control 22: 728-740, 1977.

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List of Tables:

357

Table 1 Patient and treatment data

358

Table 2 Parameters to be estimated

359

Table 3 Descriptive Statistics of estimates

360

Table 4 Nested one-way ANOVA of ABV (L)

361

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362

List of Figures:

363 364

Figure 1: Schematic diagram of the variable volume two-compartment, intravascular blood-water model.

365

Figure 2: Estimation details for patient AF300 at the first injection (RMSE=0.02 kg/kg). Left panel shows

366

estimated BWC of central compartment (dashed line), peripheral compartment (dashed-dot line)

367

and measurement (solid line). The variation in BWC due to the indicator dilution shows up at

368

measurement site with a time delay, tdelay . Right panel shows administered indicator dilution

369

profile (solid line), and inter-compartment flows q1 (t ) ( dashed line) and q2 (t ) (dashed-dot line).

370 371 372 373 374 375

Figure 3: Overview of measured water content (dashed line with circle symbol) and model estimation (solid line) during HDF session for AV patient AF300 and CV patient ST011 Figure 4: Overview of measured water content (dashed line with circle symbol) and model estimation (solid line) for AV patient KH110 and CV patient FR170 Figure 5: Intratreatment variability of ABV estimates (mean+/-SD). Physiologically motivated VVKM model (red line), classic mono back-extrapolation method (blue dashed line)

376

Figure 6: Forward sensitivity analysis for AV patient AF300 at first dilution experiment; central

377

compartment volume V1  t0  (solid line), peripheral compartment volume at V2  t0  (dashed line),

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Blood exchange between compartments q01 (dashed-dotted line), refilling/filtration q f  t0  and

379

qr / f V1

t  t0

(triangle, dotted line), time delay tdelay (rectangle, dashed line)

380

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