tests (3-5) within each of several (2-6) HD treatments. ... piece of information, cannot inferred from RBV â indeed, patients with differing ABVs can exhibit similar.
1
A Variable-Volume Kinetic Model to Estimate Absolute Blood
2
Volume in Dialysis Patients Using Dialysate Dilution Protocol
3 4
Abstract:
5
Long- and short-term adverse outcomes in hemodialysis (HD) have been associated with
6
intradialytic hypotension, a common HD complication and significant cause of morbidity. It has been
7
suggested that knowledge of absolute blood volume (ABV) could be used to significantly improve
8
treatment outcomes. Different dilution-based protocols have been proposed for estimating ABV, all relying
9
on the classic mono-exponential back-extrapolation algorithm (BEXP). In this paper, we introduce a
10
dialysate dilution protocol and an estimation algorithm based on a variable volume, two-compartment,
11
intravascular blood water content kinetic model (VVKM). We compare these two algorithms in a study
12
including 3 arteriovenous (AV) and 3 central-venous (CV) access patients, and multiple bolus injection
13
tests (3-5) within each of several (2-6) HD treatments. Investigation of the distribution of differences
14
between the two methods showed a negligible systematic difference between the mean values of ABVs
15
estimated from the BEXP and VVKM algorithms, however, the VVKM estimates were 53% and 42% more
16
precise for the CV and AV patients, respectively.
17 18
1. Introduction:
19
Volume management plays an important role in renal replacement therapies. Removing too much
20
fluid by ultrafiltration triggers intradialytic hypotension, a significant cause of long- and short-term adverse
21
outcomes, while removing too little fluid causes edema, left ventricular hypertrophy and heart failure
22
Knowing a patient’s ABV at the start of ultrafiltration would better allow clinicians to return patients to
23
their dry weight and significantly improve such outcomes
1
1-3
1-3
.
. Isotope dilution, the gold standard for
24
measuring ABV, is invasive, expensive, time consuming and impractical for routine clinical application 4.
25
A practical technique for estimating ABV is needed.
26
Current HD machine technology provides sensors such as the Crit-LineTM and the blood volume
27
monitor (BVM) that measure a patient’s hematocrit (Crit-LineTM) and blood water content (BWC). From
28
these measurements and assumptions for a single compartment, one can compute changes in a patient's
29
intravascular blood volume - referred to as relative blood volume (RBV). However, ABV, the crucial
30
piece of information, cannot inferred from RBV – indeed, patients with differing ABVs can exhibit similar
31
RBVs 5.
32
Recently, a technique that uses blood water content measurements to make estimates of ABV was
33
introduced in 6. In this technique, a bolus injection of ultra-pure dialysate was administered and the back-
34
extrapolation algorithm (BEXP) algorithm used to estimate the initial blood water concentration at the time
35
of injection. This estimate together with the size of the bolus injection was then enough to estimate ABV
36
at the time of injection.
37
pharmacokinetic approach 7 which assumes that the indicator dynamics can be sufficiently represented by
38
a single-compartmental model with constant coefficients.
39
distribution of an indicator is not uniform within the bloodstream due to blood flow
40
have considered models consisting of more than one compartment. to better reflect such distribution 10-14.
41
Multi-compartmental modelling has been studied, including fixed-volume
42
parallel and series compartment configurations 12. Applications of such models include the distribution of
43
indicators in solute kinetics
44
distribution in blood
45
models such as
46
impossible, estimation problem.
10-16
16
Fitting an exponential function to a measured indicator is a standard
10-12
, hemodialysis
, and urea kinetics
15
However, studies have shown that the
10-13
, and researchers
variable-volume
, β2-microglobulin kinetics
13
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
47
In this paper, we present a new, physiologically motivated, variable-volume, two-compartment
48
model as the basis for estimating ABV corresponding to the technique in 6. Absolute blood volume
2
49
estimates derived from this model are compared with estimates from the classic mono-exponential back-
50
extrapolation algorithm.
51 52
2. Materials & Methods
53
A Fresenius 4008H-HDF machine equipped with a BVM and dedicated data acquisition software 17
54
(Fresenius Medical Care, Bad Homburg, Germany)
provided hemodiafiltration (HDF) therapy and
55
measurement of hematocrit and blood water content, the latter of which was used to calculate RBV changes.
56
Dialysate was delivered at a flow of either 500 or 800 ml/min, and at 36 degrees C. Blood flows, dialysate
57
[Na+] and HDF infusion volumes and the pre or post-dilution configuration were set as prescribed in 6.
58
Indicator dilutions were administered using the bolus function in the HDF machine. This function
59
delivers ultrapure dialysate in multiples of 30 ml at a constant infusion rate of approximately 150 ml/ min
60
during the HDF session. This bolus volume was delivered with an accuracy of better than ±1.5% 6.
61 62
Patients
63
The study included 3 arterio-venous (AV9 and 3 central-venous (CV) access patients, and multiple
64
(3-5) indicator dilution experiments within each of several (2-6) HD treatments. Patients consented to
65
participate as approved by the Ethics Committee of the Medical University of Graz, Austria. Table 1
66
summarizes the patient and treatment data.
67 68
Modeling
69
Following
10-16
, we modeled the intravascular circulatory system by two compartments loosely
70
termed central and peripheral, reflecting vessels with high and low blood flow rates, respectively (Figure
71
1). The water mass and fluid volume constituted the state for each compartment.
72
The following assumptions are used:
73
Ultrafiltration removes fluid from the central compartment at the prescribed rate qufr .
3
74
The indicator fluid is injected into the central compartment at a rate of qind . Instantaneous mixing
75
is assumed within each compartment. Following the injection, the indicator fluid is assumed to
76
arrive at the measurement site with a fixed time delay after circulating throughout the body.
77
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
79
the blood flow from the central to peripheral compartments, q2 is the blood flow from peripheral
80
to central compartments and V1 and V2 are the fluid volumes for the central and peripheral
81
compartments respectively.
82
V1 (t )
and
q2 t
78
V2 (t )
are constants. Here q1 is
We assume that the fluid exchange between the interstitial and intravascular spaces, referred to as
83
refilling/filtration, occurs between the interstitial and peripheral compartments. For simplicity,
84
we take this nonlinear exchange q f as an affine function of the central volume as in q f q f 0 V1 .
85 86
Here we assume that q f depends only on the central volume since the interstitial volume is much
87
larger than the volume of fluid removed by ultrafiltration within the simulation time period. The
88
coefficient models the sensitivity of q f to the lymphatic flow rate and the nonlinear Starling
89
mechanism describing microvascular refilling/filtration flow into the peripheral compartment 18,
90
19
91
.
We assume that the fluid removed from intravascular space by ultrafiltration and filtration have
92
the same density and water content as the diluted indicator (i.e. ultra-pure dialysate) 20. The water
93
content and density of dilution are 0.991 kg/kg and 1.0 kg/L, respectively.
94 95
Under these assumptions, we can write mass balance equations for the indicator fluid (water) and blood
96
in each compartment in our model:
97
Central compartment: 4
98
Indicator mass balance: dmw,1 (t )
99 100
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
101 102
Peripheral compartment:
103
Indicator mass balance: dmw, 2 (t )
104 105
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
108
i 1/ 27404 4.933 104 T 0.26378Wi 1.812 104 T Wi
109
the water content. Blood and water mass define the state for each compartment. Subscript i 1, 2 denotes
110
central compartment and peripheral compartment, respectively, and subscripts ufr and ind denote
111
ultrafiltration and indicator dilution, respectively. For example, Wind denotes the water content of the
112
indicator injection and Wufr is the water content of fluid removed by UFR. In the above equations,
113
Wind ind qind (t ) and W f f q f (t ) equal the rate of water mass added by indicator dilution and refilling/filtration,
114
respectively, and Wufr ufr qufr (t ) is the rate of water mass removed by ultrafiltration.
115 116
mw,1 (t ) V1 (t )
Vi
denotes fluid volume,
i is fluid density given by
107
mw,i
denotes water mass,
(4)
21
, 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.
117
The output is the measured water content defined as water mass over blood mass. In this study, we
118
measure water content of blood Wm in the arterial line of extracorporeal circulation. Subscript m refers to 5
119
the measurement. In AV patients, arterial blood from the fistula/graft enters the extracorporeal circulation
120
with high flow rate before passing the arterial measuring site. We therefore assumed that Wm measures the
121
central compartment’s water content. For CV patients, venous blood from the superior vena cava, a mix of
122
blood from both compartments, enters the extracorporeal circulation before passing the measuring site.
123
Therefore, Wm comprises a mix of water contents from each compartment.
124 125
Parameter Estimation, Observability, and Identifiability
126
The feasibility of obtaining reasonable estimates depends on several factors including model
127
structure and model complexity relative to what is measured. A dynamic system is said to be observable if
128
the initial states can be determined from system’s measured outputs
129
condition for parameter identification, but is not a sufficient condition for identifiability 23-25. Our analysis
130
based on linearization (see Appendix) shows that the model described by Eqs. (1)-(4) is unobservable when
131
the output is an equal mix of water content of both compartments.
22
. Observability is a necessary
132
In the Appendix, we show that parameters of our two-compartment model for CV patients are not
133
identifiable because the measurement Wm is an unknown function of W1 mw,1 1V1 and W2 mw,2 2V2 . To
134
overcome this limitation, we assume that the states of central compartment and peripheral compartment are
135
equal to each other (i.e. mW ,1 mW ,2 and 1V1 2V2 ) . This assumption transforms the unobservable two-
136
compartment model into an observable, single-compartment model.
137
A list of model parameters to be estimated is given in
138
Table 2. The parameter estimation is conducted using the nonlinear least squares with the “trust-
139
region-reflective” algorithm 26 in MATLAB, in which the parameters are identified to minimize the root-
140
mean-square error (RMSE) between the water content measurements Wm and the water content estimates
141
Westimates obtained by our algorithm
6
N
RMSE
142 143 144
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 .
145
Figure 2 summarizes estimation results for an AV patient. The left panel shows the variation of
146
BWC in each compartment throughout the indicator dilution protocol and the right panel shows the
147
variation in flow rates between compartments within the dilution protocol 6. The spike in the measured
148
variation of BWC occurring at t 74 min is due to automatic transmembrane pressure tests (TMP) from
149
the Fresenius on-line HD/HDF machines. These spikes are repeated every 15-minute and each spike affects
150
measurements for 3 minutes. Since dilution starts immediately after these TMP tests, a 5-min period prior
151
to dilution is required to ensure that spike-free data were collected. Since the two-compartment model
152
equilibrated after an injection in about 10 minutes, we found a 15 min sampling period was a good
153
compromise between practicality and the model’s approximation of the actual nonlinear and time-varying
154
phenomena.
155
Finally, the estimate of ABV at any time of interest V(t) is derived from the sum of the two
156
estimated compartments (central and peripheral V t0 V1 t0 V2 t0 ) at time of start of dilution t0 and
157
measured relative blood volume ( RBVt , vol/vol) at injection time and at time of interest 6: RBV t0
158 159
RBV t
V t0 V t
.
Note that at the start of HD treatment RBV (0) 1 .
160
In the next section, we discuss and compare ABV estimates from our model with ABVs estimates
161
obtained using the classic back-extrapolation algorithm. In obtaining ABV estimates using BEXP we
162
followed 6. We found this estimation is very sensitive to the period of time used for back-extrapolation. For
163
consistency with the results in [6], in all cases we used the time period of 4-10 minutes after the injection
164
to estimate ABV.
7
165 166
Statistical Analysis
167
We assessed the differences between the algorithms with an approach motivated by Bland and
168
Altman???, namely, using a two-sided statistical tolerance interval (TI) (confidence level 95%)
169
27
170
analysis of variance (ANOVA) 28 is used to compute and compare the intratreatment variability of estimates
171
in the two algorithms.The ANOVA provides a more sophisticated comparison between the variability of
172
the BEXP and VVKM algorithms. Patients were chosen as the main factor while treatments (within
173
patients) were taken as the nested factor. Normally distributed results are reported using mean (SD),
174
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
175 176
3. Results
177
A total of 85 bolus dilution tests (60 to 210 mL) of ultrapure dialysate were performed over 21 HD
178
treatments in 6 patients using multiple indicator dilutions within each treatment. The descriptive statistics
179
of the estimation results are given in Table 3. Figure 3 shows measured water content and the estimation
180
using the VVKM algorithm for AV patient AF300 and CV patient ST011. Good agreement was observed
181
between measured and estimated BWC with RMSNE less than 0.02 kg/kg (2%) and 0.03 kg/kg (3%) for
182
AV and CV patients respectively. The largest RMSNE values were at the fourth indicator dilution of KH110
183
(RMSNE=0.03 kg/kg) for AV patients and at the second indicator dilution of FR170 (RMSNE=0.05 kg/kg)
184
for CV patients (Figure 4).
185
A normal probability plot (not shown) and a Shapiro-Wilk test for AV patients indicated that the
186
differences between the ABV estimates of the BEXP and VVKM algorithms were normally distributed
187
with mean of 0.02L, standard deviation (SD) of 0.52L, and with a 95% tolerance interval from −1.27L to
188
1.32L. For CV patients, the differences are also normally distributed with mean of − 0.09L, SD of 0.42L,
189
and with 95%TI factor from −1.10L to 0.91L. Thus, the systematic difference between the two algorithms
190
is negligible. 8
191
Since a patient could have different blood volumes on different treatment days, it is appropriate to
192
compare the results of the two algorithms at each treatment day (intratreatment variability). Figure 5
193
provides such a comparison. Within each treatment, three to five indicator dilutions were administered. The
194
ABV estimates at the start of HD treatment obtained from dilutions within the same treatment are
195
summarized as mean+/-SD. Results show that our algorithm has much better reproducibility by virtue of
196
lower SDs in all 11 dilution tests in CV patients, and in 8 out of 9 instances in AV patients.
197
The results of nested one-way analysis of variance are presented in Table 4. Intratreatment SDs
198
for the BEXP estimates were 0.51L and 0.47L for CV and AV patients respectively; and the corresponding
199
SDs for VVKM estimates were 0.24L and 0.27L for CV and AV patients, indicating significant reductions
200
in variability by 53% and 42% respectively. The AV and CV intratreatment coefficients of variation were
201
0.080 and 0.128 for BEXP, and 0.046 and 0.062 for VVKM.
202 203
4.
Discussion
204
This study shows that ABV during an HD treatment can be successfully estimated using an
205
indicator dilution protocol and a new physiologically-motivated compartmental model. The dilution
206
protocol delivers boluses of ultra-pure dialysate using the bolus function of a modern HDF dialysis
207
machine. When compared to other solutions such as normal saline, this ultra-pure dialysate has the
208
advantage of being readily available at the proper temperature and osmotic concentration. Though not
209
shown here, the VVKM algorithm can be extended for hemoglobin and hematocrit measurements available
210
within almost all HD machines.
211
In this study, our central compartment was assumed to model the high-blood flow in organs
212
including the heart, central veins and arteries, lungs, brain, and GI tract 10. This compartment is where the
213
dilution indicator mixes with blood at a high rate. The estimated inter-compartmental flow rate is lower
214
than typical cardiac output (1.64 +/- 0.39 L/min) that suggests that the positioning of the central
215
compartment in our model differs from other models where the systemic blood circulation alone is
216
considered as central compartment. It appears that without additional measurements beyond blood water 9
217
content at a single site, it not be possible to evaluate the accuracy of estimated individual parameters except
218
for ABV. The potential of a faster sampling frequency to allow estimation of these parameters is a topic
219
for future research.
220
The fidelity of our parameter estimation scheme was studies by analyzing the sensitivity of the
221
model’s output to changes in the model’s parameter. To this end, we used forward sensitivity analysis (FSA)
222
to compare sensitivities at each sampled point in time 29, 30. It is convenient to multiply forward sensitivity
223
function by the model parameter to define the un-normalized forward sensitivity function in cases where
224
magnitude order of parameters differs considerably. The un-normalized sensitivity is equal to 1 if 1%
225
fractional change in the model parameter changes the output by 1%. The un-normalized forward sensitivity
226
function of output Wm with respect to a model parameter is given by29-31 Si pi
227
Wm t , p , pi
228
where p is the vector of model parameters pi
Roughly speaking, it is said that convergence of an
229
identification algorithm increases with larger sensitivity value
230
sensitivity analysis for AV patient AF300. The plot is divided into three regions: region I captures dynamics
231
prior to dilution, region II captures dynamics immediately after dilution which is dominated by dilution
232
mixing between compartments, and region III captures post-mixing dynamics referred to as the elimination
233
phase, starting 4 minutes after dilution. Figure 6 shows that the model output Wm has much lower sensitivity
234
to model parameters in region I compared to the other regions. The output is dominated by refilling/filtration
235
prior to dilution in region I, by time delay and compartmental volumes during mixing in region II, and by
236
central compartment volume and refilling/filtration during the elimination phase (region III). The output is
237
most sensitive to the central compartment’s volume in region II with sensitivity dropping significantly in
238
region III. We observe that the shape of curves becomes similar to each other moving from region II to
239
region III. Since the BEXP algorithm is limited to modelling only the elimination phase (region III), it is
240
less likely to uniquely identify parameters.
10
30
. Figure 6 shows an example of this
241
Figure 6 shows that our model has a higher sensitivity for estimates of central compartment volume
242
V1 t0 compared to peripheral compartment volume V2 t0 , and lower sensitivities for other model
243
parameters such as blood exchange between compartments. Sensitivity analysis would suggest higher
244
variabilities in the estimates of these parameters which is consistent with actual estimation results. These
245
low sensitivities are consistent with the results in
246
from expected value. Separate sensitivity analysis using a modified model which has ABV as a state (work
247
not shown here for brevity) showed good ABV t0 sensitivity. Indeed, for example, for our AV patients,
248
intratreatment SDs of estimates for V1 t0 and V2 t0 are 0.23L and 0.32L, while the SD of
249
ABV t0 V1 t0 V2 t0 is only 0.27L.
13
where some of the estimated parameters are different
250
It is worth nothing that sensitivity function is discrete in time and can indirectly indicate how to
251
select sample points in time to enhance information extraction from the measurement since more
252
information can be extracted from a sample point with high sensitivity. Note that absolute blood volume
253
estimates in this study, similar to 6, include the added extracorporeal circulation volume estimated to be
254
around 300 ±10 mL 6. This volume needs to subtracted from our estimates to obtain actual absolute blood
255
volume.
256
Study limitations include small patients sample size, lack of validation against gold-standard
257
methods, and the assumption that the so-call F-cell ratio 32 is fixed. The change in F-cell ratio during HD
258
may not be as large as previously assumed 32.
259
In conclusion, the dilution protocol and the new VVKM-based estimation algorithm offer a
260
noninvasive, inexpensive, safe, and practical approach for ABV estimation in routine HD settings. The
261
estimation of ABV estimates is significantly more precise when compared with estimates derived from the
262
classical BEXP algorithm. This ABV information can be the basis for hypothesis generating studies aimed
263
at achieving better fluid balance management resulting in improved HD outcomes.
264 265
Appendix 11
266
Analysis of observability in nonlinear systems requires detailed theoretical considerations which 33
267
are beyond the scope of this work
. However, we can discuss this important issue which affects
268
identifiability by using a linearized version of the model. A general linear (time-invariant) two-
269
compartment model can be described in state-space form as x Ax Bu
270
y Cx
,
271
where x , u , y are respectively the state, input and output vectors. A, B and C denote the state matrix,
272
input matrix, and output matrix, respectively. The rows of the state matrix of a two-compartment model are
273
always symmetric with a negative sign as: a11 A a11
274
275
a12 . a12
The linear two compartment model is said to be observable if and only if
276
C rank 2 CA . c11 c12 C c21 c22
277
Application to the linearized model with y1 being the output, result in the following condition for
278
observability:
279
c11 c12 det 0 c11 c12 a11 c11 c12 a12 c11 c12
280
When c11 c12 , we have a situation where the output is an equal mixture of the states of both compartments
281
and the system is not observable. In such cases, since the states are unobservable, the model parameters
282
become unidentifiable
283
occurs in nonlinear model.
23-25
. Our numerical simulations suggested that a similar loss of observability also
12
284
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14
356
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
15
362
List of Figures:
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Figure 1: Schematic diagram of the variable volume two-compartment, intravascular blood-water model.
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Figure 2: Estimation details for patient AF300 at the first injection (RMSE=0.02 kg/kg). Left panel shows
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estimated BWC of central compartment (dashed line), peripheral compartment (dashed-dot line)
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and measurement (solid line). The variation in BWC due to the indicator dilution shows up at
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measurement site with a time delay, tdelay . Right panel shows administered indicator dilution
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profile (solid line), and inter-compartment flows q1 (t ) ( dashed line) and q2 (t ) (dashed-dot line).
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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)
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Figure 6: Forward sensitivity analysis for AV patient AF300 at first dilution experiment; central
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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
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qr / f V1
t t0
(triangle, dotted line), time delay tdelay (rectangle, dashed line)
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