Comparison of three renal hemodialysis models

A comparison of three renal hemodialysis models

Jose M. Gonz´lez
a
Msc supervisors: Joan Sanchez and Marta Net

Departament de F´
ısica Aplicada, Universitat Polit`cnica de Catalunya. 08034 Barcelona,
e
Spain.

September 25, 2012

JM Gonz´lez, Joan Sanchez and Marta Net
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A comparison of three renal HD models September 25, 2012 1 / 23

Overview

1 Introduction
Previous models

2 Model and Method

3 Numerical results
Important results
Minimization function

4 Conclusions

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Introduction

Qi Filter
Dialysis types and prescription: Dialysate

Dialysis types are hemodialysis Qb
o
Filter
(HD), hemofiltration (HF), Qd i
hemodiafiltration (HDF)
Qd
Prescription based on the o

frequency, the length of each Qb
i

treatment, the flow rates (Qb , Qd = Qd + Qi + Qw
o i
Qw , Qd ) and sometimes the Qb = Qb + Qw
i o

dialysis solutions is changed.

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One pool model
Fick’s Law:
d(Ce Ve )
= −Kd Ce , (1)
dt

Figure: Scheme of one pool model from M. Zi´lko and J. A. Pietrzyk and J.
o
Grabska-Chrzastowska. Accuracy of hemodialysis modeling

KT/V prescription method:
Ve
T = ln(C (0)/C (t)) (2)
Kd

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Two pools model

Figure: Scheme of two pools model from M. Zi´lko and J. A. Pietrzyk and J.
o
Grabska-Chrzastowska. Accuracy of hemodialysis modeling

dCe Kc (Ce − Ci ) Kd
=− − δ Ce , (3)
dt Ve Ve
dCi Kc (Ce − Ci )
= (4)
dt Vi
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Extended model

d
(Ve Ce ) = − Kc (Ce − Ci ) − δKd Ce + Ge , (5)
dt
d
(Vi Ci ) = Kc (Ce − Ci ) + Gi (6)
dt
New phenomena are included:
Generation term added.
Total water volume variation.

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Generation and volume considerations

Generation expression:

Ce (0) − Ce (tr )
G i = fG .
tf − tr
Total water volume:
V = 0.57DW
Fraction of external and internal compartment:

Ve = αV & Vi = (1 − α)V

Volume rate change during and after HD:

βd = (1 − Vtd /Vt0 )/td , & βp = βd td /(tf − td )

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Others considerations

Assuming that Ce (0) = Ci (0), then:

C e = Ce /Ce (0)

C i = Ci /Ce (0)
Clearances:
K d = Kd /Vt0
K c = Kc /Vt0

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Solved system

Then the resultant system is:

dC e dV t
αV t = −K c (C e − C i ) − δK d C e + G e − α Ce (7)
dt dt
dC i dV t
(1 − α)V t = K c (C e − C i ) + G i − (1 − α) Ci, (8)
dt dt

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Fitting

The parameters to ﬁt are:

K d , K c and α

Error function to minimize:
m
f (K d , K c , α) = (C j − Cje )2 (9)
j=1

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Data available from Hospital Cl´
ınic

The data available were the weight (pre and post-HD) and the
concentrations for:
urea (60Da),
creatinine (96Da),
phophate (113Da),
β2 -microglobulin (11818Da),
myoglobin (17200Da)
and prolactin (23000Da)
measured each hour during the dialysis and 3 measures more
(post-dialysis). Also the HD time of 4 or 8 h and the infusion rate (Qi ) of
50 or 100ml/min.

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Three models compared

M. 1: Two compartmental model with α free parameter.
M. 2: Two compartmental model with α free parameter, Bergtr¨m
o
and Wehle correction and no change in the Volume.
1
fB&W =
1 + ∆DW (t)/0.2DW

M. 3: Two compartmental model with α ﬁxed at 0.4.

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HD, post HD and generation

Creatinine
1.4
td=8h Qi=100
td=8h Qi=050
1.2 td=4h Qi=100
td=4h Qi=050
1
Ce (t) /Ce (0)

0.8

0.6

0.4

0.2

0
0 500 1000 1500 2000 2500 3000
t (min)

Exponential extraction, rebound phenomena and generation. No great
differences between models.
Overlapping of all HD types, filtration not affected by the value of Qi .
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Low molecular weight

BUN
1.4
td=8h Qi=100
td=8h Qi=050
1.2 td=4h Qi=100
td=4h Qi=050
1
Ce (t) /Ce (0)

0.8

0.6

0.4

0.2

0
0 500 1000 1500 2000 2500 3000
t (min)

No rebound phenomena in Urea, one compartment is good. Reﬂected in
Kc results.
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Phosphate

Phosphate
1.4
td=8h Qi=100
1.3 td=8h Qi=050
td=4h Qi=100
1.2 td=4h Qi=050
1.1
Ce (t) /Ce (0)

1
0.9
0.8
0.7
0.6
0.5
0.4
0 500 1000 1500 2000 2500 3000
t (min)

Low fraction parameter (α < 0.1), low molecular weight.
Importance of the α Strange behaviour, errors added due to dialyzate.
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The Qi importance

β2-microglobulin
1.4
td=8h Qi=100
td=8h Qi=050
1.2 td=4h Qi=100
td=4h Qi=050
1
Ce (t) /Ce (0)

0.8

0.6

0.4

0.2

0
0 500 1000 1500 2000 2500 3000
t (min)

Qi has an eﬀect in the eﬃciency which is shown in the result parameters.
¯ ¯
Kd ∼ 5E − 3 with Qi =100ml/min and Kd ∼ 3E − 3 with Qi =50ml/min
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Measurement problem

Prolactin
1.3
td=8h Qi=100
td=8h Qi=050
1.2 td=4h Qi=100
td=4h Qi=050
1.1

1
Ce (t) /Ce (0)

0.9

0.8

0.7

0.6

0.5
0 500 1000 1500 2000 2500 3000
t (min)

The presence of prolactin in blood is difficult to measure. Qi has an
important effect in the efficiency which is shown in the result parameters.
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M.1 Vs M.2 Vs M.3

Myoglobin

Model 1 Model 2 Model 3
1.3 1.3 1.3
td=8h Qi=100 td=8h Qi=100 td=8h Qi=100
1.2 td=8h Qi=050 1.2 td=8h Qi=050 1.2 td=8h Qi=050
td=4h Qi=100 td=4h Qi=100 td=4h Qi=100
1.1 td=4h Qi=050 1.1 td=4h Qi=050 1.1 td=4h Qi=050
1
1 1
Ce (t) /Ce (0)

Ce (t) /Ce (0)

Ce (t) /Ce (0)
0.9
0.9 0.9
0.8
0.8 0.8
0.7
0.7 0.7
0.6
0.6 0.5 0.6
0.5 0.4 0.5
0.4 0.3 0.4
0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000
t (min) t (min) t (min)

Including the ﬁtting of α allows a better adjusting in f (K d , K c , α).
Model 1 Model 2 Model 3
Qi =100, td =8 1.30e-3 1.10e-3 35.58e-3
Qi =100, td =4 2.43e-3 2.0e-3 44.81e-3
Qi =50, td =8 0.23e-3 0.16e-3 10.38e-3
Qi =50, td =4 0.59e-3 0.42e-3 14.75e-3

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Error function

The minimization of the error function gives f ∼ 1E − 3 except:
Phosphorus f ∼ 1E − 0
Prolactin f ∼ 5E − 3

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Rebound estimator

From equations 7, 8 we ﬁnd:

C i (t) − C e (t) = (C i (td ) − C e (td ))e −K c t/α(1−α) ,

treb = ln(q)α(1 − α)/K c .
q is the fraction in which the diﬀerence C i (t) − C e (t) is reduced.

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Conclusions

From the three models, the importance of the third parameter to
which the HD has access.
The impact of Qi on the efficiency of high molecular weight.
Relation of the rebound time established from parameters
(Intercompartmental clearance and fraction of external volume).
Possibility to quantify the effects of the HD time, infusion flow Qi
and the total mass of solutes extracted.

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Future work

Study with the tools developed data from new ﬁlters.
Study the distribution volume if the data is available in the future.
Further study is needed to use the results in the prescription of
individualized HD.

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Thank You
for your attention

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Comparison of three renal hemodialysis models

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