A DYNAMICAL MODEL FOR PULSATILE FLOW ESTIMATION IN

A LEFT VENTRICULAR ASSIST DEVICE

Abdul-Hakeem H. AlOmari, Andrey V. Savkin

School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, Australia

Dean M. Karantonis

§

, Einly Lim

†

and Nigel H. Lovell

†

§

Ventracor Limited;

†

Graduate School of Biomedical Engineering, The University of New South Wales, Sydney, Australia

Keywords:

Implantable rotary blood pumps (iRBPs), Noninvasive pulsatile ﬂow estimation, Left ventricular assist device

(LVAD).

Abstract:

In this paper, we propose a dynamical model for pulsatile ﬂow estimation of an iRBP. Noninvasive measure-

ments of the motor power (VI) and pump impeller rotational speed (ω) were acquired from the pump controller

and used together with blood hematocrit (HCT) values as inputs to the model. A circulatory mock loop was

operated with different aqueous glycerol solutions, mimicking different values of viscosities equivalent to the

range of 20 - 50% of human blood HCT, to generate pulsatile ﬂow data. Linear regression between estimated

pulsatile ﬂow (Q

est

) and measured ﬂow (Q

meas

) obtained from the mock loop resulted in a highly signiﬁcant

correlation (R

2

= 0.957) and mean absolute error of e = 0.364 L/min. Also, R

2

= 0.902 and e = 0.317 L/min

were obtained when our model was validated using six sets of ex vivo porcine data. Furthermore, in steady

state, the solution of the presented model can be described by a previously designed and veriﬁed static model.

1 INTRODUCTION

The problem of noninvasive estimation of ﬂow rate

has attracted the attention of many research groups

(for examples, see (Bertram, 2005)). It has been

shown that ﬂow rate can be accurately estimated un-

der steady-state conditions (Malagutti et al., 2007;

Ayre et al., 2003; Funakubo et al., 2002; Kitamura

et al., 2000; Tsukiya et al., 2001).

Pulsatile ﬂow estimation of iRBPs has been much

less frequently studied. This may be due to the large

number of factors that need to be taken into consid-

eration. Such factors may include the use of char-

acteristics curves of the pump where these are sensi-

tive to many physiological and mechanical parame-

ters like: changing blood viscosity, impeller inertia of

the pump.

In the present study, we used noninvasive feed-

back measurements of ω and VI to estimate the pul-

satile pump ﬂowusing a newdynamicalmodel. When

operated in steady state conditions, our model can

provide an accurate estimate of the ﬂow which agrees

with those obtained from the static model developed

in our laboratory under non-pulsatile conditions in

Malagutti et al. (2007). Another practically important

requirement for our dynamical model is its stability.

The proposed model will allow us to accurately study

and estimate the transient response and the dynamics

of the pulsatile ﬂow.

2 MATERIALS AND METHODS

2.1 In vitro Pulsatile Experiments

A VentrAssist

T M

(Ventracor Limited, Sydney, Aus-

tralia) iRBP was connected in a circulatory mock loop

in a pulsatile environment. The loop was composed

of venous and arterial reservoir tanks, a silicone bag

which represented the left ventricle chamber (Karan-

tonis et al., 2007). Ventricular contraction was simu-

lated by periodically compressing the mock ventricle

with pneumatic pistons at a ﬁxed rate. Arterial pres-

sure, central venous pressure, pump inlet and outlet

pressure were measured. Pulsatile ﬂow was measured

by an Ultrasonic ﬂow probe (Transonics Systems Inc.,

Ithaca, NY, USA). The instantaneous ω, motor volt-

402

H. AlOmari A., V. Savkin A., Karantonis D., Lovell N. and Lim E. (2009).

A DYNAMICAL MODEL FOR PULSATILE FLOW ESTIMATION IN A LEFT VENTRICULAR ASSIST DEVICE.

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing, pages 402-405

DOI: 10.5220/0001775604020405

Copyright

c

SciTePress

age and current were accessed as feedback signals

from the controller. Speed was varied from 1800 to

3000 rpm in a stepwise increment of 100 rpm last-

ing for 30 s. For each step, systemic resistance was

also changed to cover the range of desired ﬂows. The

sampling rate was 4 kHz for data recordings, but, in

further analysis, the data were down-sampled to 50

Hz.

2.2 Ex vivo Animals Experiments

The VentrAssist

T M

was acutely implanted in six

healthy pigs. In each pig, the inﬂow pump can-

nula was inserted into the apex of the left ventricle

while the outﬂow cannula was anastomosed to the as-

cending aorta (Karantonis et al., 2006). Indwelling

catheters (DwellCath, Tuta Labs, Lane Cove, NSW,

Australia) and pressure transducers (Datex-Ohmeda,

Homebush, NSW, Australia) were instrumented to the

pig’s native heart to measure left ventricle, left atrial,

aortic, and pump inlet pressures. Pump and aortic

ﬂows were measured by ultrasonic ﬂow probes from

the outlet cannula of the pump. Besides these sig-

nals, the noninvasive ω, motor current (I), and supply

voltage (V) were acquired from the pump controller.

The experimental data were sampled at 200 Hz. For

further information about data acquisition, see Karan-

tonis et al. (2006).

2.3 Dynamical Modeling

2.3.1 Static Flow Model

A noninvasive, steady-state average ﬂow (Q

ss

) esti-

mator was designed in a non-pulsatile environment

for the iRBP. The estimator was based on VI, and ω.

The static equation for the ﬂow estimator is based on

the work of Malagutti et al. (2007) and Ayre et al.

(2003) and is of the following form:

Q

ss

= a + bVI + cVI

2

+ dVI

3

+ eω+ pω

2

, (1)

where a, e, and p, are constants and the power coefﬁ-

cients b, c, and d were found to have a linear relation-

ship with the HCT (Malagutti et al., 2007).

2.3.2 Pulsatile Flow

In this section, we describe a dynamical model for

the iRBP. The main requirement for the dynamical

model is that any steady-state solution of the dynami-

cal model be a solution of the static equation (1). Fur-

thermore, we want any steady-state solution of the dy-

namical model to be stable. We introduce a variable

f(t) as follows:

f(t) = g(VI(t), ω(t)), (2)

where

g(VI(t), ω(t)) = a+ bVI(t) + cVI

2

(t) + dVI

3

(t)

+eω(t) + pω

2

(t). (3)

Here t = kh, h > 0 is the sampling interval, k =

0, 1, 2, 3, . . .. In other words, the variable f(t) repre-

sents the right-hand side of the static equation (1), de-

scribing the pump ﬂow in steady-state. We introduce

a dynamical model of the form:

(A(∇) + B(∇))Q

est

(t) = B(∇) f(t), (4)

where Q

est

(t) is the output of the system which rep-

resents the estimated instantaneous values of the pul-

satile ﬂow (Q(t)), f (t) is the input to the dynamical

system model, ∇ is the shift operator, A(∇), B(∇) are

polynomials deﬁned as follows:

A(∇)Q

est

(t) =

n

∑

i=0

a

i

Q

est

(t − i+ 1), (5)

B(∇) f(t) =

m

∑

j=1

b

j

f(t − j+ 1). (6)

Here n is the model output order, and m is the model

input order satisfying the condition m ≤ n.

Now we describe all steady-state or constant so-

lutions of the equation (3). Let Q

est

(t) ≡ Q

0

and

f(t) ≡ f

0

for all t = 0, 1, 2. . ., then, we obtain from

(4) that

(A(1) + B(1))Q

0

= B(1) f

0

. (7)

We assume that A(1) = 0 and B(1) 6= 0. This yields

to the following conditions on parameters coefﬁcients

of the model:

n

∑

i=0

a

i

= 0, (8)

m

∑

j=1

b

j

6= 0. (9)

Under the assumptions (8) and (9), it immediately fol-

lows from (7) that

Q

0

= f

0

. (10)

Since f(t) is deﬁned by (2), (3), (10) it implies that

Q

0

is the corresponding solution of the equation (1).

Hence, steady-states of the dynamical model are de-

scribed by the static model.

Furthermore, if the system is stable i.e; all z :

A(z)+B(z) = 0 belong to | z |< 1 (all poles of the sys-

tem are inside the unit disk) (Ogata, 1995), then the

solution of the dynamical system (4) with any initial

conditions and with a constant input f

0

will converge

to the constant output Q

0

satisfying (10).

A DYNAMICAL MODEL FOR PULSATILE FLOW ESTIMATION IN A LEFT VENTRICULAR ASSIST DEVICE

403

2.3.3 Data Analysis

Data was divided into two sets, one set was used

for identiﬁcation and training and the other to val-

idate the model. The off-line least squares method

(Ljung, 1999) was used to estimate the model coefﬁ-

cients. Values of parameter coefﬁcients were chosen

so that the error between estimated Q

est

(t) and mea-

sured ﬂow Q

meas

(t) was minimized. Also, parameters

should fulﬁll the assumptions deﬁned in equations (8)

and (9). Mean absolute error (e), and correlation coef-

ﬁcient (r) were used to check the performance of the

model as follows:

e =

1

N

N

∑

i=1

(Q

meas

(t) − Q

est

(t))

2

, (11)

r =

∑

N

i=1

(Q

meas

(t) − Q

meas

)(Q

est

(t) − Q

est

)

q

∑

N

i=1

(Q

meas

(t) − Q

meas

)

2

∑

N

i=1

(Q

est

(t) − Q

est

)

2

.

(12)

Here N is the length of data. Q

meas

, and Q

est

are the

mean values of the measured and estimated ﬂows re-

spectively.

3 RESULTS

Least e and highest r between the estimated and mea-

sured ﬂow in both mock loop and animal experiments

were obtained with system model orders of n = 3 and

m = 2.The resulting model is described as follows:

a

0

Q

est

(t) + (a

1

+ b

1

)Q

est

(t − 1) + (a

2

+ b

2

)Q

est

(t − 2)

+a

3

Q

est

(t − 3) = b

1

f(t − 1) + b

2

f(t − 2), (13)

where a

0

= 1, a

1

= −2.25, a

2

= 1.49, a

3

= −0.24,

b

1

= 0.27, b

2

= −0.25, Q

est

is the estimated pulsatile

ﬂow, and f(t) is the input signal. The poles-zeros plot

of the system shown in ﬁgure 1 demonstrates that the

model is stable. The dashed line in ﬁgure 2 shows part

of the estimated pulsatile and measured ﬂow obtained

from the mock loop.

Linear regression analysis between Q

est

(t) and

Q

meas

(t) obtained from the mock loop experiments is

illustrated in ﬁgure 3. A highly signiﬁcant correlation

between estimated and measured ﬂow was obtained

with a small mean absolute error value.

During animal experimentation the values of HCT

were not measured. Instead, for each animal we

used the values of HCT that gave the best ﬁt between

the estimated and measured ﬂows in the steady state

model described in equation (1). Additionally, two

sets of data were removed from the analysis due to

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imaginary Part

Zeros

Poles

Unit Disk

Figure 1: Poles-zeros plot of the system model described by

equation (13).

6094 6095 6096 6097 6098 6099 6100

4.5

5

5.5

6

6.5

7

7.5

Time (s)

Flow (L/min)

Measured Flow (Q

meas

)

Estimated Pulsatile Flow (Q

est

)

Figure 2: Estimated pulsatile ﬂow compared with measured

ﬂow obtained from mock loop.

−2 0 2 4 6 8 10 12

−2

0

2

4

6

8

10

12

14

Measured Flow (L/min)

Estimated Flow (L/min)

Q

est

= 1.080*Q

meas

+ 0.807

R

2

= 0.957

Figure 3: Linear regression plot between estimated versus

measured ﬂow obtained from mock loop.

excessively noisy power signals. The comparison be-

tween estimated pulsatile and real ﬂow for one of the

six animal experiments is shown in ﬁgure 4. Correla-

tion between estimated and real ﬂows was highly sig-

niﬁcant (R

2

= 0.902), with small mean absolute error

e = 0.317 L/min.

The response of the proposed model to abnor-

mal ﬂow conditions associated with different pump-

ing states such as ventricular collapse (VC) and aortic

valve not opening (ANO) were accurately estimated

by the model (not shown).

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404

942 944 946 948 950

2

3

4

5

6

Time (s)

Flow (L)/min

Measured Flow (Q

real

)

Estimated Flow (Q

est

)

Figure 4: Estimated ﬂow compared with measured one ob-

tained from pigs experiments.

4 DISCUSSION

In the present study, a dynamical model for pulsatile

ﬂow estimation was successfully designed. In the pro-

posed model, the level of HCT was assumed to be

known. This is the major limitation of the presented

model.

Using an autoregressive with exogenous input

(ARX) model, Yoshizawa et al. (2002) developed a

pulsatile ﬂow estimator (Yoshizawa et al., 2002). A

Mean absolute error of 1.66 L/min, and a correlation

coefﬁcient of 0.85 were obtained when another ARX

model has used to compensate for HCT. Tsukiya et

al. (2001) showed that the non-pulsatile ﬂow rate es-

timator was able to monitor the instantaneous ﬂow

(Tsukiya et al., 2001). To compare, our proposed

model resulted in a high correlation coefﬁcient R

2

=

0.957 with e= 0.364 L/min in mock loop. Also R

2

=

0.902 and e= 0.317 L/min were obtained using ex vivo

animals data.

Ayre et al. (2003) were successfully estimated

an average ﬂow for non-pulsatile and pulsatile ﬂow

(Ayre et al., 2003). More recently, pulsatile ﬂow was

accurately estimated by Karantonis et al. (2007). Al-

though these studies produced acceptable results, they

did not study the stability of the transient response of

the pump ﬂow which is one of the outcomes of the

present study.

5 CONCLUSIONS

A dynamical model for an iRBP has been presented

and shown to accurately estimate the pulsatile ﬂow

using noninvasive measurements of power and speed.

Furthermore, the proposed model is stable and its set

of steady states is identical to the set of solutions of

the previously derived static model.

ACKNOWLEDGEMENTS

This work was supported by The Australian Research

Council.

REFERENCES

Ayre, P. J., Lovell, N. H., and Woodard, J. C. (2003).

Non-invasive ﬂow estimation in an implantable rotary

blood pump: a study considering non-pulsatile and

pulsatile ﬂow. Physiol Meas, 24:179–89.

Bertram, C. (2005). Measurement for implantable rotary

blood pumps. Physiol Meas, 26:R99–R117.

Funakubo, A., Ahmed, S., Sakuma, I., and Fukui, Y. (2002).

Flow rate and pressure head estimation in a centrifugal

blood pump. Artif Organs, 26:985–90.

Karantonis, D. M., Cloherty, S. L., Mason, D. G., Ayre, P. J.,

and Lovell, N. H. (2007). Noninvasive pulsatile ﬂow

estimation for an implantable rotary blood pump. In

Proceedings of the 29th Annual International Confer-

ence of the IEEE EMBS.

Karantonis, D. M., Lovell, N. H., Ayre, P. J., Mason, D. G.,

and Cloherty, S. L. (2006). Identiﬁcation and classi-

ﬁcation of physiologically signiﬁcant pumping states

in an implantable rotary blood pump. Artif Organs,

30:671–9.

Kitamura, T., Matsushima, Y., Kono, S., Nishimura, K.,

Komeda, M., Yanai, M., Kijima, T., and Nojiri, C.

(2000). Physical model-based indirect measurements

of blood pressure and ﬂow using a centrifugal pump.

Artif Organs, 24:589–93.

Ljung, L. (1999). System Identiﬁcation: Theory for the

user. Prentice Hall PTR, USA, 2nd edition.

Malagutti, N., Karantonis, D.M., Cloherty, S. L., Ayre, P. J.,

Mason, D. G., Salamonsen, R. F., and Lovell, N. H.

(2007). Noninvasive average ﬂow estimation for an

implantable rotary blood pump: a new algorithm in-

corporating the role of blood viscosity. Artif Organs,

31:45–52.

Ogata, K. (1995). Discrete-Time Control Systems. Prentice

Hall Inc., USA, 2nd edition.

Tsukiya, T., Taenaka, Y., Nishinaka, T., Oshikawa, M.,

Ohnishi, H., Tatsumi, E., Takano, H., Konishi, Y., Ito,

K., and Shimada, M. (2001). Application of indirect

ﬂow rate measurement using motor driving signals to

a centrifugal blood pump with an integratedmotor. Ar-

tif Organs, 25:692–96.

Yoshizawa, M., Sato, T., Tanaka, A., Abe, K., Takeda, H.,

Yambe, T., Nitta, S., and Nose, Y. (2002). Sensor-

less estimation of pressure head and ﬂow of a con-

tinuous ﬂow artiﬁcial heart based on input power and

rotational speed. ASAIO Journal, 48:443–48.

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