1 of 4
Computationally Efficient Velocity Profile Solutions for Cardiac
Haemodynamics
C. E. Hann1, J. G. Chase1, B. W. Smith3, G. M. Shaw2
1
Bioengineering Centre, Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
2
Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand
3
Centre for Model-based Medical Decision Support, Aalborg University, Denmark
Abstract—This paper reformulates the non-linear
differential equations associated with time varying resistance in
minimal cardio-vascular system models into a system of linear
equations with an analytical solution. The importance of
including time varying resistance is shown for a single chamber
model where there is a 17.5% difference in cardiac output
when compared with a constant resistance model. However,
the increased complexity has significant extra computational
cost. This new formulation provides a significant
computational saving of 15x over the previous method. This
improvement enables more physiological accuracy with
minimal cost in computational time. As a result, the model can
be used in clinical situations to aid diagnosis and therapy
selection without compromising on physiological accuracy.
Keywords—Time varying resistance, Non-linear, Linear,
Computationally efficient, Analytical solution
dynamics can be considered if this otherwise simple model
is used in therapeutic decisions or testing therapy choices.
II. METHODOLOGY
For a rigid pipe of constant cross-sectional area the
velocity u in the x direction is described by the differential
equation
u
1  P     u( r, t ) 
(1)
r



t
  x r  r 
 r 
where  is density and  is the fluid viscosity (    ).
Equation (1) is derived from the Navier Stokes equations
and can be integrated over the cross-sectional area and along
the length of the pipe in the x direction to give:
I. INTRODUCTION
Cardiovascular system (CVS) models in the literature
either take a complex finite element (FE) approach, or the
simple and more flexible lumped parameter (LP) approach.
Here the focus is on the LP approach because it is simple,
flexible and not as computationally intensive as FE methods.
However, to capture greater detail in CVS dynamics, more
physiologically accurate equations and variables must be
added to LP models [1]. The addition of these equations
and variables increases the complexity of CVS models
making them more computationally expensive to solve.
Complexity costs computational power and time, and should
therefore only be added where significant benefits are
obtained over a simpler method.
Typical LP model equations governing arterial flow rate
come in two forms, either including or not including inertial
effects. In both cases, resistance to blood flow is assumed
constant under varying flow velocity and acceleration. It
was shown in [1] that around heart valves the inclusion of
time varying resistance produces results that are
significantly different
(17.5%) in terms of major
haemodynamic response metrics from those generated by
constant resistance models. However, this time varying
resistance and velocity profile (VProf) method, as presented
in [1] has significant extra computational cost.
An alternative method for setting up the model is given
which turns the non-linear problem into a linear problem,
simplifies the implementation and results in a significant
computational saving with little loss in accuracy. Clinically,
improved computational cost ensures these important
l dQ
2 l  u
 P1  P2 
( r0 )
 r02 d t
r0  r
(2)
where Q is the flow, l is the artery length and r0 is the
radius. P1 and P2 are the upstream and downstream pressures.
Equation (2) can be compared to the constant resistance
(CR) version of the equation [2] from a purely LP model
dQ
(3)
L
 P1  P2  RQ ,
dt
where L is inertance and R is the constant resistance.
However,  u ( r0 ) in (2) is unknown. Therefore, to find the
r
flow rate Q , (1) is used with Q  0 0 u(r, t )rdrd [2].
For a single chamber model of a given elastance with
constant upstream and downstream pressures the resulting
equations are defined:
2
r0
dV
 Qin  Qout
dt
uin
1  P  P      uin ( r, t ) 
r

   2 1 
t
  l  r  r 
 r 
uout
1  P  P      uout ( r, t ) 
r

   3 2 
t
  l  r  r 
r

P2  e(t ) Ees (V  Vd )  (1  e(t )) P0 (e  (V V ) )
0
e( t )  e
80( t  0.375)
(4)
(5)
(6)
(7)
(8)
2 of 4
where e(t ) is the model cardiac driver function [1,2], E es is
elastance, Vd is volume at zero pressure and P0 ,  , and
V0 define gradient, curvature and volume at zero pressure of
the EDPVR curve, and Qin and Qout are the flow rates at the
inlet and outlet respectively. The single chamber model is
shown in Fig 1.
Equation (5) is solved during the filling stage, ( uout  0 )
and (6) is solved during ejection stage ( uin  0 ). This model
therefore has an open on pressure, close on flow valve law
using Heaviside step functions to determine change in state
during the cardiac cycle [2]. Equations (4)-(6) are solved by
a finite element method where the radius r0 is equally spaced
into N nodes and the derivatives u r and urr are approximated
by finite differences. These equations are coupled and nonlinear.
After simulation of the model with different parameters
it was found that the volume could represented by parabolas
and straight lines, as shown in Fig 2. The time points
t 3 , t 4 and t 7 , T (the heartbeat period) and V1 ,V2 are patient
specific and can be measured. The intervals [t 3 , t 4 ] and
[t 7 , T ] represent iso-volumetric contraction and expansion
where the volume stays constant. V1 and V2 are the minimum
and maximum volumes. The results of the simulation
suggest values of t1, t 2 and t 5 , t 6 defined:
t1 
2
4
1
4
t 3 , t 2  t 3 , t5  t 4  (t 7 - t 4 ), t 6  t 4  (t 7  t 4 )
3
5
2
5
(9)
where the constant coefficients are based on the given
patient specific values. The analytic expression for V (t ) is
then defined:
V (t )  P1 ( H (t )  H (t  t1 ))  L1 ( H (t  t 2 )  H (t  t1 ))
 P2 ( H (t  t 3 )  H (t  t 2 ))  L2 ( H (t  t 4 )  H (t  t 3 ))
 P3 ( H (t  t 5 )  H (t  t 4 ))  L3 ( H (t  t 6 )  H (t  t 5 ))
 P4 ( H (t  t 7 )  H (t  t 6 ))  L4 ( H (t  t8 )  H (t  t 7 ))
(10)
where H (t ) is the Heaviside function, p1 ,  p4 are
parabola’s and L1 ,  L4 are straight lines. This format
gives a total of four constant unknowns, one for each
parabola. Each line is uniquely determined by the two time
points, while the parabola requires the extra, constant
parameter. Therefore, the resulting equations, which are in
terms of these constants, are more readily solved when this
definition for the volume is used.
Fig. 1. The single cardiac chamber model.
Filling
Iso-volumetric
contraction
Ejection
Iso-volumetric
expansion
Fig. 2. Representation of the volume in terms of parabolas and straight
lines with cardiac cycle phases shown.
Note that if ultra-sonography is used the volume could
be calculated for each time point using the technique of [3].
This approach would give the whole volume profile.
The advantage of setting the problem up in this way is
that the differential equations now become linear. Let the
number of nodes N  4 . Then for u  uin the differential
equations, using (10) and the assumed time values, are
defined:
 3(u3  2u2  u1 ) 2(u2  2u1 ) 3(u3  u1 )  2u2 
du1

 P(t )   


dt
( r2  r1 ) 2
( r2  r1 ) 2
r1 ( r3  r1 ) 

(11)
 u  2u2  u1
du2
u u 
 P(t )    3
 3 1 
2
dt
r2 ( r3  r1 ) 
 ( r2  r1 )
(12)
  2u3  u1

du3
u2

 P(t )   

2
dt
r3 ( r3  r1 ) 
 ( r2  r1 )
(13)
where ri are locations along the radius, u i are flow velocity
at those locations and P(t ) is the pressure gradient, defined:
1  P  P1 
r

ui (t )  u 0 i, t , i  13, P(t )    2

 l 
4 
(14)
and u4 (t )  u( r0 , t )  0 is the no slip boundary condition.
The first and second derivatives are approximated by finite
differences with O ( h 2 ) error. The resulting equations can
be rewritten:
3 of 4
1
 
r21  r2  r1 , r31  r3  r1 , F (t )  P(t )1, u(0)  0,
1
 
where A is known and
(15)
150
(16)
100
Pressure [P] (mmHg)
 7
3
 2 
r1 r3 1
 r2 1
 1
1
A 2 
r
r
2 r3 1
 21
1

2

r2 1

du
 Au  F (t )
dt
8
2
3
3 



2
2
r1 r3 1 r2 1
r1 r3 1 
r2 1
2
1
1 


2
2
r2 r3 1 
r2 1
r2 1
1
2

2

r3 r3 1
r2 1

CR
50
TR
(17)
 u1 
 
(18)
u   u2 
u 
 3
For further computational savings an analytical solution to
(15) may be used, based on the eigenvalues and
eigenvectors of A [4]. The advantage of that approach is
that A is known and the eigen-vectors and eigen-values can
be computed very quickly along with the final analytical
solution.
III. RESULTS
0
0
10
20
30
40
Volume [V] (ml)
50
60
70
80
Fig. 2. Pressure volume curve for time varying resistance (TR) and constant
resistance (CR) models.
The volume can also be approximated by parabola’s and
straight lines as in (10) to get, V approxand values for
T , t1 , t2 , t5 and t6 are given by T=0.75 and (9). These
values are substituted into (7) to form P2a p p ro x. The
differential equations (2) are solved numerically without
using the analytical solution to (15) with N  100 nodes by
replacing P2 by P2a p p ro x. Fig. 3 and 4 shows a very close
match with the results using the method in [1] with the plots
overlaid.
A single chamber model is simulated first using the
method in [1] and with constant resistance as in [2]. The
model parameter values used are shown in Table 1.
700
CR and TR plots are
overlaid
600
TABLE I
CONSTANTS USED IN SINGLE-CHAMBER SIMULATION
Blood density
Blood viscosity
Internal artery
radius
Artery length
Chamber elastance
EDPVR volume
DSPVR volume
Constant
Heart rate
Constant
Symbol
Value


1050 kg m-3
0.004 N s m-2
r0
0.0125 m
l
E es
3.56  108 Nm-5
V0
Vd


P0
Flow [Q] (ml/sec)
500
Description
300
0.2 m
200
0 m3
100
0 m3
33000 m-3
1.33 beats s-1
10 N m-2
Fig. 2 shows the resulting PV curves for constant resistance
(CR) versus time varying resistance (TR). There is a 17.5%
change in stroke volume, which shows the importance of
including the FE time varying resistance to obtain
physiologically accurate results.
Out-flow
400
0
In-flow
0
0.1
0.2
0.3
0.4
Time [t] (secs)
0.5
0.6
0.7
0.8
Fig. 3. The flows of the new method (dashed) versus old method (solid)
To measure the error, the flow in and out for both methods
is calculated at equally spaced points separated by 0.001
seconds in the intervals [0.02, 0.41] during in-flow
for Q1 , and [0.51, 0.61] during out-flow for Q2 . The relative
4 of 4
percentage error between the methods is calculated for
Q1 and Q2 , and shown in Table 2.
TABLE 2
PERCENTAGE ERRORS FOR THE FLOWS USING 100 NODES
Q1
Q2
Mean Error (%)
Standard Deviation
0.3
0.3
0.3
0.2
Table 3 compares the computational time between these
methods for 20, 40, 60, 80 and 100 nodes. The percentage
errors of Q1 and Q2 are also calculated. Note that 20 nodes
is the minimum number that the FE method in [1] must use
to have an error less than 1%. The computational speed
increase is 14-16.6x, which is significant for large models of
the full CVS as in [2]. The error between this approximate
method and the more exact VProf approach in [1] is never
greater than 1.8% and averages less than 0.5 %
above N  30
TABLE 3
COMPARING THE COMPUTATIONAL TIME OF THE OLD
METHOD TO THE NEW METHOD
Nodes
20
30
40
60
80
100
CPU time (seconds)
Old
method
2.1
2.4
2.6
3.4
4.4
5.3
New
Method
0.15
0.16
0.18
0.24
0.3
0.32
Speed
Increase (  )
14
15
14.4
14.2
14.7
16.6
Mean error (%)
Q1
Q2
0.4
0.3
0.2
0.3
0.3
0.3
1.8
0.7
0.4
0.3
0.3
0.3
IV. DISCUSSION
The very small mean relative percentage errors in Table
3 show that the new method is more than sufficiently
accurate. In particular, these errors are well within the
measurement error using any technique. There is a 17.5%
difference in stroke volume between the constant resistance
and time varying resistance models. Figure 2 shows the
importance of including time varying resistance and better
mass flow approximations than (3), if it can be done
computationally cheaply. On average for nodes between 20
and 100 there is a 15x speed increase over the FE method
showing significant computational savings for the new
method. Note that no use was made of the analytical
solution in (20) which would give further significant
reductions in computation as eigen-vectors and eigen-values
are very quick to compute even for large matrices.
V. CONCLUSION
An alternative formulation for the VProf model of [1] is
investigated, which turns the non-linear differential
equations into linear differential equations with an analytical
solution. This is approach is shown to produce solutions
very close to the original method with a significant ~15x
computational saving. The new method is important as it
enables more physiological accuracy with minimal cost in
increased computation, enabling greater clinical application
of these models.
REFERENCES
[1] Smith, B. W., Chase, J. G., Nokes, R. I., Shaw, G. M. and David,
T. (2003). “Velocity profile method for time varying resistance
in minimal cardiovascular system models.” J. Phys. Med. Biol,
Vol. 48, pp 3375-3387.
[2] Smith, B. W., Chase, J. G., Nokes, R. I., Shaw, G. M. and Wake,
G.. (2003). “Minimal haemodynamic system model including
ventricular interaction and valve dynamics.” J. Phys. Med. Eng.
Phys. Vol. , pp .
[3] Moore, C. L., Rose, G. A., Tayal, V. S., Sullivan, M. et al
(2002). “Determination of Left Ventricular Function by
Emergency Physician Echocardiography of Hypotensive
Patients.” Academic Emergency Medicine.
[4] Zill G. E. and Cullen, M. R. (1992). “Advanced Engineering
noes.Mathematics.” Boston: PWS-KENT Pub. Co.