Identification of Patient Specific
Parameters for a Minimal Cardiac Model
THE 26th ANNUAL INTERNATIONAL CONFERENCE OF THE IEEE
ENGINEERING IN MEDICINE AND BIOLOGY SOCIETY
C. E. Hann1, J. G. Chase1, G. M. Shaw2, B. W. Smith3,
1Department
of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand
3 Centre for Model-based Medical Decision Support, Aalborg University, Aalborg, Denmark
2Department
Diagnosis and Treatment
• Difficult task for medical staff, often trial and error – the “art of
medicine”
• Problem compounded by lack of complete data or knowledge
• Goal = a minimal cardiac model to identify patient parameters and
assist in diagnosis
• For example, increased resistance in pulmonary artery - suggests
blockage common to atherosclerotic heart disease
• Must be done in clinical real time (3-5 minutes) indicating a need for
computational simplicity
• This talk concentrates on mathematical and computational aspects of
parameter identification from the model.
• Eventual goal is to determine the minimal data set for useful
parameter identification
Single Chamber Model
e.g.
•
•
•
•
•
P1
P2
P3
R1
R2





Pressure in pulmonary vein (Ppu)
Pressure in left ventricle (Plv)
Pressure in aorta (Pao)
Resistance of mitral valve (Rmt)
Resistance of aortic valve (Rav)
D.E.’s and PV diagram
V  Q1  Q2
P  P  Q1 R1
Q1  1 2
L1
P  P3  Q2 R2
Q 2  2
L2
P2  e(t ) E es (V  Vd )  (1  e(t )) P0 ( e  (V V )  1),
0
e(t )  e  80( t  0.375)
2
•
Open on pressure, close
on flow valve law
•
Find parameters as
quickly and as
accurately as possible
Integral Method - Concept
x  ax  b sin( t )  c, x (0)  1
a  0.5, b  0.2, c  0.8
•
Discretised solution analogous to
measured data
(simple example with
analytical solution )
• Work backwards and find a,b,c
• Current method – solve D. E. numerically or
analytically
1.85
x (t ) 
1.8
 (ab cos t  ba 2 sin t  ca 2  c ))
1.75
1.7
- Find best least squares fit of x(t) to the data
1.65
x
- Non-linear, non-convex optimization,
computationally intense
1.6
q
1.55
1.5
P1  P2
R
• integral method
– reformulate in terms of integrals
1.45
1.4
12
1
( eat ( a  c  ab  ca 2  a 3 )
(a  1)a
2
– linear, convex optimization, minimal computation
13
14
15
16
time
17
18
19
Integral Method - Concept
Integrate x  ax  b sin( t )  c, both sides from t0 to t ( t0  4 )
•
t x dt  t (ax  b sin( t )  c) dt
t
t
0
0
 x(t )  x(t0 )  a t x dt  b t sin( t ) dt  c t 1 dt

•
t
t
t
0
0
0
x(t )  x(t0 )  a t x dt  b(cos( t0 )  cos(t ))  c(t  t0 )
t
0
Choose 10 values of t, between t0  4 and 6 seconds to form 10
equations with 3 unknowns a,b,c
a tt x dt  b(1  cos( ti ))  c(ti  t0 )  x (ti )  x (t0 ), i  1,,10
0
Integral Method - Concept
 tt x dt cos( t0 )  cos( t1 ) t1  t0  a   x (t1 )  x (t0 ) 

  

 

  b   


 t
  

x
dt
cos(
t
)

cos(
t
)
t

t
c
x
(
t
)

x
(
t
)
0
10
10
0  
0 
 10
 t
1
0
10
0
•
Linear least squares (convex problem, unique solution)
Method
Starting point
CPU time (seconds)
Solution
Integral
-
0.003
[-0.5002, -0.2000, 0.8003]
Non-linear
[-1, 1, 1]
4.6
[-0.52, -0.20, 0.83]
Non-linear
[1, 1, 1]
20.8
[0.75, 0.32, -0.91]
•
Integral method is at least 1000-10,000 times faster depending on starting point
•
Thus, very suitable for clinical application versus non-convex and non-linear
methods often used
Integrals - Single Chamber
•
D. E.’s are solved in MAPLE, Q1 and Q2 curves discretised.
Description
Symbol
Value
Discretised curves analogous to
measured data
EDPVR volume
V0
0 m3
DSPVR volume
Vd
0 m3
Constant

33000 m -3
P0
10 N m -2
Ees
3.5555x10 8 m -5

1.33 beats s-1
Heart rate
Resistance
Inertance
Pressure
•
Outflow
R1
R2
83000 N s m5
L1
L2
430000 N s2 m5
P1
P3
3 mmHG
81000 N s
Inflow
m5
480000 N s2 m5
100 mmHG
In practice Q1, Q2 can be obtained from echocardiography or from
differentiating volume data using ultrasound
Integrals – Single Chamber
•
e(t) translated  V(0)=Vmin, Q1(0)=0
(beginning of filling stage at t=0)
Filling stage - V  Q1
L1Q1  P1  P2  Q1R1
Ejection stage - V  Q2
L2Q 2  P2  P3  Q2 R2
 V (T )  V (T1 )  TT1 Q2 dT
 V (t )  Vmin  0t Q1dt
•
Choose T1, Q2(T1)=0, V(T1)=Vmax
•
Assume e(t ), V (t ), Vd , V0 , P0 ,  , Q1, Q2 are given or measured
16 values of t in filling stage
14 values of T in ejection stage

30 linear equations in 5 unknowns
Ees , R1, R2 , L1, L2
Results – Single Chamber
Optimised parameter values
True value
Optimised
value
Percentage
error
Ees
3.55555x108
3.55555x108
0.02
R1
83000
81128
2.26
R2
L1
81000
430000
81768
430876
0.95
0.20
L2
480000
479868
0.03
Parameter
PV curves for model with
optimized values versus the
model with the true values.
Model response error
with optimised values
Q1
Q2
PV
Flows in and out for the model with
optimized values versus the
model with the true values.
Mean
Standard
percentage deviation
error
0.17
0.08
0.08
0.06
0.09
0.06
•
•
Accurate parameter
identification achieved
Simulation errors all
less than 0.2%
validating parameter
identification approach
Conclusions
• Integral based optimization successfully identified patient
specific parameters for a single chamber model representative of
elements in larger such models.
• Using integrals any noise is low pass filtered
• Optimization is linear, convex, and has minimal computation
• Typically used methods are non-linear, non-convex, and require
significant computation and sometimes multiple starting points
• Avoid problem of incorrect initial conditions increasing
computational time
• D.E. is never required to be solved analytically or numerically
• Method readily extends to larger models (6+ chambers)
In summary, medical staff will have rapid data on patients assisting in
diagnosis and can trial and test therapies in clinical real time (3-5 minutes).
Acknowledgements
Engineers and Docs
Dr Geoff Chase
Dr Geoff Shaw
The Danes
Steen
Andreassen
The honorary Danes
Dr Bram Smith
Questions ???
AIC2, Kate, Carmen and Nick