Simulating Cardiac Disease From Onset with a
Minimal Cardiac Model Including Reflex Actions
THE 12th INTERNATIONAL CONFERENCE ON BIOMEDICAL
ENGINEERING
C. E. Hann1, J. G. Chase1, S. Andreassen2, B.W. Smith2, G. M. Shaw3,
1Department
of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
for Model-based Medical Decision Support, Aalborg University, Aalborg, Denmark
3 Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand
2Centre
Diagnosis and Treatment
• Cardiac disease state difficult to diagnose
- Limited data
- Reflex actions
• Minimal Cardiac Model
- primary parameters
- common ICU measurements
• Increased resistance in pulmonary artery – pulmonary
embolism, atherosclerotic heart disease
• Optimize drug treatment on computer
-e.g. Warfarin
• Require fast parameter ID
Heart Model
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 ) Ees (V  Vd )  (1  e(t )) P0 (e (V V0 )  1),
e(t )  e 80( t 0.375)
2
Reflex actions
•
•
•
•
Vaso-constriction - contract veins
Venous constriction – increase
venous dead space
Increased HR
Increased ventricular contractility
Varying HR as a function of Pao
Disease States
•
Pericardial Tamponade
- build up of fluid in pericardium
- dead space volume V0,pcd by 10 ml / 10 heart beats
•
Pulmonary Embolism - Rpul 20% each time
•
Cardiogenic shock
- e.g. left ventricle infarction, blocked coronary artery
- not enough oxygen to myocardium
- Ees,lvf, P0,lvf
•
Septic shock
- blood poisoning
- reduce systemic resistance
•
Hypovolemic shock – severe drop in total blood volume
Results
• Healthy human
Output
Value
Volume in left ventricle
111.7/45.7 ml
Volume in right ventricle
112.2/46.1 ml
Cardiac output
5.3 L/min
Max Plv
119.2 mmHg
Max Prv
26.2 mmHg
Pressure in aorta
116.6/79.1 mmHg
Pressure in pulmonary artery
25.7/7.8 mmHg
Avg pressure in pulmonary vein
2.0 mmHg
Avg pressure in vena cava
2.0 mmHg
Results
•
•
Pericardial tamponade
•
Pulmonary Embolism
Ppu – 7.9 mmHg
CO – 4.1 L/min
MAP – 88.0 mmHg
All other disease states capture physiological trends and magnitudes
Identifiability
• Add 10% noise to outputs
• Apply integral-based optimization
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 form 10 equations in
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 (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
Identifiability
•
•
Pericardial tamponade (determining V0,pcd)
Change
True value
(ml)
Optimized
Value
Error (%)
First
180
176
2.22
Second
160
158
1.25
Third
140
138
1.43
Fourth
120
117
2.50
Fifth
100
100
0
Pulmonary Embolism (determining Rpul)
Change
True value
(mmHg s ml-1)
Optimized
Value
Error (%)
First
0.1862
0.1907
2.41
Second
0.2173
0.2050
5.67
Third
0.2483
0.2694
8.50
Fourth
0.2794
0.2721
2.60
Fifth
0.3104
0.2962
4.59
Identifiability
•
•
Cardiogenic shock (determining [Ees,lvf, P0,lvf] (mmHg ml-1, mmHg)
Change
True values
Optimized
Value
Error (%)
First
[2.59,0.16]
[2.61,0.15]
[0.89,5.49]
Second
[2.30,0.19]
[2.30,0.18]
[0.34,4.39]
Third
[2.02,0.23]
[2.02,0.21]
[0.43,8.03]
Fourth
[1.73,0.26]
[1.70,0.24]
[1.48,9.85]
Fifth
[1.44,0.30]
[1.43,0.27]
[0.47,9.39]
Septic Shock (determining Rsys)
Change
True value
(mmHg s ml-1)
Optimized
Value
Error (%)
First
1.0236
1.0278
0.41
Second
0.9582
0.9714
1.37
Third
0.8929
0.8596
3.73
Fourth
0.8276
0.8316
0.49
Fifth
0.7622
0.7993
4.86
Identifiability
•
Hypovolemic Shock (determining stressed blood volume)
Change
True value
(ml)
Optimized Value
Error (%)
First
1299.9
1206.5
7.18
Second
1177.3
1103.7
6.26
Third
1063.1
953.8
10.28
Fourth
967.8
1018.9
5.28
Fifth
928.5
853.4
8.10
Conclusions
• Minimal cardiac model  simulate time varying disease states
• Accurately captured physiological trends and magnitudes
 capture wide range of dynamics
• Integral based parameter ID method
- errors from 0-10%, with 10% noise
- identifiable using common measurements
• Rapid feedback to medical staff
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