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Validation of Subject-Specific Cardiovascular
System Models from Porcine Measurements
James A. Revie*, David Stevenson*, J. Geoffrey Chase*, Christopher E. Hann*,
Bernard C. Lambermont**, Alexandre Ghuysen**, Philippe Kolh**, Geoffrey M.
Shaw***, Stefan Heldmann**** and Thomas Desaive**.
*Department of Mechanical Engineering, Centre of Bioengineering, University of
Canterbury, Christchurch, New Zealand (e-mail: [email protected]).
** Hemodynamic Research Laboratory, University of Liege, Belgium
***Department of Intensive Care, Christchurch Hospital, Christchurch, New Zealand
****Department of Mechanical Engineering, TU Darmstadt, Germany
Abstract
A previously validated mathematical model of the cardiovascular system (CVS) is
made subject-specific using an iterative, proportional gain-based identification
method. Prior works utilised a complete set of experimentally measured data that is
not clinically typical or applicable. In this paper, parameters are identified using
proportional gain-based control and a minimal, clinically available set of
measurements. The new method makes use of several intermediary steps through
identification of smaller compartmental models of CVS to reduce the number of
parameters identified simultaneously and increase the convergence stability of the
method. This new, clinically relevant, minimal measurement approach is validated
using a porcine model of acute pulmonary embolism (APE).
Trials were performed on five pigs, each inserted with three autologous blood clots of
decreasing size over a period of four to five hours. All experiments were reviewed
and approved by the Ethics Committee of the Medical Faculty at the University of
Liege, Belgium. Continuous aortic and pulmonary artery pressures (Pao, Ppa) were
measured along with left and right ventricle pressure and volume waveforms. Subjectspecific CVS models were identified from global end diastolic volume (GEDV),
stroke volume (SV), Pao, and Ppa measurements, with the mean volumes and
maximum pressures of the left and right ventricles used to verify the accuracy of the
fitted models.
The inputs (GEDV, SV, Pao, Ppa) used in the identification process were matched by
the CVS model to errors <0.5%. Prediction of the mean ventricular volumes and
maximum ventricular pressures not used to fit the model compared experimental
measurements to median absolute errors of 4.3% and 4.4%, which are equivalent to
the measurement errors of currently used monitoring devices in the ICU (~5-10%).
These results validate the potential for implementing this approach in the intensive
care unit.
Keywords: cardiovascular system, system identification, pulmonary embolism,
physiological modelling.
Introduction
Cardiovascular disease is the leading cause of intensive care unit (ICU) admission and
overall mortality in the western world and accounts for 36% of all deaths in New
Zealand [1]. However, with limited clinical data available, different disease states can
look the same on ICU monitors [2]. Hence, cardiovascular dysfunction can be
incorrectly diagnosed or mistreated due to incomplete information and complexities
involved, leading to sub-optimal use of hospital resources, increased length of stay,
and death [3-7].
Currently, cardiovascular assessment of critical care patients involves the analysis of
changes in arterial pressure, cardiac output (CO), electrocardiogram (ECG), central
venous pressure, heart rate and gas exchange measurements. Complex interactions in
these measurements and a lack of understanding of fundamental cardiac physiology
can hide the underlying pathological state so that the clinicians do not receive a clear
picture of the overall circulatory status or function. This research uses subject-specific
modelling to aggregate common ICU measurements into a clear physiological picture
to aid clinical diagnosis and decision-making.
One primary challenge is to create a realistic computational model of the heart and
circulation that is computationally light enough to be applicable at the bedside. In
particular, it should not require new, invasive monitoring or equipment not already
typically available. An additional advantage of such a model-based approach is its
effectively continuous nature because it can monitor measurements in real time. In
contrast, currently used diagnostic tests are done intermittently and thus cannot tell
clinicians what is going on as it happens, which is critical in the ICU where patient
condition can change rapidly.
In this research, the modelling methodology developed is tested on a porcine model of
pulmonary embolism [8]. Previous work by Starfinger et al. [9-12] assumed that the
maximum and minimum ventricle volumes were measured each cardiac cycle for both
sides of the heart, which is not typical but was useful to prove the concept. In
addition, heart valve resistances were constrained to population values, which is
reasonable as it was known a priori that the pigs did not have any valvular disease [8,
13]. However, it may not be valid for ICU patients. For example, aortic stenosis is an
increasingly common valvular disorder with age [14]. Hence, this work re-examines
the porcine data with a new approach using only typically available bedside
measurements. Measurements that were taken in [8], but not used to identify the
model, are used as independent “true” validation of the resulting pig-specific,
identified model.
Methodology
A.
Cardiac Model
The model discretises the cardiovascular system into six elastic chambers,
representing the left and right ventricles, aorta, vena cava, pulmonary artery, and
pulmonary vein, which capture all necessary dynamics for this study [12, 15-18].
Each pressure-volume chamber is characterised by its elastance, resistance to flow in
and out of the chamber, upstream and downstream pressures, and inertia of blood
through the valves. Myocardium activation is modelled by time varying elastance, and
interaction between the two ventricles is characterised by septum and pericardium
dynamics [16]. Notably, the model of Figure 1 differs from [12, 15-18] in that it
maintains the vena cava within the thoracic cavity, as in [9-11, 19] which is a more
physiologically accurate portrayal [20].
The model is described by Equations (1) – (24). A full list of the model parameters,
constants, and outputs are described in Tables 1, 2, and 3. Equations (1)-(4) describe
the flow, Q, through the heart valves. These first order differential equations state that
the pressure (P) difference across a valve is proportional to the sum of the resistive
(QR) and inertial effects ( LQ ) on the flow.
LavQ av  Plv  Pao  Qav Rav
(1)
LmtQ mt  Ppu  Plv  Qmt Rmt
(2)
LpvQ pv  Prv  Ppa  Q pv R pv
(3)
LtcQ tc  Pvc  Prv  Qtc Rtc
(4)
Non-valvular flow is represented using Ohm’s law (Equations (5) – (6)), where
poiseuille flow is assumed. Hence, the flow through the vasculature is proportional to
the pressure difference across the section divided by the resistance to flow.
Qsys 
Q pul 
Pao  Pvc
Rsys
Ppa  Ppu
R pul
(5)
(6)
Pressure in the non-cardiac chambers is proportional to the elastance (E) and stressed
blood volume within the chamber. In these chambers the elastance is assumed to be
constant over a heartbeat. The stressed blood volume in Equations (7) - (10) is
represented as the total blood volume (V) minus the unstressed volume (Vd).
Chambers within the thoracic cavity, as shown in Figure 1, are additionally influenced
by the intra-thoracic pressure (Pth) in the thoracic cavity.
Ppu  E pu V pu  Vd , pu   Pth
(7)
Ppa  E pa V pa  Vd , pa   Pth
(8)
Pvc  E vc Vvc  Vd ,vc   Pth
(9)
Pao  E ao Vao  Vd ,ao 
(10)
The free wall pressure of the left and right ventricles (Plvf, Prvf) are calculated using
Equations (11) – (12). These equations ignore the effects of the intra-ventricular
septum and pericardium which are taken into account later with Equations (19) – (27),
hence, the name free wall pressures. Myocardial activation of the left and right
ventricles are represented using normalised time varying elastance curves (driL and
driR) which vary in time between 0 and 1 (see Figure 6 for an example). The first
term of both these equations represent the dynamic component of the pressure due to
contraction of the heart muscles, with the second term representing the passive
pressure component due to the passive elastic recoil of the left and right ventricular
chambers. Hence, Equations (11) and (12) represent the percentage of dynamic and
passive pressure generation that is occurring in the ventricles at any point in time.

lvf Vlvf V0 ,lvf 

rvf Vrvf V0 , rvf 
Plvf  driL  Ees ,lvf  Vlvf  Vd ,lvf   1  driL   P0,lvf  e
Prvf  driR  Ees ,rvf  Vrvf  Vd ,rvf   1  driR   P0,rvf  e

(11)

(12)
1
1
The volume of blood in the non-cardiac chambers vary at a rate proportional to the
flow exiting and entering those chambers, as described by Equations (11) - (14).
Furthermore, Equations (15) and (16), representing the volume in the left and right
ventricles, take into consideration the non-linear effects of valvular flow by using
Heaviside functions. The use of Heaviside functions in these equations ensures that
there is no backwards flow through the heart valves.
Vpv  Q pul  Qmt
(13)
Vpa  Q pv  Q pul
(14)
Vvc  Qsys  Qtc
(15)
Vao  Qav  Qsys
(16)
Vrv  Heaviside Qtc Qtc  Heaviside Q pv Q pv
(17)
Vlv  Heaviside Qmt Qmt  Heaviside Qav Qav
(18)
Finally, ventricular interaction is modelled through considering the action of the intraventricular septum and the pericardium on cardiac dynamics. Equation (19) represents
all the pressures acting on the intraventricular septum, where the muscle activation of
the sepum (driS), is assumed to be equal to the average of the left and right ventricular
normalised time varying elastance curves (driL, driR). From Equation (19) the blood
volume displaced by the septum, Vspt, can be evaluated and used to calculate the real
left and right ventricular volumes (Vlv, Vrv), as shown by Equations (21) and (22).

driS  Ees ,spt  Vspt  Vd ,spt   (1  driS  P0,spt  e
spt Vspt V0 , spt 

driL  Ees ,lvf  Vlv  Vspt   1  driL   P0,lvf  e
lvf Vlv Vspt 

driR  Ees ,rvf  Vrv  Vspt   1  driR   P0,rvf  e
driS 
driL  driR
2

1 

1
rvf Vrv Vspt 

(19)
1
(20)
Vlvf  Vlv  Vspt
(21)
Vrvf  Vrv  Vspt
(22)
The pressure generated by the pericardium (Ppcd), which encases the heart, is
proportional to the sum of the left and right ventricular volumes (Vpcd) as described
by Equations (23) and (24). The total external pressure acting on the cardiac chamber
(Pperi), as seen in Equations (25)-(27), is therefore the sum of the pericardium and
intra-thoracic pressure, as the heart is located in the thoracic cavity. For a more
detailed description on the equations of the model and how they were derived please
see [15-18].

Ppcd  P0, pcd e
 pcd V pcd V0 , pcd 1

(23)
V pcd  Vlv  Vrv
(24)
Pperi  Ppcd  Pth
(25)
Plv  Plvf  Pperi
(26)
Prv  Prvf  Pperi
(27)
B.
Porcine Experiments and Data
All procedures and protocols used in the porcine experiments were reviewed and
approved by the Ethics Committee of the Medical Faculty at the University of Liege
(Belgium). Experiments were performed on 12 healthy, pure Pietrain pigs of either
sex, weighing between 20 and 30 kg. The pigs were medicated, anesthetised and
ventilated as detailed in [8, 13]. After a 30 minute stabilisation period the pigs were
randomly divided into two groups. In the first group, three autologous blood clots of
decreasing size were inserted into the external jugular vein at 0, 120 and 240 minutes
into the trial. The second group consisted of control animals.
Data acquired from the pigs include left and right ventricular pressure and volume
waveforms (Plv, Prv, Vlv, Vrv), the aortic pressure waveform (Pao) and the pulmonary
artery pressure waveform (Ppa), as shown in Table 3. Aortic and pulmonary artery
pressures (Pao, Ppa) were measured using micromanometer-tipped catheters (Sentron
pressure measuring catheter, Cordis, Miami, FL), while right and left ventricle
pressures and volumes (Plv, Prv, Vlv, Vrv) were measured using 7F, 12 electrodes (8
mm interelectrode distance) conductance micromanometer tipped catheters (see Table
3). This research uses 46 sets of data from five of the pigs (Pig 1, Pig 2, Pig 7, Pig 8,
and Pig 9) from the pulmonary embolism arm of the study. Measurements from the
sixth pig were omitted as it died very early in the trial.
C.
Model Identification
The aim of the model identification process is to create subject-specific models of the
CVS using only measurements available in a typical ICU. To personalise the model to
each subject, key parameters are identified using ICU data, as shown in Table 1.
However, the approach used in this paper is different from prior articles [9-12].
The method here identifies simplified sub-models of the CVS and then bootstraps
these simpler identified models to the more complex six-chamber model. In other
words, the parameters identified from the sub-models are combined and fixed in the
full six-chamber model to allow the other parameters of the six-chamber model to be
identified. In contrast, prior approaches identified the entire model all at once, thus
requiring a larger set of input data and assumptions. Through use of the simplified
models only a sub-set of parameters is identified at a time resulting in an increase in
convergence stability, as there is less interaction between variables as there are being
identified. A step-wise overview of the identification process is shown in Table 6.
1) Simplified Models: As an initial step to identifying the full six-chamber model, the
system is split into two simplified models representing portions of the systemic and
pulmonary circulations, as shown in Figure 2. In the six-chamber model both the vena
cava and pulmonary vein pressure waveforms (Pvc, Ppu) typically have an amplitude
less than 0.5mmHg and can be assumed to be constant. By assuming that the vena
cava and pulmonary vein pressures (Pvc, Ppu) are constant, and that initially there is no
ventricular interaction (Vspt=0, Ppcd=0), the systemic and pulmonary sides of the sixchamber model become mathematically decoupled and are separated into two smaller
models. Note that CO will be the same in both systems. Therefore, since the
identification algorithm would match the systemic and pulmonary models to the same
CO, there remains an implicit coupling. See Hann et al. [21] for details on this process
and further assumptions made in splitting the models, such as:

Inertances, Lmt, Lav, Ltc, and Lpv are set to zero, as they have little impact [16].

P0,lvf and P0,rvf are set to zero.

Vd,rvf and Vd,lvf are set to zero

Pth = 0
2) Systemic Model Identification: The first step in identifying the systemic model is
to approximate the left ventricle driver function, driL. The method of Hann et al. [22]
is used to estimate driL from features in the measured aortic pressure waveform,
Pao,true. The systemic model parameters are then identified using a proportional gain
controller which compares ratios of discrete values calculated from the model outputs
(SO) to a set of discrete measured data (SM) to optimise the model parameters (SI),
where S stands for systemic model and O = outputs, M = measured, and I = identified
model parameters, as outlined by Equations (28)-(30). The initial parameter values
used to simulate the systemic model, based off previous analysis of porcine data, are
shown in Table 4.
SI  Ppu , Rmt , E es ,lvf , Rav , E ao , Rsys , Pvc 
(28)
SO  Qmt , Plv ,Vlv,Qav , Pao ,Vao,Qsys 
(29)
dP


SM   Pao,mean,true , PPao,true , ao,max, true , SVtrue , t mt 
dt


(30)
To start the identification algorithm, the systemic model is simulated using an initial
estimate for the parameter set, SI, as seen in Table 4. At this stage only, Ppu, Eao, Rmt,
Rav, and Rsys of SI will be identified, with Pvc identified later by the pulmonary model,
and Ees,lvf identified once the rest of the parameters of the simplified models have been
fitted. Therefore, both Ees,lvf and Pvc remain at their initial value as shown in Table 4
during identification of the other systemic model parameters.
Firstly, the mitral valve resistance (Rmt), aortic elastance (Eao) and systemic resistance
(Rsys) are identified by comparing the model outputs to the measured SVtrue, PPao,true,
and Pao,mean,true to find better approximations for the parameters, as shown by:
 SVlv,approx 
 Rmt ,old
Rmt ,new  
 SVtrue 
where
SVlv,approx  max( Vlv )  min( Vlv )
(31)
 PPao,true 
 E ao,old
E ao,new  
 PP

 ao,approx 
where
PPao  max( Pao )  min( Pao )
(32)
 P

Rsys,new   ao,mean,true  Rsys,old
P

 ao,mean,approx 
where
Pao,mean 
max( Pao )  min( Pao )
2
(33)
New outputs, SO are then calculated through re-simulation of the systemic model
using the new parameter approximations for Rmt, Eao, and Rsys. These outputs are reentered into Equations (31)-(33) and the process is repeated until the model outputs,
SVlv,approx, PPao,approx, and Pao,mean,approx match the measured data to a tolerance of
0.5%.
Next, the aortic valve resistance, Rav, and pulmonary vein pressure, Ppu, are identified.
Rav and Ppu are identified separately from Rmt, Eao, and Rsys because they depend on
the accurate convergence of these parameters. Ppu trades off with Rmt ,as seen in
Equation (4), after inertial effects have been ignored, and Rav is dependent on Qav
(Equation (3)), a function of stroke volume which is matched previously using Rmt. To
calculate Ppu the mitral valve closure time, tmt is used:
Ppu  Plv (t mt )
(34)
It is assumed that at the time of mitral valve closure there is no pressure difference
across the valve and therefore Ppu is equal to the left ventricle pressure. In this study,
tmt is estimated from the approximated left ventricle driver function, driL, but would
normally be measured from end of the ‘P’ wave in the ECG or the ‘a’ wave in the
central venous pressure (CVP), both of which are normally measured.
Another important feature available from the measured data is the maximum gradient
or inflection point of the ascending section of the aortic pressure waveform,
dPao,max, true
dt
. In the systemic and six-chamber models, the parameter Rav has a
significant effect on the gradient of the ascending aortic pressure inflection point.
With all other parameters held constant, changes in Rav cause inversely proportional
changes in the maximum aortic pressure gradient. For example, as Rav decreases, flow
into the aorta (Qav) increases as shown by Equation (3). The increased Qav causes the
aortic volume to increase at a faster rate (Equation (16)), especially at the start of
ejection, resulting in a sharper increase in the aortic pressure (Equation (10)). Hence,
identification of Rav is achieved using the formula:
 dPao,max, approx 

Rav,new  
dt


where
dPao,max, approx
dt
 dPao,max, true 

 Rav,old
dt


(35)
is the maximum ascending gradient of the model output Pao. With
the new approximations for Ppu and Rav, the parameters Rmt, Eao, and Rsys are reidentified.
dPao,max, approx
dt
This
overall,
added
iterative
process
is
repeated
until
converges to the measured data and Ppu stops changing between
iterations, in both cases to a set tolerance of 0.5%.
Throughout identification of the systemic model, an estimate is used for the left
ventricle end systolic elastance (Ees,lvf), seen in Table 4, as this parameter cannot be
identified at this stage. However, this parameter is approximated later, along with the
right ventricle end systolic contractility (Ees,rvf), once the other parameters of the
pulmonary model have been identified.
3) Pulmonary Model Identification: The model inputs, model outputs, and discrete
measurements (calculated from the measured waveforms), which are used to identify
the pulmonary model, are defined by PI, PO, and PM, where P stands for pulmonary
model, as outlined by Equations (36)-(38). Table 5 shows the initial parameter inputs
for the pulmonary model.
PI  Pvc , Rtc , E es ,rvf , R pv , E pa , Rsys , Ppu 
(36)
PO  Qtc , Prv ,Vrv , Q pv , Ppa ,Vao , Q pul, 
(37)
dPpa,max, true


PM  Ppa,mean,true , PPpa,true ,
, SVtrue , ttc 
dt


(38)
Identification of the pulmonary model is achieved in a similar fashion to the systemic
model so only a brief description is given here. During this process, Ees,rvf is held
constant at its initial value (see Table 5), and is identified, along with Ees,lvf, once the
other parameters of the simplified models have been identified. First, the right
ventricle driver function, driR is identified from features in the pulmonary artery
waveform. Then Equations (39) – (40), analogous to Equations (31)-(33), are used to
identify Rtc, Epa, and Rpul:
 SVrv,approx 
 Rtc,old
Rtc,new  
 SVtrue 
where
 PPpa,true 
 E pa,old
E pa,new  
 PP

 pa,approx 
where
 Ppa,mean,true 
 R pul,old
R pul,new  
P

pa
,
mean
,
approx


where
SVrv,approx  max( Vrv )  min( Vrv )
(39)
PPpa  max( Ppa )  min( Ppa )
(40)
Ppa,mean 
max( Ppa )  min( Ppa )
2
(41)
Once these parameters converge, Pvc and Rpv are calculated using:
Pvc  Prv (ttc )
 dPpa,max, approx 

R pv,new  
dt


 dPpa,max, true 

 R pv,old
dt


(42)
(43)
where
dPao,max, approx
dt
is the maximum ascending gradient of the model output Ppa, and
these equations are analogous to Equations (34)-(35).
4) Identifying Ventricular Contractility: The final parameters to be identified for the
systemic and pulmonary models are Ees,lvf and Ees,rvf. In estimating Ees,lvf it is assumed
that: the parameter Rav has been identified, the model outputs Pao match the measured
data, and the systemic model stroke volume (SVlv,approx) has converged to the
measured stroke volume (SVtrue). By analysing ohms law for fluid flow (pressure =
flow·resistance or P=Q·R), the flow through valve (Q), a function of SVlv,approx
multiplied by the resistance (Rav) will give a good approximation of the pressure drop
across the aortic valve, ΔPav. Therefore, the model should intrinsically output a
relatively accurate systolic Plv profile, independent of Ees,lvf, as Plv ≈ Pao + ΔPav. Given
that Plv is already known, changes in Ees,lvf must trade off with left ventricle volume,
as constrained by Equation (11), ignoring the passive elastic recoil effects of the
ventricle. For example, if the identified Ees,lvf is too low then modelled left ventricle
volume, Vlv will be too large. Hence, knowledge of the true left ventricle volume can
be used to pinpoint the correct Ees,lvf. However, left ventricle volume is rarely
measured. Instead, global end diastolic volume (GEDV) is used to identify the sum of
the ventricular elastances (Ees,sum = Ees,lvf + Ees,rvf):
Ees ,sum,new 
GEDVapprox
GEDVtrue
Ees ,sum,old
where GEDVapprox  max( Vlv )  max( Vrv )
(44)
where GEDV approximately equals the sum of the left and right ventricle end
diastolic volumes. GEDV may be readily derived using a transpulmonary
thermodilution.
Inotropes indiscriminately affect both sides of the heart. To approximate the
individual left and right ventricular contractilities (Ees,lvf and Ees,rvf) it is assumed that
any inotropic effects act evenly over the whole myocardium so the ratio of the
contractilities stays constant over time. In other words, the percentage change in left
ventricle elastance (ΔEes,lvf/Ees,lvf) is equal to the percentage change of the right
ventricle elastance (ΔEes,rvf/Ees,rvf) over the same time period. Using this assumption,
Ees,lvf is split from Ees,sum using a ratio of the elastances, CE, which stays constant for
each individual. The ratio CE is identified for each measurement set using a ratio of
the afterloads and modelled vena cava pressure, Pvc:
CE 
Pao,mean,true  Pvc
Pao,mean,true  Ppa,mean,true

E es ,lvf
E es ,lvf  E es ,rvf
 const
(45)
This relationship is based on the Anrep effect [23-25] where increases in myocardial
contractility are related to increases in afterload represented by Pao,mean,true on the left
heart and Ppa,mean,true on the right heart. A mean elastance ratio, CE,mean, is averaged
from the set of CE’s found for each animal trial. In calculating CE,mean, only CE’s
greater than 0.6 (Ees,lvf/Ees,rvf > 1.5) are used to find the mean. This physiological
bound ensures that the contractility of the left ventricle is always greater than the right
ventricle contractility [26-27]. Once CE,mean is calculated Ees,lvf and Ees,rvf are derived
from Ees,sum:
E es ,lvf  C E ,mean E es ,sum
(46)
E es ,rvf  E es ,sum,new  E es ,lvf
(47)
The method for identifying the end systolic ventricle elastances is iterative and starts
with initial guesses for Ees,lvf and Ees,rvf as shown in Tables 4 and 5, which are used to
converge the systemic and pulmonary models. Once both have converged, the
modelled ventricular volumes (Vlv, Vrv) are used to calculate GEDVapprox. The
parameters Ees,lvf and Ees,rvf are then updated using Equations (44), (46), and (47) and
the parameters for the systemic and pulmonary models are re-identified. This process
is repeated until GEDVapprox has converged.
5) Calculating Ventricular Interaction and Pericardium Pressure: Initially, septum
(Vspt) and pericardium dynamics (Ppcd) are set to zero in the systemic and pulmonary
models. However, as new approximations for Ees,lvf and Ees,rvf are identified, Vspt and
Ppcd are calculated using Equations (19) and (23). Vspt and Ppcd are added to the
simplified models during the next iterative step in the identification process, thus
introducing ventricular interaction between the models. An in depth description on the
modelling and calculation of Vspt and Ppcd can be found at Smith et al. [16].
6) Identifying Valve Resistances: One of the major problems with identifying subjectspecific parameters is inter-beat variability in the measured data. This variability is
problematic when identifying the valve resistances (Rmt, Rav, Rtc, Rpv), which are
highly sensitive to small changes in the measured data. However, physiologically,
valve resistance stay constant between adjacent beats. To enforce constant valve
resistances between beats, the simplified models are identified using several different
periods of the measured data. The valve resistances identified for each set of
measured data are stored and averaged. These averaged valve resistances are then
fixed and used to re-identify the other parameters of the simplified models for each set
of the measured data. Equations (31), (35), (39), and (43) are no longer needed and
Equations (34) and (42) are replaced with:
Ppu,new 
SVlv,approx
Pvc ,new 
SVrv,approx
SVtrue
SVtrue
Ppu,old
(48)
Pvc ,new
(49)
so that the estimated tmt and ttc measurements, obtainable from ECG, are no longer
required in the identification process.
7) Identifying the Remainder of the Six-chamber Model: The six-chamber model is
the combination of the identified systemic and pulmonary models, plus two venous
chambers representing the vena cava and the pulmonary vein. To fully define the
model of Figure 1, two further parameters are required, the vena cava and pulmonary
vein elastances (Evc, Epu). The other parameters, already identified for the systemic
and pulmonary models, are fixed during identification of the six chamber model, and
remain unchanged for the remainder of the identification process. To identify Evc and
Epu the pulmonary vein pressure, Ppu, identified from the systemic model, is held
constant in the six-chamber model while the vena cava pressure, Pvc, is allowed to
vary. The six-chamber model is simulated with initial guesses for Evc and Epu.
Changes in the six-chamber model output Pvc,6 are compared to the identified
parameter, Pvc,simple, from the simplified model’s to calculate a better approximation
for Evc using:
 Pvc ,simple 
 E vc ,old
E vc ,new  

 mean( Pvc ,6 ) 
(50)
The model is re-simulated with the altered Evc to produce a new Pvc,6. As a secondary
effect of altering Evc, the simulated pulmonary volume waveform, Vpu, changes, which
is utilised to identify Epu:
E pu, new 
Ppu , simple
mean(V pu )
(51)
This process of optimising Evc and Epu is repeated until the mean six-chamber vena
cava pressure, Pvc,6 equals the Pvc,simple.
8) Summary of Identification Process: A highly iterative process, involving six
feedback loops is used to identify the parameters of the six-chamber model. This
model identification system was run on a 2.13 GHz dual core machine with 3GB of
ram. Since this research is still in the development stages it has been created using
development-orientated but relatively slow Matlab software (MathWorks, Natwick,
MA, USA). Using one processor the identification method took on average 6 minutes
and 24 seconds to identify a subject-specific model of the CVS. Preliminary tests
using C programming language, which is better suited for real time applications, have
suggested the identification process time can be reduced by a factor of 100, to
approximately 3 seconds per identified model, which is an acceptable run time in a
clinical environment.
The approach has been changed dramatically from that in Starfinger et al. [12], as the
method now only requires a minimal set of measurements that are typically available
in the ICU, whereas Starfinger et al. [12] required a larger set of invasive
measurements and assumptions based on population trends. Hence, this approach is
far more general. The step-wise process, including iterative feedback loops, of the
new algorithm is summarised in Table 6.
D.
Analyses
The model identification process was tested with 46 sets of porcine data from five
pigs with induced pulmonary embolism [8, 13]. The identification process was
validated by comparing the measured mean left and right ventricular volumes and
maximum left and right ventricular pressures to the model outputs (Plv, Vlv, Prv, Vrv).
These measurements were not used in any way to identify the CVS model. Although
they were recorded in the pig trials, they are not usually measured in critical care.
Hence, they represent true, independent validation tests.
Results and Discussion
A.
Method Results and Validation
The outputs of the six chamber model are compared to the experimentally measured
data to validate the model identification process. Firstly, the model outputs are
compared to the measurement set points used in the identification process to check the
model has converged correctly. An example of the identification process fitting the
modelled mean and amplitude of the aortic and pulmonary artery pressures to the
measured data is shown in Figure 3 for Pig 8, indicating accurate convergence.
Secondly, the model outputs are compared to measurements not used to identify the
CVS models, such as the left and right ventricular pressure and volume waveforms
(Plv, Vlv, Prv, Vrv), providing a “true” validation of the identification method. Figures 4
and 5 show the model outputs (Plv, Vlv, Prv, Vrv) predicting the measured left and right
ventricle pressures and volumes, which were not used in the identification process, for
Pig 8 at 30, 120, and 210 minutes into the trial. The simulated outputs for the median
left and right ventricle volumes (Vlv, Vrv) and maximum left and right ventricle
pressures (Plv, Prv) lie within absolute error ranges of 4.1% to 15.1%. Table 7
summarises the median absolute error results for all five subject-specific identified
models.
Table 7 shows small errors in the mean Pao and Ppa, as expected as these model
outputs were matched to the measured data during the identification process. Larger
errors are seen for the simulated maximum Plv and Prv, especially for pigs 1 and 2.
These errors arise from measured differences in stroke volume between the left and
right ventricles, which are not accounted for during identification method presented.
In particular, the identification process assumes the stroke volume of both ventricles
is equal at steady state so the systemic and pulmonary models are matched to an
averaged stroke volume. However, in the measured data for pigs 1, 2, 7, and 9 there is
a significant difference between the left and right ventricle outputs, with the ratio of
the left to right stroke volume larger than 2:1 at some stages during the trials. Such a
large ratio is not physiological and is likely due to errors in the measurement of the
right ventricle volume and right ventricle stroke volume due to its complex shape. The
large difference in stroke volumes causes the model to underestimate the left ventricle
pressure and overestimate the right ventricle pressure resulting in the errors seen in
Table 7. However, the ratio of the stroke volumes does stay relatively constant over
the duration of the porcine trials, leading to a systematic error in the Plv and Prv
maximums. Importantly, these results indicate that trends identified by subjectspecific models will still be accurate.
The variation in shape and phase difference between the modelled outputs and the
measured waveforms, as seen in Figures 3 to 5, are due to the lumped parameter
nature of the model and errors in the approximated driver functions (driL and driR)
used to describe the myocardial contraction. Figure 6 shows the difference between
the approximated and true driver functions (driR, driL) for Pig 8 at 30 minutes into
the trial, where the true driver functions are defined using measured data as:
driLtrue 
driR true 
Plv,true / Vlv,true  Vd ,lvf 
Plv,true / Vlv,true  Vd ,lvf 
Prv,true / V rv,true  V d , rvf
Prv,true / V rv,true  V d , rvf
(52)


(53)
Note that these true driver functions would not be available in a clinical scenario. The
relatively small appearing differences are enough to make significant changes in the
resulting pressures and volumes.
The experimental errors associated with the pressure measured via the high fidelity
micromanometer-tipped catheters, used in this study, are less than 1 mmHg as error
due to measurement drift is minimal due to the short time scale of the experiments.
However, there are larger errors associated with the left (~10% [28]) and right (~20%
[29]) ventricular volume experimental measurements, measured using the
conductance catheter method. In the ICU, less accurate, fluid-filled catheters are
generally used to measure arterial and pulmonary artery pressure, with a maximum
frequency response around 10 Hz. Furthermore, stroke volume can normally be
measured continuously to a percentage error of less than 10% using currently
available methods [30]. Validation tests showed that the subject-specific models are
capable of matching the measurements taken experimentally to median absolute
percentage errors of 4.3% and 4.4% for the mean ventricular volumes and maximum
ventricular pressure. These errors are comparable to the measurement errors of the
current hemodynamic devices used in the ICU. Clinically, the actual errors of the
whole system will be the combination of the errors associated with the identified
models plus the measurement errors, as the subject-specific models would be matched
to clinically available measurements including points from the arterial and pulmonary
artery pressures, ECG, CVP, GEDV, and CO. In practice, the discrete measurements
required by the system can be averaged over several heartbeats to reduce
measurement noise. However, measurement drift cannot be accounted for, but as
long as the rate of drift is slow the measurements should still reflect the change in
state of the patient over short time frames. Hence, more importantly, the model
identification process will still be able to capture the main hemodynamic trends and
acute cardiovascular changes in the patient.
The model identification process outlined in this paper improves on previous work [912] through identifying parameters using a proportional control using discrete inputs,
rather than numerical integration requiring complete measured waveforms. The
currently identified models of the CVS are matched directly to information commonly
used in the ICU like the diastolic and systolic pressures, and stroke volume, instead of
abstract features like the area under aortic pressure waveform. Hence, these models
provide a more clinically relevant picture of a patient’s cardiovascular health.
Furthermore, the use of simplified models in an intermediary identification step,
instead of identifying the whole model at once, increases the convergence stability of
the method because fewer parameters are identified at any one time.
The combination of the simplified models and proportional gain-based iteration
enables identification of the six-chamber CVS model with a minimal set of input
measurements. In the case of identifying the left and right ventricular elastances (Ees,lvf
and Ees,rvf), where little cardiac information was assumed known, an empirical relation
is used, as defined by Equations (45) and (46), to provide missing information to help
identify these parameters. Figure 7, created from identifying CVS porcine models of
pulmonary embolism and septic shock using known ventricular volumes, indicates
how the elastance ratio from Equation (45) relates Ees,lvf to Ees,sum, which is identified
using Equation (44). This empirical derived relationship (R=0.89) was used to create
Equation (46). Although applying the empirical relation requires extra iterations
during the identification process it enables accurate approximation of left and right
ventricular function without the need for highly invasive measures, such as cardiac
catheterisation combined with a vena cava occlusion manoeuvre or expensive
equipment needed for echocardiography. Importantly, these latter methods, while
accurate, do not provide continuous measurements, further limiting their utility.
The benefit of using the parameter identification method is that cardiovascular
physiology embedded in the CVS model, combined with measurements from the ICU,
can be utilised to reveal unknown information of a patient’s CVS. Hence, key
diagnostic information only available from the identified model, such as ventricular
pressure-volume loops, can be used to assist medical staff in diagnosis and therapy
selection. Furthermore, the model parameters represent physiological CVS
characteristics with diagnostic power. Thus, the model can provide a clearer picture of
circulatory status than is currently available from cardiovascular monitors.
B. Model Limitations
Currently, the six-chamber model assumes that flow through the heart valves only
occurs in the forward direction. However, during valvular insufficiency backwards
flow is possible through the effected valve of the patient. Therefore, the model at its
present state is unable to identify valvular dysfunctions, such as mitral and aortic
regurgitation.
The model also assumes Vd,lvf = Vd,rvf = 0, parameters of the left and right ventricular
end systolic pressure volume relationship (ESPVR), though in most cases this
assumption is not accurate. When the true Vd,lvf or Vd,rvf ≠ 0, Ees,lvf and Ees,rvf will not
quantitatively represent the gold standard definition of ventricular contractility as
defined by Sugawa et al. [31]. However, physiologically Vd,lvf and Vd,rvf stay relatively
constant over the range of normal loading conditions and short time periods.
Therefore, as the CVS model is a lumped parameter model the effects of a non-zero
Vd,lvf or Vd,rvf will be taken into account during the identification of Ees,lvf and Ees,rvf. In
other words, if there is a sharp increase in the contractile state of the heart, a sharp
increase will be noticed in the identified Ees,lvf and Ees,rvf. So, importantly, although the
value of modelled Ees,lvf and Ees,rvf may differ from the gold standard measurement the
qualitative trends of these parameters should still represent the changes in the
patient’s state. Preliminary testing has indicated that changes in the estimated driver
function may be used to identify Vd, which could be incorporated in future versions of
the modelling identification method.
Further limitations in identifying the ventricular contractilities arise from the
assumption that the ratio of the contractilities, CE, stays constant over the duration of
the trial. It would seem logical that if one side of the heart was badly damaged or
highly distended then the relationship between Ees,lvf and Ees,sum may no longer be
linear and the ratio may not stay greater than 0.6. It should be noted that such patients
might well be close to death. No useful literature was found comparing inotropic
effects on both the left and right ventricle simultaneously. Hence, the assumption of a
constant CE was based off measurements from the five pigs used in this study and five
pigs from an investigation into septic shock [9] where CE was found to stay relatively
constant in this range for each animal over a wide range of inotropic states and
loading conditions (see Figure 7). For use in humans, this ratio will have to be further
confirmed in trends that controllably change the inotropic state of the heart with
adrenaline or a similar infusion.
Improvements to the model could be made through use of relationships to predict the
pulmonary artery pressure (PAP) and CO. For example there is a very high correlation
between dicrotic notch and mean PAP [32]. In other words, the time point at the mean
PAP predicts the end of right ventricle ejection. This correlation is not present in the
cardiac model, so it effectively provides an extra measurement that could help
identify PAP. In additions, studies [33-34] have shown that there is strong relationship
between PAP and right ventricle ejection fraction (RVEF) which could also be used to
identify further pulmonary parameters. These relationships would enable the
identification algorithm to predict the cardiac status of wider range of patients with
fewer measurements needed.
Conclusion
An accurate method for identifying a patient’s time varying hemodynamic state has
been developed and tested on a porcine model of pulmonary embolism. Importantly,
only information typically available on existing ICU monitors is required, which is a
substantial reduction in measurements from prior work. The model can be used to
infer information on the left and right ventricle pressure volume loops, which would
be important when assessing cardiac status and the impact of inotropes or other drug
therapies. True validation comparing model outputs to measurements not used to
identify the model had median absolute errors of 4.3% and 4.4% for the mean
ventricular volumes and max ventricular pressures, which is within measurement
errors. This approach is now at a stage where it can be readily implemented in a
critical care environment, but requires validation in human trials which have begun.
Pvc
Ppu
Rsys
Qsys
Eao
Rpul
Epa
P, V
Aorta
Qpul
P, V
Pulmonary
Artery
Aortic
Valve
Rav
Qav
Pulmonary
Valve
Rpv
Qpv
Ees,rvf
Ees,lvf
P, V
Left
Ventricle
P, V
Right
Ventricle
Tricuspid
Valve
Mitral
Valve
Rmt
Qmt
Rtc
Qtc
Ppu
Pvc
(a)
(b)
Figure 2 – (a) Simplified model of the systemic system and (b) simplified model of
the pulmonary system with inertia, septum interaction, and pericardium dynamics
removed.
Pao (mmHg)
150
150
150
140
140
140
130
130
130
120
120
120
110
110
110
100
100
100
90
90
90
80
80
80
Ppa (mmHg)
70
0
0.2
0.4
t (sec)
0.6
70
0
0.2
0.4
t (sec)
0.6
70
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
0
0.2
0.4
t (sec)
30 minutes
0.6
10
0
0.2
0.4
t (sec)
120 minutes
0.6
10
0
0.2
0.4
t (sec)
0.6
0
0.2
0.4
t (sec)
0.6
210 minutes
Figure 3: Comparison of the modelled to measured aortic and pulmonary pressure
waveforms at for Pig 8 at 30, 120, and 210 minutes into the trial (the dashed line
shows the measured waveform and the solid line the CVS model output).
150
150
100
100
100
50
50
50
lv
P (mmHg)
150
0
0
0
0.2
0.4
0.6
0
0
0.2
lv
V (mmHg)
t (s)
0
90
80
80
80
70
70
70
0
0.2
0.4
60
0.6
0
0.2
0.4
60
0.6
150
100
100
100
50
50
50
70
Vlv (ml)
30 mins
80
90
0
50
60
0.4
0.6
0.2
0.4
0.6
t (s)
150
60
0
t (s)
150
0
50
0.2
t (s)
90
t (s)
lv
0.6
90
60
P (mmHg)
0.4
t (s)
70
Vlv (ml)
120 mins
80
90
0
50
60
70
Vlv (ml)
80
90
210 mins
Figure 4: Comparison of the modelled to measured left ventricle pressure waveform,
volume waveform, and pressure-volume loops for Pig 8 at 30, 120, and 210 minutes
into the trial (the dashed line shows the measured waveform and the solid line the
CVS model output).
(mmHg)
rv
60
60
40
40
40
20
20
20
P
60
0
0
0
0.2
0.4
0.6
0
0
0.2
t (s)
0.4
0.6
0
100
90
90
90
80
80
80
70
70
70
60
60
60
50
50
(mmHg)
0.4
0.6
0.4
0.6
t (s)
100
V
rv
0.2
t (s)
100
0
0.2
0.4
0.6
0
0.2
0.4
0.6
50
0
0.2
t (s)
t (s)
60
60
60
40
40
40
20
20
20
0
0
0
P
rv
(mmHg)
t (s)
60
70
80
Vrv (ml)
90
30 mins
60
70
80
Vrv (ml)
120 mins
90
60
70
80
Vrv (ml)
90
210 mins
Figure 5: Comparison of the modelled to measured right ventricle pressure
waveform, volume waveform, and pressure-volume loops for Pig 8 at 30, 120, and
210 minutes into the trial (the dashed line shows the measured waveform and the solid
line the CVS model output).
1
0.8
0.8
0.6
0.6
driL
driR
1
0.4
0.4
0.2
0.2
0
0
0.2
0.4
t(s)
0.6
0
0
0.2
0.4
t(s)
0.6
Figure 6: Comparison of the true to approximated left ventricle (left) and right
ventricle (right) driver function for Pig 8 at 30 minutes into the trial.
4
3.5
3
Ees,lvf
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
C .E
E
es,sum
Figure 7: Left ventricle end systolic elastance (Ees,lvf) compared to the product of the
elastance ratio (CE), from Equation (42), and the sum of the end systolic elastances
(Ees,sum), from CVS models identified using known ventricular volumes. The dashed
line shows the line of unity.
Table 1: Measurements used to identify the parameters.
Identified Parameters/Waveforms
Symbol
Ees,lvf
Measurements used to identify
parameters
Symbol
Description
GEDV
Global end diastolic volume
Eao
Description
Left ventricle end systolic
elastance
Aortic elastance
Evc
Vena cava elastance
Pvc
Ees,rvf
GEDV
Epa
Right ventricle end systolic
elastance
Pulmonary artery elastance
Epu
Pulmonary vein elastance
Pvc
Rmt
Rav
Mitral valve resistance
Aortic valve resistance
SV
dPao,max
Rsys
Rtc
Rpv
Systemic vascular resistance
Tricuspid resistance
Pulmonary valve resistance
dt
Pao,mean
SV
dPpa,max
PPao
PPpa
dt
Rpul
Pulmonary vascular resistance
Ppa,mean
Pvc
Vena cava pressure
ttc, SV
Ppu
Pulmonary vein pressure
tmt, SV
driL
Left ventricle normalised time
varying elastance
Pao,
GEDV
driR
Right ventricle normalised
time varying elastance
Ppa,
GEDV
Amplitude of aortic
pressure
Modelled Vena cava
pressure
Global end diastolic volume
Amplitude of pulmonary
artery pressure
Modelled vena cava
pressure
Stroke volume
Maximum ascending aortic
pressure gradient
Mean aortic pressure
Stroke volume
Maximum ascending
pulmonary artery pressure
gradient
Mean pulmonary artery
pressure
Tricuspid valve closure
time and stroke volume
Mitral valve closure time
and stroke volume
Aortic pressure waveform
and global end diastolic
volume
Pulmonary artery pressure
and global end diastolic
volume
Table 2: CVS model constants (from [16, 21]).
Symbol
P0,lvf
λlvf
Vd,lvf
P0,rvf
λrvf
Vd,rvf
Vd,spt
V0,spt
λspt
P0,spt
Ees,spt
P0,pcd
V0,pcd
λpcd
Pth
Vd,ao
Vd,vc
Vd,pa
Vd,pu
Lmt
Lav
Ltc
Lpv
Description
Parameter of left ventricle EDPVR
Parameter of left ventricle EDPVR
Parameter of left ventricle ESPVR
Parameter of right ventricle EDPVR
Parameter of right ventricle EDPVR
Parameter of right ventricle ESPVR
Parameter of septal ESPVR
Parameter of septal EDPVR
Parameter of septal EDPVR
Parameter of septal EDPVR
Septum elastance
Parameter of pericardium
Parameter of pericardium
Parameter of pericardium
Intrathoracic pressure
Aortic unstressed volume
Vena cava unstressed volume
Pulmonary artery unstressed volume
Pulmonary vein unstressed volume
Mitral valve inertance
Aortic valve inertance
Tricuspid valve inertance
Pulmonary valve inertance
Value
0
0.033
0
0
0.023
0
2
2
0.435
1.1101
48.7540
0.5003
200
0.03
0
0
0
0
0
0
0
0
0
Table 3: Modelled and measured hemodynamic waveforms.
Symbol
Vlv
Vao
Vvc
Vrv
Vpa
Vpu
Plv
Pao
Pvc
Prv
Ppa
Ppu
Qmt
Qav
Qsys
Qtc
Qpv
Qpul
Vspt
Ppcd
Description
Left ventricle volume
Aorta volume
Vena cava volume
Right ventricle volume
Pulmonary artery volume
Pulmonary vein volume
Left ventricle pressure
Aorta pressure
Vena cava pressure
Right ventricle pressure
Pulmonary artery pressure
Pulmonary vein pressure
Mitral valve flow rate
Aortic valve flow rate
Systemic flow rate
Tricuspid valve flow rate
Pulmonary valve flow rate
Pulmonary flow rate
Septum volume
Pericardium pressure
Measured
✓
✓
✓
✓
✓
✓
Table 4: Initial systemic model parameter inputs
Parameter
Initial Value
Ppu
5
Rmt
0.05
Ees,lvf
2
Rav
0.04
Eao
2.5
Rsys
2.5
Pvc
5
Table 5: Initial pulmonary model parameter inputs
Parameter
Initial Value
Pvc
5
Rtc
0.04
Ees,rvf
0.8
Rpv
0.03
Epa
2.1
Rpul
0.4
Ppu
From SI
Table 6: Overview of model identification method
Step 1: Input set of measured data.
Step 2: Approximate left and right ventricle driver functions.
Step 3: Estimate an initial set of input parameters for systemic and pulmonary
models as shown in Table 4 and 5.
Step 4: Identify systemic model of Figure 2 (a).
Step 4.1: Simulate systemic model.
Step 4.2: Identify Rmt, Eao, and Rsys with Equations (31)-(33).
Step 4.3: Re-simulate the systemic model with new parameters.
Step 4.4: If SVlv,approx, PPao,approx, and Pao,mean,approx have converged within
a tolerance of 0.5% go to Step 4.5 otherwise go back to Step
4.2.
Step 4.5: Identify Ppu and Rav with Equations (34) and (35).
Step 4.6: If Ppu and dPao. max, approx / dt have converged within a tolerance of
0.5% go to Step 5 otherwise go back to Step 4.2.
Step 5: Identify pulmonary model.
Step 5.1: Simulate pulmonary model.
Step 5.2: Identify Rtc, Epa, and Rpul for the Equations (39)-(41).
Step 5.3: Re-simulate the pulmonary model with new parameters.
Step 5.4: If SVrv,approx, PPpa,approx, and Ppa,mean,approx have converged within
a tolerance of 0.5% go to Step 5.5 otherwise go back to Step
5.2.
Step 5.5: Identify Pvc and Rpv with Equations (42) and (43).
Step 5.6: If Pvc and dPpa,max approx / dt have converged within a tolerance of
0.5% go to Step 6 otherwise go back to Step 5.2.
Step 6: Identify Ees,lvf and Ees,rvf with Equations (44), (46), and (47) and
calculate Vspt and Ppcd with Equations (24) and (26). If GEDVapprox,
Vspt, and Ppcd have converged within a tolerance of 0.5% go to Step 6
otherwise go back to Step 4.
Step 7: Repeat Steps 1 to 6 with different sets of the measured data and store
and average the identified valve resistance.
Step 8: Repeat Steps 1 to 6 using the averaged/fixed valve resistances (ie
without identifying Rmt, Rav, Rtc and Rpv) and Equations (48) and
(49).
Step 9: Identify six-chamber model.
Step 9.1: Simulate six-chamber model.
Step 9.2: Identify Evc and Epu with Equations (50) and (51).
Step 9.3: Re-simulate six-chamber model with new parameters.
Step 9.4: If Pvc has converged within a tolerance of 0.5% go to Step 10
otherwise go back to Step 9.2.
Step 10: Output six-chamber parameters and model outputs
Table 7: Median and 90th percentile absolute percentage errors of the model outputs
compared to measured data. Pao,mean and Ppa,mean are used in the identification process,
while Vlv,mean, Vrv,mean, Plv,max, and Prv,max, were not and are true validations.
Ouput
Pao,mean
Ppa,mean
Vlv,mean
Vrv,mean
Plv,max
Prv,max
Error
Median
90th
Median
90th
Pig 1
0.2
0.5
0.1
0.5
Pig 2
0.1
0.1
0.1
0.1
Pig 7
0.2
0.4
0.1
0.1
Pig 8
0.1
0.5
0.0
0.1
Pig 9
0.1
0.2
0.1
0.1
All
0.1
0.4
0.1
0.4
Median
90th
Median
90th
Median
90th
Median
90th
3.3
15.9
6.5
14.6
11.4
27.8
15.5
18.3
18.2
22.8
15.8
18.9
9.2
23.5
30.6
31.6
4.3
7.0
4.3
7.4
2.7
4.5
3.7
15.6
2.5
6.5
2.6
7.0
1.5
3.1
2.7
5.7
1.9
4.6
1.9
6.9
1.7
2.5
18.4
23.3
4.1
17.4
4.4
15.3
2.1
20.5
15.1
27.2
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