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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. Vpv Q pul Qmt (13) Vpa Q pv Q pul (14) Vvc Qsys Qtc (15) Vao Qav Qsys (16) Vrv Heaviside Qtc Qtc Heaviside Q pv Q pv (17) Vlv 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 VI. References [1] W. Rosamond, K. 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