- Email: [email protected]
A Hybrid Model of Interacting Physiological Systems
8th IFAC Symposium on Biological and Medical Systems The International Federation of Automatic Control August 29-31, 2012. Budapest, Hungary
A Hybrid Model of Interacting Physiological Systems Jörn Kretschmer*/** , Zhanqi Zhao*, Thomas Haunsberger*, Erick Drost*, Edmund Koch**, Knut Möller* *Institute of Technical Medicine, Furtwangen University, 78054 Villingen-Schwenningen, Germany (Tel: +49-7720-307-4370; e-mail: [email protected]). ** Faculty of Medicine Carl Gustav Carus, Dresden University of Technology, 01307 Dresden, Germany
Abstract: Mathematical models can be exploited to simulate physiological processes in the human body as well as predict their reaction to changes in the therapy regime of a patient. Implementing these models in Medical Decision Support Systems (MDSS) can help in optimizing therapy settings. MDSS optimizing ventilation therapy in critically ill patients should not only consider respiratory mechanics but extend simulations to also consider other parts of the human body. A previously presented framework allows combination of three model families (respiratory mechanics, cardiovascular dynamics and gas exchange) to provide a broader picture when predicting the outcome of a therapy setting. The three model families are dynamically combined to form a complex model system with interacting submodels. The framework computes the combined submodels as a tightly coupled system, i.e. an applied solver algorithm chooses a step size to fit the submodel with highest system dynamics. Tests revealed that simulation time increases rapidly with rising system complexity, i.e. the number of differential equations defining the interacting model system. This is due to very expensive computing when detailed models of cardiovascular dynamics are included. However, this simulation detail is not necessary in all scenarios. Thus, a simplified cardiovascular model that is able to reproduce basic physiological behavior is introduced. This model consists only of difference equations and does not require special algorithms to be numerically solved. The model is based on the beat-to-beat model presented by DeBoer et al. and has been extended to react to intrathoracic pressure levels that are present during mechanical ventilation. To include sensitivity to mechanical ventilation, a 19-compartment model as proposed by Leaning et al. has been analyzed and model behavior has been implemented as a simple equation into the beat-to-beat model. Tests showed, that the model is able to closely represent general model behavior compared to the 19compartment model. Blood pressures were calculated with a maximum deviation of 1.8%, leading to a simulation error of 1.7% in cardiac output. Combination with a gas exchange showed maximum simulation error of 0.4%. Therefore, the proposed model is able to be used in combinations where cardiovascular simulation does not have to be detailed. Computing costs have been decreased dramatically by factor 186 compared to a model combination employing the 19-compartment model. Keywords: Mathematical models, beat-to-beat model, physiological simulations. settings by running several simulations of the same mathematical model.
1. INTRODUCTION Mathematical models can be employed to understand the human body and its physiological processes. Physiological reactions of a patient to changes in the therapeutic regime can even be predicted by evaluating simulation results of these mathematical models. Medical Decision Support Systems (MDSS) exploit these simulations to optimize the applied therapeutic strategy. In mechanically ventilated patients a model based optimization may be employed to avoid ventilator induced lung injuries (VILI) e.g. by applying an optimized ventilation pressure. To find the optimal therapeutic setting, an MDSS needs to test different therapic
978-3-902823-10-6/12/$20.00 © 2012 IFAC
However, this optimization procedure should not be based on one single mathematical model, e.g. respiratory mechanics. In mechanically ventilated patients not only lung tissue but also gas exchange and cardiovascular dynamics are affected by the therapy. Thus, these aspects might also be of importance when evaluating the reactions of the patient to a certain therapic setting. Therefore, they need to be included in the simulation. Including several aspects of human physiology allows an MDSS to base the optimization on a broader picture. Moreover, therapic measures proposed by the MDSS may be improved if a set of models with differing simulation focus is provided. This allows individual adaptation to the
290
10.3182/20120829-3-HU-2029.00113
8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
patient’s disease further optimizing the MDSS’s performance. In order to adapt a specific model to a patient’s properties, model parameters need to be fitted to their physiological correlates. Establishing a hierarchical order of the employed model families (e.g. respiratory mechanics, gas exchange, etc.) supports this procedure (Schranz et al.). A model library containing three different model families (respiratory mechanics, gas exchange and cardiovascular dynamics) has been established before (Kretschmer et al., 2011a). A specially designed framework allows to generate different combinations of these model families and guarantees that the employed mathematical models are able to interact during simulation. These complex model systems are not precompiled, i.e. models are combined dynamically. Each of the implemented model families offers different model versions varying in simulation focus and model detail. Interfaces implemented in the framework allow to exchange parameters between models, thus enabling interaction. E.g., through these interfaces, parameter values for cardiac output ( ), intrathoracic pressure ( ), alveolar volume ( ) and air flow ( ) are exchanged between the models. Ventilator settings (inspiratory gas fractions of oxygen and carbon dioxide , , inspiratory air flow , respiratory , frequency ) are used as input to the overall system. Figure 1 shows the interfaces as implemented in the previously proposed framework. Synchronized parameter exchange is ensured by tightly coupling the interacting models, i.e. they are computed with an identical step size. This step size has to fit the model with highest system dynamics in order to keep the simulation numerically stable. Thus, models with lower dynamic behaviour are computed with a step size smaller than necessary. Tests showed that this is computationally expensive with computing time growing rapidly with the number of differential equations describing the models. An MDSS usually conducts optimization of therapy based on a number of simulations to evaluate different settings. Based on the results of these simulations the MDSS is able to find a . Vin fR Respiratory mechanics
Ventilator settings
. VA VA
Pth
Fi,O2 Fi,CO2 Gas exchange CO
Cardiovascular dynamics
Fig. 1. Model interactions and interfaces in the previously presented framework (Kretschmer et al., 2011). Interface parameters are gas fractions of oxygen and carbon dioxide ( , , ), inspiratory air flow ( ), respiratory , frequency ( ), intrathoracic pressure ( ), cardiac output ( ), air flow ( ) and alveolar volume ( ).
setting that complies most with the therapy goals defined by the clinician. Since these results need to be calculated in a reasonable time, computations need to be efficient. Thus, the MDSS needs to be able to choose a model combination that is both fitting the patient’s situation and is computationally feasible. The previously presented model family of cardiovascular dynamics (Kretschmer et al. 2011, 2012) did not include any cardiovascular model that is both reactive to intrathoracic pressure and is of low complexity. Therefore, a simple model that is reactive to intrathoracic pressure has been developed. 2. METHODS 2.1 DeBoer Model The novel model is based on the beat-to-beat model proposed by DeBoer et al. The original model is able to represent the following mechanisms: baroreflex control of heart rate and peripheral resistance, windkessel properties of the arterial tree and contractile properties of the myocardium. Moreover it shows mechanical effects of spontaneous breathing on blood pressure. The model is defined by difference equations, i.e. the characteristics of each heart beat are defined by the previous heart beat and the interbeat differences. The difference equations are linearized around operating points for systolic pressure (S), diastolic pressure (D), pulse pressure (P), peripheral resistance (R), R-R interval (I) and arterial time constant (T). Thus, e.g. systolic pressure of the n-th beat (Sn) is defined as: =
+
(1)
where sn is the deviation from the operating point S at the nth beat. Sn is the resulting systolic blood pressure at the n-th beat. Influence of respiration on blood pressure is simulated through impact on pulse pressure as shown below: =
∙
+
∙ sin (2 ∙
∙ )
(2)
Here, pn is the deviation in pulse pressure at the n-th beat, γ is a parameter describing the change in ventricular contractile force which is depending on the deviation of the R-R interval i at the previous beat. A denotes the amplitude of intrapleural pressure, fR is the respiratory frequency. Tests have shown that this model complies with results recorded from a complex 19-compartment model as proposed by Leaning et al. Figure 2 shows results for arterial pressure as simulated by the complex model compared to systolic and diastolic blood pressures calculated by the simple beat-tobeat model. Here, intrapleural pressure was set to vary between -4 and -6 mmHg. The simple model is computing blood pressures on a beat-to-beat basis, thus no continuous values can be calculated. In both results a superimposed oscillation with the same frequency as respiration rate (12/min) can be observed that reflects the influence of intrapleural pressure.
291
8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
a)
b)
c)
110 arterial blood pressure [mmHg]
arterial blood pressure [mmHg]
120 110 100 90 80
100 90 80 70 60
70 0
5
10
time [s]
0
15
Fig. 2. Arterial blood pressures in spontaneous breathing as computed by the complex 19-compartment model compared to the simple beat-to-beat model. Results of the complex model are shown in grey dashed line with systolic and diastolic pressures marked as black circles. The simple model is not calculating continuous values due to its model nature. Thus, only systolic (red crosses) and diastolic (blue crosses) pressures are shown. Results of the simple model show to be consistent with the complex model.
The simple beat-to-beat model does not include such complex interactions, thus reaction to intrathoracic pressure changes in mechanical ventilation need to be developed. For this purpose the complex 19-compartment model has been evaluated with different intrathoracic pressure amplitudes and PEEP levels. A simple equation has then been implemented into the beat-to-beat model to simulate reaction to mechanical ventilation. Equations 3-5 show the implemented model extension:
∑
= =
+
∙
,
+
∙ + ∙
∑
,
‖
‖
,
0
5 time [s]
10
0
5 time [s]
10
2.3 Implementation in the framework
Besides intrapleural pressure changes present in spontaneous breathing, the beat-to-beat model also needs to be consistent with pressure variations that appear in mechanical ventilation. In the complex 19-compartment model, intrathoracic pressure is added to all compartments that are located in the thoracic domain. When applying mechanical ventilation, intrathoracic pressure becomes positive leading to a volume shift in the compartments. This again causes blood pressure in the arterial compartments to drop if baroreflex control is disabled. FigFigure 3 shows arterial pressure as computed by the 19-compartment model for three different intrathoracic pressure levels (0-6.5mmHg, 5-16mmHg, 1026mmHg).
+
10
Fig. 3. Arterial blood pressures in mechanical ventilation at different intrathoracic pressure levels as computed with the 19-compartment model. a) 0-6.5mmHg; b)5-16mmHg; c)10-26mmHg. Baroreflex is disabled.
2.2 Reaction to intrathoracic pressure
=
5 time [s]
The developed beat-to-beat model is to substitute the continuous but computationally expensive model of the CVS in complex systems of interacting submodels. Thus, the discrete model needs to be integrated into the existing framework. A special caller algorithm is used to invoke each of the implemented submodels at every time step that is evaluated by the applied solver algorithm. The beat-to-beat model has a different time scale which is based on the R-R interval and is thus independent from the time steps that are chosen by the solver algorithm. Therefore, the caller algorithm has been adjusted to only invoke the new model when the next heart beat needs to be computed. Model combinations employing the beat-to-beat model to simulate cardiovascular dynamics thus are of hybrid nature. To enable reaction of the gas exchange model to changes in the cardiovascular model, cardiac output needs to be provided by the new model. The model does not include any simulation of blood volume and flow thus cardiac output has to be approximated using the mean arterial pressure (MAP). Equations 6-7 shows calculation of cardiac output (CO): R
,
= =
T C
(6)
,
(7)
(3)
where RA,n is the arterial resistance at the n-th beat which can be calculated with the arterial time constant (Tn) and the arterial compliance (CA), which is assumed to be constant.
(4)
2.4 Evaluation setup
(5)
Here Pth,n is the intrathoracic pressure at the n-th beat, N is the number of values computed for Pth in one breathing cycle. A is the amplitude of Pth; m, c, q are factors that are used to fit the reaction of the beat-to-beat model to the behaviour of the complex model.
The new hybrid model system was evaluated using a 3rd order respiratory mechanics model with chest wall compliance and a 4-compartment gas exchange model (Chiari et al., Benallal et al.). Here, q was set to -1.626, m to 0.9703 and c to 0.0941. Results of this model combination have been compared to one employing the complex 19-compartment cardiovascular model. All models were programmed in MATLAB® 2011b (The MathWorks® Inc., Natick, USA). Simulation was done
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8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
alveolar oxygen partial pressure [mmHg]
for 60s and computing times were measured on a standard PC (Q8200, 4x2.33GHz, 4GB RAM). 3. RESULTS Figure 4 shows results for arterial pressure (Pa) as computed by each model combination. The results depict a superimposed oscillation with a frequency equal to respiration rate set in the respiratory mechanics model (12/min). In the 19-compartment model, systolic blood pressure ranges between 107.6 and 91.8 mmHg, diastolic pressure is found to vary between 74 and 64.2 mmHg. The discrete beat-to-beat model shows systolic blood pressure between 111.2 and 92.7 mmHg and diastolic pressures between 75.6 and 64.5 mmHg. Moreover, results for a simple 3-compartment model are shown, which is not sensible to intrathoracic pressure. Here, no variations in systolic and diastolic blood pressure can be seen. Figure 5 shows results for alveolar oxygen partial pressure (PA,O2). Here, results computed by the normal model combination exhibit alternating behaviour with intermittent plateau phases in PA,O2. Zero flow phases (i.e. closed aortic valve) in the 19compartment cardiovascular model show to appear isochronal to plateau phases in PA,O2. This of course cannot be seen in the results calculated with the hybrid model combination as the beat-to-beat model does not include cardiac ejection phases. Here, only alternating behaviour of PA,O2 due to inspiration/expiration phases in respiratory mechanics model can be seen. Mean PA,O2 and PA,CO2 (alveolar carbon dioxide partial pressure) were found to match (118.5mmHg/26.5mmHg in normal model combination and 118.7mmHg/26.4mmHg in hybrid model combination). Cardiac output is also equal in both tested model systems (3.36l/min and 3.42l/min). Computing times were 104s for normal combination and 4.8s for the novel hybrid combination when employing ode113 for 130
arterial blood pressure [mmHg]
120 110 100 90 80 70 60 0
1
2
3
4
5 time [s]
6
7
8
9
10
Fig. 4. Arterial blood pressures in mechanical ventilation as computed by the complex 19-compartment model (dark grey dashed line) compared to the simple beat-to-beat model (red: systolic pressures, blue: diastolic pressures). Here, intrathoracic pressure ranged between 0.2 and 17mmHg. Dotted light grey line shows results of a simple 3-compartment cardiovascular model which is not sensible to intrathoracic pressure.
121 120.5 120 119.5 119 118.5 118 117.5 117 0
2
4
time [s]
6
8
10
Fig. 5. Alveolar oxygen partial pressure as computed employing normal (grey dashed) and hybrid (green) model combination. PA,O2 shows plateau phases when aortic valve is closed, i.e. zero flow phases in the 19-compartment model. This cannot be simulated by the beat-to-beat model. Still, both simulations result in the same mean PA,O2. solving the differential equations. Computing time of the normal model combination could be decreased to 895s when employing simple Euler integration using a step size of 0.1ms. 4. DISCUSSION An approach towards building a hybrid model combination containing a simple beat-to-beat model for simulating cardiovascular dynamics is proposed. The new model is calculating the cardiovascular properties of the human body on a beat-to-beat basis and is based on the model proposed by DeBoer et. al. The model has been extended so that reaction to intrathoracic pressure changes as typically generated during mechanical ventilation can be simulated. Results show that the beat-to-beat model is able to reproduce reaction to intrathoracic pressure changes correctly. Blood pressures are calculated with a maximum deviation of 4.7%, error in cardiac output was 1.7%. This leads to a maximum simulation error of 0.4% in the gas exchange model. Thus, the proposed hybrid model combination provides a computationally cheap way of simulating interactions of physiological processes in the human body. Still, model description of the beat-to-beat model and its implemented extensions are kept simple. Therefore, the hybrid model combination is not able to completely reproduce all simulation details that are incorporated in the normal combination employing the 19-compartment cardiovascular model. On the other hand, the simplicity of the implemented beat-to-beat model leads to a pronounced increase in computational efficiency i.e. a decrease of computing time by a factor of 186 is found on the average. The previously presented model family of cardiovascular dynamics also includes a 3-compartment model, which is a very simple solution containing differential equations. Using this simpler submodel in combination with the above mentioned respiratory mechanics and gas exchange models still requires 15.8s of computing time which is more than three times the computing costs required for the presented
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8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
hybrid combination. Moreover, this combination does not include interaction between respiratory mechanics and cardiovascular dynamics, as the 3-compartment model does not react to intrathoracic pressure. To allow comparability of simulation results in hybrid and normal model combination, the baroreflex was eliminated from both tested cardiovascular models, thus the influence of the baroreflex to induced changes in blood pressure has not been tested yet. 6. CONCLUSIONS Hybrid model combinations containing mathematical models with and without differential equations show to be computationally efficient while still being able to reproduce all physiological interactions included in the framework. Future tests need to be conducted to compare reaction to baroreflex stimulation in the new beat-to-beat model compared to the complex 19-compartment model. REFERENCES Benallal, H. and Busso, T. (2000). Analysis of end-tidal and arterial PCO2 gradients using a breathing model. Eur J Appl Physiol, vol. 83 (4 -5), pp. 402-8. Benallal, H., Denis, C. et al. (2002). Modeling of end-tidal and arterial PCO2 gradient: comparison with
experimental data. Med Sci Sports Exerc, vol. 34 (4), pp. 622-9. Chiari, L., Avanzolini, G. and Ursino, M. (1997). A comprehensive simulator of the human respiratory system: validation with experimental and simulated data. Ann Biomed Eng, vol. 25 (6), pp. 985-99. DeBoer, R.W., Karemaker, J.M. and Strackee, J. (1987). Hemodynamic fluctuations and baroreflex sensitivity in humans: a beat-to-beat model. Am J Physiol, vol. 253 (3 Pt 2), pp. H680-9. Leaning, M.S., Pullen, H.E. et al. (1983). Modelling a complex biological system: the human cardiovascular system - 1. Methodology and model description. T I Meas Control, vol. 5, pp. 71-86. Kretschmer, J., Wahl, A. and Möller, K. (2011). Dynamically Generated Models for Medical Decision Support Systems. Comput Biol Med., vol. 41, pp. 899-907. Kretschmer, J. and Möller, K. (2012). A Hierarchical Model Family of Cardiovascular Dynamics. In Jobbágy, Á. and Magjarevic, R., 5th European Conference of the International Federation for Medical and Biological Engineering vol. 37, pp. 295-8. Springer, Berlin Heidelberg. Schranz, C., Knöbel, C. et al. (2011). Hierarchical Parameter Identification in Models of Respiratory Mechanics. IEEE Trans Biomed Eng., vol. 58 (11), pp. 3234-41.
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A Hybrid Model of Interacting Physiological Systems Jörn Kretschmer*/** , Zhanqi Zhao*, Thomas Haunsberger*, Erick Drost*, Edmund Koch**, Knut Möller* *Institute of Technical Medicine, Furtwangen University, 78054 Villingen-Schwenningen, Germany (Tel: +49-7720-307-4370; e-mail: [email protected]). ** Faculty of Medicine Carl Gustav Carus, Dresden University of Technology, 01307 Dresden, Germany
Abstract: Mathematical models can be exploited to simulate physiological processes in the human body as well as predict their reaction to changes in the therapy regime of a patient. Implementing these models in Medical Decision Support Systems (MDSS) can help in optimizing therapy settings. MDSS optimizing ventilation therapy in critically ill patients should not only consider respiratory mechanics but extend simulations to also consider other parts of the human body. A previously presented framework allows combination of three model families (respiratory mechanics, cardiovascular dynamics and gas exchange) to provide a broader picture when predicting the outcome of a therapy setting. The three model families are dynamically combined to form a complex model system with interacting submodels. The framework computes the combined submodels as a tightly coupled system, i.e. an applied solver algorithm chooses a step size to fit the submodel with highest system dynamics. Tests revealed that simulation time increases rapidly with rising system complexity, i.e. the number of differential equations defining the interacting model system. This is due to very expensive computing when detailed models of cardiovascular dynamics are included. However, this simulation detail is not necessary in all scenarios. Thus, a simplified cardiovascular model that is able to reproduce basic physiological behavior is introduced. This model consists only of difference equations and does not require special algorithms to be numerically solved. The model is based on the beat-to-beat model presented by DeBoer et al. and has been extended to react to intrathoracic pressure levels that are present during mechanical ventilation. To include sensitivity to mechanical ventilation, a 19-compartment model as proposed by Leaning et al. has been analyzed and model behavior has been implemented as a simple equation into the beat-to-beat model. Tests showed, that the model is able to closely represent general model behavior compared to the 19compartment model. Blood pressures were calculated with a maximum deviation of 1.8%, leading to a simulation error of 1.7% in cardiac output. Combination with a gas exchange showed maximum simulation error of 0.4%. Therefore, the proposed model is able to be used in combinations where cardiovascular simulation does not have to be detailed. Computing costs have been decreased dramatically by factor 186 compared to a model combination employing the 19-compartment model. Keywords: Mathematical models, beat-to-beat model, physiological simulations. settings by running several simulations of the same mathematical model.
1. INTRODUCTION Mathematical models can be employed to understand the human body and its physiological processes. Physiological reactions of a patient to changes in the therapeutic regime can even be predicted by evaluating simulation results of these mathematical models. Medical Decision Support Systems (MDSS) exploit these simulations to optimize the applied therapeutic strategy. In mechanically ventilated patients a model based optimization may be employed to avoid ventilator induced lung injuries (VILI) e.g. by applying an optimized ventilation pressure. To find the optimal therapeutic setting, an MDSS needs to test different therapic
978-3-902823-10-6/12/$20.00 © 2012 IFAC
However, this optimization procedure should not be based on one single mathematical model, e.g. respiratory mechanics. In mechanically ventilated patients not only lung tissue but also gas exchange and cardiovascular dynamics are affected by the therapy. Thus, these aspects might also be of importance when evaluating the reactions of the patient to a certain therapic setting. Therefore, they need to be included in the simulation. Including several aspects of human physiology allows an MDSS to base the optimization on a broader picture. Moreover, therapic measures proposed by the MDSS may be improved if a set of models with differing simulation focus is provided. This allows individual adaptation to the
290
10.3182/20120829-3-HU-2029.00113
8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
patient’s disease further optimizing the MDSS’s performance. In order to adapt a specific model to a patient’s properties, model parameters need to be fitted to their physiological correlates. Establishing a hierarchical order of the employed model families (e.g. respiratory mechanics, gas exchange, etc.) supports this procedure (Schranz et al.). A model library containing three different model families (respiratory mechanics, gas exchange and cardiovascular dynamics) has been established before (Kretschmer et al., 2011a). A specially designed framework allows to generate different combinations of these model families and guarantees that the employed mathematical models are able to interact during simulation. These complex model systems are not precompiled, i.e. models are combined dynamically. Each of the implemented model families offers different model versions varying in simulation focus and model detail. Interfaces implemented in the framework allow to exchange parameters between models, thus enabling interaction. E.g., through these interfaces, parameter values for cardiac output ( ), intrathoracic pressure ( ), alveolar volume ( ) and air flow ( ) are exchanged between the models. Ventilator settings (inspiratory gas fractions of oxygen and carbon dioxide , , inspiratory air flow , respiratory , frequency ) are used as input to the overall system. Figure 1 shows the interfaces as implemented in the previously proposed framework. Synchronized parameter exchange is ensured by tightly coupling the interacting models, i.e. they are computed with an identical step size. This step size has to fit the model with highest system dynamics in order to keep the simulation numerically stable. Thus, models with lower dynamic behaviour are computed with a step size smaller than necessary. Tests showed that this is computationally expensive with computing time growing rapidly with the number of differential equations describing the models. An MDSS usually conducts optimization of therapy based on a number of simulations to evaluate different settings. Based on the results of these simulations the MDSS is able to find a . Vin fR Respiratory mechanics
Ventilator settings
. VA VA
Pth
Fi,O2 Fi,CO2 Gas exchange CO
Cardiovascular dynamics
Fig. 1. Model interactions and interfaces in the previously presented framework (Kretschmer et al., 2011). Interface parameters are gas fractions of oxygen and carbon dioxide ( , , ), inspiratory air flow ( ), respiratory , frequency ( ), intrathoracic pressure ( ), cardiac output ( ), air flow ( ) and alveolar volume ( ).
setting that complies most with the therapy goals defined by the clinician. Since these results need to be calculated in a reasonable time, computations need to be efficient. Thus, the MDSS needs to be able to choose a model combination that is both fitting the patient’s situation and is computationally feasible. The previously presented model family of cardiovascular dynamics (Kretschmer et al. 2011, 2012) did not include any cardiovascular model that is both reactive to intrathoracic pressure and is of low complexity. Therefore, a simple model that is reactive to intrathoracic pressure has been developed. 2. METHODS 2.1 DeBoer Model The novel model is based on the beat-to-beat model proposed by DeBoer et al. The original model is able to represent the following mechanisms: baroreflex control of heart rate and peripheral resistance, windkessel properties of the arterial tree and contractile properties of the myocardium. Moreover it shows mechanical effects of spontaneous breathing on blood pressure. The model is defined by difference equations, i.e. the characteristics of each heart beat are defined by the previous heart beat and the interbeat differences. The difference equations are linearized around operating points for systolic pressure (S), diastolic pressure (D), pulse pressure (P), peripheral resistance (R), R-R interval (I) and arterial time constant (T). Thus, e.g. systolic pressure of the n-th beat (Sn) is defined as: =
+
(1)
where sn is the deviation from the operating point S at the nth beat. Sn is the resulting systolic blood pressure at the n-th beat. Influence of respiration on blood pressure is simulated through impact on pulse pressure as shown below: =
∙
+
∙ sin (2 ∙
∙ )
(2)
Here, pn is the deviation in pulse pressure at the n-th beat, γ is a parameter describing the change in ventricular contractile force which is depending on the deviation of the R-R interval i at the previous beat. A denotes the amplitude of intrapleural pressure, fR is the respiratory frequency. Tests have shown that this model complies with results recorded from a complex 19-compartment model as proposed by Leaning et al. Figure 2 shows results for arterial pressure as simulated by the complex model compared to systolic and diastolic blood pressures calculated by the simple beat-tobeat model. Here, intrapleural pressure was set to vary between -4 and -6 mmHg. The simple model is computing blood pressures on a beat-to-beat basis, thus no continuous values can be calculated. In both results a superimposed oscillation with the same frequency as respiration rate (12/min) can be observed that reflects the influence of intrapleural pressure.
291
8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
a)
b)
c)
110 arterial blood pressure [mmHg]
arterial blood pressure [mmHg]
120 110 100 90 80
100 90 80 70 60
70 0
5
10
time [s]
0
15
Fig. 2. Arterial blood pressures in spontaneous breathing as computed by the complex 19-compartment model compared to the simple beat-to-beat model. Results of the complex model are shown in grey dashed line with systolic and diastolic pressures marked as black circles. The simple model is not calculating continuous values due to its model nature. Thus, only systolic (red crosses) and diastolic (blue crosses) pressures are shown. Results of the simple model show to be consistent with the complex model.
The simple beat-to-beat model does not include such complex interactions, thus reaction to intrathoracic pressure changes in mechanical ventilation need to be developed. For this purpose the complex 19-compartment model has been evaluated with different intrathoracic pressure amplitudes and PEEP levels. A simple equation has then been implemented into the beat-to-beat model to simulate reaction to mechanical ventilation. Equations 3-5 show the implemented model extension:
∑
= =
+
∙
,
+
∙ + ∙
∑
,
‖
‖
,
0
5 time [s]
10
0
5 time [s]
10
2.3 Implementation in the framework
Besides intrapleural pressure changes present in spontaneous breathing, the beat-to-beat model also needs to be consistent with pressure variations that appear in mechanical ventilation. In the complex 19-compartment model, intrathoracic pressure is added to all compartments that are located in the thoracic domain. When applying mechanical ventilation, intrathoracic pressure becomes positive leading to a volume shift in the compartments. This again causes blood pressure in the arterial compartments to drop if baroreflex control is disabled. FigFigure 3 shows arterial pressure as computed by the 19-compartment model for three different intrathoracic pressure levels (0-6.5mmHg, 5-16mmHg, 1026mmHg).
+
10
Fig. 3. Arterial blood pressures in mechanical ventilation at different intrathoracic pressure levels as computed with the 19-compartment model. a) 0-6.5mmHg; b)5-16mmHg; c)10-26mmHg. Baroreflex is disabled.
2.2 Reaction to intrathoracic pressure
=
5 time [s]
The developed beat-to-beat model is to substitute the continuous but computationally expensive model of the CVS in complex systems of interacting submodels. Thus, the discrete model needs to be integrated into the existing framework. A special caller algorithm is used to invoke each of the implemented submodels at every time step that is evaluated by the applied solver algorithm. The beat-to-beat model has a different time scale which is based on the R-R interval and is thus independent from the time steps that are chosen by the solver algorithm. Therefore, the caller algorithm has been adjusted to only invoke the new model when the next heart beat needs to be computed. Model combinations employing the beat-to-beat model to simulate cardiovascular dynamics thus are of hybrid nature. To enable reaction of the gas exchange model to changes in the cardiovascular model, cardiac output needs to be provided by the new model. The model does not include any simulation of blood volume and flow thus cardiac output has to be approximated using the mean arterial pressure (MAP). Equations 6-7 shows calculation of cardiac output (CO): R
,
= =
T C
(6)
,
(7)
(3)
where RA,n is the arterial resistance at the n-th beat which can be calculated with the arterial time constant (Tn) and the arterial compliance (CA), which is assumed to be constant.
(4)
2.4 Evaluation setup
(5)
Here Pth,n is the intrathoracic pressure at the n-th beat, N is the number of values computed for Pth in one breathing cycle. A is the amplitude of Pth; m, c, q are factors that are used to fit the reaction of the beat-to-beat model to the behaviour of the complex model.
The new hybrid model system was evaluated using a 3rd order respiratory mechanics model with chest wall compliance and a 4-compartment gas exchange model (Chiari et al., Benallal et al.). Here, q was set to -1.626, m to 0.9703 and c to 0.0941. Results of this model combination have been compared to one employing the complex 19-compartment cardiovascular model. All models were programmed in MATLAB® 2011b (The MathWorks® Inc., Natick, USA). Simulation was done
292
8th IFAC Symposium on Biological and Medical Systems August 29-31, 2012. Budapest, Hungary
alveolar oxygen partial pressure [mmHg]
for 60s and computing times were measured on a standard PC (Q8200, 4x2.33GHz, 4GB RAM). 3. RESULTS Figure 4 shows results for arterial pressure (Pa) as computed by each model combination. The results depict a superimposed oscillation with a frequency equal to respiration rate set in the respiratory mechanics model (12/min). In the 19-compartment model, systolic blood pressure ranges between 107.6 and 91.8 mmHg, diastolic pressure is found to vary between 74 and 64.2 mmHg. The discrete beat-to-beat model shows systolic blood pressure between 111.2 and 92.7 mmHg and diastolic pressures between 75.6 and 64.5 mmHg. Moreover, results for a simple 3-compartment model are shown, which is not sensible to intrathoracic pressure. Here, no variations in systolic and diastolic blood pressure can be seen. Figure 5 shows results for alveolar oxygen partial pressure (PA,O2). Here, results computed by the normal model combination exhibit alternating behaviour with intermittent plateau phases in PA,O2. Zero flow phases (i.e. closed aortic valve) in the 19compartment cardiovascular model show to appear isochronal to plateau phases in PA,O2. This of course cannot be seen in the results calculated with the hybrid model combination as the beat-to-beat model does not include cardiac ejection phases. Here, only alternating behaviour of PA,O2 due to inspiration/expiration phases in respiratory mechanics model can be seen. Mean PA,O2 and PA,CO2 (alveolar carbon dioxide partial pressure) were found to match (118.5mmHg/26.5mmHg in normal model combination and 118.7mmHg/26.4mmHg in hybrid model combination). Cardiac output is also equal in both tested model systems (3.36l/min and 3.42l/min). Computing times were 104s for normal combination and 4.8s for the novel hybrid combination when employing ode113 for 130
arterial blood pressure [mmHg]
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Fig. 4. Arterial blood pressures in mechanical ventilation as computed by the complex 19-compartment model (dark grey dashed line) compared to the simple beat-to-beat model (red: systolic pressures, blue: diastolic pressures). Here, intrathoracic pressure ranged between 0.2 and 17mmHg. Dotted light grey line shows results of a simple 3-compartment cardiovascular model which is not sensible to intrathoracic pressure.
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Fig. 5. Alveolar oxygen partial pressure as computed employing normal (grey dashed) and hybrid (green) model combination. PA,O2 shows plateau phases when aortic valve is closed, i.e. zero flow phases in the 19-compartment model. This cannot be simulated by the beat-to-beat model. Still, both simulations result in the same mean PA,O2. solving the differential equations. Computing time of the normal model combination could be decreased to 895s when employing simple Euler integration using a step size of 0.1ms. 4. DISCUSSION An approach towards building a hybrid model combination containing a simple beat-to-beat model for simulating cardiovascular dynamics is proposed. The new model is calculating the cardiovascular properties of the human body on a beat-to-beat basis and is based on the model proposed by DeBoer et. al. The model has been extended so that reaction to intrathoracic pressure changes as typically generated during mechanical ventilation can be simulated. Results show that the beat-to-beat model is able to reproduce reaction to intrathoracic pressure changes correctly. Blood pressures are calculated with a maximum deviation of 4.7%, error in cardiac output was 1.7%. This leads to a maximum simulation error of 0.4% in the gas exchange model. Thus, the proposed hybrid model combination provides a computationally cheap way of simulating interactions of physiological processes in the human body. Still, model description of the beat-to-beat model and its implemented extensions are kept simple. Therefore, the hybrid model combination is not able to completely reproduce all simulation details that are incorporated in the normal combination employing the 19-compartment cardiovascular model. On the other hand, the simplicity of the implemented beat-to-beat model leads to a pronounced increase in computational efficiency i.e. a decrease of computing time by a factor of 186 is found on the average. The previously presented model family of cardiovascular dynamics also includes a 3-compartment model, which is a very simple solution containing differential equations. Using this simpler submodel in combination with the above mentioned respiratory mechanics and gas exchange models still requires 15.8s of computing time which is more than three times the computing costs required for the presented
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hybrid combination. Moreover, this combination does not include interaction between respiratory mechanics and cardiovascular dynamics, as the 3-compartment model does not react to intrathoracic pressure. To allow comparability of simulation results in hybrid and normal model combination, the baroreflex was eliminated from both tested cardiovascular models, thus the influence of the baroreflex to induced changes in blood pressure has not been tested yet. 6. CONCLUSIONS Hybrid model combinations containing mathematical models with and without differential equations show to be computationally efficient while still being able to reproduce all physiological interactions included in the framework. Future tests need to be conducted to compare reaction to baroreflex stimulation in the new beat-to-beat model compared to the complex 19-compartment model. REFERENCES Benallal, H. and Busso, T. (2000). Analysis of end-tidal and arterial PCO2 gradients using a breathing model. Eur J Appl Physiol, vol. 83 (4 -5), pp. 402-8. Benallal, H., Denis, C. et al. (2002). Modeling of end-tidal and arterial PCO2 gradient: comparison with
experimental data. Med Sci Sports Exerc, vol. 34 (4), pp. 622-9. Chiari, L., Avanzolini, G. and Ursino, M. (1997). A comprehensive simulator of the human respiratory system: validation with experimental and simulated data. Ann Biomed Eng, vol. 25 (6), pp. 985-99. DeBoer, R.W., Karemaker, J.M. and Strackee, J. (1987). Hemodynamic fluctuations and baroreflex sensitivity in humans: a beat-to-beat model. Am J Physiol, vol. 253 (3 Pt 2), pp. H680-9. Leaning, M.S., Pullen, H.E. et al. (1983). Modelling a complex biological system: the human cardiovascular system - 1. Methodology and model description. T I Meas Control, vol. 5, pp. 71-86. Kretschmer, J., Wahl, A. and Möller, K. (2011). Dynamically Generated Models for Medical Decision Support Systems. Comput Biol Med., vol. 41, pp. 899-907. Kretschmer, J. and Möller, K. (2012). A Hierarchical Model Family of Cardiovascular Dynamics. In Jobbágy, Á. and Magjarevic, R., 5th European Conference of the International Federation for Medical and Biological Engineering vol. 37, pp. 295-8. Springer, Berlin Heidelberg. Schranz, C., Knöbel, C. et al. (2011). Hierarchical Parameter Identification in Models of Respiratory Mechanics. IEEE Trans Biomed Eng., vol. 58 (11), pp. 3234-41.
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