1 Introduction

Mathematical models have proven to be a valuable tool in the understanding of physiological processes in the human body. They can be used to predict reactions of a patient to changes in the therapeutic regime. Such information might help clinicians in finding individually optimized therapy settings. Mechanical ventilation, as a life saving intervention, can serve as a valuable example for the use of this technique in intensive care. Although unavoidable in many cases, the use of mechanical ventilation may be burdened with severe side effects and may even lead to further impairment of the lung tissue (so called “ventilator induced lung injury” or VILI). Inappropriate ventilator settings are suspected responsible [1–3], thus the clinician must find the best balance between benefit and risk for the patient when choosing ventilator settings. In modern ICUs the clinician neither has time to constantly monitor the patient and optimize the ventilator settings nor is able to decode the complex interactions between ventilator and the patient. Exploiting results of simulations supplied by mathematical models in combination with penalty functions allows finding the optimal ventilator settings for each patient individually. Medical decision support systems (MDSS) employ this technique to compute advice for the clinician or to set optimal ventilator settings automatically.

Physiological simulation needs to be able to adapt to changes in the patient’s disease state, i.e. different model versions should be implemented into the overall system. In order to adapt a specific model to patient individual properties, model parameters need to be fitted to their physiological counterparts. Establishing a hierarchical order of the employed model families supports this procedure [4]. Moreover, in mechanically ventilated patients, physiological effects are not limited to the mechanics of the lung but at least gas exchange and cardiovascular dynamics are influenced also. Thus, the patient model should be extended to include the major disease characteristic aspects.

We have introduced a model library containing three different model families (respiratory mechanics, gas exchange and cardiovascular dynamics) before [5]. A specially designed framework allows the generation of different combinations of models from 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. Models are allowed to interact through special interfaces exchanging parameters between models. Tests with different model combinations showed physiologically plausible results.

Various researchers have published approaches to combine models before, the most prominent being markup languages like CellML or SBML [6–9]. These platforms provide the user with tools to define models independently from specific software platforms. Moreover, they enable merging models from different researchers into complex systems [10–12]. However, all approaches of combining different models to form a complex system ultimately come down to the same challenges of computing interacting models with greatly differing system dynamics. In the previously presented 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 behavior are computed with a step size smaller than necessary. Tests revealed that this strategy is computationally disadvantageous 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 setting that complies most with the therapy goals defined by the clinician. Since these results need to be calculated in a reasonable time, model computations need to be efficient. Thus, the previously presented framework needs to provide the MDSS with a model combination that is both fitting the patient’s situation and is computationally feasible. The implemented model family of cardiovascular dynamics [13] does not include any cardiovascular model that is both reactive to intrathoracic pressure and is of low complexity. Basic cardiovascular parameters may be calculated using very simple models that do not necessarily need to be based on differential equations, but might be composed purely of simple difference equations. Such models, called beat-to-beat models, may be able to provide sufficient information about a patient’s cardiovascular status or predict reactions to changes in the therapy. The presented study is intended to provide a proof of concept for the use of such simple cardiovascular models in an interacting model system.

2 Methods

2.1 Framework

The previously presented framework [5] contains three model families: respiratory mechanics, gas exchange and cardiovascular dynamics. Each of these model families comprises models with a different simulation focus or levels of complexity, i.e. the number of variables and “free parameters” defining the respective models. We have established a hierarchical order in all model families, i.e. each model has a defined relation to its simpler predecessor and its more complex successor.

To allow simulation of physiological interactions, the models are able to exchange state signals through special interfaces. Thus, respiratory mechanics influences gas exchange through respiratory flow (\(\dot{V}_{A}\)) and alveolar volume (V A ) as well as cardiovascular dynamics by intrathoracic pressure (P th ). Moreover, gas exchange is impacted by the cardiac output (CO) computed in the cardiovascular model. The overall model system is driven by the ventilator settings, i.e. respiratory frequency (f R ), inspiratory gas flow (\(\dot{V}_{in}\)) and inspired oxygen fraction (FiO 2). Figure 1 shows the interfaces as implemented in the framework.

Fig. 1
figure 1

Model interfaces and interactions. Three internal interfaces enable model interaction. Interface signals are respiratory flow (\(\dot{V}_{A}\)), alveolar volume (V A ), cardiac output (CO) and intrathoracic pressure (P th ). An additional interface provides the system with ventilator settings (respiration frequency f R , inspiratory gas flow \(\dot{V}_{in}\) and inspired oxygen gas fraction FiO 2 ) as simulation inputs

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Each of the aforementioned state signals is exchanged using a global parameter structure which is updated with every computational step. All submodels forming the overall model system are computed with identical step sizes, enabling a synchronous model interaction. Computation step size is adapted to the model with highest dynamics, usually the cardiovascular models. This leads to expensive computation due to the uniform step sizes applied to all models. Thus, the cardiovascular models are a bottleneck for computational efficiency of the whole system, as they act as a slowing pacemaker.

2.2 Cardiovascular dynamics model family

The presented model family consists solely of closed-loop models, allowing a forward computation and thus prediction of patient reaction to changes in the therapy. The simplest model implemented in the hierarchically ordered cardiovascular model family consists of three compartments and is based on the model proposed by Parlikar and Verghese [14]. It comprises three compartments representing cardiac, arterial and venous blood flow. The model with highest complexity included in the family is the parallel structured 19 compartment model as presented by Leaning et al. [15]. Moreover, the model family comprises a 6-compartment [16] as well as a serial structured 14-compartment model [17]. By stepwise extending the basic three compartment model, or reducing the complex versions, various model versions with differing detail can be created in the model family [13]. Cardiovascular models with higher levels of detail than the 19-compartment model, e.g. as proposed by various researchers [18, 19] are not considered for the presented model family because simulation efficiency and identifiability are compromised by the level of model complexity. Below, detailed description of the three evaluated cardiovascular models—3-compartment, 19-compartment and the proposed beat-to-beat model—is given.

2.3 3-compartment model

The 3-compartment cardiovascular model used in the evaluation just for comparing computational efficiency is based on the model proposed by Parlikar and Verghese [14]. It comprises one arterial, one venous as well as one cardiac compartment, i.e. pulmonary circulation is not included. Figure 2 gives the electrical analogue of the model. Elastance of the cardiac compartment (E h ) is not constant, but alternates over time, thus simulating cardiac ejection [14]:

$$E_{h} (t_{h} ) = \left\{ {\begin{array}{lcc} {\frac{{3 \cdot \left( {E_{S} - E_{D} } \right)}}{T} \cdot t_{h} + E_{D} } & {for} & {0 \le t_{h} \le {T \mathord{\left/ {\vphantom {T 3}} \right. \kern-0pt} 3}} \\ {\frac{{6 \cdot \left( {E_{S} - E_{D} } \right)}}{T} \cdot \left( {{\raise0.7ex\hbox{$T$} \!\mathord{\left/ {\vphantom {T 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}} - t_{h} } \right) + E_{S} } & {for} & {{T \mathord{\left/ {\vphantom {T 3}} \right. \kern-0pt} 3} < t_{h} \le {T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}} \\ {E_{D} } & {for} & {{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2} < t_{h} \le T} \\ \end{array} } \right.$$
(1)

with t h denoting the time within one heart beat of length T. E S and E D are the end-systolic and end-diastolic elastances, respectively. Change in systolic filling volume (V h,s ) of the cardiac compartment is calculated as [14]:

$$\frac{{dV_{h,s} }}{dt} = \frac{{s_{1} \cdot \left( {P_{v} - P_{h} } \right)}}{{R_{v} }} + \frac{{s_{D} \cdot \left( {P_{a} - P_{h} } \right)}}{{R_{a} }}$$
(2)

s1 and sD are switching functions simulating cardiac valves. They are defined as [14]:

$$s_{1} = \left\{ {\begin{array}{ll} 0 & for\quad {0 \le t_{h} < {T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}} \\ 1 & for\quad {{T \mathord{\left/ {\vphantom {T {2 \le t_{h} < T}}} \right. \kern-0pt} {2 \le t_{h} < T}}} \\ \end{array} } \right.$$
(3)
$$s_{D} = \left\{ {\begin{array}{ll} 0 & for\quad {0 \le t_{h} < {T \mathord{\left/ {\vphantom {T {11}}} \right. \kern-0pt} {11}}} \\ 1 & for\quad {{T \mathord{\left/ {\vphantom {T {11 \le t_{h} < {T \mathord{\left/ {\vphantom {T 3}} \right. \kern-0pt} 3}}}} \right. \kern-0pt} {11 \le t_{h} < {T \mathord{\left/ {\vphantom {T 3}} \right. \kern-0pt} 3}}}} \\ 0 & for\quad {{T \mathord{\left/ {\vphantom {T {3 \le t_{h} < T}}} \right. \kern-0pt} {3 \le t_{h} < T}}} \\ \end{array} } \right.$$
(4)

P h , P a and P v are cardiac, arterial and venous blood pressures, respectively. They are calculated as a sum of their respective diastolic pressure and their systolic pressure amplitude P i,S . P h,S is defined as:

$$P_{h,s} = V_{h,s} \cdot E_{h}$$
(5)

P a,S and P v,S are calculated from [14]:

$$C_{a} \cdot \frac{{dP_{a,S} }}{dt} = \frac{{s_{D} \cdot \left( {P_{h} - P_{a} } \right)}}{{R_{a} }} + \frac{{P_{v} - P_{a} }}{{R_{c} }}$$
(6)
$$C_{v} \cdot \frac{{dP_{v,S} }}{dt} = \frac{{s_{1} \cdot \left( {P_{h} - P_{v} } \right)}}{{R_{v} }} + \frac{{P_{a} - P_{v} }}{{R_{c} }}$$
(7)

Flow out of the heart is then defined as:

$$Q_{h} = \frac{{P_{h} - P_{a} }}{{R_{a} }} \cdot s_{D}$$
(8)

To compare results of the three evaluated cardiovascular models, the model definition of Parlikar and Verghese needs to be extended by intrathoracic pressure (P th ) influence. Thus, cardiac and arterial compartments are extended to [20]:

$$P_{i} = P_{i,S} + P_{th} \quad with\,i = h,a$$
(9)
Fig. 2
figure 2

Electrical analogue of the 3-compartment cardiovascular model derived from the model proposed by Parlikar and Verghese [14]. Arterial (a) and venous (v) compartments have constant compliance; cardiac (h) compartment comprises a time-varying elastance. Cardiac valves are simulated by a switch and a diode. Arterial and cardiac compartments are modified to react to changes in intrathoracic pressure

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2.4 19-compartment model

Each compartment included in the 19-compartment model concentrates the physiological behaviour of a certain anatomical area by defining one common compliance C, resistance R, volume V, pressure P as well as unstressed volume V un . Figure 3 shows a schematic overview of the model structure. In each compartment, blood pressure is calculated as [15]:

$$P_{i} = \frac{{V_{i} - V_{i,un} }}{{C_{i} }}$$
(10)

where index i denotes the i-th compartment in the model structure. Blood flow leaving the compartment is defined as [15]:

$$Q_{i,out} = \frac{{P_{i + 1} - P_{i} }}{{R_{i} }}$$
(11)

Here, index i + 1 denotes the subsequent compartment. The principle of mass conservation dictates that change in volume is calculated from the difference between blood flows entering the compartment and blood flows leaving the compartment [15]:

$$\frac{{dV_{i} }}{dt} = \sum {Q_{in} - \sum {Q_{out} } }$$
(12)

Compliance is constant in all compartments except the ventricular and atrial compartments, where compliance is time-dependent [15]. Elastance of the right (ra) and left (la) atrial compartments is defined as:

$$E_{i} = x \cdot \left( {E_{i,S} - E_{i,D} } \right) + E_{i,D} \quad with\,i = ra,la$$
(13)

Elastance of the right (rv) and left (lv) ventricular compartments is calculated from:

$$E_{i} = y \cdot \left( {E_{i,S} - E_{i,D} } \right) + E_{i,D} \quad with\,i = rv,lv$$
(14)

In Eqs. 13–14, E S denotes the end-systolic, E D the end-diastolic elastance of the compartment. x and y are time dependent functions, defined as [15, 21]:

$$x = \left\{ \begin{array}{lcl} \sin \left( {\frac{{\pi \cdot t_{h} }}{{T_{AS} }}} \right) &for & {0 < t_{h} \le T_{AS} } \\ 0 &{for} &{T_{AS} < t_{h} \le T} \end{array} \right.$$
(15)
$$y = \left\{ \begin{array}{ll} 0 & for\quad 0 < t_{h} \le T_{AV} \\ {\sin \left[ {\frac{\pi }{{T_{VS} }} \cdot \left( {t_{h} - T_{AV} } \right)} \right]} & for\quad T_{AV} < t_{h} < \left( {T_{AV} + T_{VS}} \right) \\ 0 & for\quad \left( {T_{AV} + T_{VS} } \right) \le t_{h} \le T \\ \end{array} \right.$$
(16)

T and t h are defined identically to the 3-compartment model. T AS denotes the duration of the artrial systole, T AV describes the time between the onset of artrial systole and the beginning of the ventricular systole. T VS defines the duration of the ventricular systole.

$$T_{AS} = 0.1 + 0.09 \cdot T$$
(17)
$$T_{AV} = T_{AS} - 0.04$$
(18)
$$T_{VS} = 0.16 + 0.2 \cdot T$$
(19)

Inertial effects are omitted in this model. The model as presented by Leaning et al. has been extended to react to intrathoracic pressure variations. Thus, in all compartments, that are located inside the thoracic cavity, Eq. 10 is extended to [20]:

$$P_{i} = \frac{{V_{i} - V_{i,un} }}{{C_{i} }} + P_{th}$$
(20)

As mentioned before, differences in system dynamics between cardiovascular models and the gas exchange and respiratory mechanics models lead to high computing costs, when simulating synchronous interactions between these model families. The previously presented model family of cardiovascular dynamics does not contain models that are both reactive to intrathoracic pressure and computable with low costs. Mathematical models that calculate cardiovascular dynamics on a beat-by-beat basis require little computing efforts. Extending such models by reaction to alterations in intrathoracic pressure might provide such functionality.

Fig. 3
figure 3

Schematic representation of the 19-compartment model [15]. Each compartment concentrates the physiological behaviour of a certain anatomical area by defining a common compliance, resistance, volume, pressure as well as unstressed volume

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2.5 Beat-to-beat model

2.5.1 deBoer model

The novel model that is presented is based on the beat-to-beat model proposed by deBoer et al. [22]. The original model is able to represent baroreflex control of heart rate and peripheral resistance, windkessel properties of the arterial tree and contractile characteristics of the myocardium. Moreover, blood pressure is reactive to changes in intrathoracic pressure during spontaneous breathing. The model purely consists of difference equations, evaluating the cardiovascular status on a beat-by-beat basis. Thus, the characteristics of each heart beat are defined by the previous heart beat and the interbeat changes. The difference equations are linearized around an operating point, thus systolic pressure (S), diastolic pressure (D), pulse pressure (P), peripheral resistance (R), beat-to-beat interval (I) and arterial time constant (T) are defined as:

$$X_{n} = X + x_{n} \quad with\,X = \left[ {S,D,P,R,I,T} \right]\,and\,x = \left[ {s,d,p,r,i,\tau } \right]$$
(21)

where x n is the deviation from the operating point X at the n-th beat. X n then is the resulting value at the n-th beat. Reaction to intrathoracic pressure changes is simulated as an impact on pulse pressure [22]:

$$p_{n} = \gamma \cdot i_{n - 1} + A \cdot (2\pi \cdot f_{R} \cdot t)$$
(22)

where p n is the deviation in pulse pressure at the n-th beat; γ is a parameter defining the change in ventricular contractile force which depends on the deviation of the beat-to-beat interval at the preceding beat. Parameter A describes the amplitude of intrathoracic pressure change, f R is the respiratory frequency.

2.5.2 Model extension

The intended use of the presented simple model is to make decision support available for mechanically ventilated patients treated in ICUs. Thus, the beat-to-beat model proposed by deBoer et al. needs to be extended to be consistent with pressure variations that appear in mechanical ventilation. The complex 19-compartment model implemented in the framework computes intrathoracic pressure influence in all compartments that are located in the thoracic cavity. During quiet spontaneous breathing, intrathoracic pressure remains negative throughout the breathing cycle, usually ranging between −2 and −7 mmHg. Applying mechanical ventilation leads to positive intrathoracic pressures. This again causes a decrease of the pressure gradient between intra- and extrathoracic compartments, causing blood volume shifts into peripheral vessels. If baroreflex is disabled, it leads to a decrease in venous return, resulting in a reduced cardiac output and decreased arterial blood pressure. Figure 4 shows example results computed by the 19-compartment model for arterial pressure at three different intrathoracic pressure levels with disabled baroreflex control and omitting inertial effects. Results show a decrease of mean arterial blood pressure with an increase in end-expiratory intrathoracic pressure. Moreover, an increase of the superimposed oscillation amplitude is observed when intrathoracic pressure amplitude is increased.

Fig. 4
figure 4

Arterial blood pressures at various intrathoracic pressure levels in mechanical ventilation simulated with the 19-compartment model. Baroreflex control is disabled in this example and inertial effects are omitted. a 0–5 mmHg; b 5–15 mmHg; c 10–25 mmHg. Mean arterial pressure decreases with increasing end-expiratory intrathoracic pressure levels. Moreover, superimposed amplitude rises with increased intrathoracic pressure amplitude

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The aforementioned simple beat-to-beat model does not comprise such complex reactions, thus influence of intrathoracic pressure changes on blood pressure and cardiac output in mechanically ventilated patients may not be reproduced with that model. Therefore, the reactions of the complex 19-compartment model to different intrathoracic pressures amplitudes and end-expiratory pressure levels have been observed to extend the beat-to-beat model with a set of simple equations mimicking an analogous behaviour. The observed reactions of the 19-compartment model show a diverse influence of intrathoracic pressure on diastolic and systolic blood pressure. To account for these observations, influence of intrathoracic pressure on pulse pressure was eliminated from Eq. 22 and instead added as direct influence to Eq. 21. Equations 23–25 show the implemented model extensions. Equation 22 is shortened to:

$$p_{n} = \gamma \cdot i_{n - 1}$$
(23)

For systolic and diastolic pressures, Eq. 5 is extended to:

$$X_{n} = X + x_{n} + A_{R,X,n} \quad with\,X = \left[ {S,D} \right]\quad and \quad x = \left[ {s,d} \right]$$
(24)

where A R,n is the influence of intrathoracic pressure on the n-th beat. As mentioned above, the model does only allow computation on a beat-by-beat basis, thus synchronous and continuous influence of intrathoracic pressure is not realizable. Therefore, A R is calculated as a function of the intrathoracic pressure present at the current beat as well as maximal and minimal intrathoracic pressure of the current breath. A R is then calculated as:

$$A_{R,X,n} = \left( {\alpha_{X} \cdot \Updelta P_{th} + \beta_{X} } \right) \cdot \left( {\frac{{P_{th,n} }}{{P_{th,\hbox{max} } }} - \frac{{\overline{{P_{th} }} }}{{P_{th,\hbox{max} } }}} \right) + \vartheta_{X} \cdot \overline{{P_{th} }} \quad with\,X = \left[ {S,D} \right]$$
(25)

where ΔP th is the intrathoracic pressure amplitude, P th,max is the maximal pressure and \(\overline{{P_{th} }}\) is the mean intrathoracic pressure of the current breath. Thus, both amplitude and mean intrathoracic pressure affect systolic and diastolic blood pressure. Model parameters α, β and ϑ may be used to fit model reaction to observed patient behavior.

2.6 Framework implementation

The presented beat-to-beat model is planned to substitute the complex continuous but computationally expensive models of the cardiovascular model family in a system of interacting models. Thus, the discrete model needs to be embedded into the existing framework that is designed for continuous models. The framework incorporates an algorithm that calls each of the implemented submodels at every evaluated step [5]. The beat-to-beat model, however, has a different time scale which is solely based on the interval between two beats, thus being independent of the time steps chosen by the ODE solver algorithm. The caller algorithm has therefore been adjusted to only invoke the beat-to-beat model when the next heart beat needs to be computed. Any model combination employing the beat-to-beat model to simulate the cardiovascular system thus is of hybrid nature. Additionally, a separate algorithm extracts the thoracic pressure signal of one breath from the simulation results computed by the solver. This is done online, i.e. during simulation, to directly provide these values to the beat-to-beat model (see Eq. 25).

To use the proposed model in the overall framework effectively, i.e. to reproduce physiological interactions present in the human body, it does not only need to react to changes in intrathoracic pressure, but should also be able to provide the resulting cardiac output. A gas exchange model would then react to changes in that parameter. The beat-to-beat model does not include simulation of blood volume or blood flow, thus cardiac output needs to be approximated via different means. The presented model uses mean arterial pressure (MAP) to calculate an approximated cardiac output (CO). MAP is calculated as [23]:

$$MAP_{n} = D_{n} + \frac{1}{3} \cdot (S_{n} - D_{n} )$$
(26)

Cardiac output can then be calculated by [23]:

$$CO_{n} = \frac{{MAP_{n} }}{{R_{A,n} }}$$
(27)

where R A,n is the arterial resistance at the n-th beat, being calculated by:

$$R_{A,n} = \frac{{T_{n} }}{{C_{A} }}$$
(28)

T n is the arterial time constant at the n-th beat, C A denotes the arterial compliance, which is assumed to be constant in this model.

2.7 Evaluation

The presented beat-to-beat model was tested in a hybrid model system including 3rd order respiratory mechanics with chest wall compliance [4] and a 4-compartment gas exchange model with tidal breathing [24–26]. Model parameters α, β and ϑ have been tuned so that the beat-to-beat model reproduces the observed behavior of the 19-compartment model closely, i.e. decrease in mean arterial blood pressure and oscillation amplitude of systolic and diastolic blood pressure. Model parameters CA was modified to obtain a cardiac output close to the cardiac output computed by the 19-compartment model at all of the tested intrathoracic pressure levels. Table 1 lists the identified parameter values of α, β and ϑ. C A was set to 0.91 ml/mmHg. Simulation results of the beat-to-beat model were compared to results computed with the 3-compartment and the 19-compartment model.

Table 1 Results of parameter identification for α, β and ϑ to mimic reaction of the 19-compartment model to changes in intrathoracic pressure
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The 3-compartment model was chosen to study the reaction of simple cardiovascular models with continuous signals to alterations in intrathoracic pressure. Moreover, its simple design allows comparing computational efficiency with the presented beat-to-beat model. Model parameters ES, ED, Ra, Rv, Rc, Ca, Cv were tuned to match the observed behavior of the 19-compartment model using the Nelder–Mead Simplex Method [27]. Table 2 lists the identified parameters. Results of the 19-compartment model are computed to examine if the presented beat-to-beat is able to represent the same general model behavior. Baroreflex control was disabled in all models to ensure comparability of the results.

Table 2 Results of parameter identification for E S , E D , R a , R v , R c , C a and C v to mimic reaction of the 19-compartment model to changes in intrathoracic pressure
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Both computing time and simulation results of the hybrid model system were compared to model systems containing the 19-compartment model and the simple 3-compartment model. Table 3 lists the tested model systems. All models were programmed in MATLAB 2012a (The MathWorks® Inc., Natick, USA). Simulation was conducted for 60 s and computing times were measured on a standard PC (Q8200, 4 × 2.33 GHz, 4 GB RAM).

Table 3 Evaluated model combinations
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3 Results

3.1 Spontaneous breathing

Figure 5 shows simulation results for arterial blood pressure under spontaneous breathing conditions omitting inertial effects in the simulation. Results computed by the 3-compartment model are shown as gray dotted line; results of the 19-compartment model are shown as solid gray line with systolic and diastolic pressures marked as black circles. Due to its discrete nature, the beat-to-beat model only computes systolic and diastolic pressures being shown as cross-shaped markers. All results exhibit superimposed oscillations with a frequency equal to the respiration rate that was set in the respiratory mechanics model (12/min). The 3-compartment model computes systolic pressure with a maximum deviation of 16.8 mmHg and diastolic pressure with a maximum deviation of 10.1 mmHg compared to results computed by the 19-compartment model. The beat-to-beat model shows a maximum deviation in systolic pressure of 1 and 1.2 mmHg in diastolic pressure.

Fig. 5
figure 5

Arterial blood pressure computed by the beat-to-beat model compared to the 3-compartment and the 19-compartment model. Results computed by the 3-compartment model are shown as dotted gray line; results of the 19-compartment model are shown as solid gray line with systolic and diastolic pressures marked as black circles. Inertial effects are omitted in all models. Results computed by the beat-to-beat model are indicated by x-shaped markers. The beat-to-beat model does not provide results on a continuous time scale but computes on a discrete basis. Thus, only systolic and diastolic pressures are presented

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3.2 Mechanical ventilation

3.2.1 Reaction to changes in intrathoracic pressure

All presented cardiovascular models were tested at different levels of intrathoracic pressure. The pressure signals have been generated by a sinusoidal function ranging between the specified minimum and maximum intrathoracic pressure:

$$P_{th} = \sin \left( {2\pi \cdot f_{R} \cdot {\raise0.7ex\hbox{$t$} \!\mathord{\left/ {\vphantom {t {60}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${60}$}}} \right) \cdot {\raise0.7ex\hbox{${\Updelta P_{th} }$} \!\mathord{\left/ {\vphantom {{\Updelta P_{th} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} + {\raise0.7ex\hbox{${\Updelta P_{th} }$} \!\mathord{\left/ {\vphantom {{\Updelta P_{th} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} + P_{th,\hbox{min} }$$
(29)

Figure 6 shows arterial blood pressure for all tested models. Again, all models show superimposed pressure with an identical frequency as the respiration rate in the respiratory mechanics model. With increasing intrathoracic pressure, mean arterial pressure decreases. Moreover, oscillation amplitude rises with increasing amplitude of the intrathoracic pressure. Results calculated by the 3-compartment model show a maximum deviation of 17.2 mmHg in systolic pressure and 17.7 mmHg in diastolic pressure compared to the results of the 19-compartment model. Results calculated by the beat-to-beat model show a maximum deviation of 2.3 mmHg in systolic pressure and 1.7 mmHg in diastolic pressure.

Fig. 6
figure 6

Comparison of results for arterial blood pressures at various intrathoracic pressure levels in mechanical ventilation simulated with the 3-compartment, the 19-compartment and the beat-to-beat model. Baroreflex control is disabled and inertial effects are omitted in this example. Results of the 3-compartment model are shown as dotted gray line, results of the 19-compartment model are shown as solid gray line with systolic and diastolic pressures indicated by black circles; results of the beat-to-beat model (discrete systolic and diastolic pressures) are shown as x-shaped markers. a 0–5 mmHg; b 5–15 mmHg; c 10–25 mmHg. Mean arterial pressure decreases with increasing end-expiratory intrathoracic pressure levels in all models. Moreover, superimposed amplitude rises with increased intrathoracic pressure amplitude

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3.2.2 Model interaction

Figure 7 depicts results for arterial pressure in all three evaluated model combinations. Intrathoracic pressure computed by the integrated respiratory mechanics model ranged between 3.9 and 20 mmHg. All cardiovascular models show a decrease in mean arterial blood pressure. Mean arterial pressure computed by the beat-to-beat model (system 3) deviate from results of the 19-compartment model (system 2) by 1 mmHg. Systolic pressure amplitude computed in the beat-to-beat model has a deviation of 0.3 mmHg compared to the 19-compartment model; diastolic pressure amplitude differs by 0.2 mmHg. Results computed by the 3-compartment model (system 1) show a deviation of 2.4 mmHg in mean arterial pressure, 25.4 mmHg in systolic pressure and 19.8 mmHg in diastolic pressure, respectively.

Fig. 7
figure 7

Comparison of results for arterial blood pressure in all tested model combinations. Intrathoracic pressure ranged between 3.9 and 20 mmHg. Results of the 19-compartment model are shown as solid gray line with black circles marking systolic and diastolic pressures. Beat-to-beat model results are shown as discrete systolic and diastolic x-markers. Dotted gray line depicts results of the 3-compartment model

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Both the 19-compartment model and the 3-compartment model show distinct cardiac ejection, whereas the beat-to-beat model again only calculates discrete values. The 19-compartment model and the beat-to-beat model both show a decrease of 44 % in cardiac output in this example, while the 3-compartment model computes a decrease of 24 %.

Figure 8 shows results for arterial partial pressure of oxygen (PaO 2). All tested model combinations show alternating behaviour, i.e. an increase in PaO 2 during inspiration and a decrease during inspiratory pause and expiration. Both model system 2 employing the 19-compartment model and model system 1 including the 3-compartment model show additional local maxima in PaO 2 and during cardiac ejection phases. Model system 3 incorporating beat-to-beat cardiovascular simulation does not present such behaviour. Model systems 2 and 3 using the 19-compartment cardiovascular dynamics and beat-to-beat model, respectively, show almost identical average PaO 2 (deviation <0.1 %), while average PaO 2 in system 1 differs by 1.2 %.

Fig. 8
figure 8

Results for arterial partial pressure of oxygen. Dotted: model system 1 including 3-compartment cardiovascular dynamics; solid: model system 2 with 19-compartment cardiovascular model; dashed: model system 3 including the beat-to-beat model

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3.2.3 Computing time

Simulation of 60 s required a computing time of 15.8 s in model system 1, system 2 a computing time of 104 s, which could be decreased to 895 s when employing an ODE solver algorithm with fixed step size of 10 ms. System 3 showed a computing time of 4.8 s.

4 Discussion

Medical decision support systems (MDSS) suggest optimal therapy settings to clinicians allowing for improved patient individual treatment. For medical decision support in mechanical ventilation we propose a model based approach where models of different physiological subsystems are allowed to interact to include physiological interdependencies into the decision making process. Various models with different detailing—called model families—for each subsystem should be included enabling the MDSS to adapt the mathematical model to the current disease state of the patient. The selected model combination should always be the simplest possible combination in terms of model complexity that still is able to reproduce clinically relevant patient reactions to changes in the therapy regime. Therefore, we are presenting a simple discrete model of cardiovascular dynamics that can be incorporated into the previously published framework [5]. The new model is calculating the cardiovascular properties of the human body on a beat-to-beat basis as proposed by deBoer et al. [22]. The model has been extended to generate reactions to changes in intrathoracic pressure as they are found typically in mechanical ventilation.

Results show that the proposed model can be tuned to reproduce measured blood pressures and cardiac output at various intrathoracic pressure levels in mechanical ventilation. Blood pressures are calculated with a maximum deviation of 1.8 % in systolic pressure and 3.5 % in diastolic pressure compared to simulation results computed by the 19-compartment model. Averaged cardiac output shows a deviation of 0.3 %. The presented results have been computed with baroreflex being disabled in the cardiovascular models. Thus, simulation results are not comparable directly with physiologically observed behaviour. However, Pizov et al. [28] have reported results comparable to those presented here, i.e. decrease of mean arterial pressure when intrathoracic pressure is increased as well as influence of intrathoracic pressure amplitude on pulse pressure variation.

The adjustments in the framework now allow the computation of model combinations consisting of both continuous and discrete models. Thus the presented beat-to-beat model is able to interact with continuous models such as gas exchange and respiratory mechanics. Model system 3 containing the beat-to-beat model was demonstrated to show the same general behaviour in PaO 2 with the limitation that no local maxima are visible. This is due to the aforementioned discrete fashion of the model formulation, i.e. the model does not compute blood ejection phases but assumes a constant cardiac output during one heart beat. Still, average PaO 2 was found to differ by less than 0.1 %, which is below the noise level and therefore clinically acceptable.

The proposed hybrid model combination provides a computationally cheap way of simulating interactions of physiological processes in the human body. Nonetheless, model equations and the described extensions are kept at a simple level. Thus, the proposed model is not able to reproduce all simulation details or disease states that might be simulated in model system 2 employing the complex 19-compartment model. However, the simplicity of the proposed model allows for a significant increase in computational efficiency, i.e. a decrease of computing time by a factor of 186 is found on the average. Though also including a rather simple simulation of the human cardiovascular system, model system 1 implementing the 3-compartment cardiovascular model still requires an average computing time of 15.8 s which is more than three times the computing costs consumed by the presented hybrid model combination in system 3. Moreover, although being tuned to mimic the observed behavior of the 19-compartment model, its simple representation of the human cardiovascular system does not allow reproducing these results adequately.

Hence, we are confident to state that the proposed hybrid model combination is a computationally favourable option when simulating cardiovascular stable patients in mechanical ventilation.

To allow comparability of simulation results in the tested model combinations, baroreflex control was disabled in the cardiovascular models. Tests with enabled baroreflex should be part of further evaluation of the presented cardiovascular model. Although simulation results computed by the extended beat-to-beat model are physiologically plausible, validation with clinical data will be necessary.

5 Conclusion

Hybrid model combinations of continuous and discrete models proved to be computationally efficient while still reproducing all physiological interactions that need to be considered in mechanical ventilation. Future tests will be required to reveal agreement of cardiovascular reaction to mechanical ventilation with enabled simulation of baroreflex control and validation of simulation results with clinical data.