A Mathematical Model for the Prediction of Fluid Responsiveness


Fluid therapy is commonly used to improve cardiac output in hemodynamically instable patients in the intensive care unit. However, to predict whether patients will benefit from this intervention (i.e. are volume responsive), is difficult. Dynamic indices, that rely on heart-lung interactions, have shown to be good predictors of fluid responsiveness under strict clinical conditions, but clinical use is still limited. This is due to the lack of understanding of the complex underlying physiology since multiple quantities are involved. We present a physiologically based mathematical model of the interaction between the respiratory and cardiovascular systems incorporating dynamic indices and fluid responsiveness. Our model is based on existing models of the cardiovascular system, its control, and the respiratory system during mechanical ventilation. The model of the cardiovascular system is expanded by including non-linear cardiac elastances to improve simulation of the Frank-Starling mechanism. An original model including five mechanisms for interaction between mechanical ventilation and the circulation is also presented. This model allows for the simulation of these complex relationships and may predict the effect of volume infusion in specific patients in the future. The presented model must be seen as a first step to a bedside clinical decision support system, and can be used as an educational model.


Volume resuscitation is one of the most common therapeutic procedures in intensive care units to improve cardiac output (CO) or stroke volume index (SVI) and thus hemodynamics in critically ill patients. However, to identify patients who might benefit from this therapy by an increase in cardiac output (volume responders), is a clinical challenge. Both clinical examination and static indicators of cardiac preload (e.g. central venous pressure) have been shown to be of minimal predictive value in distinguishing volume responders from non-responders.8,16 Over the last decade, dynamic indices that rely on cardiopulmonary interactions are used to assess fluid responsiveness in ventilated patients. Examples of dynamic indices are pressure or flow fluctuations that can be observed within the peripheral arteries. They are caused by mechanical ventilation when the heart operates on the steep portion of the Frank–Starling curve instead of on the flat portion of the curve and changes in preload cause variations in stroke volume.11,15

Although dynamic indices have shown to be good predictors of volume responsiveness,12,16, their use is limited to selected populations of patients and requires specific conditions for its application.9 The reason for this limited applicability is the complex underlying physiology in which many quantities are involved (e.g. tidal volume, lung compliance, chest wall compliance and volume status), which makes the dynamic indices difficult to interpret. It is our contention that by a better understanding of the complex relationships between the involved quantities, this use can be expanded. It is our goal to develop a mathematical model that captures the dynamics of heart-lung interactions and their relation to a patient’s volume status. Such a model, that is able to simulate the highly interrelated processes of this complex physiology, could also be used for educational simulations.24,25. When adaptable to the individual patient, the model could form the basis of a decision support system by predicting the effect of any considered volume infusion in specific patients.1026

A first, and to our knowledge only step towards the identification of factors that influence the arterial pressure variations by cardiovascular modeling was made by Messerges.14 He endorsed the assumption that mathematical modeling potentially lead to more clinically relevant interpretation of dynamic indices, and introduced positive pressure ventilation, venous compression and a rightward septum shift into an existing cardiovascular model. Unfortunately, this study was not able to link dynamic indices to volume status, probably because of the use of a cardiac model with limited complexity, the absence of respiratory mechanics and an incomplete description of the interacting mechanisms. Studies related to the life-science space program also modeled external pressure influences on the cardiovascular system.6 Although they make use of similar concepts relevant for modeling heart-lung interaction and volume responsiveness, including blood volume shifts, baroreflex models, and changes in venous transmural pressure, they are intended to be used to investigate the effects of postspaceflight orthostatic intolerance so lack respiratory mechanics and specific heart-lung interactions.

The primary objective of the work presented in this paper is to develop and test a physiological model of cardiovascular function that contains the essential processes associated with the prediction of fluid responsiveness. In particular, we report on the extension of previously developed and validated models of the cardiovascular system, which were initially developed for educational purposes. In this paper, we extend these models with several aspects of the cardiovascular and respiratory system, which are relevant during mechanical ventilation and in the dynamics of the heart-lung interactions in order to be able to simulate the relation between patient characteristics, volume status and dynamic indices in mechanically ventilated patients. We will use model parameters based on literature values as much as possible. Newly introduced parameters and constants will be chosen to simulate experimental data in the best way possible and listed in Table A in Appendix B. For validation, the simulation results will be compared to clinical data reported in literature. Furthermore, a case study, based on two patients from the ICU of the Radboud University Medical Center Nijmegen will be presented to show how the model is intended to be used in the future as a decision support system to prevent patients from receiving unnessecary, and possibly harmfull fluids.

Model Description

Cardiovascular System

Our model is based on previously described models of the closed circulation and consist of multiple segments that are lumped together.229 In short, four compartments describe the heart, six vascular compartments describe the systemic circulation and three vascular compartments describe the pulmonary circulation. In the previously described models, the systemic circulation consisted of five compartments, however, we have split the intrathoracic artery compartment into an aorta and a new intrathoracic compartment. This was done in order to make it possible to differentiate between characteristics of the ascending part of the aorta and those of the more distal part of the intrathoracic arteries. Furthermore, four valves prevent backflow of blood from the ventricles into the atria during systole and from the arteries into the ventricles during diastole. The model is represented in Fig. 1.

Figure 1

Representation of the used closed, uncontrolled cardiovascular model with illustrated RA, RV, LA and LV being right atrium, right ventricle, left atrium and left ventricle together with the systemic and pulmonary arteries, capillarries and veins

The relation between the pressure (p), volume (v), and flow (f) within the cardiac and vascular compartments is described by three equations. Equation (1) results from conservation of volume, Eq. (2) from the hydraulic equivalent of the Kirchhoff’s voltage law and the descriptions of the inertance (L) and resistance (R) components, and Eq. (3) describes the compliance (C) component:

$$ \delta v(t) \cdot \delta t^{-1} = f_{\rm in}(t)-f_{\rm out}(t) $$
$$ \delta f_{\rm in}(t) \cdot \delta t^{-1} = 1 \cdot L^{-1} \cdot (p_{\rm in}(t)-p_{\rm out}(t)-R \cdot f_{\rm in}(t))\\ $$
$$ \begin{array}{ll} {p_{{\rm out}} (t) = 1\,\cdot\,C^{{ - 1}} \cdot (v(t) - V_{\rm u} )}& {{\text{if}}\,v(t)>V_{\rm u} } \\ \;\;\;\;\;\;\;\;\;\;\;{= 0} & {{\text{else}}} \end{array} $$

The inertance (L) can be negleted in many compartments since the change in blood flow over time is small, and/or the vessels cross sectional area is so big that the inductance is negligibly small, simplifying Eq. (2) to a static one. The unstressed volume (V u ) represents the intra-compartment volume where the transmural pressure is zero. The electrical equivalent of a generic compartment within the circulation model is shown in Appendix A.

The four heart compartments are modeled as elastic chambers with contractile characteristics. Switching the heart between systolic and diastolic function (e diast and e syst, respectively, with e = 1/C and e syst is a time-vaying elastance), generates the pulsatile flow, as described earlier2:

$$ e_{\rm diast}(t) = e_{\rm min} $$
$$ e_{\rm syst}(t) = e_{\rm min} + (e_{\rm max}-e_{\rm min}) \cdot \sin(\pi \cdot \frac{t_{\rm cc}-T_{\rm d}}{T_{\rm s}}), $$

where t cc is the time in the current heart cycle and T d and T s are the duration of the chamber’s diastole and systole, respectively.

The cardiac elastances however, are also are a function of the blood volume within the chamber (see Fig. 2). During diastole, while the volume of the chamber (v chamber) is below its unstressed volume multiplied by k (e.g. at V 1 in Fig. 2), diastolic elastance is constant (e min(t)). Above this value (e.g. at V 2 in Fig. 2), fibrous tissue will linearly increase the chambers elastance until v chamber reaches its unstressed volume multiplied by m:

$$ e_{\rm min}(t) = e_{\rm min} \;\;\;\;\;\;\;\;\; {\hbox{for}\ v_{\rm chamber}(t)\leq k \cdot V_{\rm u}} $$
$$ \begin{aligned} e_{\rm min}(t) = e_{\rm min} + (\Updelta e_{\rm diast}^{\rm max}(\frac{v_{\rm chamber}(t)-k \cdot V_{\rm u}}{m \cdot V_{\rm u}-k \cdot V_{\rm u}})) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {\hbox{for}\ m \cdot V_{\rm u} \geq v_{\rm chamber}(t)>k \cdot V_{\rm u}}, \end{aligned} $$

where \(\Updelta e_{\rm diast}^{max}\) is the maximum increase in diastolic elastance due to the pericardium and k and m are constants. The maximum elastance during systole (e max(t)), will follow the Frank-Starling mechanism.5 In essence, this means that (in accordance with the sliding filament theory4) the more the cardiac muscle is stretched, the larger the force of contraction is, until the muscle reaches its optimal length at v chamber(t) = k·V u . Exceeding this optimal length will reduce systolic elastance (e.g. at V 2 in Fig. 2), represented by a linear reduction in elastance, until the systolic elastance theoretically equals the diastolic elastance at v chamber(t) = m·V u :

$$ e_{\rm max}(t) = e_{\rm max} \;\;\;\;\;\;\;\;\; {\hbox{for}\ v_{\rm chamber}(t) \leq k \cdot V_{\rm u}} $$
$$ \begin{aligned} e_{\rm max}(t) = e_{\rm max} - ((e_{\rm max}-e_{\rm min})\cdot(\frac{v_{\rm chamber}(t)-k \cdot V_{\rm u}}{m \cdot V_{\rm u}-k \cdot V_{\rm u}})) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\hbox{for} \,\,m \cdot V_{\rm u} \geq v_{\rm chamber}(t)>k \cdot V_{\rm u}}. \end{aligned} $$

In patients operating on the steep part of the systolic elastance curve, the cardiac muscle is not stretched optimally, resulting in a reduction of contraction force. In these patients adding fluid will be beneficial, as the increase in preload will result in an increase in cardiac output.

Figure 2

The diastolic and systolic elastance curves illustrate the behaviour of cardiac function (e min and e max) as a function of cardiac volume during diastole and systole. Switching the heart between systolic and diastolic function generates the characteristic pressure-volume-loop as shown. The more the cardiac muscle is stretched during diastole, the larger the force of contraction is during systole, until the muscle reaches its optimal length at v chamber(t) = k·V u (Frank–Starling mechanism). Exceeding this optimal length will increase diastolic elastance and reduce systolic elastance (e.g. at V 2 in Fig. 2) until the systolic elastance theoretically equals the diastolic elastance

To be able to realistically simulate the effect of volume expansion, a regulatory set-point model of the arterial baroreflex is implemented. The used model29 aims at maintaining mean arterial blood pressure constant by dynamically adjusting heart rate (hr), heart contractility (contr), systemic vascular resistance (svr) and the unstressed volume of the venous system (v 0,ven). In this model, the inhibitory/stimulatory activity of the baroreceptor (br act) depends linearly on the difference in mean arterial pressure (map) and the set-point value of the map (MAP sp) monitored by the baroreceptors. Since the model was developed to simulate a realistic response to volume status and volume therapy, but not to study beat-to-beat variability, delays and time constants were not included, so the equations are:

$$ \begin{array}{lll} br_{\rm act} &= BR_{\rm min}- MAP_{\rm sp} & {\text{if}} \; map<BR_{\rm min} \\ &= BR_{\rm max}- MAP_{\rm sp} & {\text{if}} \; map>BR_{\rm max} \\ &\,= map - MAP_{\rm sp} & {\text{otherwise}}, \end{array} $$

with BR min and BR max the thresholds with maximal/minimal baroreceptor activity, respectively. This baroreceptor activity is multiplied by the gain from map to the regulated variable (change per mmHg deviation in map) and subsequently used to calculate the new value based on the reference value. for example:

$$ hr = (1+C_{hr/map} \cdot br_{\rm act}) \cdot HR_{0}. $$

The svrcontr and v 0,ven are calculated in a similar way (with different gains and reference values, see Appendix B).

Respiratory System

The respiratory system during mechanical ventilation is modeled as a lumped system composed of a compliant lung (with compliance C lung), an airway (with resistance R aw representing the tube and the bronchial tree) and an additional compliant vessel representing the chest wall (simplified from20). Nonlinear flow-dependent resistance to airflow in the upper airways was ignored. The ventilated subject was assumed to be sedated; hence the diaphragm and chest wall were assumed to behave passively with compliance C wall. Air forced into the lungs during positive pressure ventilation (p aw) performs work on the lung and subsequently the chest wall resulting in a change in pressure across the lung pleura, representing the thorax pressure p th. The relations between lung volume (v aw), flow (f aw), airway pressure (p aw), compliance (C) and resistance (R aw) can now be expressed as:

$$ \delta v_{\rm aw}(t) \cdot \delta t^{-1} = f_{\rm in,aw}(t)-f_{\rm out,aw}(t) $$
$$ f_{\rm in,aw}(t) = (p_{\rm aw}(t)-v_{\rm aw}(t) \cdot (1 \cdot C_{\rm lung}^{-1} + 1 \cdot C_{\rm wall}^{-1})) \cdot R_{\rm aw}^{-1} $$
$$ p_{\rm th}(t) = 1 \cdot C_{\rm wall}^{-1} \cdot v_{\rm aw}(t). $$

Heart–Lung Interaction

Since the heart, the pulmonary circulation and part of the systemic circulation are located within the thoracic cavity (see Fig. 3), they are influenced by changes in intrathoracic pressure due to mechanical ventilation. Previously, Michard15 described five different physiological mechanisms for the effect of intrathoracic pressure on the circulation (see the various panels in Fig. 3): collapse of the vena cava (1), increase in intramural pressure in the right atrium (the downstream pressure of the vena cava) (2), compression of the pulmonary capillaries (3 and 4) and intramural pressure of the left ventricle increases (5). These mechanisms are responsible for a decrease in right ventricular preload (mechanisms 1 and 2), increase in right ventricular afterload (3), increase in left ventricular preload (4) and a decrease in left ventricular afterload (5). The increase and decrease in preload results in an increase and decrease in cardiac output, respectively, according to the Frank Starling mechanism as described earlier (Fig. 2). The increase and decrease in afterload on the other hand, results in a decrease and increase in cardiac output, respectively, because of the change in driving pressure. This continuous alternation of increase and decrease in cardiac output due to the mechanical ventilation-induced fluctuations in thorax pressure results in typical arterial waveform variations, as illustrated in Fig. 4.

Figure 3

Representation of the combined cardiovascular and respiratory system with the five mechanisms playing a role in the heart-lung interactions

Figure 4

Illustration of the fluctuations in arterial pressure (upper part, with marked maximum and minimum systolic pressure (SP max & SP min), pulse pressure (PP max & PP min) and stroke volume (SV max & SV min) caused by ventilator-induced variations in airway pressure (lower part)

The extent to which the described mechanisms are interacting with the circulation is highly depended on the volume status of the patient. In contrast to normo- or hyper- volemic patients, hypovolemic patients usually operate on the steep portion of the Starling curve (see Fig. 2). Consequently, they will show a relatively large arterial pressure variation due to the changing preload. In addition, due to the relatively low intravascular pressure in hypovolemic patients, the vena cava will collapse and the pulmonary capillaries are compressed during mechanical inspiration.

In the presented model, variation of intra- and trans- mural pressures is modeled by changing the surrounding pressure of the compartments within the thoracic cavity according to the thoracic pressure (see Fig. 3). This allows the influence on the preload of the right heart and afterload of the left heart (mechanisms 2 and 5 respectively) to be taken into account. The collapsibility of the vena cava (mechanism 1) is simulated by an increase in flow resistance r vc as a function of the pressure difference between the intra- and extra- mural pressure (p vc and p th),

$$ r_{\rm vc}=R_{\rm vc,0}+C_{\rm r,vc} \cdot(p_{\rm th}-p_{\rm vc}), $$

where R vc,0 is the normal resistance of the vena cava and C r, a constant.

Compression of the pulmonary capillaries due to an increase in transmural pressure and lung volume, causes an increase in the right ventricular afterload (mechanism 3) and left ventricular preload (mechanisms 4). The effect due to the increase in transmural pressure is modeled by changing the unstressed volume (v 0, pc) proportionally to the transmural pressure (in this case, the difference between the intramural pressure of the capillaries (p pc + p th) and the airway pressure (p aw)), see Eq. (16). Additionally, the effect caused by the stretching of the capillaries as a result of the increasing lung volume, is modeled by changing the inflow resistance (r pc ) as a function of the lung volume (v lung), see Eq. (17),

$$ v_{\rm 0,pc}=V_{\rm 0,pc,0}-C_{\rm v0,pc} \cdot(p_{\rm aw}-(p_{\rm pc}+p_{\rm th})) $$
$$ r_{\rm pc}=R_{\rm pc,0} + \log_{10} (1+v_{\rm lung}) \cdot C_{\rm r,pc} , $$

where V 0,pc,0 and R pc,0 are normal values of the unstressed volume and the inflow resistance of the pulmonary capillaries, respectively. C v0,pc and C r,pc are constants.

Dynamic Indices

For the quantification of the variation in arterial pressure and stroke volume in the model, reflecting the volume status of the patient, dynamic indices are used which include the pulse pressure variation (PPV), systolic pressure variation (SPV) and stroke volume variation (SVV). PPVSPV and SVV are defined by the relative difference in maximum and minimal pulse pressure (PP max − PP min), systolic pressure (SP max − SP min) and stroke volume (SV max − SV min) over one respiratory cycle respectively,17

$$ QV(\%)=100 \cdot \frac{Q_{\rm max}-Q_{\rm min}}{(Q_{\rm max}+Q_{\rm min})/2}, $$

with Q = PP, SP, SV for PPV, SPV and SVV, respectively (see Fig. 4). In the model, the dynamical indices are calculated breath-by-breath and subsequently averaged over the last four respiratory cycles.

Simulation Results

In order to validate the model, the simulation results are compared with data obtained from ventilated ICU patients, both from the literature and from data obtained in our ICU. First, basic hemodynamics are assessed during steady state conditions under mechanical ventilation and compared with reference values. Secondly, simulations were carried oud at two different levels of volume status and three different ventilatory settings and compared with data from literature. This second part focusses on the heart lung interaction with respect to the dynamic indices. To illustrate how the model can be used in clinical practice to help guide fluid therapy, we used the model to simulate fluid responsiveness in two patients and compared the results with the development of two patients undergoing fluid therapy on our ICU. These two patients were monitored as part of a clinical study regarding fluid responsiveness which was approved by the institutional review board of our institution.

Basic Hemodynamics

Results of the simulation of basic hemodynamic variables during mechanical ventilation are shown in Table 1. All values are within the physiological range,13 except for the pulmonary systolic pressure, which was a bit higher, but this is not considered a clinically relevant difference. The cardiac volumes, as illustrated in Fig. 5 are also within targeted values, together with the ejection fractions.

Figure 5

Volumes of the left and right ventricle (vlv, vrv, respectively) and the left and right atria (vla, vra, respectively) during two cardiac cycles

Dynamic Indices

Target values for the dynamic indices, as a result of the interaction between the ventilation and circulation are derived from a clinical study,23 where the influence of the depth of tidal volume (TV) on dynamic indices, both during the state of fluid responsiveness and after fluid loading, was systematically investigated. They found that in addition to intravascular volume status, dynamic indices were significantly affected by the depth of TV under mechanical ventilation when patients are ventilated with 5, 10 and \(15\,{\text{mL}} \cdot \,{\text{kg}}^{-1}. \) Hemodynamic and ventilation data before and after fluid loading of 20 patients, including 32 fluid challenges, are presented in Table 2 (in vivo data). Both before and after volume loading, dynamic indices at TV of 5 and \(15 \,{\text{mL}} \cdot \,{\text{kg}}^{-1}\) differed significantly from those at \(10 \,{\text{mL}} \cdot \,{\text{kg}}^{-1}. \) As a result of volume loading, dynamic indices at the respective TV were significantly lower than the values prior to the volume loading of \(6 \,{\text{mL}} \cdot \,{\text{kg}}^{-1}. \) All data from the model (Table 2, model data), except for the central venous pressure (CVP), is comparable with the mean ± standard deviation of the clinical data. The CVP however, is within the normal values as presented in Table 1.

Table 1 Basic hemodynamic variables and cardiac performance during steady state conditions
Table 2 Comparison of the development of dynamic indices from the simulation-model with human (in vivo) data published in literature as a result of varying tidal volumes and respiratory rates (RR)23

Prediction of Volume Responsiveness

As an example of how the model can be used in mechanically ventilated patients in the ICU, to distinquish volume responders (change in CO>12%) from non-responders, the characteristics from two patients admitted on our ICU are displayed in Table 3. Both patients were diagnosed as hypovolemic patients by the attending physician because of low blood pressure and urine output, and were therefore given intravascular volume to improve hemodynamics. Notice that both patients have a PPV higher than 12%, which is the clinical treshold for fluid responsiveness.18 However, the non-responder in Table 3 is ventilated with a relatively high tidal volume and low respiratory rate, which causes the patient’s PPV to rise without actually being hypovolemic. As a result, the SVI of the patient on the left (responder) rises due the fluid infusion, in contast to the SVI of the non-responder. To use the model to simulate this response, first of all, patient characteristics like weight and length should be set correctly into the model. Furthermore, the right cardiovascular parameters and variables of the patient must be entered in combination with the ventilator settings. The cardiovascular variables include e.g. the heart rate, central venous pressure, mean arterial pressure and the value of the dynamic indices. Required ventilator settings include tidal volume, respiratory rate and airway pressure (to derive respiratory compliance). Simulation results are also presented in Table 3, showing a predicted response in SVI of 20 and 7% for the volume responder and non-responder, respectively, and would thereby have predicted volume responsiveness correctly.

Table 3 Example of how the model can be used in the ICU to distinquish volume responders from volume non-responders. Despite the fact that both patients have a pulse pressure variation above the clinical treshold of 12%, one behaves like a responder and one as a non-responder. This is partly due to the influences of the ventilatory setting (e.g. respiratory rate and tidal volume) on the dynamic indices


Reliable prediction of volume responsiveness in ICU patients is essential in daily clinical practice since hypovolemia may result in inadequate organ perfusion while inappropriate fluid administration can lead to organ disfunction and contributes to increased mortality.21 Because of the lack of understanding of the underlying physiology of heart lung interaction, dynamic indices are difficult to interpret. This makes the dynamic indices only applicable in a selected population of patients. The proposed model is the first step towards a bedside clinical decision support system, and can be used as an educational tool as well.

In the present study, we developed a physiologically based mathematical model of the interaction between the respiratory and cardiovascular systems incorporating dynamic indices and fluid responsiveness. The model is able to simulate the highly interrelated processes of the relevant physiology and acts in a realistic way compared to clinical data regarding variations in volume status and tidal volume. Dynamic indices calculated by the model are comparable with those clinically measured in patients during two different levels of intravascular volume and three different ventilatory settings. By using models in combination with bedside monitoring of the cardiovascular and respiratory system, which is already done in patients receiving mechanical ventilation, models can be used for individual patients. While there are various model-based clinical decision support systems that use patient specific characteristics,7,19,27 to our knowledge this is the first model aimed at the decisions regarding volume management at the ICU.

Although we illustrated that the presented model is able to discriminate a responder from a nonresponder despite the fact that both have comparable values of dynamic indices, more effort should be made to make the model patient-specific before it can be used as a bedside decision support system. This can be done, for example, by incorporating pulse contour analysis. Thereby, the patient’s peripheral vascular resistance can be calculated and from inspiratory volumes and pressures the respiratory compliances could be derived which also would make it possible to simulate individual patients more realistically. Implementation of these patient-specific physiological characteristics will make it possible to simulate individual patients more realistically and will improve the model’s reliability concerning the prediction of volume responsiveness.

Besides serving as a decision support system, the model can also be used for educational purposes. Connecting the model to a simulator that incorporates a user interface for adjusting variables and parameters and an output monitor to visualize model output, will allow the different effects of respiratory parameters or patient specific physiology on the dynamic indices to be discriminated and thereby gaining insight into their contribution on the dynamic indices. This ability makes the simulator suitable as a supportive training tool for the understanding of the influences of the patient specific physiology and ventilatory parameter on, and thereby the interpretation of, dynamical indices regarding fluid responsiveness when accompanied by an educational program.

We also wish to address several limitations of our study. First, in humans, the exact contribution of each of the five mechanisms in the heart-lung interaction is not known. In the current model, their contribution is a weighted estimation based on clinical observations. Here, additional clinical data is needed to quantify the relevance of the various mechanisms in more detail. Second, we assumed the arterial and venous compliances to be constant. This, of course, is a simplification of reality because the volume-pressure relationship of a vessel is not linear. In general, the compliance decreases at higher pressures and volumes. On the other hand, in comparison with venous compliance, arterial compliance is reasonably constant, so introducing a constant arterial compliance will not introduce a large error. In addition, venous compliance at low pressures is also near to linear. Because of the hypovolemic conditions in which the simulated patients will mostly be situated in, and because of the fact that an increase in intrathoracic pressure will decrease transmural venous pressure, low transmural pressures are likely to exist. Furthermore, the vascular smooth muscle contraction influences vascular compliance. This contraction of the smooth muscles in the venous compartment, which is particularly important for the regulation of venous pressure and cardiac preload, is modeled by adjusting the unstressed volume of the venues by the baroreflex. In this way, smooth muscle contraction also influenced the pressure-volume relationship. Therefore, we think it is appropriate to assume a constant compliance for the various vessels. Third, a more detailed lung model could be implemented in the future for a more realistic simulation of airway pressures. In the current model, only three compartments describe the respiratory system. Finally, recent research shows that increased intra-abdominal pressure also plays a role in the heart-lung interaction.22 Since the current model does not include an intra-abdominal cavity, this influence can not be taken into account. To do so, an intra-abdominal cavity should be implemented together with intra-abdominal arteries and veins that are influenced by this surrounding pressure.


The presented model is an improved version of original cardiovascular models, which describe the non-linear cardiac elastances, and extended with a respiratory system and several mechanisms describing the interaction between both. It is able to realistically simulate the relation between patient characteristics, ventilatory settings, volume status and dynamic indices which are relevant in predicting volume responsiveness. The model is able to discriminate volume responders from nonresponders but has to be developed further to make it generally applicable. The model can also be used as a didactical tool, allowing physiologists and clinicians to understand these complex relationships.


  1. 1.

    Beneken, J. E., and V. C. Rideout. The use of multiple models in cardiovascular system studies: transport and perturbation methods. IEEE Trans. Biomed. Eng. 15(4):281–289, 1968

    Article  Google Scholar 

  2. 2.

    Beneken, J. E. W. A mathematical approach to cardio-vascular function; the uncontrolled human system, ser. Inst. Med. Phys. [Internal] report no. 2-4-5/6. Utrecht: Plenum Press, 1965.

  3. 3.

    Goodwin, J. A., W. L. van Meurs, C. D. Sa Couto, J. E. Beneken, and S. A. Graves. A model for educational simulation of infant cardiovascular physiology. Anesth. Analg. 99(6):1655–1664, 2004, table of contents.

    Google Scholar 

  4. 4.

    Gordon, A. M., A. F. Huxley, and F. J. Julian. The variation in isometric tension with sarcomere length in vertebrate muscle bres. J. Physiol. 184(1):170–192, 1966.

    Google Scholar 

  5. 5.

    Guyton, A. C. Textbook of Medical Physiology, 7th ed. Philadelphia: Saunders, 1986.

  6. 6.

    Heldt, T., E. B. Shim, R. D. Kamm, and R. G. Mark. Computational modeling of cardiovascular response to orthostatic stress. J. Appl. Physiol. 92(3):1239–1254, 2002.

    Google Scholar 

  7. 7.

    Kabal, J., and B. Lagerman. Computer-aided clinical decision system: differential diagnosis and treatment of essential hypertension by a novel noninvasive hemodynamic analyzer. Cardiovasc. Eng. 5(2):14, 2005

    Google Scholar 

  8. 8.

    Kramer, A., D. Zygun, H. Hawes, P. Easton, and A. Ferland. Pulse pressure variation predicts fluid responsiveness following coronary artery bypass surgery. Chest 126(5):1563–1568, 2004

    Article  Google Scholar 

  9. 9.

    Lansdorp, B., J. Lemson, M. J. van Putten, A. de Keijzer, J. G. van der Hoeven, and P. Pickkers. Dynamic indices do not predict volume responsiveness in routine clinical practice. Brit. J. Anaesth. 108(3):395–401, 2012.

    Article  Google Scholar 

  10. 10.

    Luo, C., D. L. Ware, J. B. Zwischenberger, and J. W. Clark. Using a human cardiopulmonary model to study and predict normal and diseased ventricular mechanics, septal interaction, and atrio-ventricular blood flow patterns. Cardiovasc. Eng. 7(1):17–31, 2007.

    Google Scholar 

  11. 11.

    Magder, S. Clinical usefulness of respiratory variations in arterial pressure. Am. J. Respir. Crit. Care Med. 169(2):151–155, 2004.

    Article  Google Scholar 

  12. 12.

    Marik, P. E., R. Cavallazzi, T. Vasu, and A. Hirani. Dynamic changes in arterial waveform derived variables and fluid responsiveness in mechanically ventilated patients: a systematic review of the literature. Crit. Care Med. 37(9):2642–2647, 2009.

    Article  Google Scholar 

  13. 13.

    Marino, P. The ICU book, 3rd ed. Baltimore: Lippincott Williams and Wilkins, 2006.

  14. 14.

    Messerges, J. Modeling systolic pressure variation due to positive pressure ventilation. Conf. Proc. IEEE Eng. Med. Biol. Soc. 1:1806–1809, 2006.

    Google Scholar 

  15. 15.

    Michard, F. Changes in arterial pressure during mechanical ventilation. Anesthesiology 103(2):419–428, 2005.

    Article  Google Scholar 

  16. 16.

    Michard, F., and J. L. Teboul. Predicting fluid responsiveness in icu patients: a critical analysis of the evidence.Chest 121(6):2000–2008, 2002.

    Article  Google Scholar 

  17. 17.

    Michard, F., D. Chemla, C. Richard, M. Wysocki, M. R. Pinsky, Y. Lecarpentier, and J. L. Teboul. Clinical use of respiratory changes in arterial pulse pressure to monitor the hemodynamic effects of peep. Am. J. Respir. Crit. Care Med. 159(3):935–939, 1999.

    Google Scholar 

  18. 18.

    Michard, F., S. Boussat, D. Chemla, N. Anguel, A. Mercat, Y. Lecarpentier, C. Richard, M. R. Pinsky, and J. L. Teboul. Relation between respiratory changes in arterial pulse pressure and fluid responsiveness in septic patients with acute circulatory failure. Am. J. Respir. Crit. Care Med. 162(1):134–138, 2000

    Google Scholar 

  19. 19.

    Neal, M. L., and J. B. Bassingthwaighte. Subject-specific model estimation of cardiac output and blood volume during hemorrhage. Cardiovasc. Eng. 7(3):97–120, 2007.

    Article  Google Scholar 

  20. 20.

    Niranjan, S. C., A. Bidani, F. Ghorbel, J. B. Zwischenberger, and J. W. Clark. Theoretical study of inspiratory flow waveforms during mechanical ventilation on pulmonary blood flow and gas exchange. Comput. Biomed. Res. 32(4):355–390, 1999.

    Google Scholar 

  21. 21.

    Payen, D., A. C. de Pont, Y. Sakr, C. Spies, K. Reinhart, and J. L. Vincent. A positive fluid balance is associated with a worse outcome in patients with acute renal failure. Crit. Care 12(3):R74, 2008.

    Article  Google Scholar 

  22. 22.

    Renner, J., M. Gruenewald, R. Quaden, R. Hanss, P. Meybohm, M. Steinfath, J. Scholz, and B. Bein. Influence of increased intra-abdominal pressure on fluid responsiveness predicted by pulse pressure variation and stroke volume variation in a porcine model. Crit. Care Med. 37(2):650–658, 2009.

    Article  Google Scholar 

  23. 23.

    Reuter, D. A., J. Bayerlein, M. S. Goepfert, F. C. Weis, E. Kilger, P. Lamm, and A. E. Goetz. Influence of tidal volume on left ventricular stroke volume variation measured by pulse contour analysis in mechanically ventilated patients. Intensive Care Med. 29(3):476–480, 2003.

    Google Scholar 

  24. 24.

    Rideout, V. C. Cardiovascular system simulation in biomedical engineering education. IEEE Trans. Biomed. Eng. 19(2):101–107, 1972

    Article  Google Scholar 

  25. 25.

    Rossow, H. A., and J. H. Stern. Teaching comparative metabolism using a graphic computer model, virtual tissue.Adv. Physiol. Educ. 35(1):99–102, 2011.

    Article  Google Scholar 

  26. 26.

    Sud, V. K., R. Srinivasan, J. B. Charles, and M. W. Bungo. Effects of lower-body negative pressure on blood flow with applications to the human cardiovascular system. Med. Biol. Eng. Comput. 31(6):569–575, 1993.

    Article  Google Scholar 

  27. 27.

    Tsuruta, H., T. Sato, M. Shirataka, and N. Ikeda. Mathematical model of cardiovascular mechanics for diagnostic analysis and treatment of heart failure: part 1. model description and theoretical analysis.Med. Biol. Eng. Comput. 32(1):3–11, 1994.

    Article  Google Scholar 

  28. 28.

    van Meurs, W. L., E. Nikkelen, and M. L. Good. Pharmacokinetic-pharmacodynamic model for educational simulations. IEEE Trans. Biomed. Eng. 45(5):582–590, 1998.

    Article  Google Scholar 

  29. 29.

    van Meurs, W. L. Modeling and simulation in biomedical engineering: applications in cardiorespiratory physiology. New York: McGraw-Hill Professional; London : McGrawHill [distributor], 2011.

  30. 30.

    van Oostrom, J. H., S. Kentgens, J. E. Beneken, and J. S. Gravenstein. An integrated coronary circulation teaching model. J. Clin. Monit. Comput. 20(4):235–242, 2006.

    Article  Google Scholar 

  31. 31.

    Warwick, R., M. Pullan, and M. Poullis. Mathematical modelling to identify patients who should not undergo left ventricle remodelling surgery. Interact. Cardiovasc. Thorac. Surg. 2010.

Download references

Conflicts of Interest


Author information



Corresponding author

Correspondence to Benno Lansdorp.

Additional information

Associate Editor Ajit P. Yoganathan oversaw the review of this article.


Appendix A

Figure 6

The electrical equivalent of one compartment, with transmural pressure (p, mmHg), blood flow (f, mL s−1), volume (v, mL), resistance (R, mmHg s ml−1), Fluid inertance (L, mmHg s2 ml−1), unstressed volume (V u , mL) and compliance (C, mL mmHg−1)

Appendix B

Table A List of the used parameters and constants which are adjusted or can not directly be found in the refered literature

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lansdorp, B., van Putten, M., de Keijzer, A. et al. A Mathematical Model for the Prediction of Fluid Responsiveness. Cardiovasc Eng Tech 4, 53–62 (2013). https://doi.org/10.1007/s13239-013-0123-0

Download citation


  • Simulation
  • Modeling
  • Cardiovascular system
  • Hemodynamics
  • Heart-lung interaction