For our 6-mo-old infant, an acute blood loss of 50 mL corresponds to an approximately 7% loss in blood volume. The simulated vital signs demonstrate an appropriate clinical response (Table 1; “50-mL Blood loss” column). Note that the “Baseline” column in Table 1 represents the vital signs of the system with baroreflex control in its baseline operating point. These vital signs are identical, by design, to the vital signs of the uncontrolled full circulation.
From the perspective of educational simulations of clinical scenarios, the main differences between the cardiovascular system of an infant and an adult are quantitative rather than qualitative. Therefore, modeling the cardiovascular physiology in this context does not require formulating a new model structure, but it does require the redefinition of a complete set of parameters. We describe an existing simulation model and the derivation of a new parameter set for the infant cardiovascular system. Several parameters were derived through proportionality constants or adjustments. The simulated vital signs were within the target hemodynamic variables, and the system reacts appropriately to blood loss. Arterial pressure wave forms and left ventricular pressure volume curves were, in our opinion, realistic enough for educational simulations. From the simulation of aortic stenosis, we concluded that through only a few simple and intuitive parameter changes, we can manipulate our model to realistically reflect essential aspects of aortic stenosis.
A simulation engine of a medical simulator typically includes many other models, and other simulator functionality requires further processing time. Our simulation results show that run time is not a limiting factor to the complexity of this model. However, increasing model complexity would further complicate the already extensive parameter-estimation procedure and make future manipulation by clinical instructors to simulate other patients, pathologies, and incidents virtually impossible.
Two fundamental physiologic aspects complicate both teaching and parameter estimation of the cardiovascular system. The first aspect is the circular nature of the system. A change in any part of the circulation will affect blood volumes, pressures, and flow rates in all parts. The second aspect is control by the baroreflex. This powerful control system also spreads out the effect of a local change to many parts of the circulation and masks the dependency of system variables on system parameters. We solved both problems by initially considering the left/systemic and right/pulmonary sides of the uncontrolled cardiovascular system independently. The ability to work on a smaller set of parameters, and without the circular effect of a parameter change, did facilitate identifying and adjusting a problem with the contractility of the right heart. After combining the two sides of the circulation, we then incorporated a model of basic aspects of the baroreflex.
This model can form the basis for a screen-based teaching tool. The model may be used to demonstrate the many clinically important differences between normal infant and adult cardiovascular physiology. Clinical scenarios may be created to demonstrate specific learning objectives. In addition, this model may be used as the underlying model for the uptake and distribution of respiratory and anesthetic gases, as well as simulation of the effects of anesthetics. Models for the simulation of the myocardial oxygen balance and electrophysiologic phenomena can easily be coupled to it, with the ultimate goal of simulation of a six-month-old infant in full scale.
J. S. Gravenstein and M. L. Good provided encouraging and guiding comments for this study.
Appendix 1: A Model for Adult Cardiovascular Physiology
The uncontrolled cardiovascular model (Figure 1) accepts as inputs blood volume changes and intrathoracic pressure and generates as outputs systemic and pulmonary artery blood pressures, central venous and all pulmonary artery catheter blood pressures, and cardiac output. Note that in the 1965 model, the pressure decreases over the systemic and pulmonary beds are represented by a single resistance each; this model does not represent the accumulation of blood in tissue, nor does it reflect distinct parallel vascular beds.
For each model compartment, the variables blood pressure, inflow rate, and volume change are computed. The compartment equations are coupled because the inflow rate of a compartment also depends on the pressure of the upstream compartment, and the volume change results from the difference between inflow and outflow rates. The relationships between variables are governed by resistance, elastance, and unstressed volume. The elastances of the heart chambers vary and reflect the contraction. Transmural compartment pressure [p(t)] is a linear function of the difference between the compartment volume [v(t)] and the unstressed volume (UV) (for v(t) > UV). For selected compartments, average intrathoracic pressure is added to the transmural pressure to obtain the absolute pressure. The elastance (E) is the second parameter in the volume-pressure relationship:
The elastances of the heart chambers are time varying, reflecting the contraction. Compartment inflow rate [f(t)] is a linear function of the pressure in the upstream compartment [pin(t) and p(t)]. The inflow resistance (R) governs this relationship:
The resistances of the atrial inflow tracts to backward flow are 10 times superior to the resistances to forward flow. The resistances of the heart valves are also nonlinear, reflecting the infinite cardiac valve resistance to back flow. The change of compartment volume is equal to the difference between f(t) and the compartment outflow rate [fout(t)]:
A single differential equation governs the inertial behavior of the blood in the arteries by using a similar notation as in Table 3, formally introduced in Appendix 2:
Beneken (6) derived the parameter values that represent a normal adult in the supine position for the uncontrolled model from a combination of clinical data and computation based on physics (Table 3).
The description of the heart is critical for the estimation of the infant parameters and will be given in detail. Figure 4A shows a typical ventricular pressure-volume loop and the clinical variables that are used to characterize it. Such a loop can be generated by a time-varying elastance model of the ventricle (7):
where p(t) is the ventricular pressure and v(t) the ventricular volume as a function of time. The relationship between these variables is determined by the unstressed volume (UV) and the elastance time profile [e(t)]. Suga et al. (14) verified this model in dogs and showed that e(t), which is characterized by its maximum value and the time of this maximum, reflects the ventricular contractility. We parameterized the ventricular elastance curve as follows:
The minimum and maximum elastances are EMIN and EMAX, respectively. Tas, Tav, and Tvs are the durations of atrial systole, atrioventricular delay, and ventricular systole, respectively. Kn is a normalization constant equal to the maximum of the time function
part of Equation A6. It can be observed that this curve is a reasonable approximation of the curve for the left ventricle calculated from measured data by Suga et al. (14). Figure 4B shows the same pressure-volume loop as in Figure 4A, now with the model parameters UV, EMIN, and EMAX. This figure also shows a second curve that is obtained under different preload conditions, but with the same model parameters. The model parameters can be derived directly when a number of measured curves are available. If only clinical descriptors, such as in Figure 4A, are available, and provided that we have an estimate for the unstressed volume UV, EMIN and EMAX can be calculated as follows:
Note that in the calculation of these slopes, we use the (average) intrathoracic pressure (PTH) as a reference, rather than 0 (atmospheric pressure). This can be made plausible by observing that the unstressed volume occurs at a zero transmural pressure across the ventricular wall, which requires a ventricular pressure equal to the intrathoracic pressure. The atrial elastance curves are identical to those proposed by Beneken (6) and are parameterized as follows:
The minimum and maximum elastances are EMIN and EMAX, respectively, and Tas is the duration of the atrium systole. The ventricle systole starts at time Tas + Tav after the initiation of the atrium systole, where Tav represents the atrioventricular delay.
Beneken gives the duration of atrium and ventricle systole as a function of heart period (HP) and a numerical value for the atrioventricular delay (in seconds):
The resistances and unstressed volumes of selected compartments depend on the baroreflex. The baroreflex also affects heart rate and contractility (maximum heart chamber elastance).
The single receptor of the Wesseling and Settels model for the baroreflex (8) is represented by a sigmoid function of mean arterial blood pressure, with a threshold of 50 mm Hg, saturation of 180 mm Hg, and maximum sensitivity around the “resting pressure” of 100 mm Hg. The baroreflex part of the model has four effector variables: heart rate, contractility, total peripheral resistance (TPR), and venous unstressed volume. Parameter data for response gain, delay, and time constant are presented by Ten Voorde (15).
For our study, we implemented a variant of this model. To facilitate parameter estimation and manipulation, we replaced the receptor function with a piecewise linear function with a slope of 1 at the mean arterial blood pressure in equilibrium. The slope is dimensionless, resulting in an output of the receptor function in millimeters of mercury. We maintained the four effector variables, substituting heart period for heart rate. Because our objective was to obtain a realistic simulated response to phenomena such as blood loss and pharmacologic interventions, but not to study beat-to-beat heart rate variability, we maintained the gains but did not include delays or the time constants of the response. Table 4 gives the numerical values for the adult for the baseline baroreflex effectors from Beneken (6), the absolute baroreflex gains of the Wesseling and Settels model in the units specified by Ten Voorde (15), and, for later reference, the derived relative gains in percentage per millimeter of mercury.
In our model, the relative gain for the maximum elastance of the left ventricle is applied to the output of the receptor function, thus affecting the maximum elastance of all four heart chambers. The baroreflex-mediated change in the RSP (resistance of the systemic peripheral vessels) parameter of the Beneken model (Table 3) is such that the total peripheral resistance change is equal to the relative gain. The relative gain for the (total) unstressed volume affects the unstressed volumes of both intrathoracic and extrathoracic compartments.
Appendix 2: Parameter Estimation for Infant Cardiovascular Physiology
The following provides explicit documentation of the steps taken for derivation of specific model parameters for the infant.
From the clinical variables describing the left ventricle (Table 2), assuming a left ventricular unstressed volume of 2.0 mL and an average intrathoracic pressure of −3.25 mm Hg (16), the model parameters for the left ventricle diastolic and maximum systolic elastances were derived by using Equations A7 and A8. The assumed and calculated parameter values are listed in Table 3. Aortic valve resistance, mitral valve resistance, and left atrial inflow tract resistance were each doubled with respect to the adult values. Left atrial unstressed volume was decreased proportionally from the adult value by comparison to the left ventricle. The left atrial minimum and maximum elastances were proportionally increased from adult values by comparison to the left ventricle, because no data concerning the infant atria were available.
Systemic vascular resistance was a calculated value and for an infant was equal to 10–20 mm Hg · L−1 · min · m2 (17). This value is a function of body-surface area and was similar in infants and adults (18). Because of the infant’s smaller body-surface area, its absolute value for systemic vascular resistance was approximately twice that of an adult. Accordingly, the infant parameters resistance of the extrathoracic arteries and veins and of the systemic peripheral vessels were obtained by multiplying adult parameters by 2.
The literature provided a value of 0.46 mL/mm Hg for the arterial compliance in infants (19). The adult systemic arterial compliance in the Beneken model is equal to (1/EITHA) + (1/EETHA) = 2.5 mL/mm Hg, where EITHA and EETHA are the elastances of the intrathoracic and extrathoracic arteries, respectively. The ratio between pediatric and adult systemic arterial elastance is therefore equal to 2.5/0.46 ≈ 5.43. Infant EITHA and EETHA were derived by proportionally increasing these values from those of the adult by using this ratio. The same ratio was used to derive the elastance of the intrathoracic and extrathoracic veins.
Arterial and venous unstressed volumes were each proportionally decreased from those of the adult model on the basis of comparison of total blood volumes for that of an infant (80 mL/kg) versus an adult (70 mL/kg) (20). For an 8-kg infant and a 70-kg adult, this leads to a proportionality constant of approximately 0.13. Note that this assumes a similar distribution of blood between arterial and venous circulations in adults and infants.
The above-mentioned parameters were then incorporated in the model software to yield the first iteration of vital signs for the uncontrolled left heart and systemic loop of the infant cardiovascular system. The blood flow inertia parameter was adjusted to optimize the pressure wave form in the intrathoracic artery compartment. Note that when this parameter is used in such an empirical fashion, it should no longer be referred to in terms of underlying physics.
In the infant, the end-diastolic volume (EDV) of the right ventricle is approximately 1.5 times the EDV of the left ventricle (21). We applied this ratio to the unstressed volume going from the left to right ventricle and did the same going from the left to the right atrium. Right atrial and ventricle minimum and maximum elastances were derived by scaling the right heart from adult to infant in the same way as the left heart. As for the left heart, all right heart resistances were multiplied by 2.
We used a pulmonary peripheral vascular resistance of 3.7 mm Hg · L−1 · min = 0.22 mm Hg · mL−1 · s, which is within the published range of 2.5–7.5 mm Hg · L−1 · min for pediatric patients (17). This value is two times the adult value in the Beneken model (1.83 mm Hg · L−1 · min = 0.11 mm Hg · mL−1 · s) and is therefore also consistent with changes made to the systemic circulation. Elastance of the pulmonary arteries and pulmonary veins were both derived by increasing these values from those of the adult, proportional to the increase in systemic arterial elastance. Similarly, pulmonary arterial and pulmonary venous unstressed volumes were both proportionally decreased from that of the adult model on the basis of comparison of total blood volume for an infant versus an adult. Note that with more numerical data on changes in pulmonary vascular tone in infancy, the value of this parameter may change.
The right heart and pulmonary circulation parameters were then incorporated in the model software to yield the first iteration of vital signs for the uncontrolled right heart and pulmonary loop of the infant cardiovascular system. Two parameter changes were made to correct a difference in right and left cardiac output. Table 3 contains the corrected parameters. Subsequently, we combined the two halves of the circulation and simulated the full uncontrolled infant system with the parameters summarized in Table 3. Throughout the simulations, we maintained a constant intrathoracic pressure of −3.25 mm Hg.
There are no published data for the baroreflex of a 6-mo-old infant. The parameters for the baroreflex were therefore derived from published data concerning the neonate (22). Drouin et al. (22) observed a spontaneous baroreflex sensitivity in full-term neonates of 10.23 ms/mm Hg to changes in systolic blood pressure. At a heart period of 465 ms (heart rate of 129 bpm), this corresponds to a relative baroreflex gain for the heart period of 2.2%/mm Hg. This relative change is similar to that of the adult (Table 4). We used the same relative gains as in the adult for all baroreflex control effectors: heart period, contractility, peripheral resistance, and venous unstressed volume.
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