computation of aortic flow from pressure in humans using a nonlinear, three-element model

Computation of aortic flow from pressure in humans using a nonlinear, three-element model

K. H. WESSELING, J. R. C. JANSEN, J. J. SETTELS, AND J. J. SCHREUDER Netherlands Organization for Applied Scientific Research, TN0 Biomedical Instrumentation, Academic Medical Centre, 1105 AZ Amsterdam; Department of Pulmonary Diseases, Erasmus University, 3000 DR Rotterdam; and Department of Anesthesiology, Academic Hospital, University of Limburg, 6202 AZ Maastricht, The Netherlands

WESSELING, K.H.,J.R.C. JANSEN, J.J. SETTELS,AND J.J. SCHREUDER. Computation of aortic flow from pressure in humans using a nonlinear, three-element model. J. Appl. Physiol. 74(5): 2566-2573, 1993.-We computed aortic flow pulsations from arterial pressure by simulating a nonlinear, time-varying three-element model of aortic input impedance. The model elements represent aortic characteristic impedance, arterial compliance, and systemic vascular resistance. Parameter values for the first two elements were computed from a published, age-dependent, aortic pressure-area relationship (G. J. Langewouters et al. J. Biomech. 17: 425-435, 1984). Peripheral resistance was predicted from mean pressure and model mean flow. Model flow pulsations from aortic pressure showed the visual aspects of an aortic flow curve. For evaluation we compared model mean flow from radial arterial pressure with thermodilution cardiac output estimations, 76 times, in eight open heart surgical patients. The pooled mean difference was +7%, the SD 22%. After using one comparison per patient to calibrate the model, however, we followed quantitative changes in cardiac output that occurred either during changes in the state of the patient or subsequent to vasoactive drugs. The mean deviation from thermodilution cardiac output was +2%, the SD 8%. Given these small errors the method could monitor cardiac output continuously.

aortic input impedance; nonlinear model; time-varying model; aortic arctangent pressure-area relationship; age dependency; simulation; stroke volume; cardiac output

CARDIAC OUTPUT is an important hemodynamic variable. It is studied in physiological experiments often in relation with changes in pressure. It is monitored in surgical and critical care patients. There is a trend in physiological research, as well as in the operating room and intensive care unit, to require essentially continuous cardiac output monitoring.

ca Pressure pulse analysis h .as been rdiac output monitorin g (9, 15)

U sed for beat-to -beat Traditionally, such

methods corn .puted cardiac stroke output from certai n characteristic s of an arterial pressure pulse, based on a

variety of simple models of the arterial system. Charac- teristics used included systolic area, heart rate, and mean and diastolic pressure. In these methods, aortic properties were assumed constant under varying distending pressures. They were not constant, however, and models were not sufficiently detailed. Consequently, pressure pulse analysis often proved to be of insufficient precision (6).

More recently, several studies have shown the suitabil- ity of a three-element model of arterial input impedance to describe the relationship between aortic pressure and flow (2, 16). Once model parameters are found, flow can be computed from measured pressure by simulating the model (2). This flow then provides a continuous measure of cardiac output. Integrated over one heartbeat it provides stroke volume; integrated over one minute it provides cardiac minute volume.

We investigated whether this model could monitor cardiac output continuously. We were fortunate in that a precise thermodilution technique is now available (4), and we anticipated that this improved technique would permit a more reliable evaluation.

METHODS

Computing Flow From Pressure

During the development we made three decisions. 1) We would compute flow by simulating the response of a three-element model of arterial input impedance to arterial pressure (Fig. 1). To do this, the parameter values of the model elements must be known. Two of the model parameters, characteristic impedance and arterial compliance, can be derived from an aortic pressure-area relationship. 2) We would use the arctangent model of aortic mechanics of Langewouters et al. (7) (Fig. 2). The use of this arctangent relationship causes the model elements to be nonlinearly dependent on pressure. The third element, total systemic peripheral resistance, is a variable. Its value is not known but is an outcome of the model

2566 0161-7567193 $2.00 Copyright 0 1993 the American Physiological Society

AORTIC FLOW COMPUTED FROM PRESSURE 2567

Mt) L /

P(1) c W

-f T RP PIA) U)

1

-

1

FIG. 1. Diagram of &element model used in this study to compute flow. ZO, characteristic impedance of proximal aorta; C,, windkessel compliance of arterial system; I$,, total systemic peripheral resistance. 2, and C, have nonlinear, pressure-dependent properties, indicated by stylized S symbol. R, varies with time, as symbolized by arrow. Q(t), blood flow as function of time; P(t), arterial pressure waveform; P,(t), windkessel pressure.

simulation. 3) We would use the current peripheral resistance value as a best estimate of that parameter in the simulation of the next beat.

The model. To compute aortic flow from pressure we base the pressure-flow relationship on the traditional (1, 20) but adequate (16) model shown in Fig. 1. This model has three elements representing the three major properties of the aorta and arterial system: aortic characteristic impedance (Z,), a dynamic property of the aorta that impedes pulsatile outflow from the ventricle; windkessel or buffer compliance (C,), the ability of the aorta and arterial system to elastically store the cardiac stroke output from the left ventricle; and peripheral resistance (R,), the Poiseuille resistance of all vascular beds together. The value of total peripheral resistance in the model is the sum of Z, and R,. The special symbols in Fig. 1 indicate the nonlinear, pressure-dependent properties of Z, and C, and the time-varying property of I$.

During systole, flow is into the model; during dia-

l- I I

I I

I I

I I P [ mmHgJ , 1 1 . ,, , 1 1 I 1 , 1 1 1 . 1

.-SO 100 150 200 PO PO + PI

70 c

60

50

40 r

stole this inflow is dissipated in the periphery. Given input pressure, aortic systolic inflow is principally determined by the time constant Z,C,. If this time constant is too short, computed stroke volume will be too small. The peripheral resistance element value is not a major deter- minant of systolic inflow. Diastolic outflow from the windkessel into the peripheral resistance and windkessel pressure decay are principally determined by the time constant R,C,. If input pressure decay equals model windkessel pressure decay, the diastolic inflow is zero. Windkessel compliance is a common factor in both time constants.

Model parameter ualues. To carry out the computations numerically, proper values for the model parameters Z,, C,, and R, must be found. We do have precise and detailed results on the viscoelasticity of the human aorta in the form of pressure-area relations (7), and Z, and C, can be computed from it.

The aortic characteristic impedance Z, can be ex- pressed as

Z, = \JpI(AC’) (1)

where p is the density of blood, A is the cross-sectional area of the aorta, and C’ is the aortic compliance per unit length. C’ is the derivative of the pressure-area relationship with respect to pressure (P)

C’ = dA/dP (2) Windkessel compliance, C,, represents the lumped compliance of the entire arterial system. We assume its value to be equal to the compliance of one unit length of thoracic aorta times an aortic effective length, 1

C - W-

LC I (3) For an adult patient we assumed for L a value of 80 cm (19). More strictly, L could depend on patient height and weight.

1 6- M 0 I - a *s-c

g 5- -- -8 .o& 0.0. 0 O* 0: l 4 4- F m-c- uo-

0 0 0

3- 0 0

2-

l- A [Yrl

0 1111111111 0 50 100

x 0 0 0

0 -

FIG. 2. A: arctangent pressure-area relationship. It fits measured pressure-area curves of human aorta well (7) and is characterized by 3 parameters: maximal area (A,,,), pressure at inflection point (p,>, and width parameter (Pi). Measurements of the 3 parameters are plotted in R-D, together with their regres- sions on age (A), as listed in Table 1. Open circles, female (F) aortas; filled circles, male (M) aortas. Pa- rameters PO and P, show only small scatter around regression, but scatter for A,,, is substantial. [Adapted from data in Langewouters et al. (7).]

2568 AORTIC FLOW COMPUTED FROM PRESSURE

TABLE 1. Summay of regression equations for arctangent aortic model parameters vs. patient age and gender (7)

Parameter F M

A max? cm2

PO, mm& PI, mmHg

4.12 5.62 72 - 0.89A 76 - 0.89A 57 - 0.44A 57 - 0.44A

F, female; M, male; A,,,, maxi ma1 area; PO, inflection pressure; P, , width parameter; A, patient age ( in yr). We used these data to obtain proper pressure-area relation for a patient, given gender and age.

We define R, as the ratio of average pressure to average flow. Its value changes only slowly (15) compared with a heartbeat interval. We therefore use its current computed value to simulate the flow of the next beat. For the first beat, at the start of the simulation, a reasonable initial value is assumed. We take the ratio of 100 mmHg mean pressure and 3 l/min cardiac output. From true mean pressure and computed mean flow the next ap- proximation is computed, and so on. R, converges from the initial value to the correct value in a few heartbeats.

Pressure-area relationship. Z, and C, have been presented in terms of the aortic pressure-area relation and its derivative. We must now define that relation. Accord- ing to Langewouters et al. (7), the thoracic aortic cross- sectional area can be described as a function of pressure by an arctangent with three parameters (see Fig. 2A)

A(P) = A,,, [

1 P - PO 0.5 + - arctan p

7r i 11 P 1 (4)

A max is the maximal cross-sectional area at very high pressure. Parameter PO defines the position of the inflection point on the pressure axis (40 mmHg in Fig. 2); P, defines the width between the points at one-half and three-quarter amplitude (50 mmHg in Fig. 2). Parame- ters PO and P, compare with the mean and the SD of a probability function.

The compliance per unit length is the derivative of A(P) with respect to P (Eq. 2) and depends on P

C’(P) = (5)

Given correct values for the three arctangent parameters A max, PO, and P, , the model parameters Z, and C, can be computed for any pressure P.

Arctangent parameters. The three arctangent parameters have been determined for 45 human thoracic aortas in vitro (7) and are presented in graphic form in Fig. 2 (B-D). It appears that PO and P, regress tightly on patient age (A), given patient gender. A,,, does not regress on age, at least not over the observed age range. Its average value is indicated separately for females and males. However, an individual’s value can deviate as much as 40% from the group average. The statistical information is in Table 1. Thus given patient gender and age, we can compute A,,, , PO, and P, using the data in Table 1 to find a proper pressure-area relationship for a subject, but with uncertainty in the maximal area. To improve accu-

racy in monitoring an individual patient, a proper value for maximal area must be obtained in another way. If no other information is available, the group average value is used as a best estimate. The error involved from this is as follows.

Importance of maximal area. Calculations show that computed model flow using the group average maximal area deviates by the same percentage amount from true flow as the group average maximal area deviates from a patient’s true area. Given the 20% scatter SD in the maximal area, model flow cannot be computed with an accuracy better than this percentage. An accurate measurement of aortic diameter can yield a patient individual value for maximal area, improving accuracy. An indica- tor dilution estimate of cardiac output can also be used, as was proposed by Warner (17). The initial population average value of A,,, is then multiplied by K, an individual patient’s ratio of thermodilution to model flow cardiac output.

Model flow computation. Model flow, finally, is computed by simulating the behavior of the model under the applied arterial pressure pulsation. The values for windkessel compliance and characteristic impedance, as a function of instantaneous pressure, are inserted in the model. Because these values depend on pressure as was shown above, the model behavior is nonlinear. Simula- tion is done digitally, and model computations are repeated for each new pressure sample taken. Left ventricular stroke volume is computed by integrating model flow during systole; cardiac output is computed by multiply- ing stroke volume with instantaneous heart rate. The systolic duration and the heartbeat interval are derived from the pressure waveform. We call this approach the model flow method.

Patients

We evaluated the model flow method using arterial pressure and thermodilution cardiac output recorded in the operating room. After obtaining informed consent, we studied eight male patients who suffered from multi- ple coronary vessel disease but had intact aortic valves. All patients were monitored in 1989, and their signals were digitized. Except for two patients whose data were accidentally destroyed, all patients studied in that year were analyzed. The patients underwent elective coronary artery bypass graft surgery. Lorazepam (5 mg) was administered as premeditation 2 h before surgery. A peripheral venous catheter, a 20-gauge radial artery cannula with a Medisize continuous flush device, and a 7-Fr Swan-Ganz pulmonary artery catheter were inserted under local anesthesia. Before induction of anesthesia a series of baseline hemodynamic measurements was performed under spontaneous breathing conditions. Anes- thesia was induced with an initial dose of 7.5 pg/kg of sufentanil and was maintained with a continuous infusion of sufentanil at a rate of 3.75 pg. kg-’ l h-l. Pancur- onium bromide (0.1 mg/kg) was given for muscle relax- ation. The patients were ventilated at 10 cycles/min with an oxygen-air mixture having an inspired oxygen frac- tion of 50%.


Experimental Techniques

In all patients, we recorded pressure in the radial artery using a standard Hewlett-Packard model 78534A arterial pressure channel with a 15-Hz built-in cutoff fre- quency. In two patients, we could additionally record as- cending aortic pressure using a Millar microtip pressure transducer, but only during series e (see Measurement Protocol), just before going on bypass. A Baxter COM-2 device computed thermodilution cardiac output. A Sie- mens model 900B servoventilator was used for mechanical ventilation. It also provided the time pulses to trigger injections of 5 ml of room temperature 5% glucose with a computer-controlled power injector. An interface box passed all monitor and control lines, providing electrical isolation. This interface comprises three buttons to flush, start, and, if necessary, break the automatic thermodilution injections. The interface was connected to an IBM-compatible personal computer of AT level control- ling the hardware system. Because aortic pressure was only incidentally available in these patients but radial pressure was available throughout the operation, the lat- ter was used as input to the model. An aortic flow pulsation signal for comparison with the computed model flow pulsation was also not available in these patients. In- stead, average model flow was compared with thermodilution estimates of average cardiac output.

To improve the accuracy of the thermodilution estimates, we used the technique of phase-controlled injections, equally spread over the ventilatory cycle (4). Using bolus injections, we estimated cardiac output four times, each in a different phase of the ventilatory cycle, and averaged the results. Each injection was delayed from the start of a ventilatory period over either 0, 25, 50, or 75% of the duration of the period. We waited -30 s between injections. We averaged model flow cardiac output over two ventilatory cycles beginning at each thermodilution injection. Each series of four provided one comparison between model flow and thermodilution cardiac output.

Measurement Protocol

At various moments during the operation, measurements were done after major changes in patient state. Surgery was suspended, and hemodynamics were usually stable during the measurement periods. We did series s during spontaneous breathing, before induction of anesthesia. Series a-k were taken with the patient maintained on the ventilator. The code letters a-k correspond to the following 11 instants: a, 3 min after induction; b, immediately after sternotomy; c-e, at 15, 10, and 5 min before bypass, respectively; f-i, at 3, 8, 13, and 18 min after bypass, respectively; j, 3 min after sternal fixation; and k, after end of surgery. Because of surgical and anes- thetic events and depending on the duration of the operation, not every series was always performed in all patients. Seven to 12 series of four thermodilution estimations were thus carried out. In one patient an extra series was done. The series a thermodilution cardiac output was always obtained and was used to calibrate model flow cardiac output. During series b, c, and e before bypass, and during series h, j, and k after bypass, sodium nitroprusside was infused in doses of between 0.5 and 1.5

pg l kg-‘. min. This drug has a strong vasodilating effect. Other vasoactive drugs were administered as needed during the operation.

We performed an off-line sensitivity analysis of the model flow method by repeating the model flow computations on the digitized data base. Patient age and pressure zero level were artificially varied. The first estimate was computed with unmodified parameters. Computa- tions were then repeated with patient age entered incor- rectly at +lO and -10 yr off true age. Age appears in the values for the arctangent parameters PO and P,. The computations were repeated two more times with the correct age but with a pressure offset of +lO and -10 mmHg. Insensitivity to the absolute level of pressure is important because the pressure transducer membrane is not always precisely at heart level. The mean difference of model flow with simultaneous thermodilution cardiac output and its SD and range were used to judge sensitivity.

Our study could have produced good results accidentally because circumstances were right or patients were stable. We decided, therefore, to use the earlier, so-called “corrected” characteristic impedance (cZ) method (19) on the same data. This method has been evaluated before (5), and its behavior is known. It computes beat-to-beat stroke volume, V,, by integrating the area under the systolic portion of the arterial pressure pulse, dividing the area by the CZ method’s characteristic impedance calibration factor, Z,,

V 1

Z= Z s [P(t) - P,ldt (6)

ao T ,

A correction is then applied that accounts for changes in mean pressure (P,) and heart rate (f). This results in corrected stroke volume estimates, V,,

V cz = [0.66 + 0.005f - O.OlA(O.O14P, - 0.8)]Vz (7)

where P, is arterial end-diastolic pressure, T, is the ejec- tion period, and A is the age of the patient. The method is calibrated for each individual patient similarly to the model flow method, but adjusting Z,,.

RESULTS

Figure 3 presents an example of measured aortic and radial arterial pressure and computed model flow pulsations from each pressure signal. Compared with aortic pressure, radial pressure appears delayed and distorted. Model flow computed from aortic pressure shows the typical characteristics of an aortic waveform: a steep up- stroke, a slow downstroke, a steep end-systolic downstroke terminated in a sharp dicrotic wave, and a flat near-zero diastolic flow. The flow wave computed from the distorted radial pressure is also distorted. In particu- lar, peak flow is exaggerated.

Table 2 lists patient age, height, and model flow calibration factor K, pressure, pulse rate, and thermodilution cardiac output ranges. K indicates by which factor the patient aortic cross-sectional area is different from the population average. For our eight patients, K averages 107 t 22%. Th’ is is not significantly different from 100% (t test, Z-sided, at P = 0.05). The 22% SD is not signifi-


0’ I

0 0.5 1 1.5 2 2.5 3 3.5 4

I I

0 0.5 1 1.5 2 25 3 3.5 4

t M

FIG. 3. A: simultaneous aortic (solid line) and radial pressures (dashed line). B: simulated model flow from aortic (solid line) and radial pressures (dashed line). Model flow computed from aortic pressure shows typical characteristics of an aortic flow pulsation: steep upslope at beginning of systole, a gradual down slope terminated in a steep final phase, and sharp dicrotic notch at end systole, followed by a period of almost zero flow in diastole. Model flow computed from radial pressure is delayed and distorted and shows typical characteristics less convinc- ingly.

TABLE 2. Summary of patient information

P,, mm& f, beats/min Qtd, l/min

Patient A, yr K Min Max Min Max Min Max

8 9

10 11 12 15 16 17

62 67 60 51 62 56 51 56

1.53 59 110 59 73 4.2 6.4 1.02 54 94 61 96 3.9 5.9 1.32 68 111 62 80 3.8 5.9 1.01 66 98 71 109 4.3 6.9 0.81 57 93 61 81 3.6 5.7 1.02 55 98 50 85 3.2 4.6 1.00 71 92 82 103 3.3 5.8 0.88 55 110 49 94 3.1 5.8

K, model flow calibration factor (dimensionless); P,, mean pressure; f, heart rate; Qtd, quadruple thermodilution cardiac output. Deviations of K values from 1 are indication of accuracy of model flow method before calibration. Averaged over group of 8 patients, K is 1.07 k 0.24 (SD). All patients were male.

cantly different (F test, at P = 0.05) from the 17% SD found earlier in the A,,, value for male patients (7). Pressure, heart rate, and cardiac output varied approxi- mately over a 58 range during the operations.

The results of the 76 comparisons with thermodilution cardiac output are summarized in Table 3. We used deter- mination a in each patient to compute the calibration factor K. The remaining 68 comparisons were taken into the postcalibration error statistics. Mean error ranged from -0.1 to 0.4 Vmin. Its average of 0.1 l/min (2%) is not significantly different from zero. The SD ranged from 6 to 11% with an average of 8%. By comparison, the offset of the CZ method ranged from -0.6 to 0.8 l/min with a pooled average of 0.3 l/min. Scatter for the CZ method ranged from 7 to 21% with a pooled average of 12%.

A scatter diagram of model flow vs. thermodilution cardiac output is shown in Fig. 4. It presents the individual data taken in the eight patients summarized in Table 3. We obtained from 7 to 13 quadruple measurements in a patient. After using measurement a for calibration, at least 6 and at most 12 measurements per patient re-

TABLE 3. Model flow and CZ method errors after calibration

Model Flow Error CZ Method Error

Avg ad, Mean, SD, Mean, SD, Patient l/min N l/min l/min % l/min l/min %

8 5.12 10 -0.14 0.41 8.2 0.30 0.49 9.6 9 4.89 7 0.34 0.29 5.7 0.30 0.65 13.3

10 4.67 8 0.04 0.39 8.4 0.63 0.40 8.6 11 5.42 12 0.20 0.33 6.0 T 0.58 0.35 6.5 12 4.59 10 0.04 0.34 7.3 0.79 0.40 8.7 15 4.10 7 -0.14 0.40 9.5 0.45 0.69 16.8 16 4.69 9 0.45 0.27 5.8 0.33 0.43 9.2 17 4.29 13 -0.05 0.46 10.5 0.04 0.92 21.4

Mean 4.72 9.5 0.09 0.36 7.7 0.28 0.54 11.8 +SD kO.43 kO.22 20.07 k1.8 20.42 -to.20 k5.1

cZ, corrected characteristic impedance. %, Error SD given as percentage of corresponding average cardiac output. Error figures exclude the 8 pairs of cardiac output used for calibration. The N - 1 comparisons in each patient are averaged. Bottom lines give errors pooled for group.

8

G

quad & in

I/min

. FIG. 4. Scattergram of 68 individual model flow [quadruple (quad) Qmf] vs. phase-controlled thermodilution (quad Qti) cardiac output estimates. Values are obtained after calibration of model parameters. Statistics are summarized in Table 3.

mained, representing major changes in patient state (see Measurement Protocol). Figure 5 is the errorgram of the model flow method showing the difference between the model flow and thermodilution estimates vs. their paired means. All differences except one are within 21 l/min. Except for five outliers all differences stay within -0.50 and +0.65 Vmin. The distribution of the 68 errors is also shown.

Pooled results of the sensitivity analysis for age and pressure are given in Table 4. Overall, errors do not differ greatly under the applied changes in the age parameter and in the offset pressure. Judging from the error min- ima and maxima per condition, neither do they differ greatly in individual patients.

DISCUSSION

The model flow method uses a three-element model of aortic input impedance to compute a flow pulsation from an arterial pressure pulsation (2). A new feature in this model is that two parameter values are computed (pre-


1.5

Error histogram 0 5 10 15 I,llll,,,,l,,,,J

-1.5 1 I , 1 , I , 1 { 3 4 5 6 7

t&n, + &d/2 in

I/min

FIG. 5. Scattergram of difference between individual model flow (Q,,) and thermodilution (Q,) cardiac output estimates plotted vs. their mean. Differences are independent of level of cardiac output and remain, excepting 5 outliers, within -0.50 and +0.65 l/min. Right: histogram of differences. Excepting the 5 outliers, errors are uniformly distributed (x2 test). We did not, therefore, plot +2 SD lines.

dieted) from published arctangent aortic pressure-area relationships. This computation is carried out for every pressure sample taken on the waveform, resulting in a nonlinear, pressure-dependent model. Another new feature is that the third model parameter is obtained by iteration, is linear, but varies per heartbeat.

Aortic pressure-area relations were obtained from unrelated, in vitro measurements on segments of human thoracic aorta. The arctangent model for the aortic pressure-area relation describes the measurements with only three parameters, which depend on age and sex (7). Pa- tient age and sex were thus used as inputs. No data fitting was done, nor was any prior knowledge of a patient’s aortic properties available. Still, our results verify the validity of this approach.

The evaluation of the model flow method was performed in open heart surgical patients. This group probably presents the most taxing environment for any hemodynamic monitoring method. Our comparisons were done with surgery suspended for the duration of each quadruple measurement. The small error of the comparison of two cardiac output techniques may be due in part to this cooperation.

Arteriosclerosis

Because the method depends critically on pressure- area relationships, a logical question is, How does arteriosclerosis affect it? Langewouters et al. (8) analyzed their aortic measurements (7) for effects of sclerosis. If an aorta showed only fatty streaks and spots and fibrous plaques, it was classified as “only intima affected, mild or atherosclerotic.” If, in addition, atheromas, hemorrhage, ulcerations, and calcium deposits were found, it was classified as “also media affected, severe or arteriosclerotic.” There was about an equal number of aortas in each group. It appeared that the severely sclerotic group had larger A,,, values by 9% compared with the pooled average, whereas the mildly affected aortas had smaller areas by 7%. The position of the inflection point of the pressure-area relation, PO, did not depend on the degree of sclerosis. Width parameter, P,, was 4 mmHg lower for the severely than for the mildly affected aortas. As a re-

sult, the aortic compliance did not depend on degree of sclerosis for pressures in the physiological range.

Apparently, the increased aortic wall stiffness with in- creasing degree of sclerosis is effectively compensated by the larger diameter. For the patient group studied we may expect arteriosclerosis to be present and thus may find aortic areas larger than average. This we did find (Table 3, K = 1.07) but could not prove statistically (SD = 0.24).

Adequacy of Three-Element Model

The three-element model was proposed in 1930 by Broemser and Ranke (1,20). Burkhoff et al. (2) state that the three-element model provides a reasonable represen- tation of left ventricular afterload for predicting stroke volume but significantly underestimates peak aortic flow in dogs. McDonald and Nichols (lo), using a pressure gradient method, similarly found that peak flow is more difficult to compute. We could not do a peak flow comparison because pulsatile flow was not measured in these patients. For most applications, however, possibly incor- rect peak flows would have little consequence as long as beat-to-beat stroke volume and cardiac output are correct.

Sensitivity Analysis

The arctangent parameters PO and P, regress tightly on age (see Fig. 2) but not without scatter. This scatter can be interpreted as some patients having apparently older aortas for their age and some having younger aortas for their age. The possible effect this has on model flow-computed cardiac output was studied with our sensitivity analysis. We found that varying the patient age t10 yr hardly affected the error figures. We conclude that any effect of difference between calendar age and physiological age on aortic mechanical properties is small and does not affect significantly the precision of the model flow method.

During surgery it is often difficult to monitor the height of the arterial pressure transducer with respect to heart level. Awake subjects may change their position, anesthetized subjects may be tilted for surgical reasons, or the table may be changed in height. This may cause errors in the pressure level. We assumed that this effect would be less than t10 mmHg and computed the subsequent model flow error. Table 4 shows that there is little effect. Relative position of the pressure transducer, there-

TABLE 4. Summary of sensitivity analysis

Error Offset, l/min Error Scatter, l/min

Mean SD Min Max Mean SD Min Max

Correct 0.09 0.22 -0.14 0.45 0.36 0.07 0.27 0.46 A - 10 yr 0.17 0.16 -0.06 0.35 0.42 0.17 0.27 0.68 A + 10 yr 0.07 0.29 -0.36 0.47 0.38 0.06 0.31 0.50 P - 10 mmHg 0.09 0.15 -0.12 0.34 0.37 0.11 0.26 0.57 P + 10 mmHg 0.10 0.23 -0.21 0.44 0.36 0.04 0.33 0.42

Situations: correct, pooled errors of Table 3 given for comparison; A + 10 yr, all patient ages were entered 10 yr too young or too old; P + 10 mmHg, pressure transducer simulated 10 mmHg (13 cm) too high or too low with respect to heart level.


fore, is not critical for a correct flow computation in these subjects. We conclude that the flow computation is ro- bust with respect to age and mean pressure level.

Use of Radial For Aortic Pressure

Because aortic pressure was not available routinely in our patients, radial pressure had to be used instead. This pressure wave is distorted with respect to aortic pressure (11). The effect is that the computed aortic flow wave is also distorted (see Fig. 3). The error analysis shows that cardiac output estimates from radial pressure, neverthe- less, are rather precise.

Comparison With cZ Method

Pressure pulse analysis has been used to obtain beat- to-beat cardiac output monitoring (9). Under stable hemodynamic circumstances these techniques performed quite well, but when blood pressure, heart rate, peripheral resistance, or vascular tone changed, deviations occurred. An earlier method, the CZ method, tried to correct for this aberrant behavior (Eqs. 6 and 7), achieving improved precision under varying hemodynamic conditions and drug regimens (5, 13, 14, 19). Recently published errors for the CZ method are near the ones reported here in Table 3; older results indicate larger errors. This may be due to a less precise reference cardiac output method (18).

Precision of Thermodilution Estimates

The thermodilution technique is usually associated with large errors. Jansen et al. (4) have shown that, under conditions of mechanical ventilation, these errors are due in part to flow modulation and follow a system- atic pattern. Averaging several randomly injected estimates reduces this error. Averaging a series of phase- controlled estimations, however, reduces the error much more. In a previous study (4), averaging four phase-controlled injections reduced the errors from 14 to 3%. Thus our thermodilution errors are probably limited to no more than 5% SD.

Error Analysis

The postcalibration error mean and SD (Table 3) thus represent the differences between two imprecise techniques. Averaging over a large number of estimations, thermodilution probably has a mean error near zero (3). This has not been demonstrated for the phase-controlled injection technique in patients. We assume, however, that any mean error is small enough to be ignored. The model flow mean error, after calibration with thermodilution, is also near zero (Table 3). Thus only the scatter errors of the thermodilution and model flow methods remain. They are statistically independent errors because the methods are based on different physical principles. With certainty we can state, therefore, that the upper

limit of scatter error for each technique must be below the 8% observed in this study. For the phase-controlled thermodilution technique we estimated an error SD of maximally 5%. Thus part of the 8% error in the comparison is due to errors in the reference method. We sub- tracted this error as explained elsewhere (18). The resulting error of the model flow method alone becomes 6%; that of the CZ method, 11%.

This 6% error includes five outlier errors. The outliers all occurred immediately after cardiopulmonary bypass. We checked that outliers were not correlated with sodium nitroprusside infusion or with changes in pressure or pulse rate. It has been reported (12) that radial artery pressure immediately postbypass can become unreliable with both mean and pulse pressure in error. We tend to consider this a probable cause for the outliers. These outlier errors should not be ignored; after all they are part of routine clinical procedure. The important fact is that even the outlier errors are not enormous. The model flow method apparently uses a model with the nonlinearity control needed for a precise stroke volume computation when hemodynamics change.

Conclusion

The model flow method extracts precise cardiac output information from arterial pressure, a signal that is often already recorded for other reasons. The nonlinear three- element model, representing the three major characteristics of arterial input impedance, allows a precise computation of stroke volume and cardiac output. The arctangent aortic pressure-area relation is a useful ap- proximation of aortic nonlinearity. Radial artery pressure can be used to compute beat-to-beat values, al- though aortic pressure provides a visually better model flow waveform. For highest accuracy and precision a calibration of the model parameters is required.

Address for reprint requests: K. H. Wesseling, TN0 Biomedical In- strumentation, Academic Medical Centre, Suite LO-002 Meibergdreef 15, 1105 AZ Amsterdam, The Netherlands.

Received 27 January 1992; accepted in final form 2 October 1992.

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computation of aortic flow from pressure in humans using a nonlinear, three-element model

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