A pharmacodynamic model of respiratory rate and end-tidal carbon dioxide values during anesthesia in children

Abstract

It is essential to monitor the end-tidal carbon dioxide (ETCO₂) during general anesthesia and adjust the tidal volume and respiratory rate (RR). For the purpose of this study, we used a population pharmacodynamic modeling approach to establish the relationship between RR versus ETCO₂ data during general anesthesia in children, and to identify the clinical variables affecting this relationship. A prospective observational study was designed to include 51 patients (aged ≤ 12 years), including users of antiepileptic drugs (levetiracetam, valproic, or phenobarbital (n = 21)) and non-users (n = 30), scheduled to receive general anesthesia during elective surgery. When the ETCO₂ was at 40 mmHg, the RR was adjusted 1 breath per every 2 min until the ETCO₂ was 30 mmHg and recovered to 40 mmHg. Pharmacodynamic analysis using a sigmoid E_max model was performed to assess the RR-ETCO₂ relationship. As RR varied from 3 to 37 breaths per minute, the ETCO₂ changed from 40 to 30 mmHg. Hysteresis between the RR and ETCO₂ was observed and accounted for when the model was developed. The C_e50 (RR to achieve 50% of maximum decrease in ETCO₂; i.e. 35 mmHg) was 20.5 in non-users of antiepileptic drugs and 14.9 in those on antiepileptic drug medication. The values of γ (the steepness of the concentration-response relation curve) and k_eo (the first-order rate constant determining the equilibration between the RR and ETCO₂) were 7.53 and 0.467 min^-1, respectively. The C_e50 and ETCO₂ data fit to a sigmoid E_max model. In conclusion, the RR required to get the target ETCO₂ was much lower in children patients taking antiepileptic drugs than that of non-user children patients during the general anesthesia.

Introduction

During general anesthesia, it is essential to control and monitor the ventilation status by using the respiratory rate (RR), tidal volume, end-tidal carbon dioxide (ETCO₂), and oxygenation.

Several studies have tried to find the optimal tidal volume during mechanical ventilation to protect the lung in various clinical situations [1, 2]. With the determined tidal volume, there is variation in the RR required to maintain normocarbia, regardless of the preoperative RR measured in the ward during the resting period, and it is relatively liberally controlled.

ETCO₂ monitoring with capnography is a non-invasive and a useful surrogate parameter for the partial pressure of arterial carbon dioxide (PaCO₂) [3, 4], despite its inaccuracies [5]. The ETCO₂ monitor has been important because it can confirm an endotracheal intubation or ventilation, and can also be used to control the cerebral blood flow. Anesthesiologists have managed the cerebral blood flow with ETCO₂, which is representative of the PaCO₂ and has a great effect on the control of cerebral vessels.

Usually, the RR is adjusted based on the ETCO₂ or PaCO₂ to maintain the normocarbia, which reflects the minute ventilation. However, we have observed that patients with hydrocephalus or antiepileptic drugs show unexpectedly low ETCO₂ levels compared to other pediatric patients with the determined tidal volume and RR in mechanical ventilation during general anesthesia. Therefore, it would be useful if we established a pharmacodynamic model that incorporates a population approach with covariates such as age, sex, history of medication, or clinical status to describe the minute ventilation ETCO₂ dose-response relationship.

In this study, we attempted to identify the clinical variables affecting the relationship between RR and ETCO_2, and establish the population pharmacodynamic modeling approach to RR and ETCO₂ data during general anesthesia in children.

Materials and methods

Patient recruitment and anesthetic methods

The study was approved by the Institutional Review Board of Seoul National University Hospital (1602-079-740) and registered at http://cris.nih.go.kr (KCT0001859). Each participant was given a verbal explanation and had the opportunity to ask questions about the study methods and purposes, and informed consent was acquired from one parent. Verbal consent was acquired from the participants. All procedures followed the principles of the Declaration of Helsinki and its revisions.

A total of 56 patients (ages between 1 month and 6 years), who underwent surgery under general anesthesia and were or were not taking a chronic ( > 7 days) antiepileptic drug, were enrolled in this study. The exclusion criteria were acutely increased intracranial pressure, moyamoya disease, or other conditions that could affect PaCO₂.

All the patients complied with the preoperative fasting time according to the practice guidelines from the American Society of Anesthesiologists. All the recruited patients did not receive premedication. Standard monitoring, including electrocardiogram (ECG), heart rate (HR), non-invasive blood pressure (BP) in 1-min intervals, peripheral oxygen saturation (SpO₂), and ETCO₂ were monitored. Anesthesia was induced with thiopental sodium (5–6 mg/kg under 3-year-old) or propofol (2–2.5 mg/kg over 3-year-old), and the patients were manually ventilated with a mixture of 8% (vol) sevoflurane and 100% oxygen. Rocuronium was administered to facilitate the endotracheal intubation. Anesthesia was maintained using 1 MAC of sevoflurane with a remifentanil infusion, and the inspired oxygen fraction was 0.4. Tidal volume was 8 mL/kg. The initial RR was controlled to a goal ETCO₂ of 40 mmHg. An esophageal stethoscope was inserted to monitor the temperature. Body temperature was maintained over 35.5 °C during the study period.

Study protocol

After the stabilization of ETCO₂ at 40 mmHg for 10 min after the start of surgery, RR was increased stepwise by 1 breath every 2 min, down to a minimal ETCO₂ of 30 mmHg, with a fixed tidal volume. Subsequently, the RR was decreased stepwise by 1 breath per every 2 min, up to an ETCO₂ of 40 mmHg. During the study period, all the vital signs (ECG, HR, BP, SpO₂, and temperature) were closely monitored. After the study period, attending anesthesiologists liberally adjusted the RR to maintain an ETCO₂ between 30 and 40 mmHg.

Data analysis and model selection

The RR and corresponding ETCO₂ data with demographic characteristics were used to develop a pharmacodynamic model with NONMEM (nonlinear mixed-effects modeling) software (ver VII; GloboMax, Hanover, MD, USA). The model provides estimates of the population’s mean parameters, inter-individual random variabilities, and residual random effects. The pharmacodynamic model was run using the first-order conditional estimation method with inter-individual-residual interaction (FOCEI) to determine parameter estimates. To account for the delay between RR measurements and the ETCO₂ at the effect site (lung), an effect compartment (C_e) was modeled. It was assumed that RR was linearly linked to C_e, which was estimated by the following equation:

$$\mathrm{d}{C}_{\mathrm{e}}∕\mathrm{d}t=\left(\mathrm{RR}-C_{\mathrm{e}}\right)\times k_{\mathrm{e0}},$$

where k_e0 is the first-order rate constant determining the equilibration between the RR and ETCO₂. The C_e over time was calculated as the convolution of the RR over time with the disposition function of the effect site. The convolution was based on a “connect-the-dots” approach previously used by Schnider et al. [6]. The k_e0 was estimated by minimizing the area of the hysteresis loop of the ETCO₂ data versus the RR. One individual k_e0 value was calculated for each patient on the basis of his or her particular respiratory ramp. The relationship between RR and ETCO₂ was modeled using a sigmoid E_max model:

$$E_{\max }\mathrm{model}:E=E_0+\left(E_{\max }-E_0\right)\times C_{\mathrm{e}}^{\lambda }∕\left(C_{\mathrm{e50}}^{\lambda }+C_{\mathrm{e}}^{\lambda }\right),$$

where E₀ is the initial ETCO₂ = 40 mmHg, E_max is the ETCO₂ = 30 mmHg, C_e50 is the RR required to get 50% of maximum decrement of ETCO₂ (35 mmHg of ETCO₂ in this study), and γ is the steepness of the concentration-response relation curve. Inter-individual variability in E₀, E_max, C_e50, γ, and k_e0 was modeled using an exponential error model. Residual intra-individual variability was modeled using the additive error model. NONMEM computed the minimum objective function value (OFV), a statistical equivalent to the –2 log-likelihood of the model. A level of 0.05, which corresponds to a reduction in the OFV of 3.84 (chi-square distribution, degrees of freedom = 1, P < 0.05), was used to distinguish between the hierarchical models. First, we developed the basic population model without the covariates. Then, we explored additional covariates of sex, age, height, weight, and group (according to the history of antiepileptic drug medication) successively to determine their impact on estimates of the model parameters. The control code for developing of model was shown in Appedix 1. Bootstrap analysis internally validated the models (fit4NM 4.6.0, Eun-Kyung Lee and Gyu-Jeong Noh; http://fit4NM.org) [7]. Predictive checks were performed using fit4NM 4.6.0 [7].

Statistical analysis

Statistical analyses were performed using SPSS 23.0 for Windows (SPSS Inc., Chicago, IL, USA). Data are expressed as the mean ± standard deviation for normally distributed continuous variables, the median [IQ: 25–75%] for non-normally distribute continuous variables, and counts and percentages for categorical variables.

Results

Demographic data are shown in Table 1. A total of 56 patients were recruited, and 51 completed the study. Data for five individuals in the antiepileptic drug group were excluded because of a surgery time that was too short to get data. Fifteen children took levetiracetam, two took valproic acid, one took phenobarbital, and three used combination therapy.

Table 1.

Demographic characteristics

	Normal (n = 30)	Antiepileptic drug (n = 21)	P-value
M/F	19/11	13/8
Age (year)	2.2 [0.77–3.0]	2.4 [0.60–5.0]	0.807
Height (cm)	89.2 [74.2–98.0]	85.5 [63.0–105.5]	0.711
Weight (kg)	14.0 [9.7–16.7]	14.4 [8.0–20.0]	0.977
Diagnosis
		Hydrocephalus (12)
		Intractable seizure (3)
		Cerebral palsy (1)
		Brain tumor (3)
		Others (2)

During the study period, there were no hypertension, hypotension, bradycardia, tachycardia, or desaturation events. RR was adjusted by 1 breath every 2 min. ETCO₂ was changed later compared to the changes in RR, and this showed hysteresis. As a result, although RR was decreased to increase the ETCO₂, after an ETCO₂ of 30 mmHg was achieved, ETCO₂ decreased more to less than 30 mmHg. Therefore, the lowest ETCO₂ was 28 mmHg. Then, the reverse process was imposed. RR was quite variable and ranged from 3 to 37 breaths per minute. The time to reach the minimal ETCO₂ was compared in each group, and there was no significant difference between the two groups (20.0 [10.0−22] vs. 19.26 [14−24] min, P = 0.708).

Figure 1 shows the time course of the RR and the changes in ETCO₂ in all the recruited patients. An increased RR was matched with a decreased ETCO₂. Figure 2 shows the plot of RR versus ETCO₂ in time. As the RR increased, the ETCO₂ decreased, as expected.

Fig. 1.

Time courses of respiratory rate (RR, a) and end-tidal carbon dioxide (ETCO₂, b) during mechanical ventilation under general anesthesia in all recruited patients. Although the initial ETCO₂ was set to 40 mmHg and the minimal target ETCO₂ was 30 mmHg, the minimal target ETCO₂ was less than 30 mmHg because of hysteresis between RR and ETCO₂

Fig. 2.

The relationship between respiratory rate (RR) and end-tidal carbon dioxide (ETCO₂) during mechanical ventilation under general anesthesia in all recruited patients

For the development of the model, the total measurement of 1108 pairs of ETCO₂ and RR were used. Antiepileptic drug medication history was a significant covariate for the C_e50, and it resulted in a greater improvement in the OFV (Δ17.8, P < 0.05, degree of freedom = 1) than the basic model.

The C_e50 (20.5 vs. 14.9) differed between children who were users and non-users of antiepileptic drugs. The RR during 1 min that was needed to get an ETCO₂ = 35 mmHg (50% reduction of highest ETCO₂ [40 mmHg] to lowest ETCO₂ [30 mmHg]) in the patients taking an antiepileptic drug was approximately five less than non-users. Other demographic parameters, such as age, sex, height, and weight were not assumed as covariates. The estimates of all parameters for the sigmoid E_max model are summarized in Table 2. Diagnostic plots (goodness-of-fit plots) for the final model are shown in Fig. 3, which shows that the model had a satisfactory performance and no significant bias. Predictive checks of the final model are presented in Fig. 4. In total, only 1.0% of the data were distributed outside of the 95% prediction intervals of the predictive check.

Table 2.

Pharmacodynamic model parameters; effect of respiratory rate on end-tidal carbon dioxide

Parameters	Estimate (RSE, %)			Median [2.5–97.5%]
	Base model	Final model
E0, mmHg	39.8 (0.16)	39.8 (0.15)		39.8 [39.7–39.9]
Emax, mmHg	28.1 (0.87)	28.1 (0.92)		28.1 [27.8–28.4]
Ce50, time−1	18.1 (3.92))	Normal	20.5 (3.48)	20.3 [19.3–21.4]
		Antiepileptic drug	14.9 (6.77)	15.2 [14.1–16.3]
γ	7.6 (5.30)	7.53 (5.35)		7.6 [7.2–8.1]
keo, min−1	0.47 (8.58)	0.47 (8.60)		0.47 [0.42–0.52]
OFV	1138.8	1121.0 (Δ17.8)

E₀ = baseline ETCO₂ (ETCO₂ = 40 mmHg); corrected QT interval in absence of sevoflurane; E_max = maximum decreased ETCO₂ (ETCO₂ = 30 mmHg); C_e50 = respiratory rate required to achieve 50% of maximum decrease of ETCO₂ (ETCO₂ = 35 mmHg); k_e0 = first-order rate constant determining the efflux from the effect site; γ = slope of sigmoid relation between respiratory rate and ETCO₂.

Fig. 3.

Goodness-of-fit of the population (a) and individual prediction (b) of endo-tidal carbon dioxide of the final model. Dotted line represents the line of identity

Fig. 4.

Predictive checks of the final pharmacodynamic model. A black + indicates the observed end-tidal carbon dioxide (ETCO₂). The solid red line and shaded areas indicate the 50% prediction line and 95% prediction intervals, respectively. Baseline ETCO₂ was set to 40 mmHg. The time shows the total study period needed for changing ETCO₂ (40–30–40 mmHg)

Discussion

To our knowledge, this study was the first to apply the pharmacodynamic model to describe the effect of RR on ETCO₂ in mechanical ventilation under general anesthesia in children. This study showed that the RR required to change the ETCO₂ from 40 mmHg to 35 mmHg was lower in patients taking antiepileptic drugs compared to those who were not. This result was in line with the clinical findings.

To develop the pharmacodynamic model, first, we created the basic model and then added covariates such as age, sex, height, weight, and history of antiepileptic drugs for the final model. We fixed the initial and maximal reduced ETCO₂ because those values are the clinically acceptable range (30–40 mmHg of ETCO₂). Therefore, E_max or E₀ could not be different between the two groups. There was a hysteresis between RR and ETCO₂; there were several cases that were either over or less than the target ranges of ETCO₂. Only C_e50, γ, and k_e0 were different between the two groups. During the development of the final model, adding the covariates and history of taking antiepileptic drugs reduced the OFV more than the basic model; there was a difference in C_e50 between the two different clinical situations (Table 2). Age, sex, height, or weight did not reduce the OFV.

ETCO₂ is a useful and accurate parameter for estimating the PaCO₂ in mechanically ventilated patients [4]. During general anesthesia, the minute ventilation was usually adjusted by changing the RR with a fixed tidal volume; ETCO₂ is a guideline used to adjust the RR. Usually, the initial RR was set to the RR while the patient was on the ward or empirically with a fixed tidal volume (8 mL/kg in this study), and then gradually changed to maintain normocarbia, hypocarbia, or hypercarbia according to the clinical situation. There was a reference for RR during sleep based on age [8]. Considering the recruited patients in this study were aged from 3 months to 6 years, the ranges of the median RR were 20 to 29 breaths per minute; this was compatible with the observations of this study. The ranges of RR during anesthesia were 3 to 37 breaths per minute to maintain the ETCO₂ between 30 and 40 mmHg. The lowest RR matched with 40 mmHg of ETCO₂ was seen in a 9-month-old boy taking an antiepileptic drug; the highest RR required to achieve an ETCO₂ of 30 mmHg was in a 10-month-old boy who was not taking any antiepileptic drugs.

Overall, the RR in patients taking antiepileptic drugs was definitely lower than that of non-users with the same age and similar weight to get the desired ETCO₂. From the results of this study, we conclude that the increment of RR required to get halfway between the highest and lowest ETCO₂ in a patient taking antiepileptic drugs was less than that of a non-user. This finding was compatible with a previous clinical observation, and our newly developed pharmacodynamic model could explain this finding.

There are several categories of current antiepileptic drugs based on the foremost targets at their therapeutic concentrations. The suggested target and mechanisms are as follows: Na⁺ channel [9], Ca²⁺ channel [10], K⁺ channel [11], GABA-related actions [12], or synaptic vesicle protein 2 A [13]. Meanwhile, the mechanisms of action of volatile anesthetics are not fully known yet. Several hypotheses, such as the involvement of the GABA receptor [14], NMDA receptor [15], Ca²⁺ channel [16], protein kinase C [17], Na⁺ channel [18], and K⁺ channel [19] have been suggested. Therefore, the many targets of antiepileptic drugs overlap with the targets of volatile anesthetics. If antiepileptic drugs and volatile anesthetics are used together, inhibition of excitatory activity or exaggeration of inhibitory activity of the targets of the two drugs might show an additive or synergic effect. Therefore, in this study, the combined use of volatile aesthetic and antiepileptic drugs could generally inhibit or suppress the neuronal activity and result in reduced metabolism. As a result, CO₂ production, which is a waste product of metabolism, could be reduced with coadministration of volatile anesthetics and antiepileptic drugs.

Most of the anesthetics, including sevoflurane, reduced regional cerebral blood flow and oxygen consumption. Therefore, the metabolic rate was reduced [20]. In addition, a hyper-excitable state of neurons in epilepsy has been shown, and the neurons are fueled by glucose, which results in a hypermetabolic state [21]. Therefore, antiepileptic drugs work by reducing the brain activities and the resulting brain metabolism. Under general anesthesia, brain activities are decreased, and with the antiepileptic drugs, these activities might be profoundly decreased and result in a reduced CO₂ production because CO₂ is the waste product of metabolism. Therefore, we hypothesize that the patient taking an antiepileptic drug under general anesthesia would have a suppression of brain metabolism and reduction of CO₂ production. This assumption could be explained by the lower RR needed in the patients taking antiepileptic drugs to get the same ETCO₂ compared to non-users with a fixed tidal volume.

Many studies have tried to determine the optimal tidal volume, but few studies have determined the optimal RR because of relative liberal control. In adults, RR did not vary much during the mechanical ventilation under general anesthesia. RR might not be the most important respiratory parameter in general anesthesia, especially in adults. However, RR is one of the relatively important respiratory parameters used to control the minute ventilation in children because they are developing and have a variety of RRs, and the results of this study may be important to pediatric anesthesiologists. Basically, anesthesiologists adjust the RR with a fixed tidal volume using a target ETCO₂ [22]. Therefore, the calculated RR might be important for the target ETCO₂, and the result of this study should help explain why.

There were several limitations to this study. First, we did not distinguish the antiepileptic drugs according to the antiepileptic drug categories. However, all the antiepileptic drugs have basically shown similar effects on the brain, and the results from patients using different categories of antiepileptic drugs might not be different. Next, we changed the RR every 2 min; this duration is relatively short, and the results obtained do not show a clean stepwise change in ETCO₂. Furthermore, this does not reflect the real clinical practice. Although it does not mimic a real clinical situation, the model could provide a difference between the two groups, and this result would help notify pediatric anesthesiologists to change the RR to adjust the ETCO₂ during general anesthesia.

In conclusion, the adjusted RR needed to set the ETCO₂ between 40 and 35 mmHg was much lesser in patients taking antiepileptic drugs compared to patients not taking antiepileptic drugs while on mechanical ventilation under general anesthesia.

Appendix 1

NONMEM control code

$PROB RUN# 60101 (Direct PD fit for RR vs CO₂)

$DATA AED_2.csv IGNORE = #

$INPUT ID TIME CP DV GROUP SEX AGE HT WT; TIME: min; CP: respiratory rate; DV: end-tidal CO₂; Group: 0 = normal group, 1 = antiepileptic group; SEX: 0 = male, 1 = female; AGE: yr; HT: cm; WT: kg

$PRED

TH1 = THETA(1)

TH2 = THETA(2)

TH3 = THETA(3)

TH4 = THETA(4)

TH5 = THETA(5)

TH6 = THETA(6)

E0 = TH1*EXP(ETA(1))

EMAX = TH2*EXP(ETA(2))

CE50 = (TH3*GROUP + TH6*(1-GROUP))*EXP(ETA(3))

GAM = TH4*EXP(ETA(4))

KE0 = TH5*EXP(ETA(5))

IF (TIME.EQ.0.OR.CP.EQ.0) THEN

CE = 0

PTIME = 0

PCP = 0

PCE = 0

ENDIF

DT = TIME-PTIME

IF (DT.EQ.0.OR.CP.EQ.0) THEN

DT1 = 1

ELSE

DT1 = DT

ENDIF

IF(CP.GT.0.AND.CP.GE.PCP) THEN

SLOPE = (CP-PCP)/DT1

DELT = DT1*SLOPE + (KE0*PCP-SLOPE)*(1-EXP(-KE0*DT1))/KE0

ENDIF

IF(CP.GT.0.AND.CP.LT.PCP) THEN

SLOPE = (LOG(CP)-LOG(PCP))/DT1

DELT = PCP*KE0/(KE0 + SLOPE)*(EXP(DT1*SLOPE)-EXP(-KE0*DT1))

ENDIF

IF (CP.GT.0.AND.DT.GT.0) THEN

CE = PCE*EXP(-KE0*DT) + DELT

ELSE

CE = PCE

ENDIF

IPRED = E0 + (EMAX - E0)*CE**GAM/(CE**GAM + CE50**GAM)

PTIME = TIME

PCP = CP

PCE = CE

W = 1

IRES = DV - IPRED

IWRES = IRES / W

Y = IPRED + W*EPS(1)

$THETA;#6

(20, 40); E0

(10, 30); EMAx

(1, 15); CE50_antiepileptic group

(1, 3); GAM

(0, 0.1); KE0

(1, 15); CE50_normal group

$OMEGA;#5

0.2; IIV_E0

0.2; IIV_EMAX

0.2; IIV_CE50

0.2; IIV_GAM

0.2; IIV_KE0

$SIGMA;#1

16; EPS

$ESTIMATION NOTBT NOOBT NOSBT SIGL = 3 NSIG = 1 MAXEVAL = 9999 PRINT = 5 METHOD = 1 INTER MSFO = 60101.MSF NOABORT

$COVARIANCE PRINT = E

$TABLE ID ETA(1) ETA(2) ETA(3) ETA(4) ETA(5)

FILE = 60101.ETA NOPRINT FIRSTONLY NOAPPEND

$TABLE ID E0 EMAX CE50 GAM KE0

FILE = 60101.PAR NOPRINT ONEHEADER NOAPPEND

$TABLE ID TIME GROUP IPRED IWRES CWRES

FILE = sdtab60101 NOPRINT ONEHEADER

$TABLE ID E0 EMAX CE50 GAM KE0 ETA(1) ETA(2) ETA(3) ETA(4) ETA(5)

FILE = patab60101 NOPRINT ONEHEADER NOAPPEND

$TABLE ID AGE HT WT

FILE = cotab60101 NOPRINT ONEHEADER NOAPPEND

$TABLE ID SEX GROUP

FILE = catab60101 NOPRINT ONEHEADER NOAPPEND

Competing interests

The authors declare no competing interests.