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Physiological Modelling of Agitation-Sedation Dynamics Including
Endogenous Agitation Reduction
A. D. Rudge1, J. G. Chase1, G. M. Shaw2, D. Lee3
1
Centre for Bioengineering, Department of Mechanical Engineering, University of
Canterbury, Christchurch, New Zealand
2
Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand,
University of Otago School of Medicine and Health Sciences
3
Centre for Bioengineering, Department of Mathematics and Statistics, University of
Canterbury, Christchurch, New Zealand
Centre for Bioengineering
Department of Mechanical Engineering
University of Canterbury
Private Bag 4800
Christchurch
New Zealand
Correspondence to Andrew Rudge
Email: a.rudge@mech.canterbury.ac.nz
Telephone: +64 21 147 5582
Fax: +64 3 364-2078
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ABSTRACT
Sedation administration and agitation management are fundamental activities in any intensive
care unit. A lack of objective measures of agitation and sedation, as well as poor
understanding of the underlying dynamics, contribute to inefficient outcomes and expensive
healthcare. Recent models of agitation-sedation pharmacodynamics have enhanced
understanding of the underlying dynamics and enable development of advanced protocols for
semi-automated sedation administration. However, these initial models do not capture all
observed dynamics, particularly periods of low sedative infusion. A physiologicallyrepresentative model that incorporates endogenous agitation reduction (EAR) dynamics is
presented and validated using data from 37 critical care patients. High median Relative
Average Normalised Density (RAND) values of 0.77 and 0.78 support and minimum RAND
values of 0.51 and 0.55 for models without and with EAR dynamics respectively show that
both models are valid representations of the fundamental agitation-sedation dynamics present
in a broad spectrum of ICU patients. While the addition of the EAR dynamic increases the
ability of the model to capture the observed dynamics of the agitation-sedation system, the
improvement is relatively small and the sensitivity of the model to the EAR dynamic is low.
Although this may represent a limitation of the model, the inclusion of EAR is shown to be
important for accurately capturing periods of low, or no, sedative infusion, such as during
weaning prior to extubation.
Keywords
Physiological models, Non-linear dynamics, Dynamic modelling, Agitation, Sedation
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1.0 INTRODUCTION
Effective delivery of sedation in the intensive care unit (ICU) is fundamental to providing
comfort and relief to the critically ill. A Midazolam and Morphine combination, given by
intermittent bolus or infusion, is the mainstay of most ICU regimens [1]. Midazolam is a
sedative agent used to induce a state of conscious sedation, and Morphine is a powerful opioid
analgesic with additional mild sedative effects.
Insufficient sedation exacerbates anxiety and agitation, and increases the risk of selfextubation. Over-sedation is a common outcome, and is damaging to patient health, and
increases length of stay and cost [2]. While sedation is administered to maintain patient
comfort, most sedation in the ICU is administered in response to patient agitation [3]. Hence,
the target, or control, metric for regulating sedation in critical care is minimal agitation, rather
than a given level of consciousness.
Several recent studies have highlighted the cost and healthcare benefits of drug delivery
protocols based upon agitation-sedation assessment scales [4-6]. Very simple sedation control
protocols aimed at minimizing over-sedation have reduced length of stay 28–35%, total drug
requirements 46–57%, and testing for altered mental status 67% [2, 7]. Therefore, controlling
agitation without over-sedation offers significant potential. The key to achieving such control
is accurate models that include all fundamental clearance pharmacodynamic behaviours.
Agitation-sedation cycling describes oscillations between states of agitation and over-sedation
often observed in critically ill patients. The underlying non-linear dynamics of the agitationsedation cycle are not well understood and many complex interactions contribute to observed
patient behaviour. Therapeutic treatment methods rely heavily upon the experience and
intuition of the medical staff, introducing variability and inconsistency. Computerized
sedative infusion protocols that enable consistency of care and minimize fluctuations in
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treatment could therefore improve patient care, simplify administration, minimize drug
consumption and staff effort, and reduce costs.
In spite of these advantages, current computer assisted infusion systems utilising feedback
control in the ICU are still in their infancy [8-10]. Target Controlled Infusion (TCI) systems
deliver drugs to maintain target plasma concentrations, using a pharmacokinetic model. This
approach is well suited to anaesthesia where short periods of reduced consciousness and wellknown pharmacology are common. However, infusion systems that regulate the infusion rate
to maintain target agitation levels, thus regulating the primary metric for long-term sedation,
are the goal for improving care in the ICU.
Although conscious sedation in the ICU utilises hypnotic drugs similar to those used in Total
Intra-Venous Anaesthesia (TIVA), the drug dose and consciousness levels are distinctly
different, as are the patients and the environment. More importantly, the overall goal of the
therapy is significantly different. Anaesthesia applications aim to induce reduced
consciousness for short periods. However ICU sedation seeks to simultaneously minimise
both agitation and over-sedation over long periods of time. Hence, critical care sedation
management is a very different problem that seeks the best trade-off between sedative dose
and patient agitation. Therefore, while similarities between the two fields may provide
insight, the differences prevent simple application of anaesthesia delivery methods,
measurements and protocols to long-term ICU sedation administration.
The primary limitations to the development of optimized sedative infusion protocols are the
lack of an objective, physiologically-based, quantified agitation scale and limited
understanding of the underlying system dynamics. The subjective measures of agitation
currently employed introduce significant variability between assessors and inconsistency of
care [5, 11]. While no gold-standard agitation-sedation scale exists, the Riker Sedation
Agitation Scale (SAS) is widely accepted [12]. Quantitative agitation sensors being developed
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[13-16], offer the potential to significantly improve agitation management when coupled with
dynamic models and control protocols [17]. This research further develops the
physiologically-based models required to develop agitation feedback protocols for medical
decision support systems and eventually automated sedation administration.
Previous attempts to improve agitation management in the ICU have been limited to clinical
trials employing fixed sedative protocols using subjective agitation assessments [2, 4, 7, 18].
The use of quantitative modelling to enhance understanding of the system and provide a
simulation platform is a recently developed tool in this area [10, 17, 19]. Initial models [17,
20] were built upon a simple 2-compartment pharmacokinetic framework and treated the
concomitant administration of Morphine and Midazolam as one drug for pharmacokinetic
purposes. The assumption of a linear relationship between plasma drug concentration and
drug effect led to underestimation of the patient’s sedative requirements at higher doses. More
complex dynamics including separate pharmacokinetics, effect saturation and drug synergism
were added upon further development of the model [21, 22].
It is proposed that time-varying parameters and the inclusion of an endogenous agitation
reduction (EAR) dynamic may improve the fit of the agitation-sedation model with recorded
data. This research incorporates these additional non-linear dynamics to create more
physiologically representative models of the agitation-sedation system. These models are then
statistically validated against prior work and clinical data to assess the importance of these
additional dynamics.
2.0 METHOD
2.1 Physiological Modelling
The model presented utilizes separate pharmacokinetic (PK) models for Midazolam and
Morphine. Displayed schematically in Fig. 1, it is a closer representation of the actual
physiological system than other works [17, 19, 20-22], and includes delayed distribution, drug
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synergism, effect saturation and endogenous agitation reduction. The model is defined in
three main portions:
I. Pharmacokinetics of Morphine:
Vco
dCco
o
 ( K CL
 K ceo  K cpo )Cco  P oU  K eco Ceo  K opcC op
dt
V po
Veo
dC op
(1)
  K opcC op  K cpo C co
(2)
dCeo
  K eco Ceo  K ceo Cco
dt
(3)
dt
II. Pharmacokinetics of Midazolam:
Vcs
dCcs
s
 ( K CL
 K ces )Ccs  P sU  K ecs Ces
dt
Ves
dCes
  K ecs Ces  K ces Ccs
dt
(4)
(5)
III. Pharmacodynamics of Morphine and Midazolam:
t
dA
 w1S  w2 (t ) KT  EComb ( )e KT ( t  ) d  w3 A
dt
0
(6)
where Cc, Cp and Ce are the drug concentrations (mg/L) in the central, peripheral and effect
compartments, Vc, Vp and Ve are the distribution volumes (L) of the central, peripheral and
effect compartments, U is the intravenous infusion rate (mL/min), A is an agitation index, S is
the stimulus invoking agitation, Kij is the transfer rate (L/min) from compartment i to
compartment j, KCL is the drug clearance (L/min), KT is the effect time constant (min-1), and Po
and Ps are the proportions of Morphine (‘o’) and Midazolam (‘s’) per unit volume of solution
respectively (mg/mL). Time is represented by t (min),  is the variable of integration, and the
terms w1 and w2(t) are the relative weighting coefficients between stimulus and sedative
sensitivity. Similarly, w3 is the coefficient associated with the endogenous reduction of patient
agitation, or agitation reduction without sedative. Finally, EComb is the combined
pharmacodynamic effect of the individual effect site drug concentrations of Morphine and
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Midazolam determined using response surface modelling as defined in Minto et al. [23] and
o
s
detailed in the lower portion of Fig. 1. In Fig. 1 C50
and C50
represent the concentrations at
which Morphine or Midazolam would have 50% effect if administered alone.
This model is intended to be the simplest necessary to capture the essential dynamics of the
agitation-sedation system, matching patient observations and published literature with a
physiologically representative model. Equations (1)–(2) represent the pharmacokinetics (PK)
of the infusion and distribution of Morphine, and Equation (3) represents transport of
Morphine to the effect site. Similarly, Equation (4) represents the pharmacokinetics of the
infusion and distribution of Midazolam, and Equation (5) represents transport of Midazolam
to the effect site.
The non-linear pharmacodynamic (PD) Equation (6) is based on physiological observations of
patient behaviour, and simply states that the rate of change of agitation depends upon the
relative magnitude of the stimulus to the cumulative sedative effect and endogenous agitation
reduction. Stimulus in this context refers to the combined effect of inherent pain, distress, or
loss of inhibition caused by the diseased/injured state of the patient, and the therapeutic and
diagnostic procedures performed by medical staff [21].
2.1.1 Pharmacokinetic (PK) Modelling
Many pharmacological models exist for the delivery of Morphine or Midazolam
independently. Typically, Morphine and Midazolam are administered as a fixed ratio solution.
Hence, this paper utilizes separate compartmental PK equations for each drug, so the
combined drug infusion rate results in accurate effect site concentrations. The central PK
compartment in Fig. 1 represents the infusion site and local blood vessels, such as the heart
and lungs. The peripheral compartment can be thought of as the peripheral parts of the body
to which blood flows, such as the legs and arms, and incorporating the fatty tissues into which
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these drugs and/or metabolites can be deposited. The effect site concentration is that region in
which the drug exerts its primary effect. For drugs affecting the central nervous system, such
as Morphine and Midazolam, the cerebro-spinal-fluid [24], or the brain [25, 26], is an
acceptable representation of the effect site.
Clinical trials investigating the PK of Morphine show that concentration profiles in healthy
and ICU subjects are best approximated by a 3-compartment model [24, 27]. These studies
attempt to model the PK of intravenous (I.V.) Morphine incorporating the effect of
metabolites such as Morphine-3-glucuronide (M3G) and Morphine-6-glucuronide (M6G) by
adding additional compartments. However, the analgesic and sedative effects of these
metabolites are not easily quantified and the details of their pharmacological effect are not yet
fully understood [25, 28, 29]. Further, metabolite concentrations have been shown to be small
when administration techniques bypassing the first-pass effect are used, such as I.V.
administration [28]. Therefore, this portion of the model uses three compartments and does
not model the formation, distribution or secondary effect of Morphine metabolites.
Clinical trials investigating the PK of I.V. Midazolam show that concentration profiles in
healthy and ICU subjects are best approximated by a 2-compartment model [30-32]. While
the activity of the major metabolite of Midazolam, alpha-hydroxy-midazolam (α-OH
Midazolam) has received a lot of attention, the effect has not yet been fully defined [1, 3335]. Therefore, this portion of the model uses two compartments and does not model the
formation, distribution or secondary effect of metabolites.
The overall model defined in Equations (1)-(6) consists of several PK and PD components.
The PK parameters form the basis for the drug distribution and elimination half-life, while the
PD parameters define the shape of the response surface and drug sensitivities. It is commonly
accepted that significant inter-patient variability is observed in the pharmacology of sedatives
in the critically ill and elderly [6, 36]. However, studies have also shown that inter-patient
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variability appears to be due primarily to variations in PD parameters, such as drug
sensitivity, rather than PK parameters, such as drug clearance or volume of distribution [20,
37-39]. Therefore, identical PK parameters representative of a typical ICU patient were
obtained from the literature [24, 27, 31, 32] and applied across all patients.
2.1.2 Pharmacodynamic (PD) Modelling
Midazolam is a commonly used sedative agent that can be used to induce anaesthesia or
induce conscious sedation, depending on selected dose [31, 36]. Morphine, while primarily an
analgesic, is also a mild sedative [6, 40]. However, Morphine and Midazolam, administered
concomitantly, synergistically have an overall combined effect greater than the simple sum of
the two individual effects [41, 42]. Furthermore, the effects of Morphine and Midazolam are
typically not linearly proportional to drug concentrations, and instead behave like the wellknown sigmoid concentration-effect relationship [23, 43, 44].
The combined PD effect of these drugs is modelled in Equation (6) using a response surface
for drug interactions [23], incorporating synergism and effect saturation, as shown in Fig. 1.
The sedative effect on the vertical axis lowers awareness, relieving anxiety and reducing
agitation. Finally, Equation (6) captures the cumulative sedative effect of the drugs on the
brain over time, and provides the relationship between stimulus invoking agitation and the
sedative agents employed to manage agitation.
The non-linear pharmacodynamic (PD) Equation (6) is based on physiological observations of
patient behaviour. It states that the rate of change of agitation depends upon the relative
magnitude of the stimulus to the cumulative sedative effect and endogenous agitation
reduction. Observed agitation typically falls upon increased infusion of sedative agents.
Similarly, patients become more agitated by increased stimulus if infusion rates are not
increased. Patient agitation is therefore primarily reduced by the cumulative impact of current
and prior sedation administration, as modelled by the convolution integral in Equation (6).
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The final term in Equation (6) represents the effect of the endogenous opioid biochemical
compounds, endorphins. Abbreviated from “endogenous morphine”, endorphins are a form of
natural analgesic produced in response to pain and physical stress [45, 46]. An agitated patient
may therefore experience a reduction in agitation due to the natural sedative effect of
endorphins produced as result of agitation itself, modelled by the EAR term, -w3A, in
Equation (6). This term is unique in comparison to prior works and represents a dynamic by
which agitation can decline without the presence of exogenous sedative. Such a dynamic is a
stabilising factor and would be significant during periods of low or no exogenous sedative
input.
o ,s
PD parameters, such as w2(t), w3, and C50
, can vary widely between patients, and are fitted
using recorded drug infusion profiles for each patient. The general shape of the PD response
surface in Fig. 1 [23] is approximated by information in the literature [37, 38]. In particular,
the response surface is defined to capture the synergistic sedative effects observed when
Morphine and Midazolam are administered concomitantly [41, 42], the mild sedative effect of
Morphine alone [6, 40], and saturation dynamics.
C50 represents the concentration at which the drug, administered alone, would have 50%
effect. Using the PK model parameters and employing the recorded infusion rate, Equations
(1)-(5) yield Morphine and Midazolam effect-site concentration profiles. These profiles can
be used to estimate a patient-specific C50 value, assuming that clinical effect site
concentration rarely becomes completely saturated. Natural initial estimates for C50 would be
either the average, or 50% of the maximum, effect-site drug concentration from Equations (3)
and (5). However, the impact of the synergistic effect surface means that the total combined
effect of the drugs is higher than the simple sum of the individual drug effects, more regularly
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resulting in effect saturation. As a result, setting C50 to be 80% of the max effect site
concentration provides an effective estimate.
An integral-based fitting method is adapted from Hann et al [47], to obtain the patientspecific, time-varying, sedative sensitivity parameter, w2(t), and a patient-specific, timeinvariant w3 EAR parameter from clinical data. Initial studies investigating the sensitivity of
the model to changes in w3 indicated that small changes (i.e. less than an order of magnitude)
had no observed effect on performance metrics. Therefore, the EAR parameter, assumed to be
patient-specific, is selected for each patient from an array of values w3=[0 0.00001 0.0001
0.001]. All remaining model parameters used in this paper are taken from previous work [17,
19-21].
2.2 Model Verification
Equations (1)–(6) are implemented in conjunction with a validated nurse sedation control
model that accurately captures the basic nursing sedation administration response to patient
agitation using a simple derivative-weighted Proportional-Derivative controller [17, 19]. This
simple nurse controller captures the basic psychology behind subjective assessments [20, 48],
and closes the feedback loop between clinically observed or simulated agitation and sedation
delivery, matching current clinical practice. Comparison of recorded infusion data with
infusion profiles from simulations using fitted parameters provides a basis for statistically
validating the model presented.
Infusion data were recorded using an electronic drug infusion device [10, 49] for all ICU
patients admitted to the ICU during a nine month observation period and requiring more than
24 hours of sedation. Infusion data containing less than 48 hour of continuous data, or data
from patients whose sedation requirements were extreme, such as those with severe head
injuries, were excluded. A total of 37 ICU patients met these requirements and were enrolled
in this study. Approval was obtained from the Canterbury Ethics Board for this research.
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The unknown nature of pain and anxiety, combined with disease state, makes direct recording
of stimulus profiles impossible. However, the input for the semi-automatic sedation infusion
system implemented in the Christchurch ICU [10, 49] is the bedside medical staff’s indication
of observed agitation, and therefore demand for additional drug. These recordings indicate
times where the patient’s agitation level increased enough to warrant additional sedative,
implying the presence of stimulus. It therefore forms the basis for a surrogate measure of
stimulus for model validation [17]. The 4-hour moving average of this record retains the
underlying structure of the recording, while creating a smooth stimulus profile congruent with
recorded data. Note that clinical implementation of agitation feedback control of sedation
would not require this input profile, requiring only measured agitation to determine the
infusion needed. Hence, this is a model validation tool only.
One approach to model verification uses kernel smoothing of the recorded data to create a
probability band using Chebychev’s inequality, and compares the simulated infusion profile
to the probability band [17]. The higher the percentage of time that the simulated infusion
profile lies within the band, the better the model is considered to capture the essential
dynamics of the underlying system for that patient. While this approach is graphically useful,
it lacks an objective numerical measure of how close the simulated infusion profile is to the
empirical data. The percentage time in band (TIB) does not serve this purpose because it
simply quantifies visual closeness by means of artificial hard boundaries, and ignores the fact
that the in-band region does not have the same probabilistic importance everywhere.
This paper utilises the statistical measure Relative Average Normalised Density (RAND),
which indicates whether the simulated infusion profile coincides with a region of high
probability determined from the recorded empirical data. RAND takes values between 0 and
1, where a value close to 1 means that the simulated infusion profile is, on average, in a high
probability region and hence in good agreement with the empirical data. This measure
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effectively replaces the hard boundaries of the probability band by soft boundaries defined by
probabilistic importance determined from the empirical data. Specific details on the
development of the RAND metric can be found in the relevant literature [17, 20, 50-54].
RAND measures how probabilistically similar the model outputs are to the smoothed data,
and hence the degree of compatibility between the model and the empirical data. For example,
a RAND of 0.6 may be interpreted as the model outputs being 60% similar, on average, to the
smoothed data. Since the model is deterministic, its outputs do not come from the same
probabilistic mechanism that generated the recorded data. Hence, RAND is an extremely
stringent measure, and consistently high RAND values close to 1 are not expected. A
reasonable and practical threshold for adequate model performance is RAND ≥ 0.5, which
says that the model outputs are more similar than not, to the smoothed data. If comparing two
or more models to the data, the model with the higher RAND value would be selected. In this
case, with one model, the threshold is 0.5.
Finally, Relative Total Dose (RTD) expresses the total dose administered in the simulation as
a percentage of the actual total recorded dose [17]. Ideally, this metric would approach 100%,
indicating that the simulated total drug dose is identical to the recorded total drug dose. RTD,
TIB and RAND are different objective measures of the ability of the model to capture the
essential dynamics of the agitation-sedation system. Together, they create a clear picture of
the model’s performance.
2.2 Analyses
The results of simulations with and without the EAR dynamic in Equation (6) are analysed
using all of these performance measures. The goal is to investigate the impact of including
EAR. The first goal is to ascertain the physiological range of the EAR parameter, w3, by
broadly testing a range of possible values over all patients. Secondly, the effect of EAR
dynamics on modelled drug sensitivity, w2(t), and the resulting fit with recorded infusion data
are studied.
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3.0 RESULTS
The w3 value corresponding to the best fit between the simulated and recorded data was
w3=0.0001 for all patients. The performance metrics in Table 1 show high values for the
model with (w3=0.0001) and without (w3=0) EAR. The RAND values for simulations without
EAR have a median of 0.77 with standard deviation 0.08 and range [0.51, 0.89]. TIB without
EAR has a median of 0.87 with standard deviation of 0.05 and range [0.78, 0.97], while RTD
has a median of 98.7 with standard deviation 2.1 and range [93.1, 101.4]. Similarly, the
RAND values for simulations including EAR have a median of 0.78 with standard deviation
0.07 and range [0.55, 0.91]. TIB including EAR has a median across all patients of 0.89 with
standard deviation of 0.04 and range [0.81, 0.97], and RTD has a median of 98.6 with
standard deviation 2.1 and range [92.5, 101.0]. For w3 =0.00001 slightly lower values of the
performance metrics were observed, and for w3 = 0.001 much lower values were observed.
These high reported RAND, RTD and TIB values correspond to very close fits of the
simulated infusion rate to the recorded infusion rate, as shown in Figures 2-4. In these figures,
the upper plot shows the time-varying nature of the fitted w2(t), while the lower plot shows
the resulting fit of the simulated and recorded infusion rates. In each of these figures, the dark
solid line shows the results without the inclusion of EAR, and the dark dotted line shows the
results including EAR. The light solid line in the lower plots shows the recorded infusion rate,
and its 99% probability band is the grey band around it.
4.0 DISCUSSION
The statistical model validation metric, RAND, complements and completes the other
statistical tools (RTD and TIB) previously employed [17, 19-21] for model validation of this
system. The probability band with hard boundaries developed previously allows visual
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assessment and numerical TIB evaluation, which is useful during model development and
refinement. The RTD value simply represents a global assessment of the similarity of the
resulting infusion rates to clinical data. In contrast, RAND provides an objective, calibrated
measure of statistical compatibility between the simulated infusion profile and the recorded
data. It thus provides a statistical measure of how well the model captures the essential
dynamics of the agitation-sedation system. Together, all three metrics (RAND, TIB and RTD)
cover a range of model validation criteria.
The best fit between the simulated and recorded data was obtained using w3=0.0001 for each
and every patient, which shows the low inter-patient variability and low sensitivity to the
EAR parameter, and indicates that w3 can be assumed constant across all patients. The
sensitivity of the model to the value of w3 is reduced further by the fitting process for w2(t),
which can compensate for potentially slightly incorrect selection of w3 [47]. This insensitivity
in the EAR parameter is also seen for the similar term in the glucose-insulin system modelling
[55]. Hence, it might be expected.
The impact of EAR can be seen by comparing columns 2 and 3 to columns 5 and 6 in Table 1.
The median and upper and lower quartiles of the RAND and TIB for the model including
EAR are all higher than those without EAR, and the standard deviations of these metrics are
reduced for the model including EAR. These results indicate that the addition of the EAR
dynamic improves the ability of the model to capture the observed dynamics of the agitationsedation system.
The upper plot in Figures 2-4 show that while the difference is small, the inclusion of EAR
results in a slight decrease in w2(t) throughout the entire profile, especially where infusion
rates in the lower plot are very low. This result should be expected since the EAR dynamic is
another form of agitation decrease thus reducing the sedative sensitivity required to match the
clinical data when it is included. Note that when infusion rates are very low the primary
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means of agitation reduction would be EAR. Hence, the model that includes EAR has the
greatest effect during periods of low sedation infusion where there is less exogenous agitation
reduction. Such periods of low infusion, where the EAR dynamic is most important, would
most notably include sedative weaning prior to extubation.
Visually, the lower plots on Figs. 2–4 show that the current model produces infusion profiles
that are a close approximation to the recorded infusion profiles. The fact that the solid and
dotted dark lines on the bottom plot in these figures are difficult to distinguish indicates that
the simulated infusion rates are very similar whether EAR is included or not. High median
RAND values for both models of 0.77 (without EAR) and 0.78 (with EAR), support this
visual finding with a statistically-based objective measure. Minimum RAND values of 0.51
and 0.55 for the two respective models reinforce this result. These results support both models
as a valid representation of the fundamental agitation-sedation dynamics present in a broad
spectrum of ICU patients.
While the addition of the EAR dynamic increases the ability of the model to capture the
observed dynamics of the agitation-sedation system, the improvement is relatively small when
assessed globally across entire records. Further, the sensitivity of the model to the w3
parameter is low, a feature also found for non-drug mediated endogenous removal
mechanisms in similar dynamic systems such as the glucose-insulin system [47, 55]. This
result offers the question of whether the EAR dynamic should be included at all. It may be
possible that the errors and assumptions in the development of the model and the performance
metrics are larger than the difference in performance between the model with and without
EAR. Although this issue may represent a limitation of the model, it is important to note that
the inclusion of EAR is important for accurately capturing periods of low, or no, sedative
infusion, such as weaning. In addition, visual inspection of results show that in periods of low
or no infusion the impact is greater than elsewhere.
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The performance parameters summarized in Table 1 were achieved using very few patientspecific PD parameters and identical PK parameters across all patients. This result supports
the idea that interpatient pharmacological variability is due primarily to PD differences rather
than PK differences, as reported in several studies [20, 38, 39]. However, the insensitivity of
the w3 parameter and its smaller impact on results indicate that EAR is not a significant PD
parameter in this case. However, EAR becomes important when simulating low infusion
rates, such as during weaning. Therefore, although EAR is not always a significant dynamic,
it is important during specific clinical periods. Because it has no negative impact during other
periods it should be retained.
For some patients high w2 values are observed immediately after periods of relatively constant
w2, as seen in Figure 2-4. Although this feature is sometimes located centrally in recorded
data, in many cases this feature is observed at the end of recorded data and may correspond to
weaning off sedation. These observations may be the result of a change in sedative sensitivity
as the patient’s health improves prior to them leaving the ICU.
However, the magnitudes of the changes are in some cases quite large, which indicates that
the change in observed sedative sensitivity may also be the result of a delayed release of drugs
stored in fatty tissue or the effects of active metabolites. Although the peripheral compartment
represents the fatty tissues into which these drugs and/or metabolites can be deposited, this
dynamic may be considerably more prominent than that currently modelled, requiring an
additional separate storage compartment. Because benzodiazepines are lipid soluble, longterm infusions can lead to depositions of large amounts of the administered drug in fatty
tissues [56]. When the sedative administration stops, the stored drug is released back into
circulation [2, 6, 56]. Alternatively, these effects may be due to the prolonged action of active
metabolites. If these dynamics were present during the recordings, the effect would be an
inflated observed sedative sensitivity, w2.
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To determine if a relationship exists between the improvement resulting from the EAR
dynamic and the duration of the infusion, a correlation analysis was undertaken. The
correlation between the improvement resulting from inclusion of the EAR dynamic and the
duration of infusion was determined using the non-parametric bootstrap (Efron et al, 1993),
which does not require a parametric distribution for the data. The correlation coefficient was
found to be r=-0.007, with 95% confidence interval bounds of (-0.246 , 0.398), and P=0.968,
indicating that there is almost 97% chance that correlation exists. Although there is a lack of
correlation, this does not imply that delayed release of drugs stored in fatty tissue or the
effects of active metabolites are not present. Further investigation is required to determine the
effect of an additional storage dynamic and active metabolites.
Finally, the model incorporates physiological drug effect saturation dynamics, seen by the
dual-sigmoid surface in Fig. 1. Physiological metabolism and excretion of Morphine and
Midazolam is limited by renal and hepatic clearance capacity. However, the maximum
clearance rate and plasma concentrations at which the dynamics change from first-order to
saturated zero-order kinetics are not easily obtained, particularly in the ICU population. A
lack of available parameter values therefore makes the immediate implementation of more
representative Michaelis-Menton saturation dynamics difficult. Clinical trials with quantified
agitation sensors [13, 14] and measured plasma drug concentrations could provide the data to
improve these model parameters and add any necessary additional dynamics.
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5.0 CONCLUSIONS
The physiological model presented captures the essential dynamics of the agitation-sedation
system, both with and without the Endogenous Agitation Reduction (EAR) term. High
median RAND values of 0.77 (without EAR) and 0.78 (with EAR) support and minimum
RAND values of 0.51 and 0.55 for the two respective models show that both models are valid
representations of the fundamental agitation-sedation dynamics present in a broad spectrum of
ICU patients.
While the addition of the EAR dynamic increases the ability of the model to capture the
observed dynamics of the agitation-sedation system, the improvement is relatively small and
the sensitivity of the model to the w3 parameter is low. Although this may represent a
limitation of the model, the inclusion of EAR is important for accurately capturing periods of
low, or no, sedative infusion, such as weaning.
Page 20 of 32
6.0 ACKNOWLEDGEMENTS
The authors wish to acknowledge the funding for this research, provided by the New Zealand
Foundation for Research, Science and Technology through a Bright Futures Top Achiever
Doctoral Scholarship, and the Todd Foundation through the Award for Excellence.
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FIGURE & TABLE CAPTIONS
Fig. 1.
Representation of the agitation-sedation system model, showing separate
compartmental pharmacokinetics (upper portion), the pharmacodynamic effect
surface with the associated defining equation [21].
Fig. 2.
Plot showing the effect of endogenous agitation reduction (EAR) on sedative
sensitivity,w2(t),(upper plot), and fit to recorded infusion rate (lower plot) for
Patient 2. The dark solid line shows the results without the inclusion of EAR, and
the dark dotted line shows the results including EAR. The light solid line in the
lower plots shows the recorded infusion rate, and its 99% probability band is
indicated by the grey band.
Fig. 3.
Plot showing the effect of endogenous agitation reduction (EAR) on sedative
sensitivity,w2(t),(upper plot), and fit to recorded infusion rate (lower plot) for
Patient 37. The dark solid line shows the results without the inclusion of EAR, and
Page 27 of 32
the dark dotted line shows the results including EAR. The light solid line in the
lower plots shows the recorded infusion rate, and its 99% probability band is
indicated by the grey band.
Fig. 4.
Plot showing the effect of endogenous agitation reduction (EAR) on sedative
sensitivity,w2(t),(upper plot), and fit to recorded infusion rate (lower plot) for
Patient 36. The dark solid line shows the results without the inclusion of EAR, and
the dark dotted line shows the results including EAR. The light solid line in the
lower plots shows the recorded infusion rate, and its 99% probability band is
indicated by the grey band.
Table 1. Validation metrics for 37 patients for simulations with and without EAR
Page 28 of 32
Morphine
Midazolam
C co
C cs
o
K CL
K opc
C op
K eco
o
cp
K
K ceo
C
o
e
s
K CL
K ecs
K ces
C
s
e
EComb
C eo
o
C50
C 50s
C es
 ( )
EComb
 CO  C S 


C50 ( ) 

 E0  [ E max ( )  E0 ]
 ( )
 CO  C S 

1  
 C50 ( ) 
Fig. 1. Representation of the agitation-sedation system model showing
separate compartmental pharmacokinetics (upper portion) and the pharmacodynamic
effect surface with the associated defining equation [21]. Cc, Cp and Ce are the drug
concentrations (mg/L) in the central, peripheral and effect compartments respectively,
o
s
C50
and C50
represent the concentrations at which Morphine or Midazolam would have 50%
effect if administered alone, and Kij is the transfer rate (L/min) from compartment i to
compartment j.
Page 29 of 32
x 10
-3
2
1.8
1.6
1.4
w2(t)
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
0
20
40
60
80
100
80
100
4.5
4
Infusion Rate (mL/h)
3.5
3
2.5
2
1.5
1
0.5
0
60
Time (Hours)
Fig. 2. Plot showing the effect of endogenous agitation reduction (EAR) on sedative
sensitivity,w2(t),(upper plot), and fit to recorded infusion rate (lower plot) for Patient 2. The
dark solid line shows the results without the inclusion of EAR, and the dark dotted line shows
the results including EAR. The light solid line in the lower plots shows the recorded infusion
rate, and its 99% probability band is indicated by the grey band.
Page 30 of 32
0.01
w2(t)
0.008
0.006
0.004
0.002
0
0
10
20
0
10
20
30
40
50
60
40
50
60
Infusion Rate (mL/h)
15
10
5
0
30
Time (Hours)
Fig. 3. Plot showing the effect of endogenous agitation reduction (EAR) on sedative
sensitivity,w2(t),(upper plot), and fit to recorded infusion rate (lower plot) for Patient 36. The
dark solid line shows the results without the inclusion of EAR, and the dark dotted line shows
the results including EAR. The light solid line in the lower plots shows the recorded infusion
rate, and its 99% probability band is indicated by the grey band.
Page 31 of 32
x 10
-3
3.5
3
w2(t)
2.5
2
1.5
1
0.5
0
0
20
40
0
20
40
60
80
100
120
80
100
120
18
16
Infusion Rate (mL/h)
14
12
10
8
6
4
2
0
60
Time (Hours)
Fig. 4. Plot showing the effect of endogenous agitation reduction (EAR) on sedative
sensitivity,w2(t),(upper plot), and fit to recorded infusion rate (lower plot) for Patient 37. The
dark solid line shows the results without the inclusion of EAR, and the dark dotted line shows
the results including EAR. The light solid line in the lower plots shows the recorded infusion
rate, and its 99% probability band is indicated by the grey band.
Page 32 of 32
Table 1. Validation metrics for 37 patients for simulations with and without EAR
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Max
UQ
Median
LQ
Min
STD
RAND
0.79
0.80
0.69
0.69
0.83
0.82
0.77
0.74
0.73
0.78
0.71
0.73
0.79
0.51
0.76
0.71
0.86
0.80
0.88
0.89
0.79
0.69
0.75
0.77
0.67
0.86
0.70
0.79
0.60
0.87
0.83
0.77
0.78
0.73
0.75
0.64
0.73
0.89
0.80
0.77
0.71
0.51
0.08
No EAR, w3=0
TIB
RTD (%)
0.97
100.3
0.89
98.3
0.94
101.3
0.88
95.8
0.85
96.5
0.92
98.5
0.86
93.4
0.85
98.6
0.81
101.4
0.87
97.4
0.87
101.2
0.89
99.0
0.87
96.9
0.90
98.7
0.87
100.7
0.83
99.7
0.89
97.8
0.83
97.1
0.94
100.0
0.94
99.1
0.84
96.4
0.80
98.6
0.88
93.1
0.91
98.7
0.78
101.1
0.90
99.5
0.80
97.2
0.87
99.4
0.93
93.9
0.90
98.7
0.92
99.9
0.85
99.0
0.88
99.3
0.82
100.3
0.82
100.3
0.79
97.9
0.88
95.6
0.97
101.4
0.90
99.9
0.87
98.7
0.84
97.2
0.78
93.1
0.05
2.1
EAR, w3=0.0001
RAND
TIB
RTD (%)
0.79
0.97
99.7
0.84
0.92
97.7
0.74
0.95
99.7
0.69
0.89
95.7
0.84
0.86
96.9
0.83
0.92
98.5
0.79
0.86
92.6
0.77
0.85
98.8
0.75
0.82
101.0
0.82
0.89
97.7
0.75
0.89
100.8
0.76
0.91
98.4
0.82
0.88
97.3
0.55
0.90
97.3
0.77
0.88
100.5
0.73
0.84
100.0
0.86
0.89
97.6
0.84
0.84
97.6
0.89
0.94
100.1
0.91
0.94
99.2
0.81
0.85
96.0
0.72
0.83
98.3
0.76
0.89
93.1
0.78
0.91
98.8
0.69
0.81
100.6
0.88
0.92
98.8
0.77
0.83
97.4
0.82
0.88
99.6
0.70
0.94
92.5
0.87
0.90
98.6
0.86
0.93
99.9
0.79
0.87
99.2
0.81
0.89
99.5
0.77
0.84
100.0
0.78
0.84
100.0
0.74
0.84
96.8
0.76
0.89
96.0
0.91
0.97
101.0
0.83
0.91
99.7
0.78
0.89
98.6
0.75
0.85
97.3
0.55
0.81
92.5
0.07
0.04
2.1