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Modeling for critically ill patients: An introduction for beginners
Modeling for critically ill patients: an introduction for beginners Emmanuel Lafont MD, Saik Urien MD, PhD, Joe-Elie Salem MD, Nicholas Heming MD, PhD, Christophe Faisy MD, PhD PII: DOI: Reference:
S0883-9441(15)00462-1 doi: 10.1016/j.jcrc.2015.09.002 YJCRC 51939
To appear in:
Journal of Critical Care
Please cite this article as: Lafont Emmanuel, Urien Saik, Salem Joe-Elie, Heming Nicholas, Faisy Christophe, Modeling for critically ill patients: an introduction for beginners, Journal of Critical Care (2015), doi: 10.1016/j.jcrc.2015.09.002
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ACCEPTED MANUSCRIPT Modeling for critically ill patients: an introduction for
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beginners
Emmanuel Lafont, MD a, Saik Urien, MD, PhD b, Joe-Elie Salem, MD c, Nicholas Heming, MD,
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a
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PhD d, Christophe Faisy, MD, PhD a,
Medical Intensive Care Unit, Hôpital Européen Georges Pompidou, Assistance Publique–
b
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Hôpitaux de Paris, Université Paris Descartes Sorbonne Paris Cité, Paris, France. Centre d’Investigation Clinique-0991 INSERM, Université Paris Descartes Sorbonne Paris Cité,
Paris, France.
Centre d'Investigation Clinique-1166 INSERM, Hôpital La Pitié-Salpêtrière, Assistance
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c
Publique–Hôpitaux de Paris, Université Pierre et Marie Curie, Paris, France. Medical Intensive Care Unit, Hôpital Raymond Poincarré, Assistance Publique–Hôpitaux de
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d
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Paris, Université Versailles–Saint Quentin, Garches, France. Corresponding author: Christophe Faisy, MD, PhD, Medical Intensive Care Unit, European
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Georges Pompidou Hospital, 20, rue Leblanc, 75908 Paris Cedex 15, France. Tel: 33 156093220. Fax: 33 156093202. E-mail: [email protected]
Conflict of interest: The authors declare that they have no conflict of interest.
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ACCEPTED MANUSCRIPT ABSTRACT Models are mathematical tools used to describe real world features. Therapeutic interventions in
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the field of critical care medicine may easily be translated into such models. Indeed, numerous
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variables influencing drug pharmacokinetics and pharmacodynamics are systematically
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documented in the intensive care unit over time. Organ failure, fluid shifts, other drug administration, and renal replacement therapy may cause changes in physiological values, such
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as body weight and composition, temperature, serum protein levels, arterial pH and renal or hepatic function. Trials assessing the efficacy and safety of novel drugs usually exclude critically
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ill patients and guidelines regarding drug dosage rarely apply to such patients. Modeling in the critically ill may allow physicians to inform decisions related to therapeutic interventions,
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particularly relating to infectious diseases. However, few clinicians are familiar with these
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methods. Here, we present a current overview of population pharmacokinetic and pharmacodynamic models applicable in critically ill patients aimed at non-specialists, and then
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unit.
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emphazize recent potential of modeling for optimizing treatments and care in the intensive care
Key words: Predictive model; Critically ill patients; Pharmacokinetics; Pharmacodynamics; Bayesian network; Multicompartmental model
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ACCEPTED MANUSCRIPT 1. Introduction Pharmacokinetics (PK) discribes the relationship between the administered dose of a drug
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and its concentrations in the plasma or at its target site, whereas pharmacodynamics (PD)
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describes the relationship between drug concentration and the observed effect on the organism
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[1]. PK/PD models help identify sources of variability in the dose-concentration-effect relationship of a drug (Fig. 1). In the critically ill, PK/PD parameters are altered through a host of
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factors including fluid overload, increased capillary permeability, hypoalbuminemia, renal failure, renal replacement therapy (or other assist devices), as well as poorly perfused tissues.
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Usually recommended drug dosages are therfore not suited to critically ill patients. Modeling may help identify relevant factors related to clinical outcome, characterize and quantify their
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impact. Modeling may also help physicians make better informed predictions and clinical
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decisions, as well as improve the design of drug development studies, assess the efficacy of
regimen.
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therapeutic interventions, and optimize medication use by designing rational dosage forms and
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2. Fixed individual effect models Parameters used to define the pharmacokinetic properties of a drug are its bioavailability (fraction of the drug that is systematicaly absorbed and thus available to produce a biological effect), the area under the curve (AUC) of drug concentration in blood over time, its half-life (time for the drug to lose half of its biological activity), its volume of distribution (theroretical volume of blood that the administered drug would have to occupy), plasma protein binding, and plasma clearance (Fig. 1). The disposition of a molecule depends on its clearance (CL) and volume of distribution (V):
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dC Ke V C CL C dt
(1) 3
ACCEPTED MANUSCRIPT dC the first derivative of concentration with respect to time, dt i.e., the change in concentration over time, and Ke (CL/V) the elimination rate constant. Drug PK is altered by various conditions commonly encountered in the intensive care unit (ICU): 1) Volume of distribution may rise due to increased capillary permeability or fluid administration and consequently plasma concentration of a molecule will be reduced; hypoalbuminemia may arise, leading to increased free plasma concentration of the drug; 2) Hemodialysis increases the clearance of some drugs such as antibiotics. Of note, the propensity of a drug to be cleared through hemodialysis is expected to decrease when a drug is highly bound to plasma protein. Pharmacodynamic modeling strives to determine the effect of a drug expressed in terms of efficacy and potency (Fig. 1). In critically ill patients, the interaction between a drug and its receptor can be altered due to drug-drug interactions or to other extreme physiological changes such as hypoxemia, acidosis, ionic or temperature shifts.
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where C is drug concentration,
3. Population models
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In a population approach, model parameters are simultaneously estimated for all individuals. In fact, estimates do not refer to each individual but relate to the mean parameters of the
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population along with their between-subject and residual variabilities (random effects). Table 1
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summarizes the potential applications and limitations of commonly used modeling approaches [29].
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3.1. Basic elements for modeling a therapeutic effect
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A population is composed of patients exposed to the same treatment. In population models, the study sample is the population rather than the individual. Population modeling involves five main components: 1) Assessing treatment exposure; 2) Assessing treatment response; 3) Determining relevant covariates; 4) Characterizing the effect (random, fixed or mixed); 5) Determining the mathematical relationship between these elements. One must be aware that Items 1) and 2) constitute the structural model. The response to treatement may be affected by the development of tolerance, by a plecebo effect or by disease progression. A random effect model defines which parameters vary randomly between and within individuals (residual variability), and what the statistical distribution of their variances is. The response predicted by
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ACCEPTED MANUSCRIPT the model for the jth observation, yj, is determined with an error (j) which measures the difference between the predicted value (PRED) and the observed value (OBS) (2)
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yjOBS = yjPRED + j
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where denotes residual variability. Similarly, between-subject variability () can be modeled as
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follows: Pi = PPOP + i.
(3)
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part the variability in the response to treatment.
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where P denotes a structural parameter. Covariates such as patient characteristics, may explain in
Non-linear mixed effects model (NONMEM) is widely accepted model used for analyzing
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PK/PD data. NONMEMs help estimate model parameters as well as different sources of variability. The contribution of each individual covariate is assessed, explaining the between-
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subject variability, i.e., if a covariate has a significant effect on a structural parameter, the
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corresponding is decreased. Observed data are then compared to model predicted results in order to validate the model. The final population model is composed of mathematical equations
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including, when necessary covariates submodels which may modify structural parameters. NONNEMs may be applied at the bedside, i.e., for calculating drug dosages. 3.2. Allometry Allometry refers to the relationship between body size and physiological functions [2]. Specifically, body weight (BW) is related to volume of distribution (V) and clearance (CL): BW V V 70
V
(4) and CL
BW CL CL (5) where 70 θV and θCL are typical volume or clearance per 70 kg, and βCL and βV are the power exponents which modulate the influence of BW (typical values are ¾ for CL and 1 for V).
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ACCEPTED MANUSCRIPT 3.3. Compartmental pharmacokinetic models Compartmental models describe exchanges of material between different compartments of
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the body. A compartmental model is described using several parameters, including transfer rate
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constant, volume of distribution, elimination clearance, state-variables changing over time, independent variables (dose, time) and dependent variables (plasma concentration, effect) [3].
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Transfer rate constants are used to calculate the flow of material. Compartemental models are
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described as catenary (compartments form a chain) or mammillary (central compartment connected to peripheral compartments, all of which are independent of each other). Simple
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mammillary models are widely used in PK studies (Fig. 2). Blood and fully perfused organs (liver, kidneys, and lungs) belong to the central compartment. The elimination rate constants are
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related to the central compartment because the main excretory organs are the liver and kidneys.
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The mathematical description of between compartment-exchanges is usually achieved by
dx kx dt
(6)
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the equation:
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differential equations. It is also assumed that the law of mass conservation applies. For example
with initial conditions starting at time t0 described as: xt0 x0
(7)
resolves into: xt x0e k (t t0 ) (8) where x is the amount of material in the compartment, e equals 2.71, and k is the first order constant. When a drug is intravenously infused, the mathematical model becomes:
dx kx R dt R is the infusion rate. Concentration C is readily obtained by the equation:
(9) where
x (10) In the V case of discontinous drug administration, these models enable to accurately adjust dosage in C
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ACCEPTED MANUSCRIPT order to obtain a predicted concentration. Based on modeling, one can simulate the flow of molecules across a dialysis membrane or the transit time and diffusion of the drug into different compartments. Drug exposure is defined as the area under the concentration curve (AUC) (Fig. 1).
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Transit compartments are used to model drug distribution delay [4]. For instance, the time between administration of the drug and its distribution to the central compartment can be
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modeled by a series of n compartments connected to each other by a first order transfer rate contant, 1/ [5]. The amount of drug versus time in each compartment can be written as follows:
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dAn 1 (11) where An1 An dt An is the amount of drug inside the nth compartment and 1/ has the dimention of a transit time for each compartment. Transit compartment models assume a stochastic approach where only the mean response expected (drug concentration or effect) is observed (Table 1). When the PK model is linear, time to reach steady-state plasma concentrations can easily be determined (five elimination half-lives for one compartment model). This point is of particular importance when determining the effect of a drug which acts on the brain because the delay depends on the time needed to cross the blood-brain barrier. For instance, mathematical models are routinely used in operating theaters in order to accuratly predict the target concentration in the brain of an anesthesic agent [10]. Multicompartmental models predict the contextual half-life of sedative drugs, or the time needed for blood plasma concentrations to be devided by half after discontination of an intravenous infusion (Table 1). 3.4. Models for discrete data
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Clinical outcomes are often described as events: ICU death or recovery, recurrence or progression of disease, nosocomial infection, adverse event, and resistance to treatment. These outcomes may may be quantified as time-to-event, number of events per time interval (rates), their severity grade, or a combination of all three. Discrete data arise when the outcome variables are the number of individuals classified into categories. Such data require specific modeling structures and methods [6]. In case of a binary response, the most appropriate model is logistic regression. Log-linear models are easy to develop using standard statistical software and are used to analyze categorical data with three or more discrete valued variables. The aim of logistic regression is to model the functional form of the binomial probability p(x) by assuming that
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ACCEPTED MANUSCRIPT log {p(x)/(1 − p(x))} = β’x
(12)
β’x is a linear function of the covariates or risk factors that are believed to influence the binary
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response also called Y. The term log {p(x) /(1 − p(x))} is also called the logit or log-odds of p(x).
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Solving for p gives p(x) = e(β’x)/(1 + e(β’x)). In fact, logistic regression is a particular case of loglinear models. These models can be interpreted in terms of interactions between the various
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factors in multidimensional tables and are easily generalized to higher dimensions. In all cases,
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the link function is the logit. The statistical significance of regression coefficients has the same meaning as in linear models. PK/PD models were initially developed for continuous data which
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could have an infinite number of values in a given interval, such as blood pressure. Clinical outcomes or toxic effects are often measured as discrete data with a limited number of states.
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There are four types of discrete data: binary data (for example, alive/dead); categorical nominal
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data (state of vigilance); count of events per time of interval (number of epileptic seizure per week); categorical ordered data consisting of severity grades (Coma Glasgow Score). All those
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outcomes can be linked to drug exposure (AUC) but specific modeling approaches are required
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for each type of discrete data (Table 1). These models may be used during drug development in order to select the best drug candidate, optimize the dosing schedule by maximizing the clinical benefit to harm ratio, or to define the target exposure in different populations. 3.5. Bayesian networks Bayesian networks use observed data (e.g. symptoms) to determine the probability of unobserved data (e.g. pathology). Bayesian networks are probabilistic graphical models build on expert knowledge and observed data. They are used in the medical field, for diagnosis and risk analysis. A Bayesian network is an acyclic graph in which the connecting points between related elements represent random variables (nodes) or influences between variables (links). Arrows
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ACCEPTED MANUSCRIPT represent probabilistic or deterministic relationships between variables (Fig. 3). In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the true value
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is uncertain [7,8]. The key element of Bayesian conjecture is that prior beliefs, in the form of a
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probability distribution, may be combined with observed data, in the form of a likelihood function, to yield a posterior distribution. When more data becomes available, one may calculate
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the posterior distribution using Bayes' formula. Given the observed data D and the hypothesis H
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regarding the occurrence of an event, the cardinal rule is: p(H|D) = p(H|D) p(D|H) × p(H)
(13)
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where p(H) is the prior probability of H, p(H|D) is the posterior probability of H, means proportional, and p(D|H) represents a function of H called likelihood of H. The likelihood ratio
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p(D|H1)/p(D|H2) between two hypotheses (or two models) H1 and H2 is a measure of their
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relative merit and is also called Bayes factor. The log(likelihood ratio), measured in decibans, is frequently used to estimate the weight of evidence. The Bayesian information criterion (BIC)
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helps to select the finest model among a set of models by introducing a penalty term for the
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number of parameters in the model, thereby limiting the risk of overfitting. The Akaike information criterion (AIC) is another popular method to select the finest model. Advanced evaluation tools (normalized prediction distribution errors and visual predictive check) help to validate the Bayesian predictions. As detailled above, the principle of incremental learning in Bayesian networks is based on the a priori knowledge contained in the network at any given time, making the model timedependent. Therefore, a Bayesian network is a means to predict system behavior by combining existing and new data from a database. Bayesian networks are built following three steps: 1) Identifying the most influencial variables and the range of their possible values; 2) Defining the
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ACCEPTED MANUSCRIPT structure of the Bayesian network (identifying causal links between variables); 3) Defining the law of joint probability of the variables (the list of nodes that have a probabilistic dependence
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with the target variable). During the initial phase of model building, a uniform distribution of
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probabilities are assigned to each variable; probabilities are therafter fine-tuned. Covariates of interest are included into the model if they improve (i.e., decrease) the BIC. Difference in BIC
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>10 between two models indicates substantial evidence and BIC >20 represents a decisive
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evidence. One can therafter simulate of the behavior of the system by entering input variables (causes) in order to observe the resulting probability distribution on the effects. Bayesian
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approaches may be used to calculate sample sizes for clinical trials or to select logistic regression models [11,12].
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4. From mathematical models to randomized controlled trials: example of
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acetazolamide
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Acetazolamide, a carbonic anhydrase inhibitor, has been used for decades as a respiratory stimulant in chronic obstructive pulmonary disease (COPD) patients with metabolic alkalosis,
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especialy during the weaning period. However, the optimal dosage of acetazolamide to be administered to alkalotic COPD patients in the ICU remained unclear. The PD of acetazolamide in 68 mechanically ventilated COPD patients who received the drug during weaning from mechanical ventilation was assessed [29,30]. Population pharmacodynamics was modeled using a NONMEM approach. The main covariates of interest which influenced the effect of acetazolamide were the Simplified Acute Physiology Score II (SAPS II) at ICU admission, coprescription of furosemide or corticosteroids, and serum concentration of chloride. According to the model, higher acetazolamide dosage was required to significantly reduce serum bicarbonate concentration in the presence of high levels of serum chloride or co-prescription of systemic
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ACCEPTED MANUSCRIPT corticosteroids or furosemide. The model also predicted that 45% of treated patients would exhibit a decrease of PaCO2 >5 mmHg with doses of 1000 mg/d, indicating that COPD patients
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might benefit from the respiratory stimulant effect after the administration of higher doses
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(>1000 mg daily) of acetazolamide. Respiratory responses to acetazolamide in mechanically ventilated COPD patients were also modeled and the effect of increased amounts of the drug on
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respiratory parameters were simulated in silico. These finding are being prospectively verified by
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a large randomized controlled trial (NCT01627639), assessing the effect of higher doses of acetazolamide. This trial has recently been completed, the results of which are pending. This
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example highlights the relevance of modeling for critically ill patients using the following sequence: 1) Observations based on real facts; 2) Modeling; 3) In silico simulation and dose
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(randomized controlled trial).
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personalization based on the model; 4) Validation of the model by new observations
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5. Other areas of application for modeling in the ICU Mathematical models have great potential for optimizing drug therapies in the field of critical
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care medicine (Table 2). Below, we underline a couple of such exemples. 5.1. Anti-microbial drugs Risk factors for beta-lactam under-dosing in ICU patients are intermittent infusion and increased creatinine clearance [13]. Lodise et al. showed that piperacillin-tazobactam administered over 4 h rather than 30 min increased survival rates [14]. In critically ill patients, Cmax/MIC or AUC24h/MIC ratios are linked to outcome for concentration dependent killing agents [15]. AUC/MIC is the indicator that best characterizes vancomycin PD. In methicillinresistant Staphylococcus aureus (MRSA) bacteremia occurring in ICU patients, risk factors for vancomycin treatment failure are MIC >2 mg/l and an AUC/MIC of <420 [16]. In a basic one-
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ACCEPTED MANUSCRIPT compartment PK model, it was shown that age, serum creatinine clearance, serum albumin and disease severity at ICU admission accounted for >65% of vancomycin clearance variability [17].
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Vancomycin PK/PD models for patients with methicillin-resistant staphylococcal post-
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sternotomy mediastinitis have shown that high plasma vancomycin concentrations (>35 mg/l) might optimize the probability of in-ICU cure without impact on renal function [18]. In critically
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ill patients with refractory septic shock, NONMEM was used to describe population PK and
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dosing simulations helped to define effective vancomycin doses for different haemofiltration settings [19]. Simulation indicated that after a loading dose, doses of vancomycin should be
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individualized according to haemofiltration exchange rate. Regarding aminoglycosides, the finest PD parameter is the Cmax/MIC ratio. A Cmax/MIC of >10 during the first 48 h of aminoglycoside
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treatment has been shown to increase the cure rate of severe pneumonia caused by Gram-
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negative pathogens [20]. Following Bayesian modeling of amikacin PK in critically ill patients with sepsis, it was shown that only severity at ICU admission, creatinine clearance, and the use
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of catecholamines or positive end-expiratory pressure affected the volume of distribution and the
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clearance of amikacin [21]. NONMEM modeling indicated that height and creatinine clearance explained much of the inter-individual variability of tobramycin distribution in ICU patients [22]. Simulations based on this model showed that tobramycin concentration peak and AUC target could not be reached simultaneously in >45% of the patients, confirming that plasma concentration monitoring is required to manage the efficacy and toxicity of tobramycin in the ICU. Based on NONMEM modeling, simulations estimated the relevance of various ciprofloxacin dosage regimens in ICU patients [23]. Less than 20% of patients reached the desired drug concentration range, predicting the emergence of drug resistant mutants, questioning the use of ciprofloxacin for the treatment of Pseudomonas aeruginosa and
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ACCEPTED MANUSCRIPT Acinetobacter baumannii in ICU patients. A multicompartmental model was developed using a NONMEM approach to investigate the PK of oseltamivir in H1N1infected patients requiring
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extracorporeal membrane oxygenation (ECMO) [24]. It was found that the systemic diffusion of
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oseltamivir following enteral administration in adults under ECMO was comparable to that of ambulatory patients and far exceeded concentrations required to inhibit the neuraminidase
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activity of the H1N1 virus. Dosage adjustment for patients undergoing ECMO appeared
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unnecessary, but oseltamivir dosage should be reduced in patients suffering from renal
5.2. Host-pathogen interactions
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dysfunction.
Bayesian frameworks and compartmental models were used to assess the effectiveness of
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barrier precautions and hand washing compliance for reducing the transmission of multidrug
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resistant bacteria [2527]. Structural equation modeling (method where the model is based on the structure of the covariance matrix of the measures) showed that prior nurse contact with
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colonized/infected patients was a significant predictor of transmission of nosocomial pathogens,
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except for Pseudomonas spp., while general health and invasive surgery were significant predictors of Candida and Klebsiella spp. transmission, indicating that isolation and bed movement as a strategy to manage MRSA infections might impact the incidence of transmission of other multidrug resistant pathogens [28]. 5.3. Sedation Multicompartmental approaches (taking into account drug distibution delays due to the the blood-brain barrier) might be usefull for modeling the effects of sedative drugs in critically ill patients. PK/PD studies of sedative drugs remain scarce in patients treated by moderate hypothermia. Bastiaans et al. investigated the PK of midazolam in resuscitated patients treated
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ACCEPTED MANUSCRIPT by moderate hypothermia in comparison to normothermic and non-resuscitated patients [31]. Using NONMEM, the PK of midazolam was found to be similar in resuscitated patients treated
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by moderate hypothermia and their controls.
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5.4. Nutrition and metabolism
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Predictive multi-parameter models help to estimate energy expenditure of critically ill patients [3236]. Body cell mass (the metabolically active part of fat-free mass) of ICU patients
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has been modeled using variables obtained at the bedside [37, 38]. Nutritional screening of the critically ill also benefited from modeling. Novel scoring tools, such as the NUTRIC score, may
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help discriminate ICU patients who will benefit from aggressive nutritional assistance [39]. Nachimuthu et al. developed a dynamic Bayesian network model to describe various clinical
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parameters and their relationships to insulin and glucose homeostasis [40]. The predictions based
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on this Bayesian model were at least as effective or better than those of the rule-based protocol
5.5. Shock
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(eProtocol-insulin) currently used in ICUs.
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Using an allometric model, the PK of epinephrine in patients with septic shock was investigated; epinephrine clearance was only dependent on BW and severity of illness at ICU admission [41]. Moreover, PK of epinephrine was described as linear and no saturation at high doses was predicted in septic shock patients. In critically ill children with post-operative low cardiac output, NONMEM modeling indicated that BW and age affected the PK/PD of epinephrine, predicting the effects of a given dosage of epinephrine on heart rate, mean arterial pressure, and plasma lactate or glucose levels [42]. 5.6. Neurological disorders
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ACCEPTED MANUSCRIPT Using population and logistic regression models, the PK/PD of clazosentan was investigated in patients suffering from aneurysmal subarachnoid hemorrhage. Clazosentan
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clearance was influenced by age, sex, ethnic origin, disease status at baseline and increased over
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time [43]. However, clazosentan exposure was not found to have any relationship with efficacy endpoints while being associated with increased risks of pulmonary complications.
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5.7. Assessing ICU costs
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Modeling the impact of establishing a dedicated weaning unit in an intensive care department showed that such a unit may reduce acute bed occupancy by 10% and could reduce
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overall treatment costs [44]. As longevity increases in the population, the prevalance of chronic diseases increases, adding strain to health care facilities and to ICUs. A computerized model was
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used as a capacity analysis tool to assess ICU bed occupancy profile [45]. Lastly, a discret-event
with heart failure [46].
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simulation model indicated that a telemonitoring system could reduce costs for treating patients
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5.8. Making better informed decisions in the ICU
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By using a learning Bayesian network, the probability that a septic ICU patient would require a certain range of fluid can be predicted [47]. Another dynamic Bayesian network was applied to model Sequential Organ Failure Assessment score changes in adult critically ill patients in order to predict the most probable sequences of organ failures over the first week after admission [48]. Similarely, modeling the effect of strain imposed on ICUs and patient outcome revealed that strain particularely affected ICUs using closed rather than open intensivist staffing models [49].
6. Conclusions
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ACCEPTED MANUSCRIPT Models are tools used to describe natural or man-made phenomenons by simplifying a complex reality. Models are refuted by experience, but they are essential for scientists. George
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Box’s approach summarizes this duality: "All models are wrong! Some are useful". Critical care
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medicine is a promising field for the mathematical description of reality in order to optimize
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effectiveness of care and to minimize costs.
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Acknowledgements:
This work was funded by the Medical Intensive Care Unit, Hôpital Européen Georges
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Pompidou, Assistance Publique – Hôpitaux de Paris, Université Paris Descartes Sorbonne Paris
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Cité, Paris, France.
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ACCEPTED MANUSCRIPT References
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1. Pérez-Urizar J, Granados-Soto V, Flores-Murrieta FJ, Castaneda-Hernández G.
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Pharmacokinetic-Pharmacodynamic Modeling: Why? Arch Med Res 2000;31:53945 2. Agutter PS, Tuszynski JA. Analytic theories of allometric scaling. J Exp Biol
SC
2011;214:105562
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3. Blomhøj M, Kjeldsen TH, Ottesen J. Compartment models. Latest version Available at: http://www4.ncsu.edu/~msolufse/Compartmentmodels.pdf. Accessed December 2014
MA
4. Sun YN, Jusko WJ. Transit Compartments versus Gamma Distribution Function To Model Signal Transduction Processes in Pharmacodynamics. J Pharm Sci 1998;87:7327
ED
5. Paule I, Girard P, Freyer G, Tod M. Pharmacodynamic Models for Discrete Data. Clin
PT
Pharmacokinet 20012;12:76786
6. Zelterman D. Models for discrete data, 2th ed. New York: Oxford University Press; 2006.
CE
7. Spiegelhalter DJ, Myles JP, Jones DR, Abrams KR. Bayesian methods in health technology
AC
assessment: a review. Health Technol Assess 2000,4:1130 8. Rosner GL, Müller P. Bayesian population pharmacokinetic and pharmacodynamic analyses using mixture models. J Pharmacokinet Biopharm 1997;25:20933 9. Pressman AR, Avins AL, Hubbard A, Satariano WA. A comparison of two worlds: How does Bayes hold up to the status quo for the analysis of clinical trials? Contemp Clin Trials 2011;32:561–8 10. Casati A, Fanelli G, Casaletti E, Colnaghi E, Cedrati V, Torri G. Clinical assessment of target-controlled infusion of propofol during monitored anesthesia care. Can J Anesth 1999;46:2359
17
ACCEPTED MANUSCRIPT 11. Wagner H, Duller C. Bayesian model selection for logistic regression models with random intercept. Comput Stat Data Anal 2012;56:1256–74
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12. Zhang X, Cutter G, Belin T. Bayesian sample size determination under hypothesis tests.
RI P
Contemp Clin Trials 2011;32:393–8
13. De Waele JJ, Lipman J, Akova M, Bassetti M, Dimopoulos G, Kaukonen M, et al. Risk
SC
factors for target non-attainment during empirical treatment with β-lactam antibiotics in
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critically ill patients. Intensive Care Med 2014;40:1340–51
14. Lodise TP, Lomaestro BM, Drusano GL. Piperacillin-tazobactam for Pseudomonas
MA
aeruginosa infection: clinical implications of an extended-infusion strategy. Clin Infect Dis 2007;44:357–63
ED
15. Owens RC, Ambrose PG. Antimicrobial stewardship and the role of pharmacokinetics–
PT
pharmacodynamics in the modern antibiotic era. Diagn Microbiol Infect Dis 2007;57:7783 16. Kullar R, Davis SL, Levine DP, Rybak MJ. Impact of vancomycin exposure on outcomes in
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patients with methicillin-resistant Staphylococcus aureus bacteremia: support for consensus
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guidelines suggested targets. Clin Infect Dis 2011;52:97581 17. Del Mar Fernández de Gatta GM, Revilla N, Calvo MV, Domínguez-Gil A, Sánchez Navarro A. Pharmacokinetic/pharmacodynamic analysis of vancomycin in ICU patients. Intensive Care Med 2007;33:27985 18. Mangin O, Urien S, Mainardi JL, Fagon JY, Faisy C. Vancomycin pharmacokinetic and pharmacodynamics models for critically ill patients with post-sternotomy mediastinitis. Clin Pharmacokinet 2014;53:84961
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ACCEPTED MANUSCRIPT 19. Escobar L, Andresen M, Downey P, Gai MN, Regueira T, Bórquez T, et al. Population pharmacokinetics and dose simulation of vancomycin in critically ill patients during high-
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volume haemofiltration. Int J Antimicrob Agents 2014;44:1637
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20. Kashuba AD, Nafziger AN, Drusano GL, Bertino JS. Optimizing aminoglycoside therapy for
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nosocomial pneumonia caused by gram-negative bacteria. Antimicrob Agents Chemother 1999;43:6239
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21. Lugo-Goytia G, Castaneda-Hernández G. Bayesian approach to control of amikacin serum
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concentrations in critically ill patients with sepsis. Ann Pharmacother 2000;34:138994 22. Conil JM, Georges B, Ruiz S, Rival T, Seguin T, Cougot P, et al. Tobramycin disposition in
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ICU patients receiving a once daily regimen: population approach and dosage simulations. Br J Clin Pharmacol 2011;71:6171
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23. Khachman D, Conil JM, Georges B, Saivin S, Houin G, Toutain PL, et al. Optimizing
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ciprofloxacin dosing in intensive care unit patients through the use of population pharmacokinetic-pharmacodynamic analysis and Monte Carlo simulations. J Antimicrob
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Chemother 2011;66:17981809 24. Mulla H, Peek GJ, Harvey C, Westrope C, Kidy Z, Ramaiah R. Oseltamivir pharmacokinetics in critically ill adults receiving extracorporeal membrane oxygenation support. Anaesth Intensive Care 2013;41:6673 25. Kypraios T, O'Neill PD, Huang SS, Rifas-Shiman SL, Cooper BS. Assessing the role of undetected colonization and isolation precautions in reducing methicillin-resistant Staphylococcus aureus transmission in intensive care units. BMC Infect Dis 2010;10:29
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ACCEPTED MANUSCRIPT 26. Wang J, Wang L, Magal P, Wang Y, Zhuo J, Lu X, et al. Modelling the transmission dynamics of methicillin-resistant Staphylococcus aureus in Beijing Tongren hospital. J Hosp
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Infect 2011;79:3028
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27. Temime L, Opatowski L, Pannet Y, Brun-Buisson C, Boëlle PY, Guillemot D. Peripatetic
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health-care workers as potential superspreaders. Proc Natl Acad Sci USA 2009;106:184205 28. Rushton SP, Shirley MD, Sheridan EA, Lanyon CV, O'Donnell AG. The transmission of
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nosocomial pathogens in an intensive care unit: a space-time clustering and structural
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equation modeling approach. Epidemiol Infect 2010;138:91526 29. Heming N, Faisy C, Urien S. Population pharmacodynamic model of bicarbonate response to
Crit Care 2011;15:R213
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acetazolamide in mechanically ventilated chronic obstructive pulmonary disease patients.
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30. Heming N, Urien S, Fulda V, Meziani F, Gacouin A, Clavel M, et al. Population
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Pharmacodynamic Modeling and Simulation of the Respiratory Effect of Acetazolamide in Decompensated COPD Patients. Plos One 2014;9:e86313
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31. Bastiaans DE, Swart EL, van Akkeren JP, Derijks LJ. Pharmacokinetics of midazolam in resuscitated patients treated with moderate hypothermia. Int J Clin Pharm 2013;35:2106 32. Savard J, Faisy C, Lerolle N, Guérot E, Diehl JL, Fagon JY. Validation of a predictive method for an accurate assessment of resting energy expenditure in medical mechanically ventilated patients. Crit Care Med 2008;36:1175−83 33. MacDonald A, Hildebrandt L. Comparison of formulaic equations to determine energy expenditure in the critically ill patient. Nutrition 2009;19:23339 34. Frankenfield D, Coleman A, Alam S, Cooney R. Analysis of estimation methods for resting energy metabolic rate in critically ill adults. J Parenter Enteral Nutr 2009;33:27−36
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ACCEPTED MANUSCRIPT 35. Frankenfield D, Smith JS, Cooney RN. Validation of 2 approaches to predicting resting metabolic rate in critically ill patients. J Parenter Enteral Nutr 2004;28:25964
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36. Faisy C, Guerot E, Diehl JL, Labrousse J, Fagon JY. Assessment of resting energy
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expenditure in mechanically ventilated patients. Am J Clin Nutr 2003;78:2419
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37. Royston P, Sauerbrei W. Stability of multivariable fractional polynomial models with selection of variables and transformations: a bootstrap investigation. Stat Med 2003;22:639–
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38. Savalle M, Gillaizeau F, Puymirat E, Maruani G, Bellenfant F, Houillier P, et al. Modelling
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Endocrinol Metab 2012;303:38996
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body cell mass for a relevant nutritional assessment in critically ill patients. Am J Physiol
39. Heyland DK, Dhaliwal R, Jiang X, Day AG. Identifying critically ill patients who benefit the
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most from nutrition therapy: the development and initial validation of a novel risk assessment
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tool. Critical Care 2011;15:R268
40. Nachimuthu SK, Wong A, Haug PJ. Modeling Glucose Homeostasis and Insulin Dosing in
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an Intensive Care Unit using Dynamic Bayesian Networks. AMIA Annu Symp Proc 2010;13:5326
41. Abboud I, Lerolle N, Urien S, Tadié JM, Leviel F, Fagon JY, et al. Pharmacokinetics of epinephrine in patients with septic shock: modelization and interaction with endogenous neurohormonal status. Crit Care 2009;13:R120 42. Oualha M, Urien S, Spreux-Varoquaux O, Bordessoule A, D'Agostino I, Pouard P, et al. Pharmacokinetics, hemodynamic and metabolic effects of epinephrine to prevent postoperative low cardiac output syndrome in children. Crit Care 2014;18:R23
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ACCEPTED MANUSCRIPT 43. Zisowsky J, Fuseau E, Bruderer S, Krause A, Dingemanse J. Challenges in collecting pharmacokinetic and pharmacodynamic information in an intensive care setting: PK/PD
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modelling of clazosentan in patients with aneurysmal subarachnoid hemorrhage. Eur J Clin
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Pharmacol 2014;70:40919
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44. Lone NI, Walsh TS. Prolonged mechanical ventilation in critically ill patients: epidemiology, outcomes and modeling the potential cost consequences of establishing a regional weaning
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unit. Crit Care 2011;15:R102
45. Barado J, Guergué JM, Esparza L, Azcárate C, Mallor F, Ochoa S. A mathematical model for
MA
simulating daily bed occupancy in an intensive care unit. Crit Care Med 2012;40:1098104 46. Schroettner J, Lassnig A. Simulation model for cost estimation of integrated care concepts of
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heart failure patients. Health Econ Rev 2013;3:26
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47. Celi LA, Hinske LC, Alterovitz G, Szolovits P. An artificial intelligence tool to predict fluid requirement in the intensive care unit: a proof-of-concept study. Crit Care 2008 ;12:R151
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48. Sandri M, Berchialla P, Baldi I, Gregori D, De Blasi RA. Dynamic Bayesian Networks to
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predict sequences of organ failures in patients admitted to ICU. J Biomed Inform 2014;48:10613
49. Gabler NB, Ratcliffe SJ, Wagner J, Asch DA, Rubenfeld GD, Angus DC, et al. Mortality among patients admitted to strained intensive care units. Am J Respir Crit Care Med 2013;188:8006
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ACCEPTED MANUSCRIPT Table 1 Usual methods for model building in critical care medicine [29]
Field of application
Limitation
Allometry
Weight-based allometry is used to assess relationships between
At each stage (allometrization of data, addition of covariates), one
different individuals.
must check the decrease in inter-individual variability. Allometric
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Method
models are not suitable for extrapolation to situations where the information field is significantly beyond the range of the collected
These models are used when multiple steps of drug distribution and biotransformation occur. Delayed effects in PD of a drug can be characterized by stochastic models with transit
PK/PD models in order to describe dose-response relationships. All material in compartment is considered homogeneously distributed, equations only include information about the total
help to characterize precursor/product relationships, delayed or
amount of material in the compartment. All parameters must be
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compartments and transit times. Transit compartment models
residual effetcs of a drug, and barrier effects (blood-brain
estimated either based on experiments, or obtained from values
barrier).
found in the literature. Model predictions depend on model parameters and initial conditions. Logistic regression can only be used for binary responses. As in
describe multivariate interactions and help to find explanatory
linear models, one must be careful not to extrapolate beyond the
(independent) variables using a simple model. Statistical
observed data. Logistic regression is not a linear function of the
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Logistic regression and, more generally, log-linear models
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Models for discrete data
It is essential to take into account feed-back mechanisms in
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Transit compartment models
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data.
covariates. In order to determine the effects of covariates on the
Transitional models estimate the probability of moving from
probability of the outcome variable, one must compute the log-
one stage to another, given past history, and to determine what
odds and then translate these into probabilities. Logistic regression
factors influence these probabilities. The analysis serves to
coefficients may be impossible to estimate when probabilities are
identify the exposure measure best related to the response, and
all very close to either 0 or 1.
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analysis of case-control studies makes use of log-linear models.
Bayesian networks
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describes each response as a function of drug exposure and time. Bayesian networks help to predict the behavior of a system, help
Building Bayesian networks requires time, expertise and
to determine the causes of a phenomenon observed in the
significant computational power. The quality of parameter
system, and help to simulate the behavior of the system.
estimates may be affected by the use of non-standard distributions
Bayesian approaches can be applied to many fields of medicine,
or by the presence of confunding parameters. Markov chain Monte
covering PK/PD models, individualizing drug dosage,
Carlo simulation methods help provide unbiased estimates of the
observational studies, epidemiology (gene-environment
mean and variance of the prediction. Bayesian techniques are still
interactions), and clinical trials when randomizing is ethically
far from understood and accepted by all scientific journals because
unacceptable.
there is no direct measure of a p value.
PD, pharmacodynamics; PK, pharmacokinetics.
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ACCEPTED MANUSCRIPT Table 2 Areas of application for modeling in the ICU
Examples of specific applications in the ICU [references]
Non-linear mixed effects model
Antibiotic therapy [18], [19], [22], [23], [24]
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Modeling approach
Sedation [31]
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Weaning from mechanical ventilation [29], [30]
Shock [41], [42]
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Neurogical disorders [43]
Antibiotic therapy [17], [18], [22], [23], [24]
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Compartmental model
Sedation [31] Shock [41], [42]
Model for discrete data (including logistic regression)
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Neurogical disorders [43],
Antibiotic therapy [13], [16], [20]
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Nutrition and metabolism [36], [39]
Other approaches
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Bayesian networks
Neurogical disorders [43] Assessing ICU costs [44], [45], [46]
Antibiotic therapy [21] Host-pathogens interactions [25], [26], [27] Nutrition and metabolism [40] Clinical decision in the ICU [47], [48], [49]
Classification and Regression Tree (CART) analysis
Antibiotic therapy [14], [16], [20]
Structural equation modeling
Host-pathogens interactions [28]
Multivariable fractional polynomials
Nutrition and metabolism [38]
ICU, intensive care unit. Classification and Regression Tree (CART) analysis is a classification method which uses observational data in order to classify new data. Structural equation modeling is a method which tests theoretical models by using a system of simultaneous regression equations. Multivariable fractional polynomials is a method which determines the functional form between continuous covariates and the predicted effects. See text for details.
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ACCEPTED MANUSCRIPT Figure legends Fig. 1 Common pharmacokinetic and pharmacodynamic parameters used to describe how drugs
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work in PK/PD models. AUC, area under the curve; Cmax, maximal concentration; EC50,
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concentration for 50% maximum effect (potency) the concentration producing 50% of maximum
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effect; Emax, maximum effect (efficacy); MIC, minimum inhibitory concentration defined as the lowest or minimum antimicrobial concentration of a drug that inhibits visible microbial growth
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in artificial media after a fixed incubation time; T1/2, half-life (time required to move from a drug concentration Cx to half of its concentration Cx/2); T>MIC, time above minimum inhibitory
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concentration. Drug bioavailability defines the fraction of drug absorbed into the body. The absolute drug bioavailability is the oral-to-intravenous AUC ratios.
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Fig. 2 Multicompartmental models topology. The arrows represent the different transfers
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between compartments. Explanatory (independent) variables are volume of distribution (area of
text and Table 1.
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rectangles) and clearance (output). The dependent variables are tranfer constants. For details, see
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Fig. 3 Bayesian network provides a probability of occurrence, knowing the system's characteristics, i.e., P(OccurrenceCharacteristics). For details, see text and Table 1.
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S0883-9441(15)00462-1 doi: 10.1016/j.jcrc.2015.09.002 YJCRC 51939
To appear in:
Journal of Critical Care
Please cite this article as: Lafont Emmanuel, Urien Saik, Salem Joe-Elie, Heming Nicholas, Faisy Christophe, Modeling for critically ill patients: an introduction for beginners, Journal of Critical Care (2015), doi: 10.1016/j.jcrc.2015.09.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Modeling for critically ill patients: an introduction for
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beginners
Emmanuel Lafont, MD a, Saik Urien, MD, PhD b, Joe-Elie Salem, MD c, Nicholas Heming, MD,
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a
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PhD d, Christophe Faisy, MD, PhD a,
Medical Intensive Care Unit, Hôpital Européen Georges Pompidou, Assistance Publique–
b
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Hôpitaux de Paris, Université Paris Descartes Sorbonne Paris Cité, Paris, France. Centre d’Investigation Clinique-0991 INSERM, Université Paris Descartes Sorbonne Paris Cité,
Paris, France.
Centre d'Investigation Clinique-1166 INSERM, Hôpital La Pitié-Salpêtrière, Assistance
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c
Publique–Hôpitaux de Paris, Université Pierre et Marie Curie, Paris, France. Medical Intensive Care Unit, Hôpital Raymond Poincarré, Assistance Publique–Hôpitaux de
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d
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Paris, Université Versailles–Saint Quentin, Garches, France. Corresponding author: Christophe Faisy, MD, PhD, Medical Intensive Care Unit, European
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Georges Pompidou Hospital, 20, rue Leblanc, 75908 Paris Cedex 15, France. Tel: 33 156093220. Fax: 33 156093202. E-mail: [email protected]
Conflict of interest: The authors declare that they have no conflict of interest.
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ACCEPTED MANUSCRIPT ABSTRACT Models are mathematical tools used to describe real world features. Therapeutic interventions in
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the field of critical care medicine may easily be translated into such models. Indeed, numerous
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variables influencing drug pharmacokinetics and pharmacodynamics are systematically
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documented in the intensive care unit over time. Organ failure, fluid shifts, other drug administration, and renal replacement therapy may cause changes in physiological values, such
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as body weight and composition, temperature, serum protein levels, arterial pH and renal or hepatic function. Trials assessing the efficacy and safety of novel drugs usually exclude critically
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ill patients and guidelines regarding drug dosage rarely apply to such patients. Modeling in the critically ill may allow physicians to inform decisions related to therapeutic interventions,
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particularly relating to infectious diseases. However, few clinicians are familiar with these
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methods. Here, we present a current overview of population pharmacokinetic and pharmacodynamic models applicable in critically ill patients aimed at non-specialists, and then
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unit.
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emphazize recent potential of modeling for optimizing treatments and care in the intensive care
Key words: Predictive model; Critically ill patients; Pharmacokinetics; Pharmacodynamics; Bayesian network; Multicompartmental model
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ACCEPTED MANUSCRIPT 1. Introduction Pharmacokinetics (PK) discribes the relationship between the administered dose of a drug
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and its concentrations in the plasma or at its target site, whereas pharmacodynamics (PD)
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describes the relationship between drug concentration and the observed effect on the organism
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[1]. PK/PD models help identify sources of variability in the dose-concentration-effect relationship of a drug (Fig. 1). In the critically ill, PK/PD parameters are altered through a host of
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factors including fluid overload, increased capillary permeability, hypoalbuminemia, renal failure, renal replacement therapy (or other assist devices), as well as poorly perfused tissues.
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Usually recommended drug dosages are therfore not suited to critically ill patients. Modeling may help identify relevant factors related to clinical outcome, characterize and quantify their
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impact. Modeling may also help physicians make better informed predictions and clinical
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decisions, as well as improve the design of drug development studies, assess the efficacy of
regimen.
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therapeutic interventions, and optimize medication use by designing rational dosage forms and
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2. Fixed individual effect models Parameters used to define the pharmacokinetic properties of a drug are its bioavailability (fraction of the drug that is systematicaly absorbed and thus available to produce a biological effect), the area under the curve (AUC) of drug concentration in blood over time, its half-life (time for the drug to lose half of its biological activity), its volume of distribution (theroretical volume of blood that the administered drug would have to occupy), plasma protein binding, and plasma clearance (Fig. 1). The disposition of a molecule depends on its clearance (CL) and volume of distribution (V):
V
dC Ke V C CL C dt
(1) 3
ACCEPTED MANUSCRIPT dC the first derivative of concentration with respect to time, dt i.e., the change in concentration over time, and Ke (CL/V) the elimination rate constant. Drug PK is altered by various conditions commonly encountered in the intensive care unit (ICU): 1) Volume of distribution may rise due to increased capillary permeability or fluid administration and consequently plasma concentration of a molecule will be reduced; hypoalbuminemia may arise, leading to increased free plasma concentration of the drug; 2) Hemodialysis increases the clearance of some drugs such as antibiotics. Of note, the propensity of a drug to be cleared through hemodialysis is expected to decrease when a drug is highly bound to plasma protein. Pharmacodynamic modeling strives to determine the effect of a drug expressed in terms of efficacy and potency (Fig. 1). In critically ill patients, the interaction between a drug and its receptor can be altered due to drug-drug interactions or to other extreme physiological changes such as hypoxemia, acidosis, ionic or temperature shifts.
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where C is drug concentration,
3. Population models
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In a population approach, model parameters are simultaneously estimated for all individuals. In fact, estimates do not refer to each individual but relate to the mean parameters of the
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population along with their between-subject and residual variabilities (random effects). Table 1
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summarizes the potential applications and limitations of commonly used modeling approaches [29].
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3.1. Basic elements for modeling a therapeutic effect
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A population is composed of patients exposed to the same treatment. In population models, the study sample is the population rather than the individual. Population modeling involves five main components: 1) Assessing treatment exposure; 2) Assessing treatment response; 3) Determining relevant covariates; 4) Characterizing the effect (random, fixed or mixed); 5) Determining the mathematical relationship between these elements. One must be aware that Items 1) and 2) constitute the structural model. The response to treatement may be affected by the development of tolerance, by a plecebo effect or by disease progression. A random effect model defines which parameters vary randomly between and within individuals (residual variability), and what the statistical distribution of their variances is. The response predicted by
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ACCEPTED MANUSCRIPT the model for the jth observation, yj, is determined with an error (j) which measures the difference between the predicted value (PRED) and the observed value (OBS) (2)
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yjOBS = yjPRED + j
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where denotes residual variability. Similarly, between-subject variability () can be modeled as
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follows: Pi = PPOP + i.
(3)
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part the variability in the response to treatment.
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where P denotes a structural parameter. Covariates such as patient characteristics, may explain in
Non-linear mixed effects model (NONMEM) is widely accepted model used for analyzing
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PK/PD data. NONMEMs help estimate model parameters as well as different sources of variability. The contribution of each individual covariate is assessed, explaining the between-
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subject variability, i.e., if a covariate has a significant effect on a structural parameter, the
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corresponding is decreased. Observed data are then compared to model predicted results in order to validate the model. The final population model is composed of mathematical equations
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including, when necessary covariates submodels which may modify structural parameters. NONNEMs may be applied at the bedside, i.e., for calculating drug dosages. 3.2. Allometry Allometry refers to the relationship between body size and physiological functions [2]. Specifically, body weight (BW) is related to volume of distribution (V) and clearance (CL): BW V V 70
V
(4) and CL
BW CL CL (5) where 70 θV and θCL are typical volume or clearance per 70 kg, and βCL and βV are the power exponents which modulate the influence of BW (typical values are ¾ for CL and 1 for V).
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ACCEPTED MANUSCRIPT 3.3. Compartmental pharmacokinetic models Compartmental models describe exchanges of material between different compartments of
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the body. A compartmental model is described using several parameters, including transfer rate
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constant, volume of distribution, elimination clearance, state-variables changing over time, independent variables (dose, time) and dependent variables (plasma concentration, effect) [3].
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Transfer rate constants are used to calculate the flow of material. Compartemental models are
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described as catenary (compartments form a chain) or mammillary (central compartment connected to peripheral compartments, all of which are independent of each other). Simple
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mammillary models are widely used in PK studies (Fig. 2). Blood and fully perfused organs (liver, kidneys, and lungs) belong to the central compartment. The elimination rate constants are
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related to the central compartment because the main excretory organs are the liver and kidneys.
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The mathematical description of between compartment-exchanges is usually achieved by
dx kx dt
(6)
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the equation:
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differential equations. It is also assumed that the law of mass conservation applies. For example
with initial conditions starting at time t0 described as: xt0 x0
(7)
resolves into: xt x0e k (t t0 ) (8) where x is the amount of material in the compartment, e equals 2.71, and k is the first order constant. When a drug is intravenously infused, the mathematical model becomes:
dx kx R dt R is the infusion rate. Concentration C is readily obtained by the equation:
(9) where
x (10) In the V case of discontinous drug administration, these models enable to accurately adjust dosage in C
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ACCEPTED MANUSCRIPT order to obtain a predicted concentration. Based on modeling, one can simulate the flow of molecules across a dialysis membrane or the transit time and diffusion of the drug into different compartments. Drug exposure is defined as the area under the concentration curve (AUC) (Fig. 1).
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Transit compartments are used to model drug distribution delay [4]. For instance, the time between administration of the drug and its distribution to the central compartment can be
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modeled by a series of n compartments connected to each other by a first order transfer rate contant, 1/ [5]. The amount of drug versus time in each compartment can be written as follows:
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dAn 1 (11) where An1 An dt An is the amount of drug inside the nth compartment and 1/ has the dimention of a transit time for each compartment. Transit compartment models assume a stochastic approach where only the mean response expected (drug concentration or effect) is observed (Table 1). When the PK model is linear, time to reach steady-state plasma concentrations can easily be determined (five elimination half-lives for one compartment model). This point is of particular importance when determining the effect of a drug which acts on the brain because the delay depends on the time needed to cross the blood-brain barrier. For instance, mathematical models are routinely used in operating theaters in order to accuratly predict the target concentration in the brain of an anesthesic agent [10]. Multicompartmental models predict the contextual half-life of sedative drugs, or the time needed for blood plasma concentrations to be devided by half after discontination of an intravenous infusion (Table 1). 3.4. Models for discrete data
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Clinical outcomes are often described as events: ICU death or recovery, recurrence or progression of disease, nosocomial infection, adverse event, and resistance to treatment. These outcomes may may be quantified as time-to-event, number of events per time interval (rates), their severity grade, or a combination of all three. Discrete data arise when the outcome variables are the number of individuals classified into categories. Such data require specific modeling structures and methods [6]. In case of a binary response, the most appropriate model is logistic regression. Log-linear models are easy to develop using standard statistical software and are used to analyze categorical data with three or more discrete valued variables. The aim of logistic regression is to model the functional form of the binomial probability p(x) by assuming that
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ACCEPTED MANUSCRIPT log {p(x)/(1 − p(x))} = β’x
(12)
β’x is a linear function of the covariates or risk factors that are believed to influence the binary
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response also called Y. The term log {p(x) /(1 − p(x))} is also called the logit or log-odds of p(x).
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Solving for p gives p(x) = e(β’x)/(1 + e(β’x)). In fact, logistic regression is a particular case of loglinear models. These models can be interpreted in terms of interactions between the various
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factors in multidimensional tables and are easily generalized to higher dimensions. In all cases,
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the link function is the logit. The statistical significance of regression coefficients has the same meaning as in linear models. PK/PD models were initially developed for continuous data which
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could have an infinite number of values in a given interval, such as blood pressure. Clinical outcomes or toxic effects are often measured as discrete data with a limited number of states.
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There are four types of discrete data: binary data (for example, alive/dead); categorical nominal
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data (state of vigilance); count of events per time of interval (number of epileptic seizure per week); categorical ordered data consisting of severity grades (Coma Glasgow Score). All those
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outcomes can be linked to drug exposure (AUC) but specific modeling approaches are required
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for each type of discrete data (Table 1). These models may be used during drug development in order to select the best drug candidate, optimize the dosing schedule by maximizing the clinical benefit to harm ratio, or to define the target exposure in different populations. 3.5. Bayesian networks Bayesian networks use observed data (e.g. symptoms) to determine the probability of unobserved data (e.g. pathology). Bayesian networks are probabilistic graphical models build on expert knowledge and observed data. They are used in the medical field, for diagnosis and risk analysis. A Bayesian network is an acyclic graph in which the connecting points between related elements represent random variables (nodes) or influences between variables (links). Arrows
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ACCEPTED MANUSCRIPT represent probabilistic or deterministic relationships between variables (Fig. 3). In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the true value
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is uncertain [7,8]. The key element of Bayesian conjecture is that prior beliefs, in the form of a
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probability distribution, may be combined with observed data, in the form of a likelihood function, to yield a posterior distribution. When more data becomes available, one may calculate
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the posterior distribution using Bayes' formula. Given the observed data D and the hypothesis H
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regarding the occurrence of an event, the cardinal rule is: p(H|D) = p(H|D) p(D|H) × p(H)
(13)
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where p(H) is the prior probability of H, p(H|D) is the posterior probability of H, means proportional, and p(D|H) represents a function of H called likelihood of H. The likelihood ratio
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p(D|H1)/p(D|H2) between two hypotheses (or two models) H1 and H2 is a measure of their
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relative merit and is also called Bayes factor. The log(likelihood ratio), measured in decibans, is frequently used to estimate the weight of evidence. The Bayesian information criterion (BIC)
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helps to select the finest model among a set of models by introducing a penalty term for the
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number of parameters in the model, thereby limiting the risk of overfitting. The Akaike information criterion (AIC) is another popular method to select the finest model. Advanced evaluation tools (normalized prediction distribution errors and visual predictive check) help to validate the Bayesian predictions. As detailled above, the principle of incremental learning in Bayesian networks is based on the a priori knowledge contained in the network at any given time, making the model timedependent. Therefore, a Bayesian network is a means to predict system behavior by combining existing and new data from a database. Bayesian networks are built following three steps: 1) Identifying the most influencial variables and the range of their possible values; 2) Defining the
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ACCEPTED MANUSCRIPT structure of the Bayesian network (identifying causal links between variables); 3) Defining the law of joint probability of the variables (the list of nodes that have a probabilistic dependence
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with the target variable). During the initial phase of model building, a uniform distribution of
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probabilities are assigned to each variable; probabilities are therafter fine-tuned. Covariates of interest are included into the model if they improve (i.e., decrease) the BIC. Difference in BIC
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>10 between two models indicates substantial evidence and BIC >20 represents a decisive
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evidence. One can therafter simulate of the behavior of the system by entering input variables (causes) in order to observe the resulting probability distribution on the effects. Bayesian
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approaches may be used to calculate sample sizes for clinical trials or to select logistic regression models [11,12].
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4. From mathematical models to randomized controlled trials: example of
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acetazolamide
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Acetazolamide, a carbonic anhydrase inhibitor, has been used for decades as a respiratory stimulant in chronic obstructive pulmonary disease (COPD) patients with metabolic alkalosis,
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especialy during the weaning period. However, the optimal dosage of acetazolamide to be administered to alkalotic COPD patients in the ICU remained unclear. The PD of acetazolamide in 68 mechanically ventilated COPD patients who received the drug during weaning from mechanical ventilation was assessed [29,30]. Population pharmacodynamics was modeled using a NONMEM approach. The main covariates of interest which influenced the effect of acetazolamide were the Simplified Acute Physiology Score II (SAPS II) at ICU admission, coprescription of furosemide or corticosteroids, and serum concentration of chloride. According to the model, higher acetazolamide dosage was required to significantly reduce serum bicarbonate concentration in the presence of high levels of serum chloride or co-prescription of systemic
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ACCEPTED MANUSCRIPT corticosteroids or furosemide. The model also predicted that 45% of treated patients would exhibit a decrease of PaCO2 >5 mmHg with doses of 1000 mg/d, indicating that COPD patients
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might benefit from the respiratory stimulant effect after the administration of higher doses
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(>1000 mg daily) of acetazolamide. Respiratory responses to acetazolamide in mechanically ventilated COPD patients were also modeled and the effect of increased amounts of the drug on
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respiratory parameters were simulated in silico. These finding are being prospectively verified by
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a large randomized controlled trial (NCT01627639), assessing the effect of higher doses of acetazolamide. This trial has recently been completed, the results of which are pending. This
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example highlights the relevance of modeling for critically ill patients using the following sequence: 1) Observations based on real facts; 2) Modeling; 3) In silico simulation and dose
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(randomized controlled trial).
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personalization based on the model; 4) Validation of the model by new observations
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5. Other areas of application for modeling in the ICU Mathematical models have great potential for optimizing drug therapies in the field of critical
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care medicine (Table 2). Below, we underline a couple of such exemples. 5.1. Anti-microbial drugs Risk factors for beta-lactam under-dosing in ICU patients are intermittent infusion and increased creatinine clearance [13]. Lodise et al. showed that piperacillin-tazobactam administered over 4 h rather than 30 min increased survival rates [14]. In critically ill patients, Cmax/MIC or AUC24h/MIC ratios are linked to outcome for concentration dependent killing agents [15]. AUC/MIC is the indicator that best characterizes vancomycin PD. In methicillinresistant Staphylococcus aureus (MRSA) bacteremia occurring in ICU patients, risk factors for vancomycin treatment failure are MIC >2 mg/l and an AUC/MIC of <420 [16]. In a basic one-
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ACCEPTED MANUSCRIPT compartment PK model, it was shown that age, serum creatinine clearance, serum albumin and disease severity at ICU admission accounted for >65% of vancomycin clearance variability [17].
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Vancomycin PK/PD models for patients with methicillin-resistant staphylococcal post-
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sternotomy mediastinitis have shown that high plasma vancomycin concentrations (>35 mg/l) might optimize the probability of in-ICU cure without impact on renal function [18]. In critically
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ill patients with refractory septic shock, NONMEM was used to describe population PK and
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dosing simulations helped to define effective vancomycin doses for different haemofiltration settings [19]. Simulation indicated that after a loading dose, doses of vancomycin should be
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individualized according to haemofiltration exchange rate. Regarding aminoglycosides, the finest PD parameter is the Cmax/MIC ratio. A Cmax/MIC of >10 during the first 48 h of aminoglycoside
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treatment has been shown to increase the cure rate of severe pneumonia caused by Gram-
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negative pathogens [20]. Following Bayesian modeling of amikacin PK in critically ill patients with sepsis, it was shown that only severity at ICU admission, creatinine clearance, and the use
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of catecholamines or positive end-expiratory pressure affected the volume of distribution and the
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clearance of amikacin [21]. NONMEM modeling indicated that height and creatinine clearance explained much of the inter-individual variability of tobramycin distribution in ICU patients [22]. Simulations based on this model showed that tobramycin concentration peak and AUC target could not be reached simultaneously in >45% of the patients, confirming that plasma concentration monitoring is required to manage the efficacy and toxicity of tobramycin in the ICU. Based on NONMEM modeling, simulations estimated the relevance of various ciprofloxacin dosage regimens in ICU patients [23]. Less than 20% of patients reached the desired drug concentration range, predicting the emergence of drug resistant mutants, questioning the use of ciprofloxacin for the treatment of Pseudomonas aeruginosa and
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ACCEPTED MANUSCRIPT Acinetobacter baumannii in ICU patients. A multicompartmental model was developed using a NONMEM approach to investigate the PK of oseltamivir in H1N1infected patients requiring
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extracorporeal membrane oxygenation (ECMO) [24]. It was found that the systemic diffusion of
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oseltamivir following enteral administration in adults under ECMO was comparable to that of ambulatory patients and far exceeded concentrations required to inhibit the neuraminidase
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activity of the H1N1 virus. Dosage adjustment for patients undergoing ECMO appeared
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unnecessary, but oseltamivir dosage should be reduced in patients suffering from renal
5.2. Host-pathogen interactions
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dysfunction.
Bayesian frameworks and compartmental models were used to assess the effectiveness of
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barrier precautions and hand washing compliance for reducing the transmission of multidrug
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resistant bacteria [2527]. Structural equation modeling (method where the model is based on the structure of the covariance matrix of the measures) showed that prior nurse contact with
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colonized/infected patients was a significant predictor of transmission of nosocomial pathogens,
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except for Pseudomonas spp., while general health and invasive surgery were significant predictors of Candida and Klebsiella spp. transmission, indicating that isolation and bed movement as a strategy to manage MRSA infections might impact the incidence of transmission of other multidrug resistant pathogens [28]. 5.3. Sedation Multicompartmental approaches (taking into account drug distibution delays due to the the blood-brain barrier) might be usefull for modeling the effects of sedative drugs in critically ill patients. PK/PD studies of sedative drugs remain scarce in patients treated by moderate hypothermia. Bastiaans et al. investigated the PK of midazolam in resuscitated patients treated
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ACCEPTED MANUSCRIPT by moderate hypothermia in comparison to normothermic and non-resuscitated patients [31]. Using NONMEM, the PK of midazolam was found to be similar in resuscitated patients treated
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by moderate hypothermia and their controls.
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5.4. Nutrition and metabolism
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Predictive multi-parameter models help to estimate energy expenditure of critically ill patients [3236]. Body cell mass (the metabolically active part of fat-free mass) of ICU patients
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has been modeled using variables obtained at the bedside [37, 38]. Nutritional screening of the critically ill also benefited from modeling. Novel scoring tools, such as the NUTRIC score, may
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help discriminate ICU patients who will benefit from aggressive nutritional assistance [39]. Nachimuthu et al. developed a dynamic Bayesian network model to describe various clinical
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parameters and their relationships to insulin and glucose homeostasis [40]. The predictions based
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on this Bayesian model were at least as effective or better than those of the rule-based protocol
5.5. Shock
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(eProtocol-insulin) currently used in ICUs.
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Using an allometric model, the PK of epinephrine in patients with septic shock was investigated; epinephrine clearance was only dependent on BW and severity of illness at ICU admission [41]. Moreover, PK of epinephrine was described as linear and no saturation at high doses was predicted in septic shock patients. In critically ill children with post-operative low cardiac output, NONMEM modeling indicated that BW and age affected the PK/PD of epinephrine, predicting the effects of a given dosage of epinephrine on heart rate, mean arterial pressure, and plasma lactate or glucose levels [42]. 5.6. Neurological disorders
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ACCEPTED MANUSCRIPT Using population and logistic regression models, the PK/PD of clazosentan was investigated in patients suffering from aneurysmal subarachnoid hemorrhage. Clazosentan
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clearance was influenced by age, sex, ethnic origin, disease status at baseline and increased over
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time [43]. However, clazosentan exposure was not found to have any relationship with efficacy endpoints while being associated with increased risks of pulmonary complications.
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5.7. Assessing ICU costs
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Modeling the impact of establishing a dedicated weaning unit in an intensive care department showed that such a unit may reduce acute bed occupancy by 10% and could reduce
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overall treatment costs [44]. As longevity increases in the population, the prevalance of chronic diseases increases, adding strain to health care facilities and to ICUs. A computerized model was
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used as a capacity analysis tool to assess ICU bed occupancy profile [45]. Lastly, a discret-event
with heart failure [46].
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simulation model indicated that a telemonitoring system could reduce costs for treating patients
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5.8. Making better informed decisions in the ICU
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By using a learning Bayesian network, the probability that a septic ICU patient would require a certain range of fluid can be predicted [47]. Another dynamic Bayesian network was applied to model Sequential Organ Failure Assessment score changes in adult critically ill patients in order to predict the most probable sequences of organ failures over the first week after admission [48]. Similarely, modeling the effect of strain imposed on ICUs and patient outcome revealed that strain particularely affected ICUs using closed rather than open intensivist staffing models [49].
6. Conclusions
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ACCEPTED MANUSCRIPT Models are tools used to describe natural or man-made phenomenons by simplifying a complex reality. Models are refuted by experience, but they are essential for scientists. George
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Box’s approach summarizes this duality: "All models are wrong! Some are useful". Critical care
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medicine is a promising field for the mathematical description of reality in order to optimize
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effectiveness of care and to minimize costs.
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Acknowledgements:
This work was funded by the Medical Intensive Care Unit, Hôpital Européen Georges
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Pompidou, Assistance Publique – Hôpitaux de Paris, Université Paris Descartes Sorbonne Paris
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Cité, Paris, France.
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ACCEPTED MANUSCRIPT References
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1. Pérez-Urizar J, Granados-Soto V, Flores-Murrieta FJ, Castaneda-Hernández G.
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Pharmacokinetic-Pharmacodynamic Modeling: Why? Arch Med Res 2000;31:53945 2. Agutter PS, Tuszynski JA. Analytic theories of allometric scaling. J Exp Biol
SC
2011;214:105562
NU
3. Blomhøj M, Kjeldsen TH, Ottesen J. Compartment models. Latest version Available at: http://www4.ncsu.edu/~msolufse/Compartmentmodels.pdf. Accessed December 2014
MA
4. Sun YN, Jusko WJ. Transit Compartments versus Gamma Distribution Function To Model Signal Transduction Processes in Pharmacodynamics. J Pharm Sci 1998;87:7327
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5. Paule I, Girard P, Freyer G, Tod M. Pharmacodynamic Models for Discrete Data. Clin
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Pharmacokinet 20012;12:76786
6. Zelterman D. Models for discrete data, 2th ed. New York: Oxford University Press; 2006.
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7. Spiegelhalter DJ, Myles JP, Jones DR, Abrams KR. Bayesian methods in health technology
AC
assessment: a review. Health Technol Assess 2000,4:1130 8. Rosner GL, Müller P. Bayesian population pharmacokinetic and pharmacodynamic analyses using mixture models. J Pharmacokinet Biopharm 1997;25:20933 9. Pressman AR, Avins AL, Hubbard A, Satariano WA. A comparison of two worlds: How does Bayes hold up to the status quo for the analysis of clinical trials? Contemp Clin Trials 2011;32:561–8 10. Casati A, Fanelli G, Casaletti E, Colnaghi E, Cedrati V, Torri G. Clinical assessment of target-controlled infusion of propofol during monitored anesthesia care. Can J Anesth 1999;46:2359
17
ACCEPTED MANUSCRIPT 11. Wagner H, Duller C. Bayesian model selection for logistic regression models with random intercept. Comput Stat Data Anal 2012;56:1256–74
T
12. Zhang X, Cutter G, Belin T. Bayesian sample size determination under hypothesis tests.
RI P
Contemp Clin Trials 2011;32:393–8
13. De Waele JJ, Lipman J, Akova M, Bassetti M, Dimopoulos G, Kaukonen M, et al. Risk
SC
factors for target non-attainment during empirical treatment with β-lactam antibiotics in
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critically ill patients. Intensive Care Med 2014;40:1340–51
14. Lodise TP, Lomaestro BM, Drusano GL. Piperacillin-tazobactam for Pseudomonas
MA
aeruginosa infection: clinical implications of an extended-infusion strategy. Clin Infect Dis 2007;44:357–63
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15. Owens RC, Ambrose PG. Antimicrobial stewardship and the role of pharmacokinetics–
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pharmacodynamics in the modern antibiotic era. Diagn Microbiol Infect Dis 2007;57:7783 16. Kullar R, Davis SL, Levine DP, Rybak MJ. Impact of vancomycin exposure on outcomes in
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patients with methicillin-resistant Staphylococcus aureus bacteremia: support for consensus
AC
guidelines suggested targets. Clin Infect Dis 2011;52:97581 17. Del Mar Fernández de Gatta GM, Revilla N, Calvo MV, Domínguez-Gil A, Sánchez Navarro A. Pharmacokinetic/pharmacodynamic analysis of vancomycin in ICU patients. Intensive Care Med 2007;33:27985 18. Mangin O, Urien S, Mainardi JL, Fagon JY, Faisy C. Vancomycin pharmacokinetic and pharmacodynamics models for critically ill patients with post-sternotomy mediastinitis. Clin Pharmacokinet 2014;53:84961
18
ACCEPTED MANUSCRIPT 19. Escobar L, Andresen M, Downey P, Gai MN, Regueira T, Bórquez T, et al. Population pharmacokinetics and dose simulation of vancomycin in critically ill patients during high-
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volume haemofiltration. Int J Antimicrob Agents 2014;44:1637
RI P
20. Kashuba AD, Nafziger AN, Drusano GL, Bertino JS. Optimizing aminoglycoside therapy for
SC
nosocomial pneumonia caused by gram-negative bacteria. Antimicrob Agents Chemother 1999;43:6239
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21. Lugo-Goytia G, Castaneda-Hernández G. Bayesian approach to control of amikacin serum
MA
concentrations in critically ill patients with sepsis. Ann Pharmacother 2000;34:138994 22. Conil JM, Georges B, Ruiz S, Rival T, Seguin T, Cougot P, et al. Tobramycin disposition in
ED
ICU patients receiving a once daily regimen: population approach and dosage simulations. Br J Clin Pharmacol 2011;71:6171
PT
23. Khachman D, Conil JM, Georges B, Saivin S, Houin G, Toutain PL, et al. Optimizing
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ciprofloxacin dosing in intensive care unit patients through the use of population pharmacokinetic-pharmacodynamic analysis and Monte Carlo simulations. J Antimicrob
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Chemother 2011;66:17981809 24. Mulla H, Peek GJ, Harvey C, Westrope C, Kidy Z, Ramaiah R. Oseltamivir pharmacokinetics in critically ill adults receiving extracorporeal membrane oxygenation support. Anaesth Intensive Care 2013;41:6673 25. Kypraios T, O'Neill PD, Huang SS, Rifas-Shiman SL, Cooper BS. Assessing the role of undetected colonization and isolation precautions in reducing methicillin-resistant Staphylococcus aureus transmission in intensive care units. BMC Infect Dis 2010;10:29
19
ACCEPTED MANUSCRIPT 26. Wang J, Wang L, Magal P, Wang Y, Zhuo J, Lu X, et al. Modelling the transmission dynamics of methicillin-resistant Staphylococcus aureus in Beijing Tongren hospital. J Hosp
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Infect 2011;79:3028
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27. Temime L, Opatowski L, Pannet Y, Brun-Buisson C, Boëlle PY, Guillemot D. Peripatetic
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health-care workers as potential superspreaders. Proc Natl Acad Sci USA 2009;106:184205 28. Rushton SP, Shirley MD, Sheridan EA, Lanyon CV, O'Donnell AG. The transmission of
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nosocomial pathogens in an intensive care unit: a space-time clustering and structural
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equation modeling approach. Epidemiol Infect 2010;138:91526 29. Heming N, Faisy C, Urien S. Population pharmacodynamic model of bicarbonate response to
Crit Care 2011;15:R213
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acetazolamide in mechanically ventilated chronic obstructive pulmonary disease patients.
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30. Heming N, Urien S, Fulda V, Meziani F, Gacouin A, Clavel M, et al. Population
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Pharmacodynamic Modeling and Simulation of the Respiratory Effect of Acetazolamide in Decompensated COPD Patients. Plos One 2014;9:e86313
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31. Bastiaans DE, Swart EL, van Akkeren JP, Derijks LJ. Pharmacokinetics of midazolam in resuscitated patients treated with moderate hypothermia. Int J Clin Pharm 2013;35:2106 32. Savard J, Faisy C, Lerolle N, Guérot E, Diehl JL, Fagon JY. Validation of a predictive method for an accurate assessment of resting energy expenditure in medical mechanically ventilated patients. Crit Care Med 2008;36:1175−83 33. MacDonald A, Hildebrandt L. Comparison of formulaic equations to determine energy expenditure in the critically ill patient. Nutrition 2009;19:23339 34. Frankenfield D, Coleman A, Alam S, Cooney R. Analysis of estimation methods for resting energy metabolic rate in critically ill adults. J Parenter Enteral Nutr 2009;33:27−36
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ACCEPTED MANUSCRIPT 35. Frankenfield D, Smith JS, Cooney RN. Validation of 2 approaches to predicting resting metabolic rate in critically ill patients. J Parenter Enteral Nutr 2004;28:25964
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36. Faisy C, Guerot E, Diehl JL, Labrousse J, Fagon JY. Assessment of resting energy
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expenditure in mechanically ventilated patients. Am J Clin Nutr 2003;78:2419
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37. Royston P, Sauerbrei W. Stability of multivariable fractional polynomial models with selection of variables and transformations: a bootstrap investigation. Stat Med 2003;22:639–
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59
38. Savalle M, Gillaizeau F, Puymirat E, Maruani G, Bellenfant F, Houillier P, et al. Modelling
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Endocrinol Metab 2012;303:38996
MA
body cell mass for a relevant nutritional assessment in critically ill patients. Am J Physiol
39. Heyland DK, Dhaliwal R, Jiang X, Day AG. Identifying critically ill patients who benefit the
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most from nutrition therapy: the development and initial validation of a novel risk assessment
CE
tool. Critical Care 2011;15:R268
40. Nachimuthu SK, Wong A, Haug PJ. Modeling Glucose Homeostasis and Insulin Dosing in
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an Intensive Care Unit using Dynamic Bayesian Networks. AMIA Annu Symp Proc 2010;13:5326
41. Abboud I, Lerolle N, Urien S, Tadié JM, Leviel F, Fagon JY, et al. Pharmacokinetics of epinephrine in patients with septic shock: modelization and interaction with endogenous neurohormonal status. Crit Care 2009;13:R120 42. Oualha M, Urien S, Spreux-Varoquaux O, Bordessoule A, D'Agostino I, Pouard P, et al. Pharmacokinetics, hemodynamic and metabolic effects of epinephrine to prevent postoperative low cardiac output syndrome in children. Crit Care 2014;18:R23
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ACCEPTED MANUSCRIPT 43. Zisowsky J, Fuseau E, Bruderer S, Krause A, Dingemanse J. Challenges in collecting pharmacokinetic and pharmacodynamic information in an intensive care setting: PK/PD
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modelling of clazosentan in patients with aneurysmal subarachnoid hemorrhage. Eur J Clin
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Pharmacol 2014;70:40919
SC
44. Lone NI, Walsh TS. Prolonged mechanical ventilation in critically ill patients: epidemiology, outcomes and modeling the potential cost consequences of establishing a regional weaning
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unit. Crit Care 2011;15:R102
45. Barado J, Guergué JM, Esparza L, Azcárate C, Mallor F, Ochoa S. A mathematical model for
MA
simulating daily bed occupancy in an intensive care unit. Crit Care Med 2012;40:1098104 46. Schroettner J, Lassnig A. Simulation model for cost estimation of integrated care concepts of
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heart failure patients. Health Econ Rev 2013;3:26
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47. Celi LA, Hinske LC, Alterovitz G, Szolovits P. An artificial intelligence tool to predict fluid requirement in the intensive care unit: a proof-of-concept study. Crit Care 2008 ;12:R151
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48. Sandri M, Berchialla P, Baldi I, Gregori D, De Blasi RA. Dynamic Bayesian Networks to
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predict sequences of organ failures in patients admitted to ICU. J Biomed Inform 2014;48:10613
49. Gabler NB, Ratcliffe SJ, Wagner J, Asch DA, Rubenfeld GD, Angus DC, et al. Mortality among patients admitted to strained intensive care units. Am J Respir Crit Care Med 2013;188:8006
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ACCEPTED MANUSCRIPT Table 1 Usual methods for model building in critical care medicine [29]
Field of application
Limitation
Allometry
Weight-based allometry is used to assess relationships between
At each stage (allometrization of data, addition of covariates), one
different individuals.
must check the decrease in inter-individual variability. Allometric
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Method
models are not suitable for extrapolation to situations where the information field is significantly beyond the range of the collected
These models are used when multiple steps of drug distribution and biotransformation occur. Delayed effects in PD of a drug can be characterized by stochastic models with transit
PK/PD models in order to describe dose-response relationships. All material in compartment is considered homogeneously distributed, equations only include information about the total
help to characterize precursor/product relationships, delayed or
amount of material in the compartment. All parameters must be
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compartments and transit times. Transit compartment models
residual effetcs of a drug, and barrier effects (blood-brain
estimated either based on experiments, or obtained from values
barrier).
found in the literature. Model predictions depend on model parameters and initial conditions. Logistic regression can only be used for binary responses. As in
describe multivariate interactions and help to find explanatory
linear models, one must be careful not to extrapolate beyond the
(independent) variables using a simple model. Statistical
observed data. Logistic regression is not a linear function of the
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Logistic regression and, more generally, log-linear models
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Models for discrete data
It is essential to take into account feed-back mechanisms in
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Transit compartment models
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data.
covariates. In order to determine the effects of covariates on the
Transitional models estimate the probability of moving from
probability of the outcome variable, one must compute the log-
one stage to another, given past history, and to determine what
odds and then translate these into probabilities. Logistic regression
factors influence these probabilities. The analysis serves to
coefficients may be impossible to estimate when probabilities are
identify the exposure measure best related to the response, and
all very close to either 0 or 1.
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analysis of case-control studies makes use of log-linear models.
Bayesian networks
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describes each response as a function of drug exposure and time. Bayesian networks help to predict the behavior of a system, help
Building Bayesian networks requires time, expertise and
to determine the causes of a phenomenon observed in the
significant computational power. The quality of parameter
system, and help to simulate the behavior of the system.
estimates may be affected by the use of non-standard distributions
Bayesian approaches can be applied to many fields of medicine,
or by the presence of confunding parameters. Markov chain Monte
covering PK/PD models, individualizing drug dosage,
Carlo simulation methods help provide unbiased estimates of the
observational studies, epidemiology (gene-environment
mean and variance of the prediction. Bayesian techniques are still
interactions), and clinical trials when randomizing is ethically
far from understood and accepted by all scientific journals because
unacceptable.
there is no direct measure of a p value.
PD, pharmacodynamics; PK, pharmacokinetics.
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ACCEPTED MANUSCRIPT Table 2 Areas of application for modeling in the ICU
Examples of specific applications in the ICU [references]
Non-linear mixed effects model
Antibiotic therapy [18], [19], [22], [23], [24]
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Modeling approach
Sedation [31]
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Weaning from mechanical ventilation [29], [30]
Shock [41], [42]
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Neurogical disorders [43]
Antibiotic therapy [17], [18], [22], [23], [24]
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Compartmental model
Sedation [31] Shock [41], [42]
Model for discrete data (including logistic regression)
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Neurogical disorders [43],
Antibiotic therapy [13], [16], [20]
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Nutrition and metabolism [36], [39]
Other approaches
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Bayesian networks
Neurogical disorders [43] Assessing ICU costs [44], [45], [46]
Antibiotic therapy [21] Host-pathogens interactions [25], [26], [27] Nutrition and metabolism [40] Clinical decision in the ICU [47], [48], [49]
Classification and Regression Tree (CART) analysis
Antibiotic therapy [14], [16], [20]
Structural equation modeling
Host-pathogens interactions [28]
Multivariable fractional polynomials
Nutrition and metabolism [38]
ICU, intensive care unit. Classification and Regression Tree (CART) analysis is a classification method which uses observational data in order to classify new data. Structural equation modeling is a method which tests theoretical models by using a system of simultaneous regression equations. Multivariable fractional polynomials is a method which determines the functional form between continuous covariates and the predicted effects. See text for details.
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ACCEPTED MANUSCRIPT Figure legends Fig. 1 Common pharmacokinetic and pharmacodynamic parameters used to describe how drugs
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work in PK/PD models. AUC, area under the curve; Cmax, maximal concentration; EC50,
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concentration for 50% maximum effect (potency) the concentration producing 50% of maximum
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effect; Emax, maximum effect (efficacy); MIC, minimum inhibitory concentration defined as the lowest or minimum antimicrobial concentration of a drug that inhibits visible microbial growth
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in artificial media after a fixed incubation time; T1/2, half-life (time required to move from a drug concentration Cx to half of its concentration Cx/2); T>MIC, time above minimum inhibitory
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concentration. Drug bioavailability defines the fraction of drug absorbed into the body. The absolute drug bioavailability is the oral-to-intravenous AUC ratios.
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Fig. 2 Multicompartmental models topology. The arrows represent the different transfers
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between compartments. Explanatory (independent) variables are volume of distribution (area of
text and Table 1.
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rectangles) and clearance (output). The dependent variables are tranfer constants. For details, see
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Fig. 3 Bayesian network provides a probability of occurrence, knowing the system's characteristics, i.e., P(OccurrenceCharacteristics). For details, see text and Table 1.
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