Computational Opioid Prescribing: A Novel Application of Clinical Pharmacokinetics

We implemented a pharmacokinetics-based mathematical modeling technique using algebra to assist pre-scribers with point-of-care opioid dosing. We call this technique computational opioid prescribing (COP). Because population pharmacokinetic parameter values are needed to estimate drug dosing regimen designs for individual patients using COP, and those values are not readily available to prescribers because they exist scattered in the vast pharmacology literature, we estimated the population pharmacokinetic parameter values for 12 commonly prescribed opioids from various sources using the bootstrap resampling technique. Our results show that opioid dosing regimen design, evaluation, and modification is feasible using COP. We conclude that COP is a new technique for the quantitative assessment of opioid dosing regimen design evaluation and adjustment, which may help prescribers to manage acute and chronic pain at the point-of-care. Potential benefits include opioid dose optimization and minimization of adverse opioid drug events, leading to potential improvement in patient treatment outcomes and safety.


INTRODUCTION
The expected clinical outcome measure of interventional opioid pharmacotherapy for the treatment of acute and chronic pain is analgesia. However, studies indicate that many patients in pain are prescribed inadequate doses of opioid medications to relieve their pain (1)(2)(3)(4)(5). Multiple barriers to the adequate treatment of pain have been identified (6)(7)(8)(9), but rational opioid dosing founded on pharmacokineticsbased opioid dosing regimens for individual patients has not been adequately addressed. To date, clinical pharmacokinetics-based opioid dosing regimen design and adjustment in acute and chronic pain management has been the subject of a limited number of publications (10,11), although pharmacokinetics-based anesthetic opioid dosing is routine (12)(13)(14).
Opioid maximum concentrations (C max ) and minimum concentrations (C min ) may correspond to their minimum toxic concentrations (MTCs) and minimum effective concentrations (MECs). For example, the therapeutic range of morphine for analgesia is reported to be between 9.3 and 80 ng/mL, MEC range 9.3 to 23 ng/mL (15). This range implies that concentrations above 80 ng/mL are more likely to be associated with toxicity and concentrations below 9.3 ng/mL are more likely to produce little or no analgesic effect. Therefore, in the course of pain management with morphine, it is desirable that a dosage regimen for morphine produce plasma morphine concentrations within its therapeutic range. The goal of the design of an opioid dosing regimen is thus to achieve predicted plasma opioid concentrations within or at the boundaries of targeted C max and C min values or a desired target concentrations (C target

Opioid Population Pharmacokinetic Parameter Estimation
Population pharmacokinetic parameter values are often used to estimate drug dosing regimen designs for individual patients in whom patient-specific parameter values are not available (16,17); however, opioid pharmacokinetic parameter values are scattered about in the vast pharmacology literature. Therefore, we culled a large number of studies using references (18)(19)(20)(21), the reference lists therein, the reference lists of the references therein, and literature searches using PubMed.gov and Scholar.Google.com with search terms entered: opioids pharmacokinetics, opioids pharmacodynamics, and the latter terms, substituting the word opioids for each of the 12 individual opioids studied, so as to obtain pharmacokinetic parameter values for the following 12 opioids: (1) morphine, (2) tramadol, (3) codeine, (4) meperidine, (5), hydrocodone, (6) oxycodone immediate-release (IR), (7) oxycodone controlled-release (CR), (8) hydromorphone, (9) oxymorphone, (10) methadone, (11) fentanyl, and (12)  The large number of studies analyzed allowed a statistical approach in which the mean values from each study provided single data points (independent samples). When mean group values were not reported, the published experimental study data (22) was used to derive the primary pharmacokinetic parameter or parameters using modeling (23). All studies, despite their probable variable reliability, were accorded equal weight. Values for opioid bioavailability (F) were obtained from references (18)(19)(20)(21), the references therein, or literature search. Values for the first-order absorption constant (k a ) were obtained from references (18)(19)(20)(21), the references therein, or literature search, but, when values were not available, they were computed using the equation: Opioid population pharmacokinetic parameter estimation using the bootstrap (see below) was not performed for F or k a values.
Opioid population pharmacokinetic parameter estimation was performed using the bootstrap resampling technique (24)(25)(26)(27) Equation 4 gives the rate of decline of plasma opioid concentration from C ss max to C ss min : [4] This rate is governed by the t 1/2 or k e because k e = 0.693/t 1/2 and vice-versa. Equation 4 can be solved for τ max , giving The opioid maintenance dose (D M ) is given by where C target is the targeted plasma opioid concentration. In some cases, administration of a loading dose (D L ) may be necessary: where C p represents the plasma opioid concentration. For multiple dosing, the accumulation ratio (AR), which represents the ratio of opioid in the body at steady-state relative to the amount of opioid in the body after a single dose, is given by Hence, the maximum (C ss max ) or peak plasma opioid concentration is given by [9] and the minimum (C ss min ) or trough plasma opioid concentration by C ss min (trough) = C ss max · e −k e τ × AR. [10] The average steady-state plasma opioid concentration (Equation 1) can also be obtained using Cl . [11] At any time (T), C p is given by [12]

Opioid Dosing Regimen Design Using COP
A major aim of COP is the design of an opioid dosage regimen that achieves predicted plasma opioid concentrations within a safe and effective range. COP assumes that opioid clinical pharmacokinetics can be reasonably approximated by a linear open one-compartment model with first-order absorption and first-order elimination (Figure (1)). This model has been shown to account for the pharmacokinetics of many important drugs (30), including the opioid analgesics (21). In certain circumstances, however, it may be associated with significant error in the calculation of a drug's absorption rate constant, k a , such as when k a = k e . The required data to design an opioid dosage regimen using COP is information about the pharmacokinetics of the opioid, the values reported in Table 1), and the opioid's therapeutic range (31). Pharmacokinetic parameter values calculated using COP can be expected to have an inter-individual variation of about 25%, which may be clinically acceptable.
However, the following are assumptions and limitations of using the one-compartment model (32): (1) it is assumed that the pharmacokinetic parameters remain constant during the course of treatment; (2) changes in renal and/or hepatic function may prolong the excretion of the fraction of opioid excreted unchanged in the urine, or metabolized by the liver; (3) congestive heart failure (CHF) and myocardial infarction (MI) may cause reduction in blood flow, resulting in reduced volume of distribution and clearance, thereby prolonging opioid elimination.
Qualifications for using the one-compartment model when a theoretically correct model is multicompartmental are as follows (32,33): (1)

Intravenous (IV) Bolus Dosing
Calculation of dose size and dosing interval is performed using the equations for maximum and minimum plasma drug levels at steady state, C ss max and C ss min , respectively. This method is called the limited fluctuation method or C ss max − C ss min method (32). This approach is used when, within a dosing interval, the desired steady-state opioid plasma levels to achieve do not exceed C max and do not undercut the desired C min .
First, we estimate a target average steady-state plasma morphine concentration (C ss ave ) based on morphine's therapeutic window. Equation 1 defines C ss ave based on the minimum (C ss min ) and maximum (C ss max )

Case Study 1
The patient is a 55-year-old Caucasian man newly diagnosed with cancer of the prostate and metastases to the spine with normal cardiac, hepatic, and renal function. He is experiencing 10/10 back pain. The nurse on duty caring for the patient informs the doctor that the patient has not received morphine in the past, so he is opioid naïve. The nurse is not comfortable administering intravenous morphine 25 mg every 6 hours to the patient. The doctor searches the literature and finds evidence that in cancer patients, morphine levels at or above 20 µg/L are considered to be analgesic in most patients (37). This value is about 60% of the plasma morphine concentration calculated using COP (Equation 13). Using COP, the doctor simulates a morphine dosage regimen equal to 12 mg every 6 hours to assess the effect it would have on morphine pharmacokinetics in this patient.
The doctor reestimates C ss ave to predict the new plasma morphine concentration using the practical τ of 6 hours and the intravenous morphine dose of 12 mg. The selected dose now predicts a plasma morphine concentration within the evidence-based analgesic value. To estimate the C max and C min values with the reduced dose regimen, first, the C max and C min after the first dose (C 1st max and C 1st min , respectively) are estimated. Then, using the accumulation ratio (AR), the C ss max and C ss min values are computed as follows: AR represents the ratio of opioid in the body at steady-state relative to the amount of opioid in the body after a single dose (35 In some cases, administration of a loading dose (D L ) or starting dose (D S ) may be necessary, particularly if the half-life of the opioid is long and/or rapid achievement of therapeutic opioid concentrations is important, e.g., in acute pain. In these cases, D L may be calculated using the t 1/2 by the following method: [24] The calculated dose should be adjusted for administration based on available morphine strengths.

Extravascular (Oral) Dosing
The computation of dose and dosing interval after extravascular dosing (e.g., oral administration) is slightly more complicated than after intravenous (IV) bolus dosing because the absorption rate (k a ) and F are important factors, in addition to the other four basic clinical pharmacokinetic parameters. Nevertheless, simplified clinical pharmacokinetic-based equations for COP allow easy calculation of oral dosage regimen designs (see above) that do not require k a , because it can be shown that the terminal equilibrium level under constant infusion equals the average blood level over time for the terminal steady-state under multiple IV or oral dosing (38). Also, a simplification which applies to opioids, is when the k a is much faster than the k e . Under this condition, k a may be assumed to be instantaneous for practical purposes. In general, if the k a is 5 to 7 times greater than the k e , the absorption rate may be assumed to be instantaneous. This circumstance is similar to the IV bolus administration of opioid (39), but with F less than 1 so that the equations for IV bolus dosing apply. Therefore, the equations used for IV bolus dosing above can also be used to design extravascular (oral) dosage regimens with reasonable accuracy.

Case Study 2
The patient is a 52-year-old African American woman weighing 302 lbs (137 kg), and is 2 days postoperative from right knee arthroplasty. She complains of 8/10 pain. She requests oral pain medication. Using the opioid plasma target concentration method, the doctor selects a target morphine plasma concentration equal to half of its MTC, i.e., 40 µg/L. F is set equal to its algebraic average, which is calculated as (0.15+ 0.64)/2 = 0.40. Then, the morphine dose for starting treatment (D S = D L ) is calculated using the following equation: . [25] Multiplying by 0.001 converts µg to mg, so the starting dose equals 62 mg, which can be rounded to 60 mg for administration.
The time to reach steady state for morphine is T ss = 5 × t 1/2 = 20 hours. By this time, it can be assumed that morphine's rate in = rate out. Thus, the morphine maintenance dose to achieve a steady state in T ss is given by [26] Now, selecting a dosing interval where τ = 6 hours gives D M = 120 mg every 6 hours. This is the dose required to maintain a steady-state plasma concentration of morphine of 40 µg/L. We confirm the dose as follows.
Compute the accumulation ratio using AR = τ / t 1/2 = 6/4 = 1.5, then Thus, COP demonstrates that our oral morphine dosage regimen design achieved the therapeutic goal of maintaining a steady-state plasma morphine concentration of 40 µg/L. The calculated value is slightly higher due to computer round off error. Moreover, COP shows that plasma morphine fluctuations about C ss ave remain close to the bounds of morphine's therapeutic window for analgesia. However, towards the end of the dosing interval, the concentration falls slightly below the evidence-based plasma morphine concentration for analgesia of 20 µg/L. COP indicates that an adjustment in the dosing interval can be made if the patient complains of pain towards the end of the dosing interval. For example, using Equations 12 or 17, at the 5th-hour into the 6th-hour dosing interval, plasma morphine concentration is 24 µg/L, indicating that the patient needs to be followed closely for symptoms of pain during the last hour of the dosing interval when plasma morphine concentrations fall below the evidence-based analgesic concentration.

Constant Intravenous Infusion
To determine intravenous infusion rate we need only determine the infusion rate constant (k 0 ) and do not need to compute τ . To determine a constant rate intravenous infusion of morphine, we estimate k 0 based on a desired morphine C target to achieve, and its Cl, so that k 0 = C target · Cl. We chose a value for C target within morphine's therapeutic window. If rapid achievement of steady-state morphine concentration is desired, an intravenous loading dose may be calculated using Equation 7 or simply

Opioid Dosing Regimen Evaluation
Dosing regimen evaluation using COP applies to patients who present on a dosage regimen prescribed by another prescriber or an emergency department (ED) physician. The goal is to determine the adequacy of the regimen to achieve evidence-based analgesic plasma concentrations of opioid.

Case Study 3
A 38-year-old average weight man was seen in the ED 24 hours ago and placed on an oral regimen of 10 mg morphine sulfate every 8 h for 7/10 back pain secondary to a fall from a 6-foot ladder. He was instructed to follow-up with pain management on the next day. He presented 24  Clearly, the patients dosage regimen is inadequate to meet his analgesic needs, since only morphine levels at or above 20 µg/L are considered to be analgesic (15,37). Thus, a dosage adjustment, rather than an extensive work-up, was indicated to manage this patients increasing pain.

Case Study 4
A petite 29-year-old woman weighing 110 lbs ( max · e −k e T · AR = 36µg/L × e −0.318×0.25 ×1.0 = 33µg/L. [30] Based on this computation, the doctor decides to set C target = 35 µg/L, which is the average value between C ss max and C t (30), to calculate a maintenance dose: [31] Since 24 mg every 6 hours is not practical, the doctor selects a dose of 25 mg every 4 hours for the patient. This dose will be associated with a C ss max = 44 µg/L and C ss min = 12 µg/L. One hour after a dose, the plasma morphine concentration will be 32 µg/L. By 1 hour and 15 minutes after a dose, plasma morphine concentration will be 30 µg/L, but by 1 1 2 hours it will have fallen to 28 µg/L.
The patient is seen for follow-up within 3 days. She reports being pain free up for about 1 1 2 to 2 hours after taking a scheduled morphine dose. Thus, COP predicts she requires a plasma morphine concentration ≥30 µg/L for pain relief. In addition, COP indicates the patient will need a long-acting morphine combined with rescue doses of short-acting morphine for adequate pain control.

Case Study 5
In the course of chronic pain management, if a patient with chronic pain dies and becomes a medical examiners case, opioid levels obtained at autopsy can be misinterpreted in determining cause of death (40). In February 1999, a physician was charged with three counts of murder alleged to result from lethal oxycodone doses (41). In one case, the victim died in a motor vehicle accident, but at autopsy was found to have an oxycodone blood level of 21,900 µg/L. Assuming no significant postmortem redistribution of oxycodone, using the oxycodone IR values in Table  1) with F = 0.74,

DISCUSSION
Prescribers, in general, may be concerned that the mathematics of pharmacokinetics precludes practical application at the point-of-care. This opinion may be rooted in an assumption that if a drug is not working, more should be given or, conversely, if it is producing toxicity, less should be given. This empirical approach, although not without merit, means that learning how to use any opioid safely will entail a long an inefficient period of trial and error. Indeed, we are presently witnessing a global epidemic of opioidrelated deaths due to overdose (43,44 (47). When literature references on therapeutic ranges are not available, targeted concentration ranges should be plasma concentrations that have been observed at therapeutic doses of the drug.
We cannot expect that measured plasma opioid drug concentrations will be identical to opioid drug concentration predicted using COP, because there is always inter-and intraindividual variation, measurement error, and model misspecification error. Misspecification error is due to representing the body as a single compartment. Thus, the difference between the patient and the model (interindividual variability), and the difference between the patient's "true" value, if it could be known, and the measured concentration (residual intraindividual variability) will account for the differences in the COP predictions relative to the measured drug concentration.
The authors of this contribution have made every effort to ensure the accuracy of the information provided at the time of its composition. Nevertheless, it remains the responsibility of every prescriber to evaluate the appropriateness of a particular result in the context of the actual clinical situation and to consider any new developments in the field. Although the authors have been careful to make COP adhere to current standards and responsible literature, the authors recommend prescribers consult appropriate informational sources when prescribing new or unfamiliar opioid drugs.
COP was designed as a clinical-decision support tool for prescribers at the point-of-care to provide guidance for individualizing opioid drug therapy. Although COP may be used by prescribers for dosage regimen analysis, design, and modification, caution should be exercised when applying COP results. The large between-individual patient variability in responses to opioid drug administration suggests that COP will not work accurately in all cases. Uncertainty in patient opioid dosage histories and compliance with opioid dosage regimens makes all efforts at opioid dosage regimen analysis tentative at best. Moreover, the expected variability and occasional errors in the laboratory may confound proper interpretation. These sources of variability underscore the importance of using COP in conjunction with a comprehensive patient evaluation and management approach. This includes careful attention to correct diagnosis; identification of therapeutic goals and objective therapeutic end points; correct choice of pain relievers; constant assessment and reassessment of therapeutic outcomes; and, when appropriate, measurement of urine drug levels for therapeutic drug monitoring. Proper interpretation of analytical data requires not only an understanding of the circumstances surrounding a case but also an appreciation of the circumstances under which any data cited to aid interpretation were produced (48).

CONCLUSION
COP is a novel application of clinical pharmacokinetics to opioid dosing regimen design, evaluation, and modification. COP is a quantitative clinical-decision support tool providing guidance for individualizing opioid drug therapy at the point-of-care. Application of COP may prevent iatrogenic overdoses of opioids and it may reduce prescription costs by allowing patient dosing regimens to be individualized. Finally, COP may be useful in drug abuse prevention and detection (DAPD), when it is used in conjunction with a program of quantitative urine drug monitoring, as it can allow quantification of expected urine excretion of administered opioids.

FUTURE DIRECTIONS
We are currently developing software for Apple's iOS platform for point-of-care COP. We are also working on pharmacokinetic phenotyping of individual-based opioid pharmacokinetics and incorporating pharmacogenomics information into k e -pharmacokinetics and pharmacogenomics modeling.