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\begin{document}

\title{Population pharmacokinetic-pharmacodynamic model of subcutaneous
bupivacaine in a novel extended-release microparticle formulation}



\author[1]{Ida Storgaard}%
\author[2]{Elisabeth Kjær Jensen}%
\author[3]{Søren  Bøgevig}%
\author[3]{Torben Balchen}%
\author[2]{Anders Holten Springborg}%
\author[4]{Mike Allan Royal}%
\author[5]{Kirsten Møller}%
\author[2]{Mads Utke Werner}%
\author[1]{Trine Lund}%
\affil[1]{University of Copenhagen}%
\affil[2]{Department of Anesthesia Pain and Respiratory Support Neuroscience Center}%
\affil[3]{Copenhagen University Hospitals}%
\affil[4]{Liquidia Corporation Inc}%
\affil[5]{Copenhagen University Hospital-Rigshospitalet}%


\vspace{-1em}



  
  \date{May 26, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
The objective of this study was to develop a population
pharmacokinetic-pharmacodynamic model of subcutaneously administered
bupivacaine in a novel extended release microparticle formulation for
postoperative pain management. Bupivacaine was administered
subcutaneously in the lower leg to 28 healthy male subjects in doses
from 150 to 600 mg in a phase 1 randomized, placebo-controlled,
double-blind, dose-ascending study with two different compositions of
microparticle formulations called LIQ865. Population
pharmacokinetic-pharmacodynamic models were fitted to plasma
concentration-effect-time data using non-linear mixed-effects modeling.
The pharmacokinetics were best described by a two-compartment model with
biphasic absorption as two parallel absorption processes: a fast,
zero-order process and a slower, first-order process with two transit
compartments. The slow absorption process was found to be dose-dependent
and rate-limiting for bupivacaine clearance at higher doses. Bupivacaine
clearance and the transit rate constant describing the slow absorption
process both decreased with increasing doses following a power function
with a shared covariate effect of dose on the two parameters. The
pharmacokinetic-pharmacodynamic relationship between plasma
concentrations and effect was best described by a linear function. This
model gives new insight into the pharmacokinetics and pharmacodynamics
of microparticle formulations of bupivacaine, and the biphasic
absorption seen for several local anesthetics.%
\end{abstract}%



\sloppy


TITLE PAGE

Population pharmacokinetic-pharmacodynamic model of subcutaneous
bupivacaine in a novel extended-release microparticle formulation

Ida Klitzing Storgaard, M.Sc.,\textsuperscript{1} Elisabeth Kj\selectlanguage{ngerman}ær Jensen,
M.D.,\textsuperscript{2,3} Søren Bøgevig, M.D.,
Ph.D.,\textsuperscript{4} Torben Balchen,
M.Sc.,\textsuperscript{3}Anders Holten Springborg,
M.D.,\textsuperscript{2,3} Mike Allan Royal, M.D.,
J.D.,\textsuperscript{5} Kirsten Møller, M.D.,
Ph.D.,\textsuperscript{6,7} Mads Utke Werner M.D.,
Ph.D.,\textsuperscript{2,3,8} and Trine Meldgaard Lund, M.Sc.,
Ph.D.\textsuperscript{1}

\textbf{Abbreviated title:}

PopPK/PD model for subcutaneous bupivacaine

\emph{Affiliations:}

\textsuperscript{\textbf{1}} Department of Drug Design and Pharmacology,
Faculty of Health and Medical Sciences, University of Copenhagen,
DENMARK

\textsuperscript{\textbf{2}} Department of Anesthesia, Pain and
Respiratory Support, Neuroscience Center, Rigshospitalet, DENMARK

\textsuperscript{\textbf{3}} DanTrials, Zelo Phase 1 Unit, Copenhagen
University Hospitals, DENMARK

\textsuperscript{\textbf{4}} Department of Clinical Pharmacology,
Copenhagen University Hospitals, DENMARK

\textsuperscript{\textbf{5}} Liquidia Technologies, Morrisville, NC, USA

\textsuperscript{\textbf{6}} Department of Neuroanaesthesiology,
Neuroscience Center, Copenhagen University Hospital-Rigshospitalet,
Copenhagen, DENMARK

\textsuperscript{\textbf{7}} Department of Clinical Medicine, Faculty of
Health and Medical Sciences, University of Copenhagen, Copenhagen,
DENMARK

\textsuperscript{\textbf{8}} Department of Clinical Science, Lund
University Lund, SWEDEN

\textbf{Corresponding author:}

Trine Meldgaard Lund, MSc, PhD

Department of Drug Design and Pharmacology, Faculty of Health and
Medical Sciences, University of Copenhagen

Universitetsparken 2, 2100 Copenhagen O, DENMARK

Mail: trine.lund@sund.ku.dk

Phone: +45 3533 6340

\textbf{Clinical trial number and registry URL:}

EudraCT (IEC): 2016-002420-88

ClinicalTrials.gov Identifier: NCT02982889

\textbf{Keywords:}

Bupivacaine; PKPD modeling; local anesthesia; pain management;
pharmacokinetics

\textbf{Word and Element Counts:}

Abstract: 196

Total body word count: 3,550

No. of pages: 20

No. of Figures: 5

No. of Tables: 1

No. of Supplemental Digital Content: 1

Abstract

The objective of this study was to develop a population
pharmacokinetic-pharmacodynamic model of subcutaneously administered
bupivacaine in a novel extended release microparticle formulation for
postoperative pain management. Bupivacaine was administered
subcutaneously in the lower leg to 28 healthy male subjects in doses
from 150 to 600 mg in a phase 1 randomized, placebo-controlled,
double-blind, dose-ascending study with two different compositions of
microparticle formulations called LIQ865. Population
pharmacokinetic-pharmacodynamic models were fitted to plasma
concentration-effect-time data using non-linear mixed-effects modeling.
The pharmacokinetics were best described by a two-compartment model with
biphasic absorption as two parallel absorption processes: a fast,
zero-order process and a slower, first-order process with two transit
compartments. The slow absorption process was found to be dose-dependent
and rate-limiting for bupivacaine clearance at higher doses. Bupivacaine
clearance and the transit rate constant describing the slow absorption
process both decreased with increasing doses following a power function
with a shared covariate effect of dose on the two parameters. The
pharmacokinetic-pharmacodynamic relationship between plasma
concentrations and effect was best described by a linear function. This
model gives new insight into the pharmacokinetics and pharmacodynamics
of microparticle formulations of bupivacaine, and the biphasic
absorption seen for several local anesthetics.

Glossary of Terms

PK pharmacokinetic

popPK population pharmacokinetic

PD pharmacodynamic

PLGA poly-lactic-co-glycolic acid

WDT warmth detection threshold

\emph{k} \textsubscript{tr} transit rate constant

CL/F apparent clearance

OFV objective function value

GOF goodness-of-fit

\emph{C} \textsubscript{max} maximum plasma concentration

IIV inter-individual variability

CV coefficient of variance

Q/F apparent inter-compartmental clearance

V/F apparent distribution volume of the central compartment

V2/F apparent distribution volume of the peripheral compartment

\par\null

Introduction and Background

Persistent postoperative pain following surgical procedures is still
common and leads to poor outcomes and prolonged hospitalization for
patients {[}1{]}. As the number of surgical procedures is increasing
globally, there is a growing need to improve postoperative pain
management {[}1{]}. The many well-known drawbacks of using opioids for
postoperative pain treatment make opioid-sparing approaches preferable.
The vast majority of local anesthetics for postoperative interventional
pain management have an efficacy of less than 24 hours {[}2,3{]}. In
recent years, several new prolonged-release formulations of the local
anesthetic bupivacaine have been developed or are in active development,
and two have been approved by the FDA and/or by EMA {[}3{]}. These
formulations have a reported duration of pain relief between 48 and 72
hours {[}3{]}, but their impact on clinical outcomes after surgery
appears to be limited {[}4,5{]}. This study uses data from a phase 1
randomized, placebo-controlled, double-blind, dose-ascending study,
which explores two different compositions of a novel microparticle
formulation LIQ865 containing bupivacaine for extended release using
particle replication in non-wetting templates (PRINT\textsuperscript{®})
technology {[}6{]}.

Previous studies have found biphasic absorption profiles and effect
trajectories of extended duration anesthetics such as bupivacaine and
lidocaine in humans and dogs {[}7-12{]}. A commonality of this biphasic
absorption of bupivacaine is a fast initial absorption process followed
by (or parallel to) a slower absorption of the remaining drug
{[}2,7-12{]}. There is large variability between individuals in the dose
fractions absorbed by the fast and the slow absorption processes,
respectively {[}8,9{]}, which a simple average would tend to hide.
Population pharmacokinetic (popPK) modeling is, therefore, an ideal tool
to describe the PK of locally administered bupivacaine, as the
inter-individual variability can be accounted for and quantified.
Previously developed popPK and/or pharmacodynamic (PD) models of
bupivacaine were based on central neuraxial blocks {[}7,10-12{]}. The
models developed by Doherty, Simon\emph{, et al.} (2004){[}10{]} and
Olofsen\emph{, et al.} (2008){[}11{]} all described biphasic absorption
by two parallel processes but implemented in different manners to
various degrees of success in producing the biphasic concentration-time
profile. Therefore, this study seeks to develop a popPK/PD model of
bupivacaine after subcutaneous administration based on data from 28
volunteers in a first-in-human study of the novel microparticle
bupivacaine-containing formulation LIQ865. The purpose of the model is
to describe the biphasic absorption and effect profiles and to
investigate the influence of increasing doses on the pharmacokinetic
parameters.

Materials and Methods

Pharmacokinetic and Pharmacodynamic Data

The data was collected in a phase 1 randomized, placebo-controlled,
double-blind, dose-ascending clinical study (dose safety and
pharmacodynamics is described in a separate paper submitted to Basic \&
Clinical Pharmacology \& Toxicology: `Bupivacaine in a novel
extended-release micro-particle formulation (LIQ865). A phase 1,
randomized, double-blind, placebo-controlled, dose-ascending study in
male volunteers: Dose safety and pharmaco-dynamics' by Jensen EK,
Bøgevig S, Balchen T, Springborg AH, Royal MA, Storgaard IK, Lund TM,
Møller K, Werner MU). The study protocol with updated amendments was
approved by the Research Ethical Committee for the Capital Region of
Denmark (H-16040444) and the Danish Medicines Agency (2016092814) and
was registered in EudraCT (2016-002420-88) and ClinicalTrials.gov
(NCT02982889, principal investigator: Mads U. Werner, date of
registration: 06-DEC-2016) prior to study start. The study was conducted
in accordance with the Basic \& Clinical Pharmacology \& Toxicology
policy for experimental and clinical studies {[}13{]}. Written informed
consent was obtained from all subjects included in the study.

Twenty-eight healthy, non-smoking male subjects (required age 18-45
years; BMI 18.5-30.0 kg/m\textsuperscript{2};
weight\emph{\textgreater{}} 60 kg) were divided into five cohorts
receiving subcutaneous injections of bupivacaine (doses 150, 225, 300,
450, or 600 mg) as either formulation LIQ865A or LIQ865B. The
extended-release study drugs were made using the PRINT-technology,
producing hexagonal microparticles{[}6{]}. LIQ865A (formulation A)
consisted of a mix of the PRINT bupivacaine base and
poly-lactic-co-glycolic acid (PLGA) in a ratio of 55\%/45\%, while
LIQ865B (formulation B) consisted of 100\% PRINT bupivacaine base.
Subcutaneous injections were performed from two injection points with
fan-like needle trajectories in a testing area (2.5 x 5.0
cm\textsuperscript{2}) on the lower legs, with bupivacaine formulation
on one side (active drug) and a corresponding volume of diluent on the
other (control). Diluent control was used as the baseline for PD.
Aqueous 0.5\% bupivacaine was scheduled for use as an active control in
cohort 5. However, as this cohort was discontinued due to the first
patient reporting symptoms of possible systemic side effects, a
historical control utilizing data from a similarly designed study was
used instead {[}14{]}. Due to the study being first-in-man for this
formulation, group sizes were kept small.

Blood samples (129 mL in total) were taken during Day 0 at times 0, 0.5,
1.0, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, 8, 12, and 24 hours, and once daily
on Day 2-5 post-injection. Locally, blood samples were centrifuged,
separated, frozen, and stored at -80°C. The frozen samples were then
shipped on dry ice to the analysis facility (AIT Bioscience,
Indianapolis, IN), where they were analyzed by liquid
chromatography-tandem mass spectrometry (LC-MS/MS) assay with a lower
limit of detection of 2.00 \selectlanguage{greek}µ\selectlanguage{english}g/L.

Warmth detection threshold (WDT) measurement was used as a surrogate PD
endpoint in this study, and the PD assessments were made at baseline and
1, 2, 4, 6, 8, 12, 24 hours, and at Days 2-5 post-injection. The
assessments were made in triplicate at each time point on both legs. The
temperature was increased from a baseline of 32\selectlanguage{ngerman}°C using a ramp rate of
1°C/s until a sensation of warmth was perceived. The stimulus was then
terminated by the subject using a handheld stop button device. The
cut-off limit was 50°C, set to avoid skin damage. Assessment values
exceeding 50°C were assigned the value of 51°C. A higher WDT thus
indicated lower sensibility and, therefore, a higher effect.

Population Model Development, Software, and Statistics

Data management was performed in Phoenix NLME version 8.3.295 (Certara,
USA). Models were fitted to the concentration-time data using non-linear
mixed effects modeling{[}15-18{]} in Phoenix. The first-order
conditional estimation-extended least squares (FOCE-ELS) algorithm was
used during the development of the structural and stochastic model, and
the quasi-random parametric expectation maximization (QRPEM) algorithm
was used for further modeling of covariates and PK/PD model as the
models became more complex. Models were parameterized in terms of
clearances and volumes.

One-, two-, and three-compartment PK models were fitted to the
concentration-time data. Different absorption models were applied; first
order absorption, zero-order absorption, absorption with lag time,
parallel absorption from the same or two different dose points, transit
compartment models with and without an absorption compartment, and
combinations of these. Additive, proportional, and combined additive and
proportional error models were fitted, and the best was chosen.
Different combinations of random effects (inter-individual variability)
applied on fixed effects were investigated. The influences of
formulation (A or B) and dose described either as cohort (1-5), high or
low dose group, or dose as a continuous covariate (relative to the
lowest dose) were tested as covariates on relevant population
parameters. A shared covariate effect of dose on the transit rate
constant (\emph{k} \textsubscript{tr}) and clearance (CL) was also
tested and compared to a model with separate covariate effects of dose
on\emph{k} \textsubscript{tr} and CL.

After the PK model was developed, PD data was linked to plasma
concentrations through a number of different PK/PD models without
freezing PK parameters (open model) {[}19{]}. Due to the upper value of
51°C for WDT, it was explored whether Beal's M3 or M4 methods {[}20{]}
could be used to account for the censored data above the threshold.

Criteria considered for model evaluation were objective function value
(OFV) (\emph{p} \textless{} 0.05 for the structural model and
\emph{p}\textless{} 0.001 for covariate inclusion), relative standard
errors for fixed effects (preferably \textless{} 30\%), and
goodness-of-fit (GOF) plots generated in Phoenix. Shrinkage values for
random effects reflect the reliability of individual (post hoc)
parameter estimates (with lower values indicating better reliability)
and should ideally be below 25\%.

Results

Pharmacokinetic and Pharmacodynamic Data

The final dataset for PK/PD modeling contained a total of 453 data
points for PK and 335 data points for PD from 28 healthy adult male
subjects. Data points covered measurement times from 0 (pre-dose) to 120
hours after dosing. Cohorts 1 (150 mg), 2 (225 mg), and 3 (300 mg) each
had 6 subjects, 3 given each formulation A and B. Cohort 4 (450 mg) had
9 subjects, 6 given formulation A and 3 given formulation B. Cohort 5
(600 mg) had a single subject given formulation A. Cohort 5 was
discontinued after subject \#504 reported symptoms indicating local
anesthetic systemic toxicity after injection of 600 mg LIQ865A. Three
other subjects (\#501, \#502, and \#503) already included in cohort 5
were therefore reassigned to cohort 4, expanding this cohort. Other
adverse effects were minor and local to the injection sites. These
included reversible indurations lasting 4-13 weeks experienced by 5
subjects receiving LIQ865A. The mean of the control WDT measurements in
the leg injected with diluent was 36.5°C (CV = 4.5\%).

Initial evaluations of plots of plasma concentration vs. time for
individual subjects indicated the presence of a bi-phasic systemic
absorption with two peaks apparent in the majority of plasma
concentration-time profiles. Supplementary Digital Content S1 shows
plasma concentration-time profiles and effect-time (with WDT as effect
measure) profiles for cohort 4 as an example.

The first peak occurred within the first hours after administration,
while the second peak was broader and peaked around 25-50 hours after
administration. There was high variability between subjects regarding
the presence, magnitude, and dominance of each absorption process.
Clearance appeared to slow down with increasing doses, indicating either
saturation of the elimination processes or dose-dependent absorption
rates resulting in flip-flop kinetics at higher doses. The latter option
is supported by the fact that there are no previous reports of
saturation of clearance in clinically used doses of bupivacaine. Large
inter-individual variability in bupivacaine plasma concentration
profiles was observed, as subjects given the same dose had considerably
varying magnitudes of maximum plasma concentrations (\emph{C}
\textsubscript{max}). An example of this is shown in the individual
profiles for subjects \#109 and \#112 in Figure 1, where\emph{C}
\textsubscript{max} of subject \#112 is approximately 2.5 times that of
subject \#109, though both were given the lowest dose of 150 mg. In
addition, subject \#202 did not reach the same\emph{C}
\textsubscript{max} as subject \#112 despite having received the higher
dose of 225 mg.

Pharmacokinetic Model with Biphasic Absorption

For the structural model, the PK of bupivacaine was best described using
a 2-compartment model compared to a 1-compartment model
(\emph{p}\textless{} 0.001). Absorption models with lag time, fixed
numbers of transit compartments, variable numbers of transit
compartments (Stirling approximation), zero- and first-order absorption
models, and variations and combinations thereof were evaluated but were
not able to describe the bi-phasic absorption profile well. Finally, the
following two parallel absorption phases were implemented with separate
dosing points: the fast absorption process described by zero-order
kinetics with the duration \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{0}, and the slow absorption
process described by first-order kinetics with two transit compartments
and Erlang type absorption (where the transit rate constant \emph{k}
\textsubscript{tr}equals the absorption rate constant \emph{k}
\textsubscript{a}). The fraction of the dose absorbed through the fast
process was defined as the parameter Fr, while the fraction of the dose
absorbed through the slow process was defined as (1-Fr). Fr was then
limited to a value between 0 and 1. This model was able to describe the
bi-phasic absorption well and was the overall best fit as evaluated on
goodness-of-fit plots and OFV. A schematic representation of the
population PK model is shown in Figure 2.

A proportional error model was the best fit for the data in terms of GOF
plots and standard deviations. Regarding individual parameter estimates,
random effects (inter-individual variability, IIV) were applied on the
parameters CL, \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{0}, Fr, and \emph{k} \textsubscript{tr},
with shrinkages below 10\%, indicating good reliability on individual
estimates for these parameters.

Significant covariate relationships included dose (scaled to the lowest
dose of 150 mg, since this cohort appeared to have the apparent
clearance closest to clearance after an i.v. dose based on literature
{[}2{]} and initial evaluations of data) as a covariate on CL
and\emph{k} \textsubscript{tr}. Formulation (A or B) did not show any
significant covariate effect on any parameters. CL only appeared to slow
down once \emph{k} \textsubscript{tr} decreased to a value below the
elimination rate constant, and both parameters followed approximately
the same rate of decline. This pointed to flip-flop kinetics as the
cause of the decrease in CL rather than a saturation of elimination
processes. As the observed decrease in CL followed the dose-dependent
decrease in \emph{k} \textsubscript{tr}, a model with shared covariate
effect for dose on CL and \emph{k} \textsubscript{tr} was applied. The
implementation of this covariate removed the trends in the eta vs.
covariate plots as seen for CL and \emph{k} \textsubscript{tr} (Figure
3). This model was chosen as the final PK model based on GOF plots and
OFV. With this model, it is possible to estimate population values of CL
and \emph{k} \textsubscript{tr} for any chosen dose of bupivacaine (see
Table 1 and Figure 4).

Population Pharmacokinetic-pharmacodynamic Model

The following relations between plasma concentration and effect were
tested: linear, log-linear, E\textsubscript{max} model, sigmoidal
E\textsubscript{max} model, the addition of an effect compartment
(indirect models), and effect relating to the plasma concentration of
the peripheral compartment. The linear correlation between plasma
concentration and effect was the best fit for the PK/PD model. Although
methods of letting the model estimate effects at values above the
ceiling of 51\selectlanguage{ngerman}°C were explored, these did not improve the model overall,
and the linear relation parameters between plasma concentration and
effect were the same. A random effect parameter (IIV) was included on
the slope of the linear relation. The inclusion of a random effect
parameter on the baseline of the relation was tested. However, this
resulted in a high shrinkage of 49\% and a small estimated value (CV\%
\textless{} 2\%) and, therefore, this parameter was not included.
Parameter values, relative standard errors, IIV, and shrinkage for the
final PK/PD model are presented in Table 1. The population-predicted
profiles of plasma concentration and effect (WDT) over time are
presented in Figure 5. For the full model code of the final popPK/PD
model, see Supplemental Digital Content S2. For selected GOF plots of
the final popPK/PD model, see Supplemental Digital Content S3.

Discussion

The PK profiles of individual subjects showed large variability in
maximum plasma concentrations and relative amounts absorbed through the
fast and slower absorption routes, even within the same dosing cohorts
(see Figure 1). This is reflected in the IIV on the parameters
\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{0} (51.9\%) and Fr (57.4\%), describing the duration of
and the fraction absorbed through 0-order absorption, respectively. The
model is able to capture this variability relatively well in a
descriptive manner, but it complicates the prediction of the magnitude
of plasma concentrations and effects (and, therefore, side effects).
This issue could be illuminated further by a study with larger cohort
sizes than what was ethically reasonable in this first-in-human trial.
Reasons for the large variability in \emph{C} \textsubscript{max} could
be the varying degree of inflammation observed in the subjects, as
inflammation increases the blood flow in the injection area. This, in
turn, could increase both Fr and the transit rate constant\emph{k}
\textsubscript{tr}. Another reason could be the method of preparation of
the drug suspension for injection, as the homogeneity and concentration
of drug throughout the suspension could vary.

From the observed PK profiles, CL appeared to be dose-dependent with
possible flip-flop kinetics. Covariate box plots of eta on\emph{k}
\textsubscript{tr} and CL faceted on cohorts showed a trend of decreased
\emph{k} \textsubscript{tr} and CL with higher doses (Figure 3).
Therefore, it appeared more likely that \emph{k} \textsubscript{tr}was
the dose-dependent parameter while CL was dependent on\emph{k}
\textsubscript{tr} at higher doses. As\emph{k} \textsubscript{tr}
decreases and approaches the elimination rate constant \emph{k}
\textsubscript{e}, flip-flop kinetics occur, where the rate of
elimination is limited by the rate of absorption. Covariate models with
different covariate effects for dose on\emph{k} \textsubscript{tr} and
CL were investigated, along with models where a shared covariate effect
was applied, as chosen for the final PK model. The dose-dependence of
the absorption rate could be due to saturable absorption processes or
properties of the formulation, e.g., longer diffusion distances through
the formulation before absorption is possible. With this covariate
model, it is possible to estimate CL and\emph{k} \textsubscript{tr} for
a given dose using the values in Table 1. However, as the present study
was first-in-human, the subjects were all male and of a limited weight
range, and extrapolating the model to real-world patients should be done
with caution.

The absorption of bupivacaine was best described by two simultaneous
absorption processes: a 0-order absorption process, functionally similar
to an infusion (the fast phase), and a first-order absorption with two
transit compartments with Erlang type absorption (the slower phase).
This combined absorption model was able to describe the individual PK
profiles well. A possible explanation for the simultaneous phases may be
found in the microparticle suspension formulation. A fraction of drug
molecules may be readily available for systemic absorption upon
administration, while the remainder must diffuse through the formulation
and diluent first. The type of formulation, A or B, did not show a
significant covariate effect on any parameters, although increased signs
of local inflammation were observed in subjects given the formulation
with PLGA.

Subject \#504 was the only subject given a dose of 600 mg, as the dosing
group was discontinued after this subject experienced suspected systemic
side effects including dizziness, light-headedness, pricking
paresthesia, and numbness of the tongue. These were treated as suspected
unexpected serious adverse reactions suggestive of local anesthetic
systemic toxicity. These side effects followed a biphasic trajectory, as
was seen for the PK profiles and effects in general. Several subjects
experienced a period with some recovery of sensory function between two
periods of complete local anesthesia. This breakthrough sensitivity
should be considered when using extended-release local formulations of
local anesthetics like the ones in this study.

Intravenous (i.v.) PK data were not obtained in this study, but a review
by Heppolette\emph{, et al.} (2020){[}2{]} found bupivacaine clearance
after i.v. dosing to be between 27.9 and 39 L/h. The apparent clearance
found in this study of 35.4 L/h (CV = 6.3\%, IIV = 24.5\%) for the low
dose group is in the upper end of this interval, indicating high
bioavailability. Therefore, it seems likely that the CL found for the
low dose group is either not affected or only affected to a negligible
degree by flip-flop kinetics. The apparent volume of distribution of the
central compartment (V/F) of 347 L found in this study is relatively
high, as literature values for distribution volumes are generally below
200 L {[}2{]}. Distribution volumes in literature are found following
i.v., epidural, peripheral nerve block (femoral or sciatic)
administration, and not subcutaneous administration as in this study
{[}2{]}. This study found a 2-compartment model to be the best fit for
the data; Olofsen\emph{, et al.} (2008){[}11{]} likewise describe a
model with distribution to peripheral compartments. Overall, the model
is structurally similar to previously developed models of epidurally
administered bupivacaine, with some exceptions, particularly the
structure of the absorption model, which is due to the different
administration routes and formulations of bupivacaine.

In the PK/PD model, a linear relationship between plasma concentration
and effect (measured as WDT) best described the data. There was a
ceiling effect for the PD data, as applying higher temperatures than the
maximum of 50\selectlanguage{ngerman}°C would have carried the risk of skin damage. The ceiling
effect influences the quality of effect predictions using the linear
relation, and the variability of the linear relationship is also
reflected in the relatively high IIV on the slope (56.7\%). The
intercept (baseline) corresponded to the mean of the PD control
measurements. Schnider\emph{, et al.} (1996){[}12{]} found an
E\textsubscript{max} type model to be the best fit for PD data, but this
model type did not provide a better fit in this study. Interestingly,
the local anesthetic effect seems to follow systemic plasma
concentrations. This indicates that the plasma concentration
proportionally is proportional to the drug concentration at the effect
site, with no observable lag in the distribution between plasma and
effect site.

In conclusion, we developed a population PK/PD model to describe the
biphasic systemic absorption of subcutaneously administrated bupivacaine
in a microparticle formulation by two parallel absorption processes: a
fast process with 0-order absorption and a slow process with two transit
compartments and first-order, Erlang-type absorption. The model had two
compartments with first-order elimination and a linear relation between
plasma concentration and effect measured as warmth detection threshold.
The highest inter-individual variability (\textgreater{}50\%) was seen
for the duration of the fast absorption process, the fraction of dose
absorbed by the fast process, and the slope of the linear relation
between plasma concentration and effect. The slow absorption process was
dose-dependent and limited apparent clearance (CL/F) at larger doses
(flip-flop kinetics), as seen by implementing dose relative to the
lowest dose of 150 mg as a covariate on the transit rate constant
(\emph{k} \textsubscript{tr}) and CL/F with shared covariate effect.
With this model, \emph{k} \textsubscript{tr} and CL/F can be estimated
for any dose by multiplying the population estimates of 0.117
h\textsuperscript{-1} and 35.4 L/h, respectively, with the factor
(Dose/150 mg)\textsuperscript{-0.43}. This model may be used for further
development of extended-release local anesthetics with caution given to
the large variability on certain parameters and the homogeneity of the
subject population.

Acknowledgements

\textbf{Financial disclosures/Funding statement:}

The study was financed by Liquidia Technologies, Morrisville, NC, USA.

\textbf{Conflicts of interest:}

The authors declare no competing interests.

\textbf{Data availability statement:}

Research data are not shared due to confidentiality.

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Table 1. Population parameter values for the intermediate (int.)
population pharmacokinetic model, the population pharmacodynamic model,
and the final population pharmacokinetic-pharmacodynamic model for
bupivacaine. Inter-individual variability (IIV) and shrinkage are shown
for the final model.\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
\emph{Parameter} & \emph{Description} &
\emph{Estimate\textsubscript{int.} (CV\%)} &
\emph{Estimate\textsubscript{final} (CV\%)} & \emph{IIV{[}\%{]} (CV\%)}
& \emph{Shrinkage}\tabularnewline
\midrule
\endhead
\emph{k}\textsubscript{tr}\textsuperscript{a}
{[}h\textsuperscript{-1}{]} & Transit rate constant for a 150 mg dose &
0.114 (12) & 0.117 (13) & 31.2 (33) & 9.7\%\tabularnewline
\(\tau_{0}\) {[}h{]} & Duration of 0-order absorption process &
1.33 (15) & 1.24 (14) & 51.9 (30) & 6.4\%\tabularnewline
Fr & Fraction of dose absorbed through 0-order process & 0.270 (15) &
0.318 (13) & 57.4 (27) & 0.99\%\tabularnewline
CL/F\textsuperscript{a} {[}L/h{]} & Apparent total clearance for a 150
mg dose & 35.2 (8) & 35.4 (6.8) & 25.3 (29) & 3.3\%\tabularnewline
Q/F {[}L/h{]} & Apparent intercompartmental clearance & 58.2 (51) & 65.1
(47) & - & -\tabularnewline
V/F {[}L{]} & Apparent distribution volume of the central compartment &
274 (7.1) & 347 (6.3) & - & -\tabularnewline
V2/F {[}L{]} & Apparent distribution volume of the peripheral
compartment & 154 (16) & 144 (16) & - & -\tabularnewline
\emph{k}\textsubscript{tr}CL\textsubscript{Dose}\textsuperscript{a} &
Covariate effect of dose on \emph{k}\textsubscript{tr} and CL & -0.473
(21) & -0.430 (24) & - & -\tabularnewline
BL {[}\selectlanguage{ngerman}°C{]} & Baseline effect; intercept of the linear correlation
between plasma concentration and effect & 36.7 (0.5) & 36.7 (0.7) & - &
-\tabularnewline
SL {[}\selectlanguage{ngerman}°C\selectlanguage{english}[?]L\selectlanguage{english}[?]\selectlanguage{greek}µ\selectlanguage{english}g\textsuperscript{-1}{]} & Slope of linear correlation
between plasma concentration and effect & 0.0421 (11) & 0.0418 (13) &
56.7 (29) & 3.5\%\tabularnewline
ERR\textsubscript{C} & Proportional error for the observed plasma
concentrations & 0.195 (3.6) & 0.204 (4.0) & - & -\tabularnewline
ERR\textsubscript{E} & Proportional error for the observed effect
measures & 0.0548 (4.1) & 0.0550 (4.9) & - & -\tabularnewline
\emph{Secondary parameters} & \emph{Secondary parameters} &
\emph{Secondary parameters} & \emph{Secondary parameters} &
\emph{Secondary parameters} &\tabularnewline
\emph{k}\textsubscript{e} {[}h\textsuperscript{-1}{]} & Elimination rate
constant = CL / V & - & 0.102 (8.6) & - & -\tabularnewline
MTT {[}h{]} & Mean transit time = 2 /
\emph{k\textsubscript{t}}\textsubscript{r} & - & 17.1 (13) & - &
-\tabularnewline
t\selectlanguage{ngerman}\textsubscript{½} {[}h{]} & Elimination half-life = ln(2) /
\emph{k}\textsubscript{e} & - & 6.80 (8.6) & - & -\tabularnewline
\emph{k}\textsubscript{0} {[}h\textsuperscript{-1}{]} & 0-order
absorption rate constant = 1 / \(\tau_{0}\) & - & 0.808 (14) & - &
-\tabularnewline
\bottomrule
\end{longtable}

\textsuperscript{\textbf{a}} \selectlanguage{ngerman}\textbf{The model estimates CL/F
and\emph{k}\textsubscript{tr} for a dose of 150 mg (the reference). CL/F
and \emph{k}\textsubscript{tr} for other doses are calculated using the
covariate effect (\emph{k}\textsubscript{tr}CL\textsubscript{Dose}):
CL/F = 35.4 L/h × (Dose/150 mg)\textsuperscript{-0.430}
and\emph{k}\textsubscript{tr} = 0.117 h\textsuperscript{-1} × (Dose/150
mg)\textsuperscript{-0.430}}

Figure legends

Fig. 1. Selected individual profiles of individually predicted (dashed
lines) and observed values (dots) for plasma concentrations and effect
(warmth detection threshold), respectively, over the first 50 hours
after dose. Subjects \#109 and \#112 showed large variation in
bupivacaine plasma levels even with the same dose, and subject \#202 had
lower plasma levels than \#112 despite the higher dose. This phenomenon
was present in all cohorts with more than one subject. Note the
inter-individual variation in the presence and dominance of the fast and
slow absorption peaks.

Fig. 2. Schematic representation of the population
pharmacokinetic-pharmacodynamic model for bupivacaine. The fraction of
total dose absorbed through the fast process (Fr) is limited to a value
between 0 and 1. \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{0} is the duration of the 0-order
absorption process (the fast phase),\emph{k} \textsubscript{tr} is the
transit rate constant, Q/F is the intercompartmental clearance, CL/F is
the apparent clearance, V/F is the apparent distribution volume of the
central compartment, and V2/F is the apparent distribution volume of the
peripheral compartment.

Fig. 3. Boxplots of inter-individual variability for apparent clearance
(CL/F, blue) and transit rate constant (\emph{k} \textsubscript{tr},
green) by dose for the base PK model (left) and the final PK model
(right). As the final model accounts for the influence of dose on CL/F
and \emph{k} \textsubscript{tr}, the trend in the inter-individual
variability disappears.

Fig. 4. Apparent clearance (CL/F) as a function of dose, as predicted by
the model. Points (formulation A) and triangles (formulation B) are
individually estimated values of CL/F for the subjects in this study.
Diamonds are means of the individually estimated values of CL/F and
error bars represent standard deviation. The solid line represents the
function CL/F = 35.4 L/h \selectlanguage{ngerman}× (Dose/150 mg)\textsuperscript{-0.430} (see
Table 1).

Fig. 5. Population predicted profiles of plasma concentration (left) and
effect as warmth detection threshold (right) over time after dose in
hours. Solid lines represent population predicted values, dashed lines
are individually predicted values, and points represent observed values.\selectlanguage{english}
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