Multilinear Calibration of Lattice Boltzmann pMDI Spray Simulation
Sorin Mitran, Anthony Hickey2 and Jay Holt3
Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
2Eshelman School of Pharmacy, University of North Carolina, Chapel Hill, NC, USA
3Cirrus Pharmaceuticals, Durham, NC, USA
KEYWORDS: pressurized metered dose inhaler (pMDI), computational fluid dynamics
(CFD), lattice Boltzmann method (LBM), spray particle statistics, multifactor analysis
INTRODUCTION
Metered dose inhalers are in widespread use as delivery systems for lung medication.
Treatment efficacy depends on particle sizes produced by the propellant release into
the pMDI nozzle [1], and subsequent entrainment in airway flows, a complicated
physico-chemical process involving droplet formation, evaporation, and interaction with
inspiration airflow. The environmentally-mandated change from chlorofluorocarbon
(CFC) propellants to hydrofluoroalkanes has increased interest in establishment of
general methods for systematic evaluation of formulation and device variables upon the
efficacy of pMDI products.
Standard approaches are semi-empirical in nature, typically based upon parametric
probability density function (PDF) estimation, establishing parameters hypothesized to
belong to some predetermined family of statistical distributions [2] by processing of
experimental results. Even if some general arguments can be made to suggest the
relevance of, say, log-normal distributions to spray formation processes dominated by
drift and diffusion [3, 4], the strongly non-equilibrium nature of pMDI spray formation
leads to significant deviation from the assumed PDF form [2]. Furthermore, the limited
number of PDF parameters (i.e. two parameters for log-normal distributions), cannot be
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assumed to capture the effect of multiple device variables (co-solvents, surfactants,
nozzle geometry) upon particle size distributions. An alternative approach is to attempt
first-principle computational simulation based upon continuum [5] or kinetic descriptions
[6, 7] of the multiphase flow processes. Though effects of all relevant device variables
could be captured by such an approach, the computational cost of conventional
implementations on central processing units (CPUs) is prohibitive. The geometric
complexity of human airways adds another level of complexity to realistic computational
models of spray formation, transport, airway surface deposition.
In this study a hybrid approach is introduced based upon:
1) A computationally efficient graphics processing unit (GPU) implementation of a
lattice Boltzmann method (LBM) for multiphase transport processes [8];
2) A multilinear multifactor statistical analysis of device performance in terms of
fundamental physical quantities characterizing the formulation.
The approach can be interpreted as a statistical calibration of parameters within an LBM
simulation, incorporating fundamental physical principles through the model formulation,
but accounting for missing information by processing of experimental results.
METHODS
The Boltzmann equation for time evolution of the PDF f (s) (t,x,v) , with f (s) (t,x,v)dxdv
signifying the probability of finding a molecule of some chemical species s at time t in
phase-space volume dxdv is given by
¶ f (s)
+ v ×Ñx f (s) + g ×Ñv f (s) = W(s ) ( f , f ) ,
¶t
2
and is discretized on a regular three-dimensional (3D) spatial lattice [9], with velocities
constrained to belong to a finite set v Î{ca ,a = 0,..,18} , oriented along lattice directions
(the D3Q19 model was used [8], and g denotes external acceleration fields). The
discrete velocity approximation leads to representation of the PDF f (s) (t,x,v) by the set
of directional PDFs { fa(s) (t,x),a = 0,..,18}. The collision integral along direction a for
species s , Wa(s) ( f , f ) involves PDFs from all species in the mixture, collectively denoted
as f , but is approximated here by a single-species multirelaxation time model [7]
.
Time advancement is carried out through the “stream and collide” algorithm [9], and
phase separation is induced by a forcing term [10] leading to the discrete evolution
equation
.
The phase separation forcing term
Sa(s )=
(ca - u)×(F(s,I ) + r g) (s,eq)
fa (r ,u)
r RT
models intermolecular forces through F I , and r ,u are the bulk density and velocity
obtained as zeroth and first moments of the velocity distribution over all species
r = å å M (s ) fa(s) , r u = å å M (s) fa(s)ca ,
s
a
s
a
with M (s) the molar mass of species s . The intermolecular force is represented as a
contribution from the non-ideal part of the equation of state and a surface tension
F(s,I ) = -Ñ(P - r RT ) + k (s) rÑÑ2 r ,
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with the surface tension given as s = k (s) ò [ n ×Ñr ] dn .
2
GPU implementation of the above LBM approach allows efficient computation of spray
formation and subsequent transport, on the order of 10-15 minutes, thus permitting
systematic exploration of the parameter space.
DATA ANALYSIS
The above LBM model contains parameters that capture molecular-level interactions
(s )
between species present in the formulation such as the relaxation parameters Lab
or
(s)
surface tension coefficients k (s ) , parameters collectively denoted as U = {Lab
,k (s) ,...} .
These parameters could be extracted (at considerable computational cost) from
molecular dynamics simulations [11], but are, even then, subject to uncertainty due to
assumed forms of the intermolecular potentials. Rather than considering such detailed
molecular simulation, the approach taken here is to extract best fits of the parameter set
from a multifactor statistical analysis of spray particle size distributions extracted from
experimental measurements, e.g. laser diffraction analysis (Fig. 1).
The multifactor analysis is carried out based on a multilinear extension to tensor
decompositions [12] of the standard two-factor decomposition of a matrix D into a
column space U1 and a row space U 2 through the singular value decomposition (SVD),
D = U1SU2T . Let D denote a N rank tensor with N -1 the number of parameters within
the vector U . The generalization of the matrix SVD is D = Z ÄU1 ÄU2 Ä...ÄU N , with D
denoting the available data that is decomposed into a core tensor Z (typically full unlike
the diagonal matrix S arising in the SVD) that captures the interaction between the
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factor matrices U1,...,U N -1 . Each factor matrix is obtained by a SVD applied to a
flattening of the data tensor along the dimensions corresponding to a particular factor
[12]. The multilinear formalism allows disentangling the effect of individual parameters
upon the data of interest. Here, the data tensor is obtained from particle density
distributions as functions of particle diameter and time of passage at a fixed position
downstream of the nozzle (Fig. 1) for various values of the components within the
parameter vector U .
Figure 1. (Top,Left) Typical spray particle size distribution extracted from laser
diffraction analysis of a Proventil spray. (Top,Right) Synthetic particle size distribution
generated from LBM simulation. (Bottom) Density contours at nozzle exit.
CONCLUSIONS
A simulation method based upon fundamental physical principles with parameters
calibrated from multilinear, multifactor analysis of experimental spray particle size
distributions has been introduced. The mathematical model allows identification of the
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effect of molecular-level quantities (e.g. surface tension, intermolecular forces) upon
pMDI performance. The mathematical formalism has been successfully tested against
synthetic data. Work is currently in progress on processing of experimental data to allow
systematic evaluation of how pMDI formulation parameters affect device performance.
ACKNOWLEDGMENTS
This study was supported by U.S. FDA through grant 1U01FD004943-01.
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