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EVALUATION OF PROBABILITY DENSITY FUNCTIONS TO APPROXIMATE PARTICLE SIZE DISTRIBUTIONS

*Present address:Advanced Inhalation Research,Inc.,840Memorial Drive,Cambridge,MA 02139,U.S.A.R For convenience,the dispersed phase of the aerosol is referred to as particles throughout the

text.

J .Aerosol Sci .Vol.31,No.7,pp.813}831,2000 2000Elsevier Science Ltd.All rights reserved

Printed in Great Britain

0021-8502/00/$-see front matter

PII:S0021-8502(99)00557-1

EVALUATION OF PROBABILITY DENSITY FUNCTIONS TO APPROXIMATE PARTICLE SIZE DISTRIBUTIONS OF REPRESENTATIVE PHARMACEUTICAL AEROSOLS

Craig A.Dunbar *and Anthony J.Hickey

Dispersed Systems Laboratory,School of Pharmacy,Beard Hall CB C 7360,UNC-CH,Chapel Hill,

NC 27599-7360,U.S.A.

(First received 7January 1999)

Abstract *The purpose of this work was to evaluate the application of probability density functions (PDFs)and curve-"tting methods to approximate particle size distributions emitted from four pharmaceutical aerosol systems characterized using standard methods.The aerosols were produced by a nebulizer,pressurized metered dose inhaler (pMDI),dry powder inhaler (DPI)and nasal spray.PDFs selected for analysis were (i)log-normal,(ii)upper-limit,(iii)Nukiyama }Tanasawa (iv)Rosin }Rammler and (v)modi "ed Rosin }Rammler.Two curve-"tting methods were used to estimate the adjustable parameters of the PDFs:linear least-squares "t of the cumulative distribution function (Method A)and non-linear least-squares "t of the probability density function (Method B).Large truncation of the pMDI and DPI particle size distributions obtained by cascade impaction resulted in poor "ts of the PDFs.The nebulizer and nasal spray were not a !ected by truncation and were well represented by the Rosin }Rammler and log-normal PDFs,respectively (Method B only).Probability distribution functions (Method B)were "tted without bias from linear coordinate transformation and produced signi "cantly better "ts than Method A for each aerosol system (p (0.05).Considerable caution must be used when estimating representative parameters from cumulative or probability distributions. 2000Elsevier Science Ltd.All rights reserved

INTRODUCTION

The log-normal probability density function has been used extensively to represent particle size distributions of aerosols.R This may be justi "ed by the observation of a close approxi-mation of empirical size distributions and presentation of the function in a convenient mathematical form for graphical analysis (Raabe,1971;Hinds,1982).However,the assump-tion that the particle size distributions of pharmaceutical aerosols can be accurately and exclusively represented by the log-normal probability density function has not been re-ported.The purpose of this work was to evaluate various probability density functions and curve-"tting methods to approximate particle size distributions produced by representative pharmaceutical aerosol systems characterized by pharmacopoeial and regulatory methods.The complex nature of atomization and dispersion produces a distribution of particle sizes.Graphical description of the particle size distribution is obtained by dividing the range of sizes into suitable intervals and producing a characteristic distribution or histogram.The ordinate can represent any size characteristic of the particles,e.g.,number,volume,mass,etc.The mass of particles is directly related to the dose being delivered (mass of drug),which is of speci "c interest for the characterization of pharmaceutical aerosols.Normalizing the ordinate by the total mass of particles (M)yields the relative mass distribution.The probability distribution is obtained by dividing the relative mass by the particle size interval ( D G

):

p K G "m G M 1 D

G

,(1)

813

where p K

G is the probability density of the i th interval (with units of m \ ).A smooth curve

can be drawn through the data at the size interval mid-points to give the probability distribution curve.The probability that a selected mass of particles will be less or greater than a particular size can be calculated from the area under the curve.Integration of the probability distribution yields the cumulative mass distribution;

F K

G "

"G

p (D )K

d D ,

(2)

where F K G is the mass fraction of particles with diameters less than D G

.

Particle size distributions can be represented in mathematical form by probability density functions (PDF),or by integration of the PDF,cumulative distribution functions (CDF).Probability density functions are expressed in terms of an independent variable (D )and two or more adjustable parameters representing particle size (

)and the distribution of particle

sizes (

);

p (D )"f (D ; ,

).

(3)

A complete statistical description of the particle size distribution can be obtained by de "nition of suitable adjustable parameters.Estimation of the adjustable parameters by graphical analysis and calculation of other representative size and distribution parameters are advantages of mathematical representation of particle size distributions.Description of the distribution by a single parameter set can only be applied to distributions with strong central tendencies,i.e.,data that falls under a single peak or mode.A fundamental description of the complex atomization processes producing the distribution of particle sizes has not been developed.The majority of probability density functions describing aerosol particle size distributions have,therefore,been derived empirically or mathemat-ically.

Four general categories of device are used for the delivery of aerosols to the respiratory tract.These are the nebulizer,pressurized metered dose inhaler (pMDI),dry powder inhaler (DPI)and nasal spray.The nebulizer,pMDI and DPI produce particles that are suitable for drug delivery to the lungs,characterized by a mass median aerodynamic diameter (MMAD)(5 m (Task Group on Lung Dynamics,1966;Heyder et al .,1986).The nasal spray delivers particles to the nasal mucosa,producing a mass median diameter '20 m (Task Group on Lung Dynamics,1966;Dunbar et al .,1998).Standard pharmaceutical methods for characterizing the particle size distributions produced by these devices include cascade impaction (nebulizer,pMDI and DPI)and laser di !raction (nasal spray).Graphical inter-pretation of cascade impaction data is based on the assumption that the particle size distribution may be represented as a log-normal function "tted to the cumulative mass distribution (Graseby Andersen,1985;Hinds,1986;USP 16012,1999).

PROBABILITY DENSITY FUNCTIONS

Five probability density functions have been selected to characterize the wide range of aerosol particle size distributions selected for analysis.These are:(1)log-normal,(2)upper limit,(3)Nukiyama }Tanasawa,(4)Rosin }Rammler,and (5)modi "ed Rosin }Rammler.The probability density functions,cumulative distribution functions,and the linear transforma-tion of the CDF are summarized for each distribution in Table 1.Derivation of the linear transformations are presented in the appendix.

The log-normal and upper-limit probability density functions were developed from the normal distribution.The general form of the normal mass probability density function is given as

p (z )K "d F K d z "

1(exp ! z

2

,(4)

814 C.A.Dunbar and A.J.Hickey

T a b l e 1.S u m m a r y o f p r o b a b i l i t y d e n s i t y f u n c t i o n s

L i n e a r t r a n s f o r m a t i o n o f C F F (y "m x #c )

D i s t r i b u t i o n

P r o b a b i l i t y d e n s i t y C u m u l a t i v e d i s t r i b u t i o n f u n c t i o n

f u n c t i o n

y

x m c

L o g -n o r m a l

1 e x p !z 2

z " y ;y "l n

D

D

K

; "l n \

1

(

X \

e x p !z 2

d z l n (D ) \ (F K )l n (

)l n (D K

)U p p e r l i m i t

1 e x p !z 2

z " y ;y "l n

a D D

!D

; "l n \ 1

( X \

e x p !

z

2

d z l n D D !D

\ (F K )l n (

)l n

D K D !D K

N u k i y a m a }T a n a s a w a

b B (6/ )D e x p !+b D B ,1! (6/ ,b D B ) (6/ )

Q \ 1!F K 12

D B 0.5

12

D B K

R o s i n }R a m m l e r

q D O \ X O e x p !D X

O 1!e x p !

D X O

l n l n

11!F K l n (D )

q

!q l n (X )

M o d i "e d R o s i n }R a m m l e r

q D l n O \ D l n O X e x p ! l n D l n X

O 1!e x p !

l n D l n X

O

l n l n 11!F K

l n l n (D )

q

!q l n l n (X )

Evaluation of probability density functions

815

where

z " y ,y "D !D K

, " \ ,

(5)

D K

is the geometric mass mean diameter of the distribution and is the standard deviation.Since the normal distribution is symmetrical,D K

can be replaced by the mass median

diameter (D K

).The independent variable of the log-normal distribution,y ,is de "ned as

y "ln

D D

K

(6)

and the distribution parameter, ,

"ln \

(7)

where

is the geometric standard deviation.The log-normal distribution has gained wide

acceptance for characterizing aerosol particle size distributions due to its relatively conve-nient form for graphical analysis and the facility with which any representative diameter may be calculated by transforming the expressions (Hatch and Choate,1929;Hinds,1986).Mugele and Evans proposed a modi "cation to the log-normal distribution by introduc-ing a maximum diameter,D

,into the de "nition of the independent variable,as follows:

y "ln

aD

D

!D

,

(8)

where a is a dimensionless constant related to dispersion (Mugele and Evans,1953);

a "

D K D

!D K

.(9)

The maximum diameter was introduced to emphasize the existence of a maximum diameter

rather than assume that extremely large diameters occurred with very low frequencies.An expression for D

was developed from the geometry of the cumulative distribution,

D D K "D K (D K #D K )!2D K D K D K !D K D K

,(10)

where D K and D K

are the 10th and 90th percentiles of the cumulative mass distribu-tion,respectively (Mugele and Evans,1953).

Nukiyama and Tanasawa developed an empirically based PDF to characterize the particle sizes produced from twin-#uid atomizers,given by

p (D )"

b B

D exp !+bD B ,,(11)

where b (de "ned by equation (A11)in the appendix)and are the size and distribution parameters,respectively (Nukiyama and Tanasawa,1939).The complete gamma function of the argument,x ,is de "ned by the integral;

(x )"

t V \ exp \R d t ,(12)

where t is the variable of integration.

An empirical function was developed to characterize the size distribution of powdered coal,given by

p (D )K "qD O \ X O exp !

D X

O

(13)

816 C.A.Dunbar and A.J.Hickey

Table 2.Description of pharmaceutical aerosol system components

Device Formulation Speci "cations

Manufacturer Nebulizer

Sodium chloride H 0.9%w/v NaCl

CIS-US,Inc.,Distilled water 40psig operating pressure Bedford,MA

5ml volume

Albuterol sulphate 63 l metering valve Allen and Hanburys Ltd.,pMDI

CFC 11/12

50 g per actuation

Uxbridge,U.K.

Oleic acid (surfactant)CFC 11/12(28:72%w/w)10%w/w surfactant/drug DPI Fluticasone propionate (FP)10 g mg \ FP/lactose Glaxo Wellcome,Inc.,Lactose monohydrate 250 g per actuation RTP,NC Nasal spray

Distilled water

63 l metering valve Valois of America,PR232actuator

Greenwich,CN

H Non-therapeutic model compound.

where X and q are the size and distribution parameters,respectively (Rosin and Rammler,1933).This distribution has been widely used in the "eld of atomization and sprays (Lefebvre,1989).The Rosin }Rammler distribution has a simple mathematical form suitable for graphical analysis and calculation of other representative diameters.

Analysis of size distribution data produced by pressure-swirl atomizers revealed that a modi "ed form of the Rosin }Rammler PDF provided a better "t to the experimental data,capturing the deviation produced by the larger particle diameters (Rizk and Lefebvre,1985).The modi "ed Rosin }Rammler PDF is given by

p (D )K "q D ln O \ D ln O X exp ! ln D ln X

O

.

(14)

EXPERIMENTAL METHODS

Speci "cations of the four pharmaceutical aerosol systems are summarized in Table 2.The

nebulizer,pMDI and DPI probability distribution data were obtained from previous studies using an Andersen eight-stage,non-viable,ambient sampler (Graseby-Andersen,Smyrna,GA)(Dunbar and Hickey,1999;Fults et al .,1991;Dunbar et al .,1999).The independent variable used to characterize the particle size distributions obtained by inertial impaction was the e !ective cut-o !diameter (ECD).Cumulative and probability mass distribution calculation methods are summarized in Table 3.The Andersen ambient sampler included a preseparator,that was used to prevent overloading of the upper stages,and an induction port.The ECDs for the preseparator and induction port are unreliable due to their broad collection e $ciency curves and cannot be included as discrete points in the cumulative and probability distributions (Graseby Andersen,1985;Hinds,1986).Consequently,the mass collected in the preseparator and induction port were combined with the mass collected on the "rst stage (stage 0)and the upper-limit ECD de "ned by the ECD for stage 0,as shown in Table 3.Sampling from the nebulizer and pMDI aerosols was performed at the impactors 'calibrated #ow rate of 28.3l min \ .The DPI aerosol was sampled at an air #ow rate of 60l min \ and the e !ective cut-o !diameters of the impactor stages recalculated (Van Oort et al .,1996).The ECDs of the individual impaction stages are shown in Table 3at the two sampling air #ow rates.Particle size distribution data for the nasal spray was obtained from previous studies using laser di !raction (Mastersizer-S,Malvern Instruments,Malvern,UK)(Dunbar et al .,1998).The particle size distribution generated by the nasal spray pump was larger than the operating sampling range for the inertial impactor.

Evaluation of probability density functions 817

T a b l e 3.C a l c u l a t i o n o f c u m u l a t i v e a n d p r o b a b i l i t y m a s s d i s t r i b u t i o n s f r o m i n e r t i a l i m p a c t i o n d a t a (D H "e !e c t i v e c u t -o !d i a m e t e r (E C D )o f t h e j t h s t a g e ;E C D s a t 60l m i n \ i n p a r e n t h e s e s )

S t a g e /a c c e s s o r y

D H ( m )D H

r a n g e ( m )(D H \ #D H )/2( m )

D H "D H \ !D H ( m )M a s s (m g )

F (D H )p ((D H \ #D H )/2)( m \ )

D e v i c e ****m

**I n d u c t i o n p o r t ****m **P r e s e p a r a t o r ***

*

m

*

*

S t a g e 0(j "0)

9.0(6.2)('9.06.2)

*

*

m

1!1M #G G

m G *

S t a g e 1(j "1)5.8(4.0)9.0}5.8(6.2}4.0)7.4

(5.1)

3.2

(2.2)

m

1!1M #G G m G m M #; D

S t a g e 2(j "2)4.7(3.2)5.8}4.7(4.0}3.2)5.3

(3.6)

1.1

(0.8)

m

1!1M #G

G

m G m

M #; D

S t a g e 3(j "3)3.3(2.3)4.7}3.3(3.2}2.3)4.0

(2.8)

1.4

(0.7)

m

1!1M #G G

m G m

M #; D

S t a g e 4(j "4)2.1(1.4)3.3}2.1(2.3}1.4)2.7

(1.85)

1.2

(0.9)

m

1!1M #G G

m G m

M #; D

S t a g e 5(j "5)1.1(0.8)2.1}1.1(1.4}0.8)1.6

(1.1)

1.0

(0.6)

m

1!1M #G G m G m M #; D

S t a g e 6(j "6)0.7(0.5)1.1}0.7(0.8}0.5)0.9

(0.65)

0.4

(0.3)

m

1!1M #G

G

m G m M #; D

S t a g e 7(j "7)0.4(0.3)0.7}0.4(0.5}0.3)0.55

(0.4)

0.3

(0.2)

m

1!1M #G

G

m G m M #; D

F i l t e r (j "8)0.0(0.0)0.4}0.0(0.3}0.0)0.2

(0.15)

0.4

(0.3)m

1!1M #G

G m G m M #; D

M #"G G

m G 818

C.A.Dunbar and A.J.Hickey

S D

K

represents the mass median aerodynamic diameter (MMAD)for data obtained by inertial impaction.

Representative parameters were estimated from the experimental cumulative mass distri-butions and were de "ned as the mass median diameter,S D K

,and relative span factor, .

The relative span factor provides a measure of the width of the distribution and is de "ned as

"D K !D

K D K

.(15)Other parameters calculated from the experimental data to characterize the particle size probability distributions were "ne particle fraction (FPF)and large particle fraction (LPF)of the total dose.Fine particle fraction of the total dose was de "ned as the ratio of particle mass (ECD $.$to total particle mass,where ECD $.$

is an e !ective cut-o !diameter of the

order of 5 m that is dependent on the measurement instrument (see Table 3).FPF provides an in v itro estimate of the fraction of particles that could be delivered to the lungs.The ECD $.$

for calculation of the "ne particle fractions were 5.8 m (nebulizer and pMDI),

6.2 m (DPI)and 5.2 m (nasal spray).Large particle fraction,calculated for the nasal spray only,was de "ned as the ratio of particle mass 'lower-limit diameter to total particle mass,where the lower-limit diameter is of the order of 20 m (Dunbar et al .,1998).LPF provides an in v itro estimate of the fraction of particles that could be delivered to the nasal passages.The lower-limit diameter for calculation of the large particle fraction was 20.8 m.

STATISTICAL ANALYSIS

The adjustable parameters for each probability density function were estimated by two methods.Method A used a linear least-squares "t of the cumulative distribution function to the particle size cumulative mass distribution.Method B used a non-linear least-squares "t of the probability density function to the particle size probability mass distribution.Calculation of the adjustable parameters using Method A was performed using a commer-cial spreadsheet package (Excel-97,Microsoft Corp.,Seattle,WA).Details of the linear transformation derivation of the cumulative distribution functions are given in the appen-dix.Non-linear least-squares "t of the PDFs (Method B)was performed using a commercial data "tting software package (Scientist v.2,MicroMath Scienti "c Software,Salt Lake City,UT).Representative parameters of the distributions were de "ned as the mass median diameter,D K

,and relative span factor, .

There is no absolute measure of the `goodness of "t a of a mathematical model to an experimental data set.A single model has an in "nite range of adjustable parameters,precluding the determination of an absolute measure of `goodness of "t a .However,it is possible to quantify the `likelihood a that the experimental data set could have occurred for selected adjustable parameters.Therefore,a statistical measure of the `goodness of "t a can be estimated.Agreement between the empirical data and probability density functions was measured using a maximum likelihood estimate based on the Akaike information criterion (AIC)(Akaike,1974).The maximum likelihood estimate used was the model selection criterion (MSC),de "ned as

MSC "ln

L G w G (p G !p )

L G w G (p G !p

G )

!2M N ,(16)where w G

is the weight of i th data point,M is the number of adjustable parameters and N the

number of data points (Scientist Users Manual,1994).The weights of the data points were set to unity by assuming that the measurements errors were independent and normally distributed with constant standard deviation.An increase in the MSC value represented an improvement in the `goodness of "t a .

Evaluation of probability density functions 819

Fig.1.Nebulizer probability distributions.Adjustable parameters estimated from (A)linear least-squares "t of CDF and (B)non-linear least-squares "t of PDF.PDF:**L-N;**U-L;}}}R-R;

-----MR-R;}-}-NT;*experimental data.

Table 4.Nebulizer average representative parameters and MSC values estimated from PDFs using

Methods A and B (%rsd in parentheses;n "3)Nebulizer

Method A

Method B Distribution D K ( m)

R MSC D K ( m) MSC L-N 1.0(4.3) 2.29(9.6)0.983(0.1) 1.69(35.1) 1.2(4.9) 2.68(3.2) 2.68(12.5)U-L 1.2(4.3) 1.65(4.2)0.999(0.1)* 1.2(3.8) 1.63(4.7)*N-T 1.0(11.7) 2.59(6.2)0.991(0.1)0.99(60.3) 1.2(4.9) 2.37(2.5) 3.14(13.8)R-R 1.2(4.6) 2.42(5.3)0.958(0.7) 1.65(20.9) 1.2(4.1) 1.76(1.0) 6.27(20.0)MR-R

1.1(1.2)

0.93(19.5)

0.976(1.0)

*

1.3(

2.8)

2.13(8.7)

*

Overall comparisons of the `goodness of "t a of the PDFs were made using an ANOVA model with an -level "https://www.doczj.com/doc/fc3352734.html,parisons between individual pairs were performed using a pairwise two sample t -test (Excel-97,Microsoft Corp.,Seattle,WA).Multiple pairwise comparisons were analyzed using the Tukey }Kramer method,which was an exact -level test for the equal sample sizes used in this study (JMP,SAS Institute,Inc.,Cary,NC).

RESULTS

Nebulizer

Probability distributions for the nebulizer are shown in Figs 1(A)and (B).Adjustable parameters were estimated from Method A using linear least-squares "t of the CDF,and Method B using nonlinear least-squares "t of the PDF.Average representative parameters obtained from the respective distributions are shown in Table 4with the MSC values.Also included in Table 4are the correlation coe $cients obtained from the linear least-squares "t of the CDF.MSC values for the upper-limit and modi "ed Rosin }Rammler PDFs could not

820 C.A.Dunbar and A.J.Hickey

Fig.2.Multiple comparisons of nebulizer average MSC values produced by PDFs (data points,

means and error bars shown,n "3).

be calculated due to truncation of the distributions,i.e.the PDFs did not "t the whole experimental data set.The modi "ed Rosin }Rammler PDF was truncated at D (1 m.The upper-limit PDF was truncated at D 'D

.

Individual pairwise comparisons were performed to determine signi "cant di !erences between means obtained by Methods A and B for a given PDF.The MSC values calculated for Method B were signi "cantly higher than the values calculated by Method A for the respective probability density functions (p (0.05).Multiple pairwise comparisons of the MSC values were performed to determine the rank-order of the `goodness of "t a for each estimation method and probability density https://www.doczj.com/doc/fc3352734.html,parisons of the MSC values are shown in Fig.2for the two estimation methods.The rank-order of the MSC values obtained using Method A was log-normal (1.69),Rosin }Rammler (1.65),Nukiyama }Tanasawa (0.99)(p '0.05).The rank order of MSC values obtained using Method B was Rosin }Rammler (6.27)'Nukiyama }Tanasawa (3.14),log-normal (2.68)(p (0.05).The best "t to the experimental data was obtained by the Rosin }Rammler PDF using Method B (MSC "6.27).Pressurized metered dose inhaler

Probability distributions for the pMDI are shown in Figs 3(A)and (B),with the average representative parameters and MSC values shown in Table 5.Representative parameters could not be estimated for the upper-limit PDF using Method A because a value for D

could not be obtained from the experimental cumulative distribution,which was truncated at approximately the 50th percentile.The relative span factor for the upper-limit PDF using Method B could not be calculated due to negative estimates of D

.MSC

values for the modi "ed Rosin }Rammler PDFs could not be calculated due to truncation of the distribution at D (1 m.

The MSC values calculated for Method B were signi "cantly higher than the values calculated by Method A for each probability density function (p (0.05).Multiple pairwise comparisons of the MSC values obtained by Methods A and B are shown in Fig.4.There was no statistically signi "cant di !erence between the MSC values for the log-normal (MSC "0.05)and Nukiyama }Tanasawa (MSC "0.10)PDFs using Method A (p '0.05).The log-normal and Nukiyama }Tanasawa PDFs produced signi "cantly better "ts to the data than Rosin }Rammler (p (0.05).The rank order of MSC values obtained from

Evaluation of probability density functions 821

Fig.3.pMDI probability distributions.Adjustable parameters estimated from (A)linear least-squares "t of CDF and (B)non-linear least-squares "t of PDF.PDF:**L-N;**U-L;}}}R-R;

-----MR-R;}-}-NT;*experimental data.

Table 5.pMDI average representative parameters and MSC values estimated from PDFs using

Methods A and B (%rsd in parentheses;n "3)pMDI

Method A

Method B Distribution D K ( m)

R MSC D K ( m) MSC L-N 4.7(2.0) 2.93(2.1)0.869(11.6)!0.05(451.7) 3.1(3.7) 2.22(5.7)0.23(55.8)U-L **** 2.9(4.2)*!0.05N-T 5.2(2.9) 3.52(23.2)0.979(0.8)!0.10(141.0) 3.2(5.1) 2.41(9.9)0.12(82.4)R-R 5.1(2.9) 1.83(24.0)0.841(13.3)!0.54(8.8) 3.5(22.2) 2.24(23.3)!0.22(43.0)MR-R

4.5(4.5)

44.1(9.5)

0.981(0.9)

*

2.0(2

3.1)

20.4(56.2)

*

Method B was log-normal (0.23),Nukiyama }Tanasawa (0.12),upper-limit (!0.05)'Rosin }Rammler (!0.22)(p (0.05).Dry powder inhaler

Probability distributions for the DPI are shown in Figs 5(A)and (B),with the representa-tive parameters and MSC values shown in Table 6.The MSC values and probability distributions revealed that none of the PDFs produced a good "t to the experimental data.Representative parameters could not be estimated from the upper-limit PDF and MSC values could not be calculated for the modi "ed Rosin }Rammler for the same reasons given for the pMDI https://www.doczj.com/doc/fc3352734.html,parisons of the MSC values calculated for the respective PDFs using Method A and B showed that Method B produced signi "cantly higher MSC values (p (0.05).Multiple pairwise comparisons for Method A and B are shown in Fig.6.Nasal spray

Probability distributions for the nasal spray are shown in Figs 7(A)and (B),with the average representative parameters and MSC values shown in Table 7.The MSCs obtained by Method B were signi "cantly higher than those obtained by Method A for each PDF

822 C.A.Dunbar and A.J.Hickey

Fig.4.Multiple comparisons of pMDI average MSC values produced by PDFs (data points,means

and error bars shown,n "

3).

Fig.5.DPI probability distributions.Adjustable parameters estimated from (A)linear least-squares "t of CDF and (B)non-linear least-squares "t of PDF.PDF:**L-N;**U-L;}}}R-R;

-----MR-R;}-}-NT;*experimental data.Table 6.DPI average representative parameters and MSC values estimated from PDFs using

Methods A and B (%rsd in parentheses;n "3)DPI

Method A

Method B Distribution D K ( m)

R MSC D K ( m) MSC L-N 10.4(6.3) 2.86(2.5)0.956(1.5)!0.5436.312.3(5.9) 4.03(4.2)!0.26(81.5)U-L ****12.0(4.9)*!0.62N-T 10.8(9.0) 2.61(6.6)0.993(0.2)!0.56(28.9)12.1(5.8) 3.68(5.8)!0.31(64.4)R-R 7.8(10.6) 1.22(8.3)0.922(1.5)!1.33(5.5)11.0(5.0) 2.14(6.8)!0.54(26.6)MR-R

25.1(12.9)

25.6(43.6)

0.991(0.1)

*

17.9(6.7)

14.1(19.8)

*

Evaluation of probability density functions 823

Fig.6.Multiple comparisons of DPI average MSC values produced by PDFs (data points,means

and error bars shown,n "

3).

Fig.7.Nasal spray probability distributions.Adjustable parameters estimated from (A)linear least-squares "t of CDF and (B)non-linear least-squares "t of PDF.PDF:**L-N;**U-L;

}}}R-R;-----MR-R;}-}-NT;*experimental data.

Table 7.Nasal spray average representative parameters and MSC values estimated from PDFs

using Methods A and B (%rsd in parentheses;n "3)Nasal spray

Method A

Method B Distribution D K ( m)

R MSC D K ( m) MSC L-N 65.5(2.6) 2.82(2.6)0.969(0.5) 1.97(8.1)52.9(5.1) 2.73(3.7) 4.01(1.3)U-L 78.2(2.8) 2.73(1.7)0.902(0.8)0.61(11.0)52.9(5.2) 2.73(3.7) 3.95(1.3)N-T 70.8(4.2) 3.04(4.2)0.963(0.4) 1.79(13.1)52.1(5.1) 2.43(2.9) 3.52(2.3)R-R 100.5(1.59) 1.83(1.65)0.849(0.9)0.17(15.1)50.8(4.7) 1.70(2.0) 1.89(2.7)MR-R

74.4(1.9)

2.22(2.4)

0.957(0.4)

1.09(9.6)

52.2(4.9)

2.22(4.2)

3.34(

4.0)

824 C.A.Dunbar and A.J.Hickey

Fig.8.Multiple comparisons of nasal spray average MSC values produced by PDFs (data points,

means and error bars shown,n "3).

Table 8.Average representative parameters estimated by analysis of the experimental data (%rsd in

parentheses;n "3)Device D K ( m) FPF (%)LPF (%)Truncation (%)Nebulizer 1.2(5.0) 1.77(5.2)98.6(0.9)*0.6(61.3)pMDI 3.7(8.1)*45.1(4.9)*45.2(5.3)DPI

*

*

15.4(8.6)*

79.8(1.2)Nasal spray

50.7(5.4)

2.18(7.4)

0.1

(9.3)

87.5(0.7)

*

(p (0.05).Multiple pairwise comparisons for Methods A and B are shown in Fig.8.The rank order of MSC values using Method A was log-normal (1.97),Nukiyama }Tanasawa (1.79)'modi "ed Rosin }Rammler (1.09)'upper-limit (0.61)'Rosin }Rammler (0.17)(p (0.05).The rank order of MSC values obtained using Method B was log-normal (4.01),upper limit (3.95)'Nukiyama }Tanasawa (3.52),modi "ed Rosin }Rammler (3.34)'Rosin }Rammler (1.89)(p (0.05).

Graphical analysis of the experimental cumulative mass distributions

The average representative parameters estimated by graphical analysis of the experi-mental cumulative mass distributions for each aerosol system are summarized in Table 8.Both representative parameters could not be estimated for the pMDI and DPI due to truncation of the cumulative mass distributions at approximately the 50th and 80th percentiles,respectively.Fine particle fractions and large particle fraction (nasal spray only)of each device are also shown in Table 8,along with the percentage truncation at the upper cut-o !diameter of the measurement instrument.The percentage truncation could not be calculated for laser di !raction,as this instrument does not capture the whole aerosol.However,the value was considered to be low due to the large upper-limit cut-o !diameter (approximately 814 m).

DISCUSSION

Truncation of the pMDI and DPI particle size distributions obtained by cascade impaction resulted in poor "ts of the PDFs due to a loss of information for large fractions of

Evaluation of probability density functions 825

826 C.A.Dunbar and A.J.Hickey

the particle size distributions(50and80%,respectively).The Andersen eight-stage cascade

impactor was not suitable for accurately characterizing the particle size distribution

produced by the pMDI and DPI.It is suggested that cascade impactors with e!ective

cut-o!diameters with a minimum range of90%of the particle size distribution should be

used to characterize these systems.The bimodal tendency of the DPI probability distribu-

tion,produced by the binary mixture of the formulation(FP drug particles of the order of

3 m,and lactose carrier particles of the order of90 m),breached the central tendency

requirement.Dry powder inhalers that do not employ carrier particles may conform with

one or more of the PDFs discussed.

Non-linear least-squares"t of the probability density function(Method B)produced

a signi"cantly better"t to the experimental particle size probability distributions than

linear least-squares"t of the cumulative distribution function(Method A)for each PDF

and aerosol system(p(0.05).Probability distribution functions were"tted without bias

from linear coordinate transformation.However,transformation of the cumulative distri-

bution function into linear coordinates distorted the scales of the cumulative mass

distribution.For example,transforming the cumulative mass fraction to a normal probabil-

ity compressed the scale around the median and expanded the scale on the periphery(tails)

of the distribution.The tails of the distribution were the regions of highest variability in the

experimental data,and this variability was magni"ed by the linear transformation of the

cumulative distribution function.Fitting a straight line to the cumulative distribution by

linear least-squares allocated equal weight to the data points.This overemphasized the

contribution from the tails of the distributions,even though good"ts of the cumulative

distribution functions were observed,as indicated by high correlation coe$cients.An

iterative weighted least-squares procedure could be used to improve the"t to the data

(Kottler,1951).Information on the calibrated collection e$ciencies of the measurement

instrument are required to implement this method(Raabe,1978).

No single probability density function represented the particle size distributions produc-

ed by the four di!erent pharmaceutical aerosol systems.The Rosin}Rammler PDF using

Method B produced the best"t to the nebulizer probability distribution(MSC"6.27).

Comparison of the representative parameters estimated from the Rosin}Rammler PDF

and graphical analysis of the experimental cumulative mass distribution revealed no

signi"cant di!erences between the average values of(p"0.999)and (p"0.802).The

Nukiyama}Tanasawa and log-normal PDF using Method B produced good"ts to the

nebulizer probability distribution(MSC"3.14and2.68,respectively),although a signi"-

cantly wider distribution of particle sizes( "2.37and2.68,respectively)were estimated

relative to the estimate by graphical analysis( "1.77)and Rosin}Rammler PDF

( "1.76)(p(0.05).All of the PDFs produced a good"t to the nasal spray probability

distribution(MSC'3.0),the exception being the Rosin}Rammler PDF(MSC"1.89).

There were signi"cant di!erences between the representative parameters estimated from

each PDF and graphical analysis of the experimental cumulative mass distribution for the

nasal spray(p(0.05).

The modi"ed Rosin}Rammler PDF truncated the distributions for D)1 m(p(D)P0

as D P1)due to the ln(D)term in the PDF(Table1),precluding its use for representation

of the nebulizer,pMDI and DPI probability distributions.An error was found in the

application of the upper-limit PDF.Estimation of D from the geometry of the linear cumulative distribution as proposed by Mugele and Evans produced negative and zero

values(Mugele and Evans,1951).This occurred when

D K D K *D K .(17) Estimation of the adjustable parameters(D K ,D , )using non-linear least-squares"t of the upper-limit PDF produced unrealistically large,and sometimes negative,values for D .For example,the average D estimated for the pMDI probability distribution using Method B was!7.1;10 m(173%rsd).Erroneous estimates of D were a result of the maximum diameter tending to in"nity for a log-normal cumulative mass distribution.

Table9.Average representative parameters and MSC values calculated for modi"ed upper-limit PDF using estimation Methods A and B(%rsd in parentheses;n"3)

Modi"ed upper-limit PDF

Method A Method B

Device D K ( m) R MSC D K ( m) MSC

Nebulizer 1.1

(3.7)

2.87

(4.2)

(0.867

(9.5)

2.78

(13.0)

1.2

(4.9)

2.68

(3.2)

2.68

(15.5)

pMDI 4.8

(2.0)

2.64

(2.5)

0.861

(12.1)

!0.25

(86.7)

3.0

(3.5)

2.13

(6.3)

0.19

(67.0)

DPI10.2

(6.5)

2.52

(2.9)

0.967

(0.9)

!0.85

(20.2)

12.1

(5.9)

3.31

(4.4)

!0.28

(72.5)

Nasal spray73.5 2.730.9300.9452.4 2.50 3.82

(1.6)(2.1)(0.9)(7.7)(5.1)(3.2)(1.4)

This problem can be overcome by estimating a"nite maximum diameter,i.e.,D K . Hence,the independent variable is de"ned as follows;

y"ln aD D K !D ,(18)

where D K is approximated from the log-normal cumulative distribution function. Table9shows the average representative parameters and MSC values calculated for each aerosol system using the modi"ed upper-limit PDF.Improvements made to the upper-limit PDF by the modi"ed form were(1)representation of the whole distribution,(2)a stable estimate of D ,and(3)a best"t to the nebulizer probability distribution data using adjustable parameter estimation Method A(MSC"2.78).

Considerable caution must be used when estimating representative parameters from cumulative or probability distributions.Measurement errors are magni"ed by linear trans-formation of the cumulative distribution functions,producing a relatively poor"t to the probability distribution.Analysis of the probability distribution provides a more reliable method of parameter estimation.However,application of a suitable mathematical model is required to estimate representative parameters of the probability distribution.A procedure for"tting a mathematical model to a probability distribution data set is proposed as follows:

(i)Con"rm the central tendency(modality)of the probability distribution.

(ii)Con"rm that'90%of the particle size distribution was obtained by the characteriza-tion method.

(iii)Estimate the adjustable parameters using a non-linear least-squares"t of the probabil-ity distribution.

(iv)Calculate representative parameters of the particle size distribution(D K and )and their error estimates.

(v)Estimate a statistical measure of the`goodness of"t a.

CONCLUSIONS

Large truncation of the pMDI and DPI particle size distributions resulted in poor"ts of the PDFs,suggesting that the Andersen eight-stage cascade impactor was not suitable for accurately characterizing the particle size distribution produced by these systems.

Evaluation of probability density functions827

828 C.A.Dunbar and A.J.Hickey

Nonlinear least-squares"t of the probability density function(Method B)produced a signi"cantly better"t to the experimental probability distribution than linear least-squares"t of the cumulative distribution function(Method A)for each PDF and aerosol system(p(0.05).Probability distribution functions were"tted without bias from linear coordinate transformation.However,transformation of the cumulative distribution func-tion into linear coordinates distorted the scales of the cumulative mass distribution and magni"ed the measurement errors at the tails of the distribution.No single probability density function represented the particle size distributions produced by the four di!erent pharmaceutical aerosol systems.The nebulizer and nasal spray probability distributions were well represented by the Rosin}Rammler and log-normal PDFs,respectively.Probabil-ity distributions produced by the DPI breached the central tendency requirement, precluding representation by mathematical models.The modi"ed Rosin}Rammler PDF truncated distributions at D(1 m,precluding its use for analysis of the nebulizer,pMDI and DPI probability distributions.A modi"ed form of the upper-limit PDF was presented to stabilize the D estimation.Considerable caution must be used when estimating representative parameters from the cumulative or probability distributions.A generalized procedure for adoption of a mathematical"t to particle size distribution data sets was proposed.

Acknowledgements*Stevie,Samantha and Gabby are kindly acknowledged for their support during the preparation of this manuscript(CD).This work was funded in part by a research grant from Glaxo Wellcome,Inc., RTP,NC.

REFERENCES

Akaike,H.(1974)A new look at the statistical model identi"cation.IEEE Trans.Automat.Control19,716. Dunbar,C.A.,Brouet,G.and Hickey,A.J.(1998)Optical particle sizing methods for evaluation of nasal sprays. Pharm.Sci.14,S-212.

Dunbar,C.A.and Hickey,A.J.(1999)Selected parameters a!ecting characterization of nebulized aqueous solutions by cascade impaction and comparison with phase-Doppler analysis.Eur.J.Pharm.Biopharm.Technol. (in press).

Dunbar,C.A.,Morgan,B.,Van Oort,M.and Hickey,A.J.(1999)A novel method to compare in v itro dry powder inhaler performance using power.Eur.J.Pharm.Biopharm.Technol.submitted for publication.

Fults,K.,Cyr,T.D.and Hickey,A.J.(1991)The in#uence of sampling chamber dimensions on aerosols particle size measurement by cascade impactor and twin impinger.J.Pharm.Pharmacol.43,726.

Graseby Andersen,X.(1985)Operating Manual for Andersen1ACFM Non-Viable Ambient Particle Sizing Samplers. Graseby Andersen,Smyrna,GA.

Hatch,T.and Choate,S.P.(1929)Statistical description of the size properties of non-uniform particulate substances.J.Franklin Inst.207,369.

Hinds,W.C.(1982)Aerosol Technology*Properties,Beha v iour and Measurement of Airborne Particles.Wiley,New York,NY.

Hinds,W.C.(1986)Data analysis.In:Cascade Impactor*Sampling and Data Analysis(Edited by Lodge,J.P.and Chan,T.L.),p.45.American Industrial Hygiene Association,Akron,OH.

Heyder,J.,Gebhart,J.,Rudolf,G.,Schiller,C.F.and Stahlhofen,W.(1986)Deposition of particles in the human respiratory tract in the size range0.005to15 m.J.Aerosol Sci.17,811.

Kottler,F.(1951)The goodness of"t and the distribution of particle sizes.Parts I and II.J.Franklin Inst.251,499 and617.

Lefebvre,A.H.(1989)Atomization and Sprays.Hemisphere Publishing Corp.,New York,NY.

Mugele,R.A.and Evans,H.D.(1953)Droplet size distribution in sprays.Ind.Engng.Chem.43(6),1317. Nukiyama,S.and Tanasawa,Y.(1939)An experiment on the atomization of liquid Part III.On the distribution of the size of drops.Trans.Soc.Mech.Engng Jpn5(18),63.

Raabe,O.G.(1971)Particle size analysis utilizing grouped data and the log-normal distribution.Aerosol Science2, 289.

Raabe,O.G.(1978)A general method for"tting size distributions to multicomponent aerosol data using weighted least-squares.En v iron.Sci.Technol.12(10),1162.

Rizk,N.K.and Lefebvre,A.H.(1985)Drop-size distribution characteristics of spill-return atomizers.AIAA J.Propul.1(3),16.

Rosin,P.and Rammler,E.(1933)The laws governing the"neness of powdered coal.J.Inst.Fuel7(31),29. Scientist Users Manual(1994)MicroMath Scienti"c Software.Salt Lake City,UT.

Task Group on Lung Dynamics(1966)Deposition and retention models for internal dosimetry of the human respiratory tract.Health Phys.12,173.

USP16012(1999)Aerosols,Metered Dose Inhalers and Dry Powder Inhalers.United States Pharmacopoeia(10th Suppl.)4933}4949.

Van Oort,M.,Downey,B.and Roberts,W.(1996)Veri"cation of operating the Andersen cascade impactor at di!erent#ow rates.Pharm.Forum22(2),2211.

APPENDIX.DERIVATION OF THE LINEAR TRANSFORMATION

OF THE CUMULATIVE DISTRIBUTION FUNCTIONS

?og -normal distribution

The normal mass probability density function (PDF)is given by

p (z )K "d F K d z "1(

exp ! z

2

,(4)

where

z " y ,y "D !D K

, " \

(5)

D K

is the mass mean diameter (noting that the mean and median are equal for a symmetrical distribution)and is the standard deviation.The log-normal mass PDF is obtained by de "ning

y "ln D !ln D

K

, "ln \

.

(A1)

Presenting the log-normal mass PDF as a function of diameter gives

p (D )K "d F K d D "((2 D ln )\ exp !

ln (D /D K )

2ln

,(A2)

where D K is the mass median diameter ("geometric mass mean diameter)and

is the geometric standard

deviation.Integration of equation (4)gives the cumulative distribution function F (z )K

,that takes the form of the

normal probability integral ( ),

F (z )K "1( X \

exp ! z

2

d z " (z ),

(A3)

where

z "

ln D !ln D

K ln

.(A4)

The cumulative distribution function is transformed into linear coordinates by taking the inverse normal probability integral ( \ )of equation (A3)and rearranging to express ln(D )as a function of \ (F K

),

ln(D )"ln \ (F K )#ln D K

,

(A5)

;pper -limit distribution

The upper limit PDF is an extension of the log-normal PDF where the distributed quantity is given by

y "ln(aD )!ln(D

!D ),

(A6)where D

is the maximum diameter (equation (10))and a is a dimensionless constant (equation (9))related to

dispersion (Mugele and Evans,1953).This yields a mass PDF as follows:

p (D )K "D (D !D )((2 ln )\ exp ! ln (aD /D !D )

2ln

.

(A7)

Integration of equation (A7)gives the cumulative distribution function that takes the form of the normal

probability integral function,given by equation (A3),where

z "ln(aD )!ln(D !D )

ln

.(A8)

Transformation of equation (A3)into linear coordinates and rearranging gives

ln

D D !D

"ln \ (F K )#ln D

K D !D

K

,

(A9)

where ln(D /(D !D ))is expressed as a function of \ (F K

).

Evaluation of probability density functions 829

Nukiyama }1anasawa distribution

Nukiyama and Tanasawa developed a PDF to describe the frequency distribution of sprays,

p (D )" b B

D exp !+bD B ,,

(11)

where b and are size and distribution parameters,respectively,and (x )is the complete gamma function of the argument,x (equation (12))(Nukiyama and Tanasawa,1939).The size distribution parameter can be related to the mass median diameter by integrating equation (11)to obtain the cumulative distribution function:

F (D )K "1!

(6/ ,bD B )

,(A10)where (x ,y )is the incomplete gamma function.The size distribution parameter,b ,is related to D

K

,as follows:

b "

2D B K

.(A11)

Transformation of the CDF (equation (A10))into linear coordinates is achieved by equating the ratio of complete to incomplete gamma functions to the probability integral of the -distribution (Q )with degrees of freedom,

F ( " )K "1!Q ( " )"1! (6/ ,bD B )

,

(A12)

where

"12/ ,

"2bD B .

(A13)Taking the inverse of the -distribution probability integral (Q \ )of equation (A12)and rearranging to express

Q \ (1!F K

"12/ )as a function of D B gives

Q \ 1!F K 12

" (0.5"12/ )

D B

K

D B .

(A14)

Rosin }Rammler distribution

The Rosin }Rammler PDF takes the following form:

p (D )K "qD O \

X O

exp ! D X

O ,

(13)

where X and q are size and distribution parameters,respectively (Rosin and Rammler,1933).Integrating equation

(13)yields the cumulative distribution function,

F (D )K "1!exp ! D X

O

.

(A15)

The size distribution parameter,X ,is expressed in terms of D

K

as follows:

X O "D O K ln(2)

.

(A16)

The cumulative distribution function can be transformed into linear coordinates by taking the natural log of equation (A15)and rearranging to give

ln

1

1!F

K

"q ln(D )!q ln(X ),

(A17)

where ln(1/(1!F K ))is expressed as a function of ln(D ).

Modi ,ed Rosin }Rammler distribution

The Rosin }Rammler cumulative distribution function has been modi "ed as follows (Rizk and Lefebvre,1985):

F (D )K "1!exp ! ln D ln X

O

.

(A18)

830

C.A.Dunbar and A.J.Hickey

Di !erentiation of equation (A18)and expressing X in terms of D

K

produces the modi "ed

Rosin }Rammler PDF,given as

p (D )K "q D ln O \ D ln O X exp ! ln D ln X

O

,

(14)

where

ln O (X )"

ln O (D K )

ln(2)

.(A19)

Transformation of the CDF (equation (A18))into linear coordinates and rearranging gives ln ln

1

1!F

K

"q ln ln(D )!q ln ln(X ),

(A20)

where ln ln(1/(1!F K

))is expressed as a function of ln ln(D).

Evaluation of probability density functions

831

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