kuba tatarkiewicz phd, r&d global director · © © 2019 horiba, ltd. all rights reserved.2019...

© 2019 HORIBA, Ltd. All rights reserved.© 2019 HORIBA, Ltd. All rights reserved. 1

How to present and compare dataobtained by particle tracking analysis

and other related methods

Kuba Tatarkiewicz PhD, R&D Global Director

3-6-2019

© 2019 HORIBA, Ltd. All rights reserved. 2

• Measuring sizes of individual particleshydrodynamic diameter dh > d0 due to diluent drag

• Counting tracked particles in a given volumeinvestigated volume depends on size and refractive index

• Testing small volume of a sample• statistical process of estimating particle size distribution• volume tested about 100,000x smaller than sample volume

Particle tracking analysis


• Definition of CN: counts per volume [part/mL]other defs: mass CM [mg/mL] or volume CV [μL/mL]

• Typically CN is given for a range of sizes e.g. between 100 nm and 1 μm diameters

• Hence use of plots with size bins• bin widths not necessarily equal• some software do not support unequal bins (notably Excel)

Concentration measurements


• Error in counting proportional to this applies to total counts and individual bin counts

• Number of bins depends on expected errorsN=10,000 and 100 bins, average error 10 counts/bin or 10%

• To obtain “nice”, smooth distributions:• decrease number of bins and use wider bins• use narrow bins to calculate parameters of a distribution

Statistical considerations

𝑁𝑁


Example plots of size distributions

Counts Density of PSDlogarithmic bins

Cumulative

0

200

400

600

800

1,000

1,200

1,400

1,600

0 400 800 1,200 1,600 2,000

Cou

nts

per b

in [N

]

Diameter [nm]

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

10 100 1,000

Cum

ulat

ive

coun

ts

Diameter [nm]

D50=217 nm


• Definition of a bin:• d ∈ [bi, bi+1[ where i=1,…,N• equivalent definition: bi ≤ d < bi+1 (non-overlapping bins)

• typically b1 = 0 and bN = max size to be measured• Strange bins definition encountered:

Binning


• Concentration is just a rational number, also per bin• Histogram uses area to represent a distribution• With unequal bins natural measurable is:

density of particle size distribution (PSD)units: number of particles per bin width and

per investigated volume [part/nm/mL]total concentration ≡ area of a histogram

What is really plotted


• When plotting concentration vs. size,DO NOT connect points*

• Total concentration is a sum of numbers, not an integral• When plotting density of PSD, one can connect

middle of bins ☛ linear approximation• Total concentration CN is then

an area under the curve

Dimensional analysis

*There’s no data between points – by definition of a bin


• Create a list of measured sizestypically diameters of nanoparticles are given in [nm]

• Bin those sizes in specific binningtypically logarithmic binning

(bi+2 – bi+1 ) / (bi+1 – bi ) = const

• Calculate density of PSD [part/nm/mL]; V0 needed• Plot as a histogram of density of PSD• Calculate parameters of a distribution (AV, SD, D50 ...)

Practical procedure


Example of real data in logarithmic binning

Concentration from 50 nm to 700 nm = area of histogram of density of PSD

Den

sity

of P

SD


• Most standards of size are mono-dispersethere are no standards for concentration…

• Typically real life samples are poly-dispersee.g. multiple sizes or continuous distributions

• Natural samples like sea water or blood:• highly poly-disperse with dominating small particles

• Junge distribution N(d) ∼ d-x where x=3.5÷4

Real data vs. standards


• Fitting unknown distribution of sizes • Assumed binomial(s) distribution• Height (or area) of different peaks

is NOT conserved during fitting(not invariant of any fitting procedure)

• Looks great but lacks numerical accuracy• Artificial peaks can be created

Fitted data problem (NanoSight FTLA)

cf. van der Pol et al. J Thrombosis and Haemostasis, 12, 1182 (2014)

© 2019 HORIBA, Ltd. All rights reserved.

• Concentration

• Average size

• Standard deviation

• D50

13

Parametric description of distributions

Dk =1

𝑁𝑁𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡�𝑖𝑖=1

𝑘𝑘

𝑛𝑛𝑖𝑖 definition Dk = 0.5☛ k☛ bk equal D50

𝑆𝑆𝑆𝑆 =1


𝑁𝑁

𝑛𝑛𝑖𝑖 ∗ 𝑑𝑑𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑖𝑖 +𝐵𝐵𝑖𝑖2

2

Bi = (bi+1 - bi) 𝑁𝑁𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = �𝑖𝑖=1

𝑁𝑁𝑛𝑛𝑖𝑖𝐵𝐵𝑖𝑖𝐵𝐵𝑖𝑖 = �

𝑖𝑖=1

𝑁𝑁

𝑛𝑛𝑖𝑖

𝑑𝑑𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑎𝑎𝑎𝑎 =1


𝑁𝑁

𝑛𝑛𝑖𝑖 ∗ (𝑏𝑏𝑖𝑖 +𝐵𝐵𝑖𝑖2

)


D50 & mode depend on binning!

most accurate

which one is mode?which one is mode?


• Parametric description using AV and SD

is not accurate or unique!

Anscombe’s quartet – same AV and SD, various shapes

• Even higher moments do not give enough info

• Basic experimental question:How similar are two measured distributions?

Lies, damned lies, and statistics


Kolmogorov-Smirnov statistics

max

Non-parametric test:Kolmogorov-Smirnov statistics

dav = 256 nm, SD = 145 nm, CV = 0.57dav = 163 nm, SD = 68 nm, CV = 0.42

Comparing parameters:

Sheet1

DA,BalphaDA,B,αReject?

0.23350.050.0338yes


Anderson-Darling statistics

Area between two cumulatives is a good measure of distance between two or more distributions with unknown shape.


• Normalize area between cumulativesextreme case: a) particles at 10 nm, b) particles at 1000 nm

• Distance is a number from the range [0,1]same distributions have distance 0extreme case – distance 1

• If areas are calculated between distributions in same normalization, it can be a good measure

Distance between distributions

d

cum

ulat

ive

© 2019 HORIBA, Ltd. All rights reserved.

DankeБольшое спасибо

Grazie

감사합니다

Obrigado谢谢

ขอบคณุครบั

ありがとうございました

धन्यवाद

Cảm ơn

Tack ska ni ha

Thank you

Merci

Gracias

Dziękuję

Σας ευχαριστούμε

நன்ற

19

How to present and compare data�obtained by particle tracking analysis�and other related methodsParticle tracking analysisConcentration measurementsStatistical considerationsExample plots of size distributionsBinningWhat is really plottedDimensional analysisPractical procedureExample of real data in logarithmic binningReal data vs. standardsFitted data problem (NanoSight FTLA)Parametric description of distributionsD50 & mode depend on binning!Lies, damned lies, and statisticsKolmogorov-Smirnov statisticsAnderson-Darling statisticsDistance between distributionsSlide Number 19

kuba tatarkiewicz phd, r&d global director · © © 2019 horiba, ltd. all rights reserved.2019...

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