Partial Differential Equation Modeling of Flow Cytometry ... · PDF fileData Math. Model Stat....

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Partial Differential Equation Modeling of FlowCytometry Data from CFSE-based Proliferation

Assays

W. Clayton ThompsonIn collaboration with ...

Center for Quantitative Sciences in BiomedicineCenter for Research in Scientific Computation

Department of MathematicsNorth Carolina State University

01 March 2012

Clay Thompson CFSE Modeling


H.T. BanksThesis Advisor

Karyn SuttonDept. Mathematics, University of Louisiana at Lafayette

Tim SchenkelDepartment of Virology, Saarland University, Homburg, Germany

Jordi Argilaguet, Sandra Giest, Cristina Peligero, Andreas MeyerhansICREA Infection Biology Lab, Univ. Pompeu Fabra, Barcelona, Spain

Gennady BocharovInstitute of Numerical Mathematics, RAS, Moscow, Russia

Marie DoumicINRIA Rocquencourt, Projet BANG, Rocquencourt, France



1 CFSE Data Overview

2 PDE Modeling of CFSE Data

3 Data Statistical Model

4 Next Steps



Data Overview

(A. Meyerhans)



CFSE Labeling (Lyons and Parish, 1994)

Cells cultured with CFDA-SE then washed

CFDA-SE becomes protein-bound and fluorescent CFSE

Dye split between daughter cells at division

Dye naturally turns over/degrades (very slowly)

Fluorescence Intensity (FI) of CFSE measured via flowcytometry

FI linear with dye concentration ⇒ FI ∝ mass

Several advantages over other dyes/techniques



CFSE Data Set

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5x 105 CFSE Time Series Histogram Data

Label Intensity z [Log UI]

Str

uctu

red

Pop

ulat

ion

Den

sity

[num

bers

/UI]

t = 0 hrst = 24 hrst = 48 hrst = 72 hrst = 96 hrst = 120 hrs



Goals of Modeling

Cellular ‘Dynamic Responsiveness’Link cell counts with proliferation/death rates

Population doubling timeCell viabilityBiological descriptors (cell cycle time, etc.)

Uncertainty Identification, Variability Quantification...... in the experimental procedure... for estimated rates/etc

Analyze cell differentiation and division-linked changes

Investigate immunospecific extracellular signalingpathways

Comparison among donors/cell types/disease progression



Traditional Approach (curve fitting)

Fit data with gaussian curves to determine approximatecells per generationTraditional ‘semi-quantitative analysis’ pioneered by Gettand Hodgkin et al. (2000)

(A.V. Gett and P.D. Hodgkin, A cellular calculus for signal integration by T cells, Nature Immunology 1 (2000),

239–244.)



Traditional Approach (cont’d)

Gett-Hodgkin method quick, easy to implement, usefulcomparisons between data sets (e.g. stimulationconditions)

Compatible with ODE, DDE models; ‘indirect fitting’ forparameter estimationGeneralizations, extensions, and various other modelingefforts

Smith-Martin model (with generalizations)Cyton modelBranching process models



Label-Structured Model

All previous work with cell numbers determined bydeconvolutionAlternatively, we propose to fit the CFSE histogram datadirectly

Capture full behavior of the population densityNo assumption on the shape of CFSE uptake/distribution

Histogram presentation of cytometry data makesstructured population models a natural choice

Key ideas first formulated by Luzyanina et al., 2007FI (or log FI) ⇔ Division Number



Label-Structured Model (cont’d)

This model must account for (Luzyanina et al., 2007):

Dilution of CFSE as cells divide (AutoFI)

Slow decay of FI over time (CFSE turnover)

Asynchronous division times

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5x 105 Data in Logarithmic Coordinate (z)

z [Log UI]

Cel

l Cou

nts

t = 24 hrst = 48 hrst = 96 hrst = 120 hrs



Cellular Autofluorescence

0 1 2 3 4 5 6

0

3000

6000

9000

12000

15000Donor 1 CD4 Data, t = 0 hrs

z [Log UI]

Cel

l Cou

nts

PHA+CFSECFSE onlyAutoFI

Xi = X CFSEi + X Auto

⇓Xi+1 = X CFSE

i /2 + X Auto



CFSE Turnover

(C. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies, Immunology and Cell Biol. 77

(1999), 499–508.)



‘Biphasic Decay’

0 20 40 60 80 100 120 140 1601.5

2

2.5

3

3.5

4

4.5

5x 104 Donor 1 Exponential Fit,

Time (hrs)

CF

SE

Inte

nsity

dxdt = v(x) = c(x − xa)

Exponential

0 20 40 60 80 100 120 140 1601.5

2

2.5

3

3.5

4

4.5

5x 104 Donor 1 Gompertz Fit,

Time (hrs)

CF

SE

Inte

nsity

dxdt = v(t , x) = c(x − xa)e−kt

Gompertz



Fragmentation Mathematical Model

Structured density n(t , x) (cells/UI)

(Exponential) Proliferation rate α(t , x)

(Exponential) Death rate β(x)

Gompertz decay rate, v(t , x) = c(x − xa)e−kt

∂n(t , x)∂t

+∂[v(t , x)n(t , x)]

∂x= −(α(t , x) + β(x))n(t , x)

+ χ[xa,x∗]4α(t , 2x − xa)n(t , 2x − xa)



Inverse Problem

Parameters xa, c, k , α(t , y), β(y) to be determined byfitting to data.Need (finite-dimensional) parameterization of α and β.

Piecewise linear functions

Statistical properties of error currently unknown

Use OLS (independent, identically distributed, constantvariance error) for proof of concept

θOLS = arg minθ∈Θ

I∑

i=1

J(i)∑

j=1

(I[n](ti , zj ; θ)− N ji )

2 = arg min J(θ),

Forward solve with hpde by L.Shampine (Lax-Wendroff)

Use fmincon (BGFS + active set) for optimization



Time-Independent Proliferation is Insufficient

0 1 2 3−2

0

2

4

6

8

10x 104 Best−Fit Histograms, t =24 hrs

z [Log UI]

Cel

l Num

bers

DataModel

0 1 2 3−2

0

2

4

6

8

10


z [Log UI]

Cel

l Num

bers

DataModel

0 1 2 3−0.5

0

0.5

1

1.5

2

2.5x 105 Best−Fit Histograms, t =96 hrs

z [Log UI]

Cel

l Num

bers

DataModel

0 1 2 3−0.5

0

0.5

1

1.5

2

2.5

3


z [Log UI]

Cel

l Num

bers

DataModel



Time-Dependent Proliferation is Sufficient

0 1 2 3−2

0

2

4

6

8


z [Log UI]

Cel

l Num

bers

DataModel

0 1 2 3−2

0

2

4

6

8


z [Log UI]

Cel

l Num

bers

DataModel

0 1 2 3−0.5

0

0.5

1

1.5

2


z [Log UI]

Cel

l Num

bers

DataModel

0 1 2 3−0.5

0

0.5

1

1.5

2

2.5

3


z [Log UI]

Cel

l Num

bers

DataModel



Fragmentation Model Summary

Model is capable of precisely fitting the observed data

c, k , xa estimated consistently (as α and β nodes change),though subject to high experimental variability

Time-dependence of the proliferation rate is an essentialfeature of the model

Biologically relevant average values of proliferation anddeath (in terms of number of divisions undergone) areeasily computable.But...

Still cannot compute cell numbersData overlap affecting estimated rates (?)Large number of parameters necessary



Fragmentation Model Summary (cont’d)

∂n(t , x)∂t

+∂[v(t , x)n(t , x)]

∂x= −(α(t , x) + β(x))n(t , x)

+ χ[xa,x∗]4α(t , 2x − xa)n(t , 2x − xa)

Applications to protein fragmentation and aggregation

Possible generalizations to size/volume structure



Division Structure: The Compartmental Model

Use compartments (on division number) to eliminatefragmentation terms

No need for structure dependence of estimated rates

∂n0

∂t+

∂[v(t , x)n0(t , x)]∂x

=− (α0(t) + β0(t))n0(t , x)

∂n1

∂t+

∂[v(t , x)n1(t , x)]∂x

=− (α1(t) + β1(t))n1(t , x) + R1(t , x)

...

∂nimax

∂t+

∂[v(t , x)nimax(t , x)]

∂x=− βimax

(t)nimax(t , x) + Rimax

(t , x)

where Ri(t , x) = 4αi−1(t)ni−1(t , 2x − xa) for 1 ≤ i ≤ imax



Method of Characteristics Solution

n0(t , x(t ; s)) =Φ0(s)exp

(

−

∫ t

0f0(τ)dτ

)

ni(t , x(t ; s)) =Φi(s)exp

(

−

∫ t

0fi(τ)dτ

)

+

∫ t

0Ri(τ, x(τ ; s))exp

(

−

∫ t

τ

fi(ξ)dξ)

dτ

where fi(t) = αi(t) + βi(t)− ce−kt

The cell numbers can be easily computed Ni(t) =∫

ni(t , x)dx



Parameterizations

B1 βi(t) = 0 for all i and for all t

B2 βi(t) = β for all i and for all t

B3 β0(t) = β0, βi(t) = 0 for i ≥ 1

B4 β0(t) = β0, βi(t) = β for i ≥ 1

B5 βi(t) = βi for each i

A1 α0(t) = α0; αi(t) = α for all i

A2 αi(t) = αi for all t

A3 α0(t) = α0χ[t>t∗]; αi(t) = α for all i

A4 α0(t) = α0χ[t>t∗]; αi(t) = αi

A5 piecewise linear functions of time (see below)



Distributed Autofluorescence

0 100 200 300 400 500 600 700 800 900 10000

50

100

150Donor 1 CD4 Day 1 AutoFI

x [Log UI]

Cel

l Cou

nts

DataLognormal

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60


x [Log UI]

Cel

l Cou

nts

DataLognormal

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60


x [Log UI]

Cel

l Cou

nts

DataLognormal

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80


x [Log UI]

Cel

l Cou

nts

DataLognormal

AutoFI appears approximately lognormally distributed

Dynamic properties ignored (for now)

Can study effective design of intracellular dyes



Distributed Autofluorescence (cont’d)

η(t , x) = E [n(t , x ; xa)|P] =∫ xmax

a

xmina

n(t , x ; xa)dP(xa)

dPdxa

= p(xa) =1

xaσ√

2πexp

(

− (log x−µ)2

2σ2

)

where

µ = log(E [xa])−12

log(

1 +Var(xa)

E [xa]2

)

σ2 = log(

1 +Var(xa)

E [xa]2

)



Another Inverse Problem

Population density n(t , x) =∑imax

i=0 ni(t , x)

Use OLS framework again–assume constant varianceerror

θOLS(njk ) = arg min

θ∈ΘJ(θ|nj

k )

= arg minθ∈Θ

∑

k ,j

(

I[n](tj , zjk ; θ)− nj

k

)2

Need to compare different parameterizations (modelcomparison)–Akaike Information Criterion

AIC = m log(

J(θOLS)m

)

+ 2p



Best-fit, AIC-selected results

0 0.5 1 1.5 2 2.5 3 3.50

2

4

6

8

10x 104 Calibrated Model, t =24hrs

Log UI

Cel

l Cou

nts

DataModel

0 0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6

7

8

9x 104 Calibrated Model, t =48hrs

Log UI

Cel

l Cou

nts

DataModel

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5x 105 Calibrated Model, t =96hrs

Log UI

Cel

l Cou

nts

DataModel

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5x 105 Calibrated Model, t =120hrs

Log UI

Cel

l Cou

nts

DataModel



Cell Numbers

0 20 40 60 80 100 1200

2

4

6

8

10x 107 Total Cell Counts

Time [hrs]

Num

ber

of C

ells

UndividedDividedTotal

Ni(t) =∫

ni(t , x)

0 20 40 60 80 100 1200

5

10

15x 106 Total Precursors

Time [hrs]

Num

ber

of P

recu

rsor

s

UndividedDividedTotal

Pi(t) = Ni(t)/2i

Population doubling time and precursor viability easilycomputable



Model Results and Conclusions

Cell/precursor numbers (per generation) easy to computeMore complex models receive highest ranking

Highly time-dependent proliferation rates (A5)Heterogeneous death rates (B5)Distributed AutoFI is an important modeling feature

But...AIC may be biased by statistical model‘Time-dependence’ possibly a byproduct of Malthusian formCell counts between data points biased by model form



The Statistical Model

Links the mathematical model to the data

Implications for estimation procedure

N jk = I[n](tj , z

jk ; θ0) + Ekj

Currently using constant variance (CV) model,Var(Ekj) = σ2

0 (⇒ Absolute Error)

Could use constant coefficient of variance (CCV),Var(Ekj) = σ2

0I[n](tj , zjk ; θ0)

2 (⇒ Relative Error)



Residual Plots

0 1 2 3 4 5 6 7 8 9 10x 10

4

−4

−2

0

2

4x 104 Residuals vs Model, t =24

Model Solution

Res

idua

ls r

1j

0 1 2 3 4 5 6 7 8 9 10x 10

4

−10

0

10Modified Residuals vs Model, t =24

Model Solution

Mod

ified

Res

idua

ls

0 1 2 3 4 5 6x 10

4

−2

0

2

x 104 Residuals vs Model, t =48

Model Solution

Res

idua

ls r

2j

0 1 2 3 4 5 6x 10

4

−10

0


Model Solution

Mod

ified

Res

idua

ls0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 105

−5

0

5x 104 Residuals vs Model, t =96

Model Solution

Res

idua

ls r

3j

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

5

−10

0


Model SolutionM

odifi

ed R

esid

uals

0 0.5 1 1.5 2 2.5x 10

5

−5

0

5x 10

4 Residuals vs Model, t =120

Model Solution

Res

idua

ls r

4j

0 0.5 1 1.5 2 2.5x 10

5

−10

0


Model Solution

Mod

ified

Res

idua

ls



Residual Plots (cont’d)

0 0.5 1 1.5 2 2.5x 10

5

−6

−4

−2

0

2

4

6x 104 Residuals vs Model

Model Solution

Res

idua

ls

0 0.5 1 1.5 2 2.5x 10

5

−10

−5

0

5

10Modified Residuals vs Model

Model Solution

Mod

ified

Res

idua

ls



New Statistical Model

N jk ∼ N

(

λj I[n](tj , zk ), λjBbj

I[n](tj , zk )

)

λj = bj/bj

bj is the ‘true’ number of beads counted at time tj

bj is the actual number of beads counted

B is the total number of beads original placed into eachwell

‘Sampling without replacement’



New Statistical Model (cont’d)

Can be derived from counting arguments (ignoringinterdependence)

Additional parameters bj to be estimated

Explains residual variance, ‘precursor cohort problem’

Implications for estimation procedure, model comparison

0 0.5 1 1.5 2x 10

4

−5000

0

5000Residuals vs Model

Model Solution

Res

idua

ls

0 0.5 1 1.5 2x 10

4

−40

−30

−20

−10

0

10

20

30

40Modified (sqrt) Residuals vs Model

Model Solution

Mod

ified

(sq

rt)

Res

idua

ls



Model Generalizations

Examination of AutoFI distributionCell division as a fission processActivation and/or time-dependence (machine calibrationissues?)Nonparametric estimation?... or not even estimate it at all?

(Improved) biologically meaningful prolf/death ratesSmith-Martin, probabilistic mechanismsInclude stimulation/signaling mechanisms



Allgower et al.

Dynamics for cell division, CFSE quantity, and measuredFI can be decoupledAllows for fast computational solution

ni(t , x) = Ni(t , x)ni (t , x)

where

dNi

dt= −(αi(t) + βi(t))Ni (t) + 2αi−1(t)Ni−1(t)

N0(0) = N0,Ni(0) = 0

and

∂ni

∂t−

∂[v(t , x)n(t , x)]∂x

= 0

ni (0, x) = 2iΦ(2i x)/N0

Convolution operator to link CFSE content with measuredFI (hence AutoFI)



Experimental Extensions

Account for multiple cell cultures present in PBMC culture

Antigen-specific stimulation

Division-linked changes, differentiated subsets

Extracellular signaling, knockout experiments

In vitro vs in vivo differences

Linking to immune/pathogenesis models

Analyze Proliferation in Diseased vs Healthy cells



Selected Sources

D. Schittler, J. Hasenauer, and F. Allgower, A generalized population model for cell proliferation: integrating

division numbers and label dynamics, Proc. 8th Intl. Workshop on Computational Systems Biology, June2011, Zurich, Switzerland.

H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, G. Bocharov, Marie Doumic, Tim Schenkel, Jordi

Argilaguet, Sandra Giest, Cristina Peligero, and Andreas Meyerhans, A New Model for the Estimation of CellProliferation Dynamics Using CFSE Data, CRSC-TR11-05, North Carolina State University, Revised July2011; J. Immunological Methods 343 (2011), 143–160.

H.T. Banks, W. Clayton Thompson, Cristina Peligero, Sandra Giest, Jordi Argilaguet, and Andreas

Meyerhans, A Division-Dependent Compartmental Model for Computing Cell Numbers in CFSE-basedLymphocyte Proliferation Assays, CRSC-TR12-03, North Carolina State University, February 2012; MathBiosci. Eng. (submitted).

T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans, and G. Bocharov, Numerical

modelling of label-structured cell population growth using CFSE distribution data, Theoretical Biology andMedical Modelling 4 (2007), Published Online.

W. Clayton Thompson, Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based

Proliferation Assays, Ph.D. Dissertation, Dept. Mathematics, North Carolina State University (2012).


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