Modeling of Cell Proliferation with Flow Cytometry Data from CFSE · PDF fileData Frag. Model Model Summaries Stat. Model Results Modeling of Cell Proliferation with Flow Cytometry

Data Frag. Model Model Summaries Stat. Model Results

Modeling of Cell Proliferation with FlowCytometry Data from CFSE-based Assays

H.T. Banks

Center for Quantitative Sciences in BiomedicineCenter for Research in Scientific Computation

Department of MathematicsNorth Carolina State University

Raleigh, NC

H.T. Banks CFSE Modeling


Clay Thompson and Karyn SuttonCRSC Postdocs at NCSU

Tim SchenkelDepartment of Virology, Saarland University, Homburg, Germany

Jordi Argilaguet, Sandra Giest,Cristina Peligero, Andreas Meyerhans

ICREA Infection Biology Lab, Univ. Pompeu Fabra, Barcelona, Spain

Gennady BocharovInstitute of Numerical Mathematics, RAS, Moscow, Russia



1 CFSE Data Overview

2 Initial Fragmentation Model of CFSE Data

3 Model Summaries

4 Data Statistical Model

5 Results



Selected Sources and References

H.T. Banks, Frederique Charles, Marie Doumic, Karyn L. Sutton, and W. Clayton Thompson, Labelstructured cell proliferation models, Appl. Math. Letters 23 (2010), 1412–1415;doi:10.1016/j.aml.2010.07.009

H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, Gennady Bocharov, Dirk Roose, Tim Schenkel, andAndreas Meyerhans, Estimation of cell proliferation dynamics using CFSE data, CRSC-TR09-17, NorthCarolina State University, August 2009; Bull. Math. Biol., 70 (2011), 116–150.

H.T. Banks, Karyn L. Sutton, W. Clayton Thompson, G. Bocharov, Marie Doumic, Tim Schenkel, JordiArgilaguet, 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, 373 (2011), 143–160; DOI:10.1016/j.jim.2011.08.014.

H.T. Banks, W. Clayton Thompson, Jordi Argilaguet, Cristina Peligero, and Andreas Meyerhans, Adivision-dependent compartmental model for computing cell numbers in CFSE-based lymphocyteproliferation assays, CRSC-TR12-03, N. C. State University, Raleigh, NC, January, 2012; Math. Biosci.Engr., to appear.

H.T. Banks and W. Clayton Thompson, Mathematical models of dividing cell populations: Application toCFSE data, CRSC-TR12-10, N. C. State University, Raleigh, NC, April, 2012; Journal on MathematicalModelling of Natural Phenomena, to appear.

H.T. Banks and W. Clayton Thompson, A division-dependent compartmental model with cyton andintracellular label dynamics, CRSC-TR12-12, N. C. State University, Raleigh, NC, May, 2012; Intl. J. of Pureand Applied Math., 77 (2012), 119–147.




H.T. Banks, Amanda Choi, Tori Huffman, John Nardini, Laura Poag and W. Clayton Thompson, ModelingCFSE label decay in flow cytometry data, CRSC-TR12-20, N. C. State University, Raleigh, NC, November,2012; Applied Mathematical Letters, (2103), doi:10.1016/j.aml.2012.12.010.

H.T. Banks, D. F. Kapraun, W. Clayton Thompson, Cristina Peligero, Jordi Argilaguet and AndreasMeyerhans, A novel statistical analysis and interpretation of flow cytometry data, CRSC-TR12-23, N. C.State University, Raleigh, NC, December, 2012; J. Biological Dynamics, submitted.

A.V. Gett and P.D. Hodgkin, A cellular calculus for signal integration by T cells, Nature Immunology 1 (2000),239–244.

E.D. Hawkins, Mirja Hommel, M.L Turner, Francis Battye, J Markham and P.D Hodgkin, Measuringlymphocyte proliferation, survival and differentiation using CFSE time-series data, Nature Protocols, 2(2007), 2057–2067.

E.D. Hawkins, M.L. Turner, M.R. Dowling, C. van Gend, and P.D. Hodgkin, A model of immune regulation asa consequence of randomized lymphocyte division and death times, Proc. Natl. Acad. Sci., 104 (2007),5032–5037.

T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans, and G. Bocharov, Numericalmodelling of label-structured cell population growth using CFSE distribution data, Theoretical Biology andMedical Modelling 4 (2007), Published Online.

C. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies, Immunology and Cell Biol.77 (1999), 499–508.




B. Quah, H. Warren and C. Parish, Monitoring lymphocyte proliferation in vitro and in vivo with theintracellular fluorescent dye carboxyfluorescein diacetate succinimidyl ester, Nature Protocols 2:9 (2007),2049–2056.

D. Schittler, J. Hasenauer, and F. Allgower, A generalized model for cell proliferation: Integrating divisionnumbers and label dynamics, Proc. Eighth International Workshop on Computational Systems Biology(WCSB 2011), June 2001, Zurich, Switzerland, p. 165–168.

J. Hasenauer, D. Schittler and F. Allgower, Analysis and simulation of division- and label-structuredpopulation models: A new tool to analyze proliferation assays, Bull. Math. Biol., 74 (2012), 2692–2732.



Scientific Goals

Long-term collaboration between NCSU and UPFMonitor cell division and differentiation

Assess polyfunctionalityInvestigate immunospecific extracellular signaling pathwaysIdentify correlated and (ideally) causal relationshipsbetween immune mechanisms

Quantitative measure of ‘dynamic responsiveness’

Link observed (cellular) behaviors to clinical outcomes;improve clinical outcomes

The ‘central problem in immunology’ (according to G.Bocharov): to understand the ‘cellular and molecularmechanisms that control the ability of the immune systemto mount a protective response against pathogen-derivedforeign antigens, but avoid a pathological response toself-antigens’



Persistent Infections

Trade-off between immune protection and immunopathy canlead to persistent infection

(A. Meyerhans)



Persistent Infections (cont’d)

(A. Meyerhans)

‘Dynamic equilibrium’ between host immune response andmicrobe expansion

Applications in HIV, HCV, TB

Open questions:Characterization of set pointOutcome of perturbations, modification for host benefit



Goals

Observations:

Clinical outcome believed to be strongly influenced by timing andmagnitude of ‘clonal expansion’

CFSE (intracellular dye) + flow cytometry = powerful new tool fortracking cell division

Analysis of data:

Develop a mathematical model for CFSE data

Link cell counts to measures of proliferation/death rates

Population doubling time; cell cycle timeCell viability

Applications:

Analyze experimental and biological variability

Optimal experimental design

Lab experiments (more in a moment...)



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



Data Overview

(A. Meyerhans)



CFSE Labeling (Lyons and Parish, 1994)

Cells cultured with CFDA-SE (carboxyfluorescein diacetatesuccinimidyl ester) then washed

CFDA-SE becomes protein-bound and fluorescent CFSE(the fluorescent dye carboxyfluorescein succinimidyl ester)

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 Labeling (Lyons and Parish, 1994)

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

(1999), 499–508.)



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 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 (Gett and Hodgkin)

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

(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 (as in age,size, etc) except here structure label is “CFSE content orintensity”

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



Label-Structured Model (cont’d)

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

Slow decay of CFSE FI over time

Dilution of CFSE as cells divide

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



Rate of Label Decay

(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



‘Translated Coordinate’ (Banks et al, 2010)

While the structure variable z does correlate with divisionnumber, the ‘translated variable’

s(t , z) = z − ck log 10

(

e−kt − 1)

has an even stronger correlation.

Follows from the Method of Characteristics

Intuitively, s represents the FI of a cell with a hypotheticallabel which does not decay

Change of reference: Eulerian vs Lagrangian coordinates



‘Translated Coordinate’ (cont’d)

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


−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5x 105

X: 1.592Y: 3.109e+05

Data in Translated Coordinate s

s [Log UI]

Cel

l Cou

nts




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

‘Translated coordinate’ very strongly correlated withdivision number

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...(Aggregate Data/Aggregate Dynamics)

Still cannot compute cell numbers and cohort rateparametersData overlap affecting estimated rates (?)Large number of parameters necessary



Best-fit, AIC-selected results: Model A5B5Dist

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



Label Structure Fragmentation Model-Model 1

CFSE label dynamics can be described by the fragmentationequation

∂n(t , x)∂t

− ce−kt ∂[(x − xa)n(t , x)]∂x

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

n(0, x) = Φ(x)

n(t , xmax) = 0

v(t , xa)n(t , xa) = 0. (1)

where n(t , x) is the structured population density (cells per unitFI) at time t and measured FI x and the advection term (withparameters c and k) represents the Gompertz decay processfor decrease in FI resulting from intracellular turnover of CFSE.



Compart. Models with Generation Structure-Model 2

A model can be reformulated with distinct compartments foreach generation. The resulting model is a system of partialdifferential equations

∂n0

∂t− ce−kt(x − xa)

∂n0

∂x=− (α0(t) + β0(t) − ce−kt)n0(t , x)

∂n1


∂n1

∂x=− (α1(t) + β1(t) − ce−kt)n1(t , x)

+ R1(t , x)...

∂nimax


∂nimax

∂x=− (βimax

(t) − ce−kt)nimax(t , x)

+ Rimax(t , x), (2)



Compartmental Models-Model 2

Here ni(t , x) is structured population density for cells havingundergone i divisions. The recruitment terms describe thesymmetric division of CFSE upon mitosis. Given byRi(t , x) = 4αi−1(t)ni−1(t ,2x − xa). Boundary and initialconditions same as in (1).

Observe that, because the number of divisions undergone hasnow been explicitly identified by the subscripted generationnumber, no longer necessary for division and death rates todepend upon measured fluorescence intensity.




IDEA: Combine compartmental models with probabilisticstructures (so-called Cyton Models) for cell division and deathLet Ni(t), 0 ≤ i ≤ imax represent the number of cells havingundergone i divisions at time t . Assume there are N0 cells inthe population at t = 0. In its simplest form, the cyton modelrelates the number of cells in the population to the number ofcells which divide and die in a unit of time,

N0(t) = N0 −∫ t

0

(

ndiv0 (s)− ndie

0 (s))

ds

Ni(t) =∫ t

0

(

2ndivi−1(s)− ndiv

i (s)− ndiei (s)

)

ds, (3)

where ndivi (t) and ndie

i (t) indicate the numbers per unit time ofcells having undergone i divisions that divide and die,respectively, at time t .




Let φ0(t) and ψ0(t) be probability density rate functions (in units1/time) for the times to division and death, respectively, for anundivided cell. Let F0, the initial precursor fraction, be thefraction of undivided cells which would hypothetically divide inthe absence of any cell death. It follows that

ndiv0 (t) = F0N0

(

1 −∫ t

0ψ0(s)ds

)

φ0(t)

ndie0 (t) = N0

(

1 − F0

∫ t

0φ0(s)ds

)

ψ0(t). (4)




Can define probability density rate functions φi(t) and ψi(t) fortimes to division and death, respectively, for cells havingundergone i divisions, as well as the progressor fractions Fi ofcells which would complete the i th division in the absence ofcell death. Then the numbers per unit time of dividing anddying cells are computed as

ndivi (t) = 2Fi

∫ t

0ndiv

i−1(s)(

1 −∫ t−s

0ψi(ξ)dξ

)

φi(t − s)ds

ndiei (t) = 2

∫ t

0ndiv

i−1(s)(

1 − Fi

∫ t−s

0φi(ξ)dξ

)

ψi(t − s)ds. (5)




Allgower, et al., have proposed a model which is structured bythe fluorescence intensity resulting from the mass of CFSEwithin the cell, i.e., intracellular label ignoring autofluorescence.This leads to a system for the mass of fluorescence n(t , x)described by the system of equations

∂n0

∂t− ce−kt ∂[xn0]

∂x=− (α0(t) + β0(t))n0(t , x)

...

∂ni

∂t− ce−kt ∂[xni ]

∂x=− (αi(t) + βi(t))ni (t , x) + 4αi−1(t)ni−1(t ,2x)

(6)...

with boundary and initial conditions as in (1) and (2).




The major advantage of formulating the model in terms of massof FI is the very simple form of the model solution. The solutionto the model (6) can be written as

ni(t , x) = Ni(t)ni (t , x) (7)

for all i . In this representation the functions Ni(t) satisfy theweakly coupled system of ordinary differential equations

dN0

dt=− (α0(t) + β0(t))N0(t)

dN1

dt=− (α1(t) + β1(t))N1(t) + 2αi−1(t)Ni−1(t) (8)

...

with initial conditions N0(0) = N0, Ni(0) = 0 for all i ≥ 1.




The functions ni(t , x) each satisfy the partial differentialequation

∂ni(t , x)∂t

− ce−kt ∂[xni(t , x)]∂x

= 0 (9)

with initial condition

ni(0, x) =2iΦ(2ix)

N0.

Note that, by definition,

N0 =

∫

∞

0Φ(ξ)dξ.




Let ni(t , x) be the structured density relative to the measuredfluorescence (x is related to x by the addition of cellularautofluorescence). To account for autofluorescence andbecause autofluorescence may vary from cell to cell in thepopulation, this is most accurately treated by computing theconvolution integral

n(t , x) =∫

∞

0n(t , x)p(t , x − x)dx , (10)

where p(t , x) is (for each fixed time t) a probability densityfunction describing the distribution of autofluorescence in thepopulation.




So finally we consider the system of equations

∂n0


∂x=(

ndiv0 (t)− ndie

0 (t))

n0(t , x)

∂n1


∂x=(

2ndiv0 (t) − ndiv

1 (t) − ndie1 (t)

)

n1(t , x)

(11)...

where the definitions of ndiv0 (t), ndie

0 (t), ndivi (t) and ndie

i (t) aregiven in Equations (4) - (5).




This model, which is based on simple mass balance, can besolved by factorization (7) as before; the label densities n(t , x)are readily computed according to Equation (9), and the cellnumbers are now provided by the cyton model as discussedabove. Thus the new model (11) is capable of accuratelydescribing the evolving generation structure of a population ofcells while also accounting for the manner in which the CFSEprofile of the population changes in time. The model is easilyrelatable to biologically meaningful parameters (times todivision and death) and can be solved efficiently so that it iseminently suitable for use in an inverse problem.



The Statistical Model

Links the mathematical model to the data

Implications for estimation procedure (Least squares vs.weighted least squares)

N jk = I[n](tj , z

jk ; θ0) + (I[n](tj , z

jk ; θ0))

γEkj

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

0 (⇒ Absolute Error: Ykj = model + Ekj )

Also tried constant coefficient of variance (CCV),Var(Ekj) = σ2

0I[n](tj , zjk ; θ0)

2 (⇒ RelativeError:Ykj = model × (1 + Ekj))

Found in between error most closely described data,Var(Ekj) = σ2

0I[n](tj , zjk ; θ0) (⇒ SquareRoot Error:

Ykj = model +√

model × Ekj )



Residual Plots γ = 0 vs. γ = 1

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



Residual Plots

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

lsN j

k = I[n](tj , zjk ;q0) + (I[n](tj , z

jk ;q0))

γEkj with γ = 0 (left) vs.γ = 1/2 (right). And γ = 1/2 works well!!



Least Squares Estimation

Define

qWLS = arg minq∈Q

J(q; {λj})

= arg minq∈Q

∑

j ,k

(

λj I[n](tj , zk ;q)− N jk

)2

w jk

where

w jk =

λjBbj

I[n](tj , zjk ;q0), I[n](tj , z

jk ;q0) > I∗

λjBbj

I∗, I[n](tj , zjk ;q0) ≤ I∗

.

Then qWLS is a consistent estimator of q0 (Banks, Kenz,Thompson, 2012)



Fit to Data

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

Approximated Initial Condition, t =23.5 hrs

Log Fluorescence Intensity [log UI]

Cou

nted

Cel

ls

4σ RegionDataModel

Donor 2, CD4 cells, Day 1, Data Set 121



Fit to Data (cont’d)

0 1 2 3 4 5 60

1000

2000

3000

4000

5000

Fit to Data, t =46 hrs


Cou

nted

Cel

ls

4σ RegionDataModel





0 1 2 3 4 5 60

1000

2000

3000

4000

5000

Fit to Data, t =67.5 hrs


Cou

nted

Cel

ls

4σ RegionDataModel





0 1 2 3 4 5 60

1000

2000

3000

4000

5000



Cou

nted

Cel

ls

4σ RegionDataModel





0 1 2 3 4 5 60

1000

2000

3000

4000

5000



Cou

nted

Cel

ls

4σ RegionDataModel




Fit to Data, CD8 Cells

0 1 2 3 4 5 60

2000

4000

6000

8000

10000



Cou

nted

Cel

ls

4σ RegionDataModel

0 1 2 3 4 5 60

2000

4000

6000

8000



Cou

nted

Cel

ls

4σ RegionDataModel

Donors 1 (left) and 2 (right), CD8 cells, Day 4, Data Set 121



Cytons

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02Time to Division/Death for Undivided Cells

Time [hrs]

Pro

babi

lity

Den

sity

0 20 40 60 80 100 120−0.4

−0.2

0

0.2

0.4Time to Division/Death

Time [hrs]

Pro

babi

lity

Den

sity

Donor 2, CD8 cells, undivided (left) and divided (right), DataSet 121



Generation Structure

0 20 40 60 80 100 1200

2

4

6

8x 105

Time [hrs]

Num

ber

of C

ells

Cell Numbers

0123456+

0 20 40 60 80 100 1200

2

4

6

8

10x 105

Time [hrs]

Num

ber

of C

ells

Cell Numbers

0123456+

0 20 40 60 80 100 1200

1

2

3

4

5

6x 105

Time [hrs]

Num

ber

of C

ells

Cell Numbers

0123456+

0 20 40 60 80 100 1200

2

4

6

8x 105

Time [hrs]

Num

ber

of C

ells

Cell Numbers

0123456+

Donor 1 (top) and Donor 2 (bottom), CD4 cells (left) and CD8cells (right)



Experimental Extensions

Account for multiple cell cultures present in PBMC culture

Antigen-specific stimulation

Effects of cryopreservation

Extracellular signaling, knockout experiments

In vitro vs in vivo differences

Linking to immune process models

Analyze Proliferation in Diseased vs Healthy cells



Acknowledgements

National Institute of Allergy and Infectious Disease grant NIAID9R01AI071915

U.S. Air Force Office of Scientific Research under grantAFOSR-FA9550-09-1-0226


Documents

Modeling of Cell Proliferation with Flow Cytometry Data from CFSE · PDF fileData Frag. Model Model Summaries Stat. Model Results Modeling of Cell Proliferation with Flow Cytometry