Identification of age-structured models: Cell cycle …drops/ramkiPapers/186_2008_Sherer...ARTICLE Identiﬁcation of Age-Structured Models: Cell Cycle Phase Transitions E. Sherer,1

ARTICLE

Identification of Age-Structured Models:Cell Cycle Phase Transitions

E. Sherer,1 E. Tocce,1,2 R.E. Hannemann,1,3 A.E. Rundell,3 D. Ramkrishna1

1School of Chemical Engineering, Forney Hall of Chemical Engineering,

480 Stadium Mall Way, Purdue University, West Lafayette, Indiana 47907;

telephone: 765-494-4066; fax: 765-494-0805; e-mail: [email protected] and Biological Engineering Department, University of Wisconsin,

Madison, Wisconsin3Weldon School of Biomedical Engineering, Purdue University, West Lafayette, Indiana

Received 7 May 2007; revision received 19 July 2007; accepted 9 August 2007

Published online 4 September 2007 in Wiley InterScience (www.interscience.wiley.co
m). DOI 10.1002/bit.21633
ABSTRACT: A methodology is developed that determinesage-specific transition rates between cell cycle phases duringbalanced growth by utilizing age-structured populationbalance equations. Age-distributed models are the simplestway to account for varied behavior of individual cells.However, this simplicity is offset by difficulties in makingobservations of age distributions, so age-distributed modelsare difficult to fit to experimental data. Herein, the proposedmethodology is implemented to identify an age-structuredmodel for human leukemia cells (Jurkat) based only onmeasurements of the total number density after the additionof bromodeoxyuridine partitions the total cell populationinto two subpopulations. Each of the subpopulations willtemporarily undergo a period of unbalanced growth, whichprovides sufficient information to extract age-dependenttransition rates, while the total cell population remains inbalanced growth. The stipulation of initial balanced growthpermits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, aflexible transition rate representation, utilizing a series ofcubic spline nodes, finds a bimodal G0/G1 transition ageprobability distribution best fits the experimental data. Thisresolution may be unnecessary as convex combinations ofmore restricted transition rates derived from normalizedGaussian, lognormal, or skewed lognormal transition-ageprobability distributions corroborate the spline predictions,but require fewer parameters. The fit of data with a single lognormal distribution is somewhat inferior suggesting thebimodal result as more likely. Regardless of the choice ofbasis functions, this methodology can identify age distribu-tions, age-specific transition rates, and transition-age dis-tributions during balanced growth conditions.

Biotechnol. Bioeng. 2008;99: 960–974.

� 2007 Wiley Periodicals, Inc.

KEYWORDS: population balance; inverse model; cell divi-sion; BrdU; cell generations

Correspondence to: D. Ramkrishna

960 Biotechnology and Bioengineering, Vol. 99, No. 4, March 1, 2008

Introduction

The growth behavior of an individual cell depends on acomplex system of molecular interactions, feedback regula-tions, and synthesis reactions (Darzynkiewicz et al., 1996;Janossy et al., 2004; Novak and Tyson, 1993, 2004). Thiscomplexity can lead to processes that are distributedthroughout the life cycle (Liou et al., 1998), such as protein(Kromenaker and Srienc, 1991) or RNA (Arino andKimmel, 1987) production and susceptibility to certainchemotherapeutic agents (Del Bino et al., 1991). Researchersare beginning to exploit the non-uniformity of theseprocesses with the aid of predictive models (Mantzariset al., 2002; Sherer et al., 2006). The heterogeneity of theresulting cell population is often represented in models by asingle or small set of characteristic variables. Commoncharacteristic variables include: cell mass (Mantzaris et al.,1999; Subramanian et al., 1970; Subramanian and Ramk-rishna, 1971), volume (Webb and Grabosch, 1987), DNAcontent, protein content (Kromenaker and Srienc, 1991), orage (Sennerstam and Stromberg, 1995; Tyrcha, 2001;Zamamiri et al., 2002). A population balance model des-cribes the cellular dynamics based on growth and transitionrates that are functions of the characteristic variables(Fredrickson and Tsuchiya, 1963; Fredrickson et al., 1967;Fredrickson and Mantzaris, 2002).

Age, the amount of time since a cell last made a transitionfrom one phase of the cell cycle to the next, is a simple way toaccount for the varied behavior of individual cells. That is,the amount of time spent by cells in any one phase of the cellcycle is distributed, so the likelihood of a cell transitioningfrom one phase to the next is dependent upon its age. An

� 2007 Wiley Periodicals, Inc.

age-dependent transition rate can be used to describe thisvariation. For example, in the cell cycle, certain intracellulartasks must be completed before a cell will proceed to the nextphase, so it is unlikely that a very young cell will transition.Also, checkpoints or other disturbances can shift and biasthe transition age. Lognormal distributions are mostcommonly used to define transition-age distributions ofcell cycle phases (Steel, 1977). The bias is captured evenmore closely if linear combinations of distributions(Shackney and Shankey, 1999) or asymmetry factors (Baischet al., 1995) are incorporated. Reaction probability densityfunctions provide a more mechanistic view of cellulartransitions (Hjortsø, 2006) so long as the rate of each step inthe reaction sequence is known.

The advantages of utilizing age as the characteristicvariable to describe the transition rates are diminished bythe inability to easily measure the age distributions within apopulation of cells. While inverse methods are available toextract individual behavior information from populationobservations, these require quantitative measurements ofthe characteristic variables (Ramkrishna, 2000; Tyrcha,2004). Synchronizing cells would allow for age approxima-tions, but the synchronization process can lead to alteredgrowth patterns (Chiorino et al., 2001; Cooper, 2003, 2004;Liou et al., 1998; Liu, 2005). Continuous observation of cellcultures is currently the method of choice for extractingindividual age-dependent behavior (Kutalik et al., 2005), butthis method is tedious as a significant number of cells mustbe observed to produce relevant distributions. In this paper,we develop an integrated experimental and analyticaltechnique to overcome the difficulty of extracting the agedistributions and the associated age-dependent transitionrates from experimental data. Previous derivations haveestimated age distributions assuming balanced growthconditions and a fixed, constant valued transition rate(Baisch and Otto, 1993; Bernard et al., 2003; Shackney, 1973;Steel, 1977).

For a clear understanding of how models are identifiedusing our methodology, it is necessary to recognize that theapproach to balanced exponential growth of the overallpopulation is preceded by two stages: (i) physiologicalchanges in individual cells and (ii) a transient period ofpopulation adjustment to the time-invariant distributioncharacteristic of the balanced growth phase. The secondstage represents a situation when stage (i) has been com-pleted, but the population distributions in the different cellcycle phases are evolving to the constant proportion inbalanced growth. Stage (i) can be achieved by maintaining aconstant culture environment and is characterized by time-independent transition rates. Balanced growth is achievedupon completion of stage (ii) so the age densities are alsotime independent (Dyson et al., 2000; Fredrickson et al.,1967). This feature of balanced growth implies that thesystem has reached a state such that the population densitiesof the phases will maintain constant ratio to one another(Dyson et al., 2002). Herein, we show that assumingbalanced growth enables the calculation of age distributions

based only on knowledge of the transition rates. Thus, theonly requirement for attaining the age distributions asso-ciated with the cell cycle dynamics are the transition rates.

A problem arises while trying to determine the vectoralunknowns describing the age distributions associated witheach phase of the cell cycle from scalar population densitydata. We demonstrate how an experimental strategy thatlabels a subpopulation can overcome this difficulty. Inutilizing this method, cells are subjected to pulse labelingwith bromodeoxyuridine (BrdU). Only cells in the S-phaseof the cell cycle at the time of BrdU application will belabeled with BrdU (Dien and Srienc, 1991; Dolbeare et al.,1983; Sena et al., 1999). This creates two independentsubpopulations, one labeled and the other unlabeled, both ofwhich continue around the cell cycle. Two importantfeatures of this technique are: (i) that each subpopulationbegins in unbalanced growth as their age distributions arechanging with time and (ii) the BrdU does not affect thegrowth behavior of individual cells, so the total populationremains in balanced growth (Baisch and Otto, 1993;Ormerod and Kubbies, 1992). This transient behavior is afeature of how the total population densities in each ofthe cell cycle phases are evolving to the eventual constantproportion in fully balanced growth. Thus, the measure-ment of labeled and unlabelled population densities pro-vides the vectoral transient data that has the potential toidentify vectoral transition rates without explicitly measur-ing age distributions. While BrdU is used herein todetermine the age-structured cell cycle phase transitionrates, similar experimental approaches can be used todetermine cell division rates. The labeled mitosis methodusing 3H-thymidine (Steel, 1977) or the transient cellgeneration distributions from pkh dyes or carboxyfluor-escein diacetate succinimidly ester (Lyons, 1999; Nordonet al., 1999) also produce transient data that can be used todetermine the age-structured cell division rate.

In presenting the methodology for such identification, themodel and mathematical foundation are first developed andfollowed by an in vitro illustration for its use. The AnalyticalFoundation Section presents the cell growth model describ-ing BrdU labeling with a proof of extractions of transitionrates from subpopulations in unbalanced growth while thetotal populated maintains balanced growth. The Experi-mental Materials and Methods Section outlines theexperimental materials and methods used to collect invitro data. The age-dependent transition rates that resultfrom the experimental BrdU results are discussed in theResults and Discussion Section, and the fitness is comparedwith normalized Gaussian, lognormal, and skew lognormalage density derived transition rates. The ConclusionsSection summarizes the major conclusions and findings.

Analytical Foundation

This section introduces an age-structured cell cycle model(Model Structure and Derivation of Initial Conditions

Sherer et al. : Cell Cycle Phase Transitions 961

Biotechnology and Bioengineering. DOI 10.1002/bit

Sections), proves that the model predictions are sufficient toidentify age-structured transition rates using BrdU labeling(Extraction of Age-Dependent Parameters From PartitionedPopulation Data Section), and shows the simulationmethodology for comparison with the BrdU pulse-labelingexperiments (Implementation Section).

Model Structure

The division cycle of a cell is discretized into three phases inour model: (1) a combined G0 and G1 phase, (2) an S phase,and (3) a combined G2 and M phase (see Fig. 1). Newlydivided cells may either be in the quiescent G0 phase orpreparing to undergo DNA synthesis in the G1 phase. Thesetwo states were indistinguishable experimentally and arecombined into a single phase. The S phase is a period whenthe DNA and many proteins are replicated before celldivision. Cells then enter the second gap phase, G2, beforephysically dividing by undergoing mitosis, M. As the G2 andM states were also indistinguishable experimentally, they arelumped into a single phase. The cells were maintained in anutrient rich environment throughout the in vitro experi-ments so a low level of quiescence is anticipated.

The number density and transition rate of cells in theith phase can be written as ni(t, t) and Gi(t) respectivelywhere t is the age, or time elapsed since entering the currentphase, and t is the time. It is assumed that the transitionrate is stable, or time independent, in balanced growthconditions but the number density will increase with time ascells divide. Neglecting cell death, the population balanceequation can be written as

@nðt; tÞ@t

þ @nðt; tÞ@t

¼ �DoutðtÞnðt; tÞ (1)

where

nðt; tÞ ¼n1ðt; tÞn2ðt; tÞn3ðt; tÞ

24

35

and

DoutðtÞ ¼G1ðtÞ 0 0

0 G2ðtÞ 0

0 0 G3ðtÞ

24

35

Figure 1. Phases of cell cycle model. Cells transition out of the i th phase

according to an age-dependent transition rate (Gi) with cell division occurring via

the G2/M to G0/G1 transition.


The age density is subject to the initial condition

nðt; 0Þ ¼ n0ðtÞand a boundary condition that consists of cells that have just

transitioned out of a phase to become new cells in the next

phase.

nð0; tÞ ¼Z 1

0

DinðtÞnðt; tÞ dt

where

DinðtÞ ¼0 0 2G3ðtÞ

G1ðtÞ 0 0

0 G2ðtÞ 0

24

35

Under balanced growth conditions, the age distribution istime invariant so that the ith phase number density can beseparated into the product of the population density, Ni(t),and the age-density, fi(t)

niðt; tÞ ¼ NiðtÞfiðtÞ; where

Z 1

0

fiðtÞ ¼ 1 (2)

Substitution of Equation (2) into the population balanceequation (Eq. (1)) and integration over all ages gives anordinary differential equation for the population increase ineach of the phases (see Appendix)

dNðtÞdt

¼ DNðtÞ (3)

where

Dij ¼Z 1

0

Dinij ðtÞ � Dout

ij ðtÞh i

fjðtÞ dt

Only two quantities are required for simulating thissystem: transition rates and initial number densities. Asshown in the Derivation of Initial Conditions Section, thebalanced growth age distributions and associated numberdensities can be calculated using only the transition ratesunder balanced growth conditions. This implies thatknowledge of the transition rates is sufficient for completemodel characterization.

Derivation of Initial Conditions

Under balanced growth, the matrix of rate constants, D, istime invariant and its positive real eigenvalue, m, is thespecific growth rate for both the total population and eachcell cycle phase. The system of ordinary differential equa-tions can be written in terms of m and the correspondingeigenvector, h

mh ¼ Dh

where the normalized eigenvector hi=P

j hjis the fraction of

cells in the ith cell cycle phase.

To find the age-distribution within each phase, the partialdifferential equation system is first reduced to ordinary by

DOI 10.1002/bit

substitution of the separated number density (Eq. (2)) intothe population balance equation (Eq. (1)). As the totalnumber of cells in each phase increases according to thesame overall specific growth rate in balanced growth,ordinary differential equations for the change in the agedistribution with age are obtained (see Appendix)

dfiðtÞdt

¼ �X3

j¼1

ðDoutij ðtÞfjðtÞÞ þ mfiðtÞ

" #(4)

Using the normalization condition stated in Equation (2),the age distribution in each phase can be written as

fiðtÞ ¼�Z 1

0

exp

�Z t0

0

ðDoutii ðt00Þ þ mÞ dt00

�dt0

��1

exp

�Z t

0


� (5)

Figure 2. Decomposing the Jurkat cells into cell cycle phases with BrdU labeling the S

cells in each phase. a: A bivariate BrdU versus DNA content density plot is used to distinguis

below the line). b: A DNA histogram of the total population is used to determine the percen

(BrdUþ) cells. d: DNA histogram of unlabeled (BrdU�) cells.

So, during balanced growth, the age-distributed numberdensity is given as

niðt; tÞ ¼ hiPj hj

fiðtÞX

j

NjðtÞ (6)

This expression can be used as an initial condition beforea perturbation is introduced through the addition of BrdU.While the Model Structure Section describes an age-structured cell cycle model subject to the initial age-distributions derived in the Derivation of Initial Conditions,the Extraction of Age-Dependent Parameters From Parti-tioned Population Data Section addresses the question as towhether BrdU labeling experimental data is sufficient toidentify the transition rates or age-distributions.

Extraction of Age-Dependent Parameters FromPartitioned Population Data

In a balanced growth condition there is no unique set oftransition rates and initial conditions that describes the total

phase cells 1 h after labeling. ModFit software is used to determine the percentage of

h cells that incorporated BrdU (cells above the drawn line) from those that did not (cells

tage of cells in each phase for the overall cell population. c: DNA histogram of labeled



population data. Unbalanced growth data with a knowninitial age distribution and minimal fluctuations in transi-tion rates are needed for their identification, and BrdUpulse-labeling of a balanced growth population satisfies allof these requirements. BrdU labeling (Fig. 2a) partitions thebalanced growth population (Fig. 2b) into two subpopula-tions (Fig. 2c and d). Each of these subpopulations willundergo unbalanced growth while the total population con-tinues in balanced growth with unaltered transition rates.

BrdU is an analog to the DNA base thymidine, a uridinederivative, such that cells that are actively synthesizing DNA,the S phase cells, will incorporate greater amounts of BrdUthan cells in the G0/G1 or G2/M phases. By pulsing, orremoving excess BrdU after a short time period, only thosecells that were in the S phase at the time of application, t¼ 0,will incorporate the BrdU. (We found that direct removal ofall excess BrdU from the media proved difficult experi-mentally. However, labeling an aliquot of the originalpopulation, removing excess BrdU, and then recombiningwith the remainder of the original cell population does resultin pulse labeling. The BrdU unexposed cells likely absorbresidual BrdU but in small enough amounts to appear asBrdU unlabeled.) This labeling process may be representedby a pre-multiplying projection matrix, L, which concur-rently produces a second projection matrix, I� L, so thatthere are two subpopulation distributions among thedifferent phases described by Ln0(t) and (I� L)n0(t).The BrdU technique produces the following

Ln0ðtÞ ¼0

an0;2ðtÞ0

264

375;

I � Lð Þn0ðtÞ ¼an0;1ðtÞ þ ð1 � aÞn0;1ðtÞ

ð1 � aÞn0;2ðtÞan0;3ðtÞ þ ð1 � aÞn0;3ðtÞ

264

375

(7)

where a is the fraction of the total population exposed toBrdU. (In the BrdU experiments the aliquot of BrdUexposed cells is added to a population not exposed toBrdU—we found this helped achieve true pulse-labeling.The unlabeled boundary condition in Equation (7) reflectsthis mixture.) It is now of interest to follow the populationdistributions obtained by solving Equation (1) for the twoinitial conditions stated above. However, since only thetotal, rather than age-distributed, population density ismeasurable, the equations of interest are only those in thetotal population density (Eq. (3)), for the two initialconditions as shown below

dNðtÞdt

¼ DðtÞNðtÞ;

Nð0Þ ¼ LN0 labeled population

ðI � LÞN0 unlabeled population

� (8)


where the variations in age-density with time lead to atransition matrix, D, that now also becomes a function oftime.

The population at the time of labeling, N0, is in balancedgrowth, so it is contained purely in the eigenspace, Em, whichcorresponds to the specific growth rate eigenvalue, m.Another way of stating this is to say that the projection, Pm,of all three-dimensional vectors onto Em maps N0 into N0

(i.e., PmN0 ¼N0). This property clearly cannot be shared byLN0 and (I� L)N0 simultaneously as PmL 6¼ L. Thus, weconclude that the transients produced by either of the initialconditions in Equation (8) will be truly three-dimensionaland, in principle, sufficient for the identification of the threetransition rates contained in the matrix Dout(t). The pulse-labeling of cells with BrdU is required for this identificationto take place.

Implementation

After calculation of the initial number densities, transientnumber densities are computed using a variation of themethod of characteristics (Ramkrishna, 2000): the methodof successive generations (Liou et al., 1997). The numberdensity for the ith phase, ni(t, t), is discretized into a series ofk number densities, n

ðkÞi , with k representing the number of

transitions undergone or generations. The BrdU labelingfurther separates each initial number density into labeledand unlabeled portions so the initial generation (k¼ 0)number densities are given by Equation (7). For example, allcells beginning the S phase (i¼ 2, k¼ 0) are labeled and willmove to the G2/M (i¼ 3, k¼ 1) then G0/G1 (i¼ 1, k¼ 2)phases before returning to the S phase (i¼ 2, k¼ 3).Following this pattern, the total phase number density oflabeled and unlabeled cells will be

Labeled ðBrdU þ Þ Subpopulation

n1ðt; tÞ ¼ aP1k¼0

nð3kþ2Þ1 ðt; tÞ


nð3kÞ2 ðt; tÞ


nð3kþ1Þ3 ðt; tÞ

Unlabeled ðBrdU � Þ Subpopulation

n1ðt; tÞ¼P1k¼0

ðnð3kÞ1 ðt; tÞþn

ð3kþ1Þ1 ðt; tÞþð1 � aÞnð3kþ2Þ

1 ðt; tÞÞ

n2ðt; tÞ¼P1k¼0

ðð1 � aÞnð3kÞ2 ðt; tÞþn

ð3kþ1Þ2 ðt; tÞþn

ð3kþ2Þ2 ðt; tÞÞ

n3ðt; tÞ¼P1k¼0

ðnð3kÞ3 ðt; tÞþð1 � aÞnð3kþ1Þ

3 ðt; tÞþnð3kþ2Þ3 ðt; tÞÞ

(9)

The number densities for each of the generations can befound by backtracking along the characteristics until either

DOI 10.1002/bit

the initial condition, for the k¼ 0 generation, or boundarycondition, for the k> 0 generations, is reached

Initial Conditions :

t tnð0Þi ðt; tÞ ¼ n

ð0Þi ðt � t; 0Þ exp �

R t

t�t Giðt0Þ dt0� �

nðkÞi ðt; tÞ ¼ 0; k ¼ 1; 2; . . .

(

Boundary Conditions :

t > tnð0Þi ðt; tÞ ¼ 0

nðkÞi ðt; tÞ ¼n

ðkÞi ð0; t � tÞexp �

R t

0 Giðt0Þ dt0� �

; k ¼ 1; 2; . . .

(

(10)

The initial cell distribution is known, and the boundaryconditions are calculated successively as the boundary celldensity depends on the final solution in the previous phase

nðkÞð0; tÞ ¼Z 1

0

Dinðt; tÞnðk�1Þðt; tÞ dt (11)

The final number densities are calculated and the appro-priate combinations of these final number densities (Eq. (9))describe the dynamics of the subpopulations.

Experimental Materials and Methods

Materials and Methods

Jurkat cells were grown in 75 cm2 flasks (Corning 430725) at378C and 5% CO2 in 20 mL of a 90/10 mixture of Iscove’sModified Dulbecco’s Medium (Sigma I3390) and fetalbovine serum (Sigma F2442) supplemented with 1 mL L-glutamine (Sigma G2150)/10 mL medium. Exponentialgrowth was maintained for a week prior to the addition ofBrdU by passing the cells every 2 days so concentrationsremained between 105 and 106 cells/mL. One day before theaddition of BrdU, cells were passed to create two flasks at aconcentration of approximately 2 105 cells/mL in 20 mL ofmedia. Also 1 day before pulse labeling, 20 mL of completemedia was placed in the incubator to pre-warm and pre-gasin preparation for re-suspension after pulse labeling.

The BD Biosciences BrdU Flow Kit (559619) was used forthe BrdU procedures. A 10 mg/mL stock BrdU in 1 DPBSsolution was diluted to 1 mM by adding 31 mL of the stock to1 mL of 1 DPBS. For pulse-labeling, a 4 mL aliquot isexposed to 40 mL of the diluted BrdU solution (10 mM finalBrdU concentration) in a 50 mL centrifuge tube. Forty-fiveminutes after inoculation, the remaining 16 mL of cells weretransferred to the centrifuge tube (so a, the fraction ofunlabeled cells, ¼0.2) which was then spun at 1,000 rpm for5 min before resuspending the cells in 20 mL of pre-warmedand pre-gassed complete media and transferring to theoriginal flask.

Samples of 1 mL were collected every 2–4 h forapproximately 40 h beginning with the newly resuspendedcells. Samples were washed, after centrifuging for 5 minat 1,000 rpm and discarding the supernatant, in 1 mL of

Pharmingen Stain Buffer (PSB); resuspended in 100 mLof BD Cytofix/Cytoperm buffer for 15 min at room tem-perature; washed in a 1 mL of 1 Perm/Wash solution; andgiven a final wash in 1 mL of PSB. The cells were preservedby re-suspending in 1 mL of PBS and storing in a 48C freezer.Cytometry analysis was performed within a week of samplecollection.

In preparation for cytometry analysis, the samples werecentrifuged, resuspended in 100 mL Cytofix/Cytopermbuffer for 5 min, washed with 1 mL of 1 Perm/WashBuffer, and resuspended in 100 mL of a 30% 1 mg/mL DNaseand 70% 1 DPBS solution all at room temperature. Thecells are incubated for one hour at 378C. The cells werelabeled for BrdU uptake after they are washed in 1 mL ofPerm/Wash Buffer and resuspended in a solution containing1 mL of FITC and 50 mL of 1 Perm/Wash buffer for 20 min.The DNA was then labeled by washing cells in 1 mL of 1Perm/Wash buffer, and resuspending in 1 mL of propidiumiodide (PI) solution (30 mL of 0.1% (v/v) Triton X-100(Sigma 93443) in PBS, 6 mg DNAse-free Rnase A (SigmaR5503) and 600 mL of a 1 mg/mL PI stock solution (SigmaP4864)).

Samples were immediately taken to a BD-Elite flowcytometer where 10,000 cells were analyzed per sample. Anargon laser was used for excitation at 488 nm with the FITCemission collected at 525 nm and the PI emission at 675 nm.There was very little emission overlap so no compensationwas required. The labeled and unlabeled cells were gatedusing FITC emission and DNA histograms of bothpopulations were measured with the PI. This pulse labelingBrdU experiment was performed in duplicate and the twocultures, derived the same mother culture, gave extremelyconsistent results at all time points.

Sample Analysis

The percentage of cells in each phase for labeled andunlabeled cells was extracted from the cytometry data bydifferentiating phases based on the DNA content. Thebivariate BrdU incorporation against DNA content densityplot were then used to gate the BrdU labeled and unlabeledcells (Fig. 2a). The phase percentages were determined byplotting the DNA histogram of each of these subpopulations(see Fig. 2c and d) and using the ModFit DNA labelingcell cycle analysis software package to define the phaseboundaries.

Results and Discussion

Extraction of Phase Transition Rates FromJurkat Cells Using BrdU Methods

Each transition rate function was fit to experimental BrdUtotal number density data in each labeled and unlabeledcell cycle phase using the Matlab multivariate mini-



Figure 3. A series of bivariate BrdU versus DNA cell density plots illustrate the

progression of the labeled and unlabeled subpopulations progression around the cell

cycle. Ten thousand cells were collected for each figure with speckled or empty areas

representing lower densities. The line drawn through each of the plots separates the

BrdU labeled population from the unlabeled population. a: Immediately after BrdU

inoculation (t¼ 1 h). b: Nine hours after BrdU inoculation. c: Twenty-four hours after

BrdU inoculation.

mization function ‘‘fminsearch.’’ This function employs aNelder–Mead Simplex method to minimize the chosenleast-squares objective function. The least-squares error, E,is the summation over all data points collected—Nsamples

collected (13 samples) and Nphases phases in the independentsubpopulations (three unlabeledþ three labeled phases)—of the squared difference between the experimental cell cyclephase percentages, x

expi;j , and the model predicted percen-

tages, xsimi;j , that result from the proposed transition rates

E ¼XNphases

i¼1

XNsamples

j¼1

ðxexpi;j � xsim

i;j Þ2 (12)

For each independent subpopulation, the experimentalphase percentage in the ith phase and jth time sample (wherethe jth sample is collected at time tj) is the ratio ofthe number of cells in the phase to the total number of cells in thesubpopulation: x

expi;j ¼ N

expi;j =

Pk N

expk;j . Similarly, the simula-

tion phase percentages are

xsimi;j ¼ Nsim

i ðtjÞ=X

kNsim

k ðtjÞ

where

Nsimi ðtjÞ ¼

Z 1

0

niðt; tjÞ dt:

In most cases, hypotheses and models are writtenregarding the transition age probability distribution, hi(t),rather than the transition rate, Gi(t), itself. The likelihood ofchanging cell cycle phases within a certain age window isrepresented by that phase’s transition-age probabilitydistribution. While this distribution may be observableexperimentally, a kinetic model, such as the populationbalance models, use rate terms such as the rate of transitionfor a cell of a given age. The age-dependent transition ratescan be calculated from the transition-age distributions(Eakman et al., 1966)

GiðtÞ ¼hiðtÞ

1 �R t

0 hiðt0Þ dt0ð13Þ

such that

hiðtÞ ¼ GiðtÞ 1 �Zt

0

hiðt0Þ dt0

24

35; hið0Þ ¼ 0

In this study, we evaluated and compared the transition ratesderived from normalized Gaussian, lognormal, and skewedlognormal transition rates, along with a more flexibletransition rate representation utilizing series of cubic splineswith movable spline nodes locations. Ages up to 50 h wereevaluated in 0.25 h increments. Polynomials were used toapproximate all transition rates with 71 orders giving anexcellent fit over the entire age ranges for the transition ratesinvestigated.


The flow cytometry data was analyzed as described in theSample Analysis Section. The results for pulse-labeling showthat the BrdU labeled cells are initially synchronized in the Sphase (Fig. 3a). As time progresses, this plug moves to thelabeled G2/M phase and then divides into the labeled G0/G1

phase (Fig. 3b) before reentering the labeled S phase(Fig. 3c). The unlabeled cells are also initially synchronizeddue to labeling of the S phase cells, but to a lesser degree thanthe unlabeled population due to the addition of the cells notexposed to BrdU. The rate of loss of the synchrony in thelabeled and unlabeled subpopulations is dependent uponthe shape of the cell cycle phase transition rates.

Determination of Unrestricted Age-DependentTransition Rates Using BrdU Pulse-Labeling

To determine the general shape of the cell cycle phasetransition rates, a flexible representation was first implemen-ted using a series of 15 cubic splines for each of the three cellcycle phase transition rates. The transition rates weremanipulated to best fit the average of the two BrdU pulse-label experiments. (Again, these two experiments wereconducted simultaneously and cells from each experimentshare the same mother culture.) The spline representationwas flexible in that both the value of the spline (transitionrate) as well as its location (age) could be manipulated toreduce errors due to improper spline node placement. Theparameters associated with the age-dependent transitionrates were identified by fitting the cell cycle data; the fitted

DOI 10.1002/bit

Figure 4. Best fit of cell cycle phase transition rates to BrdU pulse-labeling data

using a 15 node cubic spline representation for the transition rate out of each phase.

Both the rate and age locations of each node are variable in the minimization. a: Cell

cycle phase transition rates—spline nodes and the polynomial approximation of the

cubic spline fit. b: Cell cycle phase transition age densities resulting from those

transition rates.

cell cycle phase age-dependent transition rate are shown inFigure 4a and the resulting transition age distributions inFigure 4b. The fit of the thick lines shown in Figure 5a and bto the experimental pulse label BrdU data points illustratesthe accuracy of the fitted model predictions.

Each of the phase transition-age distributions has its ownunique shape and characteristics. Both the S and G2/M phasetransition-age probability distributions display a positivebias but with relatively small standard deviations (Table II)though the S phase transition is later and slightly broaderthan the G2/M phase. This would imply that for the Sand G2/M phases: (i) cells require at least a certain amountof time to progress through the phase, (ii) cells movethrough the phase at nearly identical rates, and (iii) delaysdue to checkpoints or heterogeneity are either unlikely or

minimal. While the age of cells transitioning out of the Sphase or G2/M phase is relatively consistent, the transitionage probability distribution for the G0/G1 phase is actuallybimodal.

To assess the significance the bimodal phenomenon andensure it is not an artifact of (i) over-fitting with the splines,(ii) the pulse data itself, or (iii) the balanced growthrestrictions, the effect of each of these factors was evaluated.The importance of bimodal distributions is compared withsingle mode distributions in the Determination of Alter-native Functional Forms for the Age-Dependent TransitionRates Section and the predictions based on pulse-dataexperiments are compared with continuous labeling data inthe Comparisons With Continuous Labeling Data Section.The implication of the assumption of balanced growth wasexamined by including the initial phase percentages of eachsubpopulation as fitted parameters. (The initial phasepercentages need not sum to 1 in this case.) As seen inFigures 4 and 5a and b, neither the cell cycle phase transitionage probability distribution nor the fits to the experimentaldata were altered significantly.

Determination of Alternative Functional Formsfor the Age-Dependent Transition Rates

While the flexible spline representation of the phasetransition rates does an excellent job of matching thepulse-data, the high number of degrees of freedom may leadto over-fitting. That is, simpler or regularized representa-tions may fit nearly as well; in particular, the two transitionage modes seen the in the G0/G1 phase may not be crucial toexplaining the data. To this end, other basis functions for theage-dependent transition rates, derived from transition agedensities of the lognormal, skewed lognormal, and normal-ized Gaussian distributions were explored. Since Lagrangianinformation was not used, the only quantitative method todetermine the likelihood of a transition age distributions isthe residual fit to the experimental data. A comparison of theleast-squares errors, provided in Table I, indicates that theflexible cubic splines functional form does fit the experi-mental data better than the alternative individual functionalforms for determining the age-dependent transition rates.Table II lists the best fit parameters for the alternativefunctional forms. These differences can be seen directlywhen the model predictions using the spline and lognormalderived transition rates are compared with the experimentalphase percentages (see Fig. 5c and d). Particularly for theBrdU labeled subpopulation, the errors in the singlelognormal predictions become progressively larger as timeprogresses while the spline fit remains accurate. In light ofthe extremely tight confidence intervals on the experimentaldata, single distributions are not as adequate as the spline fit.

When normalized Gaussian distributions are used for thetransition probability with respect to age, the G0/G1

representation (Fig. 6a) becomes distorted into a physio-logically unlikely arrangement. The estimated G0/G1 normaldistribution is positioned such that most of the young cells



Figure 5. Comparison of BrdU pulse-labeling subpopulation cell cycle phase experimental data with best-fit flexible spline model predictions. The experimental data in each

phase is represented by: *, G0/G1; , S; �, G2/M. The thick lines correspond to model predictions when the cell cycle phase transition rates are fit to only the pulse data and the

initial phase percentages are defined by the transition rates. The thin lines are best-fit model predictions when (a and b) both pulse and continuous data (Comparisons With

Continuous Labeling Data Section) is fit and the initial phase percentages are minimization input parameters or (c and d) single lognormals are fit to pulse data (Determination of

Alternative Functional Forms for the Age-Dependent Transition Rates Section). a and c: Pulse-labeling dynamics for the BrdU labeled subpopulation. b and d: Pulse-labeling

dynamics for the BrdU unlabeled subpopulation.

Table I. Least-squares fit, E, when the minimization is performed for each

distributions examined along with the number of parameters manipulated

to attain that fit.

Fit to pulse data only:

restricted initial

condition

Total number of

fitted

parameters

E (103),

pulse

data

E (103),

continuous

data

Flexible cubic splines

(defines the transition rates)

90 1.9553 1.0004

Normal 6 3.2681 1.3472

Lognormal 6 2.5217 1.1682

Skewed lognormal 9 2.4832 1.1908

2 lognormals/phase 15 1.6891 1.3655

3 lognormals/phase 24 1.6893 1.3146


just entering the G0/G1 phase transition quickly while thelarge variance still enables the transition of older cells: that is,the increased variance leads to a likely over-prediction ofyounger cell division rates, even though the residual’s errorwhen fit to the data is relatively low. When a lognormal isused as the transition-age distribution (Fig. 6b), the G0/G1

phase distribution is again extremely broad. Skewedlognormal distributions were intuitively appealing becausethey can enhance the bias of distributions, but this bias is oflittle use in capturing the bimodal behavior. The skewedlognormal fit is similar to that of the lognormal (Fig. 6c). Nosingle distribution per phase can capture the cubic-splineestimated bimodal behavior of the G0/G1 phase, but rather,

DOI 10.1002/bit

Table II. Summary of the parameter sets attained from the best fits of each of the phase transition representations.

Phase Mean, m (h) Standard deviation, s (h) Weight, w, or asymmetry

Distribution: restricted initial condition

Cubic spline (calculated) G0/G1 16.2627 9.3577

S 11.2515 2.5025

G2/M 6.3182 1.7564

Single transition age distributions

Normal G0/G1 11.31 13.39

S 10.50 0.70

G2/M 5.79 0.51

Lognormal G0/G1 16.59 14.75

S 10.49 1.04

G2/M 5.76 0.80

Skewed lognormal G0/G1 16.70 14.15 1.00

S 10.04 1.07 1.08

G2/M 5.64 0.82 0.94

Lognormal basis functions’ transition age distributions

2 lognormal combination G0/G1 ½ 7:57 25:61 ½ 1:00 2:21 ½ 0:55 0:45 S ½ 10:52 15:22 ½ 1:00 1:21 ½ 0:92 0:08

G2/M ½ 6:09 6:26 ½ 0:99 1:13 ½ 0:59 0:41 3 lognormal combination G0/G1 ½ 7:75 25:39 7:93 ½ 1:15 1:62 1:29 ½ 0:50 0:45 0:05

S ½ 10:47 14:45 10:53 ½ 1:23 1:08 1:27 ½ 0:87 0:09 0:04 G2/M ½ 6:03 5:28 8:77 ½ 0:91 1:00 1:39 ½ 0:99 0:01 0:00

Distribution: unrestricted initial condition

3 lognormal combination ( f1,0þ¼ [0.00 106.33 0.23], G0/G1 ½ 25:13 8:79 5:28 ½ 1:87 1:39 0:99 ½ 0:36 0:59 0:05 f1,0�¼ [54.21 18.51 18.07], f1,0¼ [54.94 32.79 10.40]) S ½ 11:22 10:73 5:25 ½ 1:24 1:65 1:28 ½ 0:05 0:95 0:00

G2/M ½ 6:75 6:09 5:40 ½ 1:30 1:14 0:89 ½ 0:01 0:25 0:74

Empty items imply that that parameter does not influence that distribution.

this results in a more disperse distribution to accommodatethe data as indicated by the larger standard deviationsassociated with the G0/G1 phase in Table II. Regardless ofthese differences in the G0/G1 phase, the means andvariations seen in the S and G2/M phases for each of thesingle mode transition age distributions, are similar to thoseof the flexible splines (Fig. 4 vs. Fig. 6; Table II).

What effect does a broad G0/G1 phase versus abimodal G0/G1 phase have on ability of the model to fitthe BrdU pulse data? Comparing the best predictions for theBrdU labeled dynamics, the broad G0/G1 phase doesproduce oscillations of slightly lower amplitude than thebimodal fit, but this looks to be of marginal significance inrelating to the experimental BrdU pulse data (Fig. 5c). Inaddition, the predictions for the BrdU unlabeled subpopu-lation using the broad or bimodal G0/G1 phase are nearlyindistinguishable (Fig. 5d).

Lognormal Basis Functions

The predictions of the single mode distributions wereimproved upon by using a weighted combination of eithertwo or three lognormal distributions to describe thetransition age distribution in each phase where the means,standard deviation, and weights are to be determined

hlognormali ðtÞ ¼

w1hlognormali;1 þ w2h

lognormali;2

þ w3hlognormali;3 ;

Xj

wj ¼ 1

This description invites the possibility of multiple sub-

populations or more distributed transition-age distributions

using fewer parameters. We found that the combinations of

lognormals orient themselves to form transition age-

distributions that are similar in shape to the flexible spline

nodes representation (Fig. 4b vs. Fig. 7a and b). In the S

and G2/M phases, the two lognormals in each phase have

considerable overlap and the superposition forms distribu-

tions with means and standard deviations that are slightly

greater those derived using the splines (see Table II).

However, there were significant differences in the means of

the two lognormals that comprise the G0/G1 phase which

results in a transition age probability distribution that is very

similar to that found with the splines. Using three lognormal

distribution per phase results in predicted labeled and

unlabeled phase dynamics are that are similar between when

cubic-splines (Fig. 4b) and two lognormals per phase

(Fig. 7a) though the number parameters used was greater

than five times less than that required for the spline fit.

Comparisons With Continuous Labeling Data

For validation of this method, data was obtained from acontinuous BrdU label where all cells eventually becomeBrdU labeled during progression through the S phase. Thisdata explicitly shows the decay in the G0/G1 and G2/Mphases. All cell preparation and assay experimental protocolsare the same as described earlier for the pulse labeled BrdUexperiments with the exception of BrdU labeling of theentire 20 mL flask of cells. (The flask contents is transferred



Figure 6. Best fit of cell cycle transition rates to BrdU dynamic data from single transition-age distributions for each phase. a: Gaussian cell cycle phase transition-age

distributions. b: Lognormal cell cycle phase transition-age distributions. c: Skewed lognormal cell cycle phase transition-age distributions.

to a centrifuge tube and exposed to 200 mL of the dilutedBrdU solution; no cells are left unexposed.) As before, thesecontinuous BrdU labeling experiments were performed induplicate with extremely consistent results at all time points.

During continuous labeling, unlabeled cells entering the Sphase become labeled with BrdU as shown in the bivariateBrdU–DNA plots time sequence of Figure 8. The unlabeledpopulation is initially in unbalanced growth. This data isused as an independent verification for the G0/G1 and G2/Mpredictions based on the pulse data. In light of thecontinuous labeling observed in the experimental data,Equation (9), which describes the phase number densities ofthe labeled and unlabeled cells, must be modified to reflectthe accumulation of labeled cells. Equation (14) accounts for


this accumulation and can be used in lieu of Equation (9) forcontinuous labeling

Labeled ðBrdU þ Þ Subpopulation

n1ðt; tÞ ¼P1k¼2

nðkÞ1 ðt; tÞ

n2ðt; tÞ ¼P1k¼0

nðkÞ2 ðt; tÞ

n3ðt; tÞ ¼P1k¼1

nðkÞ3 ðt; tÞ

Unlabeled ðBrdU � Þ Subpopulation

n1ðt; tÞ ¼ nð0Þ1 ðt; tÞ þ n

ð1Þ1 ðt; tÞ

n2ðt; tÞ ¼ 0

n3ðt; tÞ ¼ nð0Þ3 ðt; tÞ

(14)

DOI 10.1002/bit

Figure 7. Linear combinations of lognormal distributions used to approximate

the residence time distribution of each cell cycle phase. The added lognormal

distributions give additional flexibility to the G0/G1, S and G2/M distributions.

a: Two lognormal distributions per phase. b: Three lognormals per phase.

Figure 8. A series of bivariate BrdU versus DNA cell density plots illustrate the

progression of the labeled and unlabeled subpopulations progression around the cell

cycle. Ten thousand cells were collected for each figure with speckled or empty

regions representing lower cell densities. a: Immediately after BrdU inoculation

(t¼ 1 h). b: Ten hours after BrdU inoculation. The arrow denotes movement of BrdU

unlabeled cells becoming labeling during the S phase. c: Twenty-four hours after BrdU

inoculation.

The pulse data derived transition rates for both the flexiblespline and lognormal distributions were used to predict thedynamics of the continuous BrdU. The least squares errorquantifying the predicted model dynamics fit to the con-tinuous experimental data are provided in Table I for allassumed distribution forms. As illustrated in Figure 9 thespline and lognormal distribution forms both capture thetrends of the continuous labeling experimental data.

Throughout the continuous BrdU labeling experiment,the ratio of labeled cell fluxes into and out of the S phaseremains constant hence, it impossible to determine the agestructure of the S phase transition rates using this data alone.When the pulse-data (two independent subpopulations) issupplemented with the continuous data (one subpopula-tion) for fitting spline transition rates without the restriction

of balanced growth using all available data, 33 parametersare fitted to three lognormals per phase. The identifiedtransition rates are extremely consistent (Fig. 4b) from whenthe pulse data is used alone, and the fit to the pulse data isalso similar (see Fig. 5a and b) with a least squares fit error tothe pulse data of 1.1972 103. The fit to the continuousBrdU data is also improved (Fig. 9c and d) with a leastsquares fit error of 1.08163 103.

Conclusions

A methodology has been developed that enables predictionsof cell cycle phase age-dependent transition rates based onexperimental data with BrdU labeling where the crux of theidentification relies on comparison of the BrdU data withmodel predictions based entirely on the transition rates. Formaking model predictions based on transition rates, an age-structured cell cycle model is presented in which the age-dependent transition rates are posed. Model predictionsrequire knowledge of initial age-distributions, which areunidentifiable directly, so an expression that yields initialage-distributions from transition rates is derived under theassumption of a balanced growth setting. Since the transi-tion rates and an initial condition are sufficient for modelidentification, and the transition rates imply the initialcondition, the age-dependent transition rates are the only



Figure 9. Comparison of BrdU continuous labeling subpopulation cell cycle phase experimental data with best-fit flexible spline model predictions. The experimental data in

each phase is represented by: *, G0/G1; , S; �, G2/M. The thick lines correspond to model predictions when the cell cycle phase transition rates are fit to only the pulse data and

the initial phase percentages are defined by the transition rates. The thin lines are model predictions when (a and b) single lognormals are fit to pulse data while in (c and d) both

pulse and continuous data is fit and the initial phase percentages are minimization input parameters. a and c: Continuous labeling dynamics for the labeled population. b and

d: Continuous labeling dynamics for the unlabeled population.

input for making model predictions. We then illustratehow pulse BrdU labeling, which introduces instances ofunbalanced growth while maintaining individual cells andthe total population in balanced growth, is sufficient for theidentification of the unique set of transition rates from theunbalanced growth dynamics. The identified age transitiondistributions were validated through a comparison of thepredicted dynamics with those observed for continuousBrdU labeling.

In identifying the transition rates, a general representationof the transition rates using a series of cubic splines, whereboth the spline node values as well as locations were mini-mization parameters, resulted in transition probabilitydistributions with both expected and unexpected features.Very few transitions occur before a threshold age and after


which there is a window of ages when transitions are verylikely; however, for the G0/G1 phase, the best fit transitionage probability distribution suggested multiple subpopula-tions through its bimodal form. This bimodal behavior mayindicate the presence of multiple subpopulations resultingfrom either continuous, systematic differences between cellsor individual, discrete events. With multiple distinct modes,each subpopulation is governed by an independent set of cellcycle kinetics. Or, the subpopulations could result froma random event which may or may not occur each timethrough the cycle such as a checkpoint control. While theexistence of multiple subpopulations is able to be identifiedand their behavior characterized using age-dependenttransition rates, a Lagrangian approach where individualcells are followed would be required to distinguish between

DOI 10.1002/bit

the two mechanisms. Fit of the age probability distributionto the data assuming single normalized Gaussian, lognor-mal, and skew-lognormal transition probability distribu-tions was reasonable, but not as accurate due to the inabilityto capture the G0/G1 bimodal transition age distribution.But, describing each phase transition with a convex com-bination of two or three lognormal distributions improvedthe overall fit of the BrdU data relative to the splinerepresentation while requiring only 15 or 24 parametersrather than the 90 used for the spline fit. We see thisapproach as an efficient alternative with adequate flexibilityto capture the major features of data while minimizing thenumber of parameters to be identified.

The foregoing inversion involved no regularization as theerror bounds on the data were very small. However, largererror bounds would call for the use of appropriateregularization (Tikhonov, 1977).

The authors are grateful for the assistance of the Purdue University

Cytometry Laboratories and support from the Purdue Cancer Center.

Appendix

Derivation of Initial Age-Distributions(Age-Distribution During Balanced Growth)

Balanced growth was earlier defined as the growth conditionthat satisfies two criterions: (i) intransient growth condi-tions and (ii) time-invariant cell-state distributions. Time-independent transition rates satisfy the first condition, andthe second condition implies that the age-distribution in theith cell cycle phase is described by the time-independentdistribution fiðtÞ when balanced growth is achieved. So,during balanced growth, the number density in the ith phaseis niðt; tÞ ¼ NiðtÞfiðtÞ (Eq. (2)). Substitution of thisbalanced growth number density into the populationbalance equation for the ith phase (Eq. (1))

@niðt; tÞ@t

þ @niðt; tÞ@t

¼ �Douti;i ðtÞniðt; tÞ

gives

@ðNiðtÞfiðtÞÞ@t

þ @ðNiðtÞfiðtÞÞ@t

¼ �Douti;i ðtÞNiðtÞfiðtÞ

Since the total product density is age-independent and thebalanced growth age-distribution is time-independent, thisequation simplifies to

fiðtÞdNiðtÞ

dtþ NiðtÞ

dfiðtÞdt

¼ �Douti;i ðtÞNiðtÞ fiðtÞ (15)

A further implication of the time-independent age-distributions is identical specific growth rates for each cellcycle phase during balanced growth. Returning to thepopulation balance equation, substitution of the balancedgrowth separated number density niðt; tÞ ¼ NiðtÞfiðtÞ intoEquation (1) and integrating over all ages (where

R10 fiðtÞ dt ¼ 1) will give the rates of change in the total

number density

dNiðtÞdt

þ NiðtÞZ 1

0

dfiðtÞdt

dt ¼ �NiðtÞZ 1

0

Douti;i ðtÞfiðtÞ dt

The boundary and containment conditions can be used tosimplify the termZ 1

0

dfiðtÞdt

dt ¼ fiðtÞj1�fiðtÞj0¼ �X

j

Z 1

0

Dini;jfjðtÞ dt

to give Equation (3)

dNðtÞdt

¼ DNðtÞ

where

Dij ¼Z 1

0

½Dinij ðtÞ � Dout

ij ðtÞ fjðtÞ dt

The time independent age-distribution gives a time-independent coefficient matrix D. The specific growth rateof each cell cycle phase during balanced growth will be thepositive real eigenvalue of this matrix, m, so

dNiðtÞdt

¼ mNiðtÞ

The use of this expression in Equation (15) allows thetotal number density to be removed, leaving a differentialequation (Eq. 4) for change in the age-distribution with age

fiðtÞmNiðtÞ þ NiðtÞ dfiðtÞdt

¼ �P3

j¼1

Doutij ðtÞNjðtÞfjðtÞ

dfiðtÞdt

¼ �P3

j¼1

ðDoutij ðtÞfjðtÞÞ þ mfiðtÞ

" #dfiðtÞ

dt¼ �ðDout

ii ðtÞ þ mÞfiðtÞ

The general solution of this type of differential equation is

fiðtÞ ¼ fið0Þ exp �Z t

0


�

where the regulatory conditionR1

0 fiðtÞ ¼ 0 defines fið0Þ

R10 fiðtÞ dt ¼ 1 ¼ fið0Þ

R10 exp �

R t

0 ðDoutii ðt0Þ þ mÞ dt0

� �dt

) fið0Þ ¼ 1R1

0exp �

R1

0ðDout

iiðt0ÞþmÞ dt0

h idt

So, the age-distribution in the ith cell cycle phase duringbalance growth is given by Equation (5)

fiðtÞ ¼Z 1

0

exp �Z t0

0


�dt0

� ��1

exp �Z t

0


�



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DOI 10.1002/bit

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