Balance of Cell Proliferation and Death amongDynamic Populations: A Mathematical Model

Michael W. Miller

Department of Neuroscience and Physiology, State University of New York–Upstate MedicalUniversity, 750 East Adams Street, Syracuse, New York 13210

Research Service, Veterans Affairs Medical Center, Syracuse, New York 13210

Received 30 September 2002; accepted 6 May 2003

ABSTRACT: Developmental changes in cell num-bers represent the dynamic balance between cell prolif-eration and death. One obstacle to assessing this balanceis an inability to quantify the total amount of cell death,i.e., with a positive indicator such as terminal dUTPnick end labeling (TUNEL) or caspase activity. A novelmathematical model is described wherein data on dailycell growth (the change in cell number) and cell cy-cle kinetics can be used to determine the total amount ofcell death. Two sets of data from previously publishedstudies were tested in this model; primary cultured cor-tical neurons and B104 neuroblastoma cells. These twopreparations have contrasting features: neuronal cul-tures are heterogeneous and have relatively few cellsthat are actively cycling (i.e., the growth fraction forthese cells is low), whereas B104 cells are relativelyhomogeneous cultures in which the growth fraction ishigh. In primary cortical cultures, there was a balance incell production and death. Treatment with a potent

anti-mitogen, ethanol (400 mg/dl), affected this balanceprincipally by reducing cell production, although therate of cell death was also increased. In untreated B104cells, there was eight-fold more cell production than celldeath. Growth factors such as platelet-derived growthfactor BB doubled cell production. Ethanol reduced cellproduction by >60%, and it eliminated growth factor-mediated cell production. All of these changes occurredin the absence of an effect on the amount of cell death.Thus, the model is ideal for predicting the effects of anepigenetic factor (e.g., a growth factor, toxin, or phar-macological agent) on cell development and can be use-ful in determining the consequences of a genetic manip-ulation as well. © 2003 Wiley Periodicals, Inc. J Neurobiol 57:

172–182, 2003

Keywords: apoptosis; cell cycle; cerebral cortex; fetalalcohol syndrome; growth fraction; growth factors; neu-roblastomateratology

INTRODUCTION

The number of cells in the mature central nervoussystem (CNS) is determined not only by the prolifer-ation of glial and neuronal precursors and their suc-

cessful migration to their final residence, but also byselective neuronal death. Cell proliferation is a well-described phenomenon. Be it in vitro or in vivo,proliferating cells pass through the cell cycle. Accord-ingly, the cells sequentially pass through the G1, S,G2, and M phases, during which they prepare forDNA synthesis, replicate their DNA, prepare for celldivision, and then mitotically divide, respectively.

Various methods have been used for describing thepassage of cells through the cell cycle, i.e., the cellcycle kinetics. These methods include percent labeledmitoses (e.g., Atlas and Bond, 1965; Waechter andJaensch, 1972), flow cytometry (e.g., Bohmer, 1982;

Correspondence to: M. W. Miller (millermw@upstate.edu).Contract grant sponsor: N.I.A.A.A.; contract grant numbers:

AA06916 and AA07568.Contract grant sponsor: the Department of Veterans Affairs.

© 2003 Wiley Periodicals, Inc.

DOI 10.1002/neu.10265

172

Darzynkiewicz, 1983; Gray et al., 1986; Miller andKuhn, 1995), and cumulative labeling with bromode-oxyuridine (Nowakowski et al., 1989; Miller andKuhn, 1995; Takahashi et al., 1996; Luo and Miller,1997; Jacobs and Miller, 2000, 2001). The cumulativelabeling method is particularly instructive because itprovides not only information about cell cycle kinet-ics, but also about the proportion of cells that isactively cycling, the growth fraction (GF).

Death can occur at various times during normalCNS ontogeny. It can occur among proliferating (e.g.,Blaschke et al., 1996, 1998; Reznikov, et al., 1997;Thomaidou et al., 1997; Martens et al., 2000) andpostmitotic, differentiating populations (e.g., Al-Ghoul and Miller, 1989; Oppenheim, 1991; Miller,1995a; Price et al., 1997). Determining the scope ofneuronal death has been elusive for various biologicalreasons. One reason is that dying cells can exhibitdistinct features, for example, apoptosis and necrosis(Kerr et al., 1972; Darzynkiewicz et al., 2001), or theymay rely on one of various signal transduction path-ways (e.g., a p53-dependent or independent mecha-nism). Hence, it is difficult to identify a key proteinthat is associated with all modes of cell death.

The task of assessing cell death is confoundedbecause cells may express a key feature for only atransiently defined epoch during or at the end of thedegenerative process. Only a subset of the dying cellsis identified by a particular labeling method. In addi-tion, methodological complications in standard ap-proaches to documenting cell death can make it dif-ficult to determine the amount of cell death (Saraste,1999). Thus, no method that positively identifies dy-ing cells can fully account for the totality of celldeath.

The present study describes novel mathematicalmodel based on the temporal change in cell number(cell growth) and the cell cycle kinetics [the totallength of the cell cycle (TC) and length of the S phaseof the cell cycle (TS)]. This model relies solely onmeasures of cell proliferation; it does not depend uponquantitative measures of cell death. Thus, twostrengths of the mathematical model are (1) its abilityto segregate the contributions of cell proliferation andcell death, and (2) to account for the totality of celldeath regardless of mode or mechanism.

The rigor and flexibility of the model was testedusing various pairs of opposites. For example, data fortwo types of cultured cells were used: a heterogeneouspopulation (enriched primary cultures of fetal corticalneurons) and a homogeneous population (B104 neu-roblastoma cells). In addition, the effects of epigeneticagents that alter cell growth were examined.

Two mitogenic growth factors that affect the TC

and/or GF for neural cells are (platelet-derived growthfactor (PDGF) and basic fibroblast growth factor(bFGF, a.k.a. FGF-2) (Luo and Miller, 1997; 1999a).These agents can affect multiple developmentalevents. bFGF and PDGF can promote cell survival,although such death among cultured cortical cells andB104 cells is minimal (e.g., Dreher et al., 1989;Zawada et al., 1996; Iwasaki et al., 1997; Kawabe etal., 1997; Luo and Miller, 1997; Kuzis et al., 1999).Thus, these growth factors can be considered to besolely mitogenic.

The impact of an anti-mitogenic substance, etha-nol, was examined. Various studies show that ethanolinhibits cell proliferation in vitro (Kennedy and Muk-erji, 1986; Snyder et al., 1992; Luo and Miller, 1996,1999a; Jacobs and Miller, 2001) and in vivo (Kennedyand Elliott, 1985; Miller, 1986, 1989, 1995b, 1999;Miller and Nowakowski, 1991). On the other hand,ethanol can induce cell death (in vitro: Pantazis et al.,1993; Bhave and Hoffman, 1997; Seabold et al.,1998; Jacobs and Miller, 2001; in vivo: Miller, 1995b,1996, 1999; Kuhn and Miller, 1998; Goodlett andHorn, 2001; Mooney and Miller, 2001).

METHODS

Cell Culture Preparations

Two sets of published data were fit to the model: (1) thoseon purified primary cultured cortical neurons (Jacobs andMiller, 2000, 2001), and (2) on B104 neuroblastoma cells(Luo and Miller, 1997, 1999b).

Cortical Neurons. The cultures of cortical neurons wereheterogeneous, in cell type and in behavior. They containedneurons with different chemical and morphological pheno-types (e.g., Dichter and Delfs, 1981; Kreigstein and Dichter,1983; Connors and Gutnick, 1990). More importantly, al-though most of the cells were post-mitotic, a minority(�29%) was actively cycling (Jacobs and Miller, 2000,2001). These included not only nestin-positive neural stemcells, but also cells that appeared to be neurons. Despite thisheterogeneity, the cycling population behaved as a singlecycling population that exhibited one set of cell cycle ki-netics.

Primary cultures of neocortical cells were obtained from16-day-old rat fetuses (Seabold et al., 1998; Jacobs andMiller, 2000, 2001). The cells were dissociated, plated in amonolayer, and then raised in a medium supplemented with1.0% fetal calf serum for as many as 5 days. Some cultureswere treated with ethanol (400 mg/dl). The ethanol concen-tration was stabilized (variation of �5%) by maintaining thecultures in sealed containers. The medium was changedregularly.

Mathematical Model of Cell Proliferation and Death 173

B104 Rat Neuroblastoma Cells. Cultures of B104 cells areconsidered to be homogeneous. These cells have the abili-ties to proliferate and express the morphological and phys-iological features of neurons (e.g., Bottenstein and Sato,1979; Luo and Miller, 1997; 1996b). In untreated cultures,�75% of the cells are actively cycling and as a group theybehave as a desynchronized population with a uniform setof cell cycle kinetics.

B104 cells were plated at 50% confluency and raised ina serum-free medium for up to 5 days (Luo and Miller,1997, 1999b). The cells were treated with bFGF (0 or 15ng/ml) or PDGF-BB (0 or 15 ng/ml). Half of the cultureswere additionally treated with ethanol (400 mg/dl) using themethod described above.

Quantitative Measures

Three measures were used to estimate cell production andcell death. These measures were the daily change in thenumber of cells, the total length of the cell cycle (TC), andproportion of the population that is actively cycling, thegrowth fraction (GF).

The change in cell number (�n) was determined bycounting the numbers of cells that excluded trypan blue.This was an objective assay that includes total counts ofviable cells. Detaching or floating cells were rarely detectedregardless of the treatment (e.g., Luo and Miller, 1997;Jacobs and Miller, 2001). Cell counts were made on con-secutive days beginning with the day of plating, Day 0. Allcounts were based on at least three, duplicated independenttrials per day.

The TC and the GF were determined using a cumulativelabeling procedure (Nowakowski et al., 1989). The principalof this method is that cells incorporate bromodeoxyuridine(BrdU) during the S phase. Constant bathing of the culturesallows cells that newly arrive in the S phase to becomelabeled. Thus, with time, progressively more cells haveincorporated BrdU. The increase in the ratio of labeled tototal numbers of cells (the labeling index or LI) dependsupon the cell cycle kinetics. The TC and TS can be calculatedfrom the rate of the change of the LI. Previous in vitro (Luoand Miller, 1997; Jacobs and Miller, 2001) and in vivo(Miller and Nowakowski, 1991) studies show that untreatedand ethanol-treated cycling cells behave as a single popu-lation. That is, the data are best fit to a simple linear model.

After a certain amount of time, all the cells that arepassing through the cell cycle become labeled. The LI doesnot change. This LI is the GF.

The Mathematical Model

The change in the number of viable cells (�n) over 2consecutive days (Day x 3 Day x � 1) is equal to thedifference in the cell number over these 2 days, i.e.,nDay x�1 � n Day x.

�n�Day x3Day x�1� � �nDay x�1� � �nDay x� (1)

Likewise, the daily change in cell number is equal to thedifference between the number of cells produced (p) and thenumber that died (d) on a particular day.

�n�Day x3Day x�1� � �pDay x� � �dDay x� (2)

Implicit in the model is the ability to estimate cell deathby a deductive approach. That is, the amount of cell deathcan be determined without having to count the numbers ofcells that die. To make this deduction, it is imperative tocalculate the maximal, or potential, number of cells (pot n)that can be produced by the seeded population. The numberof cells that can be produced on a particular day (in theabsence of cell death) is determined by the potential numberof cells and the actual number of cells on the previous day.

pDay x � �pot nDay x�1� � �nDay x� (3)

The potential increase in the number of cells on a par-ticular day is determined by the starting number of cells(n

Day x), the GF, and a cell cycle ratio (cc). In the ideal

situation (i.e., when growth proceeds without restraints),this potential growth progresses at an exponential rate. Thisrate is defined by the TC and the GF.

pot nDay x�1 � �nDay x��1 � GF�ccDay x (4)

ccDay x � 24 hours/TC Day x (5)

By combining equations (4) and (5), a new equation forthe potential number of cells is described as

pot nDay x�1 � �nDay x��1 � GF�24 hours/TC Day x (6)

The number of cells produced on a particular day isdescribed by a new equation based on the combination ofequations (3) and (6).

pDay x � �nDay x��1 � GF�24 hours/TC Day x � �nDay x� (7)

Thus, the amount of daily cell production can be calcu-lated inasmuch as each of the values specified on the rightside of this equation can be experimentally determined.

Two examples of the utility of equation (7) are (A) asituation in which GF is maximal (i.e., 1.0) and the TC isshort (e.g., 6.0 h), and (B) conditions in which the GF isrelatively low (e.g., 0.5) and the TC is 24 h. In example A,the equation predicts that initial population (nDay x) of twoprogenitors becomes a population of 32 cells on the nextday (nDay x�1). The daily net increase (or cell production),therefore, is 30 cells. In example B, a similar-sized seedpopulation of two cells gives rise to three cells over 24 h.Thus, the daily cell production in this example is one cell.

The amount of cell death can be determined using theinformation on daily changes in cell number and cell pro-duction. The amount of cell death can be deduced from dataobtained from equation (2). Having defined the numbers of

174 Miller

cells produced on a particular day based on the cell cyclekinetics [see equation (7)], the daily change in cell numbercan be calculated from a new formula that combines equa-tions (2) and (7).

�n�Day x3Day x�1�

� �nDay x��1 � GF�24 hours/TC Day x � �nDay x� � �dDay x� (8)

Substituting the expression for the change in cell numberper day from equation (1) and then simplifying, the numberof cells that died on Day x can be calculated as follows:

dDay x � �nDay x��1 � GF�24 hours/TC Day x � �nDay x�1� (9)

The calculated values of the numbers of cell that wereproduced or died on each day can be used to determine thedaily rates of cell production or death, respectively. The rateof cell production (P) is the quotient of the number of cellsproduced on Day x and the total number of cells in thepopulation on Day x.

PDay x � �pDay x�/�nDay x� (10)

By substituting the expression for the number of cellsproduced on Day x [from equation (7)], an expression forthe rate of cell production can be determined.

PDay x � ��nDay x��1 � GF�24 hours/TC Day x � �nDay x�/�nDay x�

(11)

This equation can be simplified to read

PDay x � �1 � GF�24 hours/TC Day x � 1 (12)

The rate of cell death (D) is equal to the ratio of thenumber of cells that died on Day x to the total number ofcells in the population on Day x.

DDay x � �dDay x�/�nDay x� (13)

By combining equations (9) and (13) and then simplify-ing, an expression for the rate of cell death can be derived.

DDay x � ��nDay x��1 � GF�24 hours/TC Day x � �nDay x�1�/�nDay x�

(14)

After simplifying, the rate of cell death can be calculatedusing the following equation.

DDay x � ��1 � GF�24 hours/TC Day x � ��nDay x�1�/�nDay x� (15)

Thus, the rate of cell death can be determined from data

on numbers of cells in the colonies on 2 consecutive daysand the cell cycle kinetics.

Check of the Model

The formulae describing the rates of cell production and celldeath can be checked by taking the special case of steadystate, when the rate of cell production is the same as the rateof cell death.

PDay x � DDay x (16)

Accordingly, equations (12) and (15) are equated. Thus,

�1 � GF�C Day x24 hours/T � 1

� ��1 � GF�24 hours/TC Day x � ��nDay x�1�/�nDay x� (17)

This equation can be simplified.

�1 � ���nDay x�1�/�nDay x� (18)

Hence,

�nDay x� � �nDay x�1� (19)

Equation (19) is valid only when there is no change incell number over a particular 24 h period. Thus, steady-stateconditions are achieved and the formulae check.

RESULTS

The results are described for two types of cell cul-tures: primary cultures of cortical cells and B104neuroblastoma cells. These two cultures represent het-erogeneous and homogeneous populations of cells.Initially, data for untreated cells are described, andthen data regarding the effects of a toxin (ethanol) andmitogenic growth factors were fit to the model. Thegoal was to assess the flexibility and utility of themathematical model.

Primary Cortical Neurons

Untreated Cells. Neurons were obtained from thecerebral cortices of 16-day-old rat fetuses and cul-tured in a medium supplemented with 1.0% fetal calfserum. The growth of the cells was monitored overthe first 5 days postplating (Fig. 1, top, and Table 1).After a initial drop over the first 24 h postplating, thenumbers of cells in the untreated cultures steadied.The cell cycle kinetics was determined on Days 1 and3 (Table 2). Based upon three pieces of data (the dailychange in cell number, the TC, and the GF), thecontributions of cell production and cell death to the

Mathematical Model of Cell Proliferation and Death 175

number of viable cells were calculated. Cell produc-tion was steady between Days 2 and 4 (Fig. 1, bot-tom).

The mean daily amount of cell production wasmatched by a similar amount of daily cell loss (Fig. 1middle). That is, cell production and death were vir-

tually identical. This contributed to the stability in theoverall numbers of cultured cells. Interestingly, themodel projected a greater amount of cell death thanthe total amount of cell death measured empirically.These empirical measurements were based on the sumof the number of cells that exhibited nuclear conden-sation and/or were TUNEL-positive.

The daily rates of cell production and cell deathwere calculated by dividing the number of cells pro-duced or the number of cells that died by the totalnumber of cells in the culture on a particular day. Inthe controls, the mean rate of cell production (0.345 0.018) was steady over the initial days postplating.This is a result of the stable cell cycle kinetics (Table2). The rate of cell death was also stable. The meanrate of cell death (0.355 0.022) was similar to themean rate of cell production. This is another exampleof the balance in cell production and cell death in theuntreated cultures.

Ethanol-Treated Cells. Ethanol altered cell produc-tion and cell death among the cultured cortical cells.Cell numbers in the ethanol-treated cultures fellsteadily between Day 1 and Day 5, the greatest de-cline occurring over the first day postplating. Thenumber of cells produced daily was calculated for theperiod between Days 2 and 4. The mean number ofcells produced daily in the ethanol-treated cultures fellon succeeding days between Days 2 and 4 (the timeduring which change in cell number in the untreatedcultures was constant; Table 1). Moreover, mean cellproduction was significantly (p � 0.05) less(�65.9%) in the ethanol-treated cultures than in thecontrols (Fig. 1, top).

The mathematical model projected that the amountof cell death between Days 2 and 4 was significantly(p � 0.05) lower in the ethanol-treated cultures thanin the controls (Fig. 1, bottom). This estimate did notconcur with the empirically determined amount of celldeath; accordingly, the measured amount was signif-icantly (p � 0.05) greater (43.3%) in the ethanol-treated cultures than in the controls.

The effect of ethanol on cell death was not as largeas its effect on cell production. The net result was thatthe balance of cell production (as indicated by theratio of the number of cells produced to the number ofcells that died) in the ethanol-treated cultures was halfthat in the controls. This imbalance is consistent withthe continual decline in total cell numbers in theethanol-treated cultures and the stable numbers in thecontrol cultures.

The daily rates of cell production and death (thenumbers of cells produced or dying each day) weresignificantly (p � 0.05) affected by ethanol. The mean

Figure 1 Cell production and death among primary cul-tured cortical neurons. (Top) The change in the number ofcells in cortical cultures over the first 5 days postplating isplotted. These data are taken from Jacobs and Miller (2001),with permission. The number of cells in the untreated cul-tures (solid circles) was stable between Days 1 and 5. Forcultures treated with ethanol (400 mg/dL; open circles),however, cell number consistently fell. (Bottom, left) Meandaily cell production between Days 2 and 4 was calculatedfor untreated cultures (solid bars) and for cultures treatedwith ethanol (open bars). (Bottom, right) The effects ofethanol on the mean amount of cell death predicted from themodel (modeled) and empirically determined (measured)are shown. The mean amount of cell death was lower inethanol-treated cells. Each bar represents the mean for threepairs of independent trials per time point. The T-bars signifythe standard errors of the means. Asterisks denote statisti-cally significant (p � 0.05) differences between the ethanol-treated and untreated cultures. The # identifies a significantdifference relative to the number of cells in the untreatedcultures as predicted by the model.

176 Miller

rate of cell production in the ethanol-treated cells(0.184 0.024) was nearly half that for untreatedcells. In contrast, the mean rate of cell death (0.397 0.033) was greater (11.8%) than that for untreatedcells, but the difference was not statistically signifi-cant. The net result was that ethanol affected thebalance of cell production and cell death. This wasexemplified by comparing the ratio of the number ofcells produced on a particular day to the number ofcells that died on the same day (Table 1). In thecontrols, this ratio was approached unity, i.e., a virtualequal balance between cell production and death. In

the ethanol-treated cultures, however, the ratio washalved. This difference was statistically significant (p� 0.05).

B104 Neuroblastoma Cells

Untreated Cells. B104 cells proliferated in a mediumthat was supplemented with fetal calf serum or wasserum-free (Luo and Miller, 1997, 1999b). Interest-ingly, when the B104 cells were raised in a mediumsupplemented with 1.0% fetal calf serum, their cellcycle kinetics was similar to that for the primary

Table 1 Cell Production and Death in Cultured Cortical Neurons

Untreated Ethanol

Daychange in cell number (�104)

[�n; from equation (1)]change in cell number (�104)

[�n; from equation (1)]13 2 �6.21 �3.1423 3 0.60 �15.233 4 2.20 �6.5443 5 �5.00 �6.33

numbers of cells produced (�104)[p; from equation (7)]

numbers of cells produced (�104)[p; from equation(7)]

1 25.7 11.72 23.5 11.03 23.8 7.494 24.5 6.00

number of cells that died (�104)[d; from equation (9)]

number of cells that died (�104)[d; from equation(9)]

1 31.9 14.82 22.9 26.23 21.6 14.04 29.5 12.3

number of cells produced

number of cells that died[p/d; quotient of data above]

number of cells produced

number of cells that died[p/d; quotient of data above]

1 0.806 0.7912 1.03 0.4203 1.10 0.5354 0.831 0.488

Data are derived from the formulae described in the Methods. The data for the Tc and GF for Day 1 and for Day 3 (taken from Table 2)were used for both Days 1 and 2 and for both Days 3 and 4, respectively.

Table 2 Cell Cycle Kinetics for Primary-Cultures of Cortical Neurons

Untreated Untreated Ethanol Ethanol

Time of initial BrdU exposure Day 1 Day 3 Day 1 Day 3GF: growth fraction (%) 28.4 1.0 27.8 0.8 26.8 0.4 27.5 0.8Tc: length of cell cycle (h) 20.0 1.5 19.6 1.5 27.7 2.1 28.4 1.9Ts: length of S phase (h) 5.8 1.3 6.2 0.8 10.1 1.6 9.1 1.2

Each of the above values represents the means (the standard errors of the means) of four measures. Data were taken from Jacobs andMiller (2000, 2001).

Mathematical Model of Cell Proliferation and Death 177

cultured neurons (cf. Tables 2 and 3). Thus, the con-spicuous difference between the B104 cells and thecultured neurons is in their GF. For the B104 cells,three-quarters of the cells were actively cycling,whereas for the primary neurons, the GF was �29%.

The data used in these analyses were for cells thatwere raised in a serum-free medium. These conditionswere used to strengthen the conclusions regarding theeffects of the mitogenic growth factors and the actionsof ethanol on these effects. Under these conditions,B104 cells grew slowly; their doubling time was 2.1

days (cf. for cells raised in a medium supplementedwith 1.0% serum, the doubling time was only 1.70days). Mean daily cell production and cell deathamong the B104 cells are shown in Figure 2. Al-though cell production was modest, the amount of celldeath among these cells was minimal. In fact, cellproduction was nearly an order of magnitude greaterthan the amount of cell death.

Effects of Mitogenic Growth Factors. Two growthfactors, bFGF and PDGF-BB, promoted cell produc-tion. Both increases were statistically significant (p� 0.05). These resulted, in part, from decreases in theTC and increases in the GF (Table 3), but they werealso attributable to a lack of change in the dailyamount of cell death. Neither the rate of cell produc-tion nor the rate of cell death was significantly af-fected by bFGF or PDGF-BB.

Figure 2 Cell production and death among B104 neuro-blastoma cells. B104 cells were untreated or treated withbFGF or PDGF-BB (15 ng/mL) for as many as 7 days. Halfof the cultures were cotreated with ethanol (400 mg/dL).The increase in cell number, length of the cell cycle, and thegrowth fraction were determined for each of the six groupsof cells. The data were used in the mathematical model andestimates for cell production and cell death were generated(left). The rates of cell production and cell death (based onthe numbers of cells in the culture on each day) were alsocalculated (right). Mean values and standard errors of themeans are plotted as bars and T-bars, respectively. Eachmean is based on six independent measures at each of fourconsecutive time points. Asterisks denote statistically sig-nificant (p � .05) differences between associated groupstreated with and without ethanol. Pound signs identify sig-nificant (p � 0.05) differences relative to the untreated cells.

Figure 3 Effect of ethanol on the balance of cell produc-tion and cell death among B104 neuroblastoma cells. Anindicator of the balance of cell production and cell death isthe ratio of the amount of cell production to the amount ofcell death. This ratio was calculated for six treatmentgroups—cells treated with a growth factor and/or ethanol.Each value represents the means (and standard errors) of sixtrials. Statistically significant (p � 0.05) differences arenoted by asterisks. * and # denote statistically significant (p� 0.05) differences between associated groups treated withand without ethanol and relative to the untreated cells,respectively.

Table 3 Cell Cycle Kinetics for B104 Neuroblastoma Cells

Untreated Ethanol bFGF bFGF � Ethanol PDGF-BB PDGF-BB � Ethanol

GF: growth fraction (%) 75.5 0.8 63.3 1.1 79.0 2.0 66.0 1.3 85.1 0.4 72.1 0.6Tc: length of cell cycle (h) 46.1 1.7 62.8 3.6 41.7 0.6 57.5 1.5 41.7 1.6 51.5 1.9Ts: length of S phase (h) 8.1 0.8 9.9 0.6 8.1 0.6 10.0 0.6 9.4 0.2 9.3 0.4

Each of the above values represents the means (the standard errors of the means) of four measures. Data were taken from Luo and Miller(1997).

178 Miller

The balance of cell production and cell death wassignificantly (p � 0.05) affected by each of thegrowth factors (Fig. 3). Both bFGF and PDGF-BBincrease the ratio of the amount of cell production tothe amount of cell death (49.3 and 42.0%, respective-ly). No significant difference was detected betweenthe two growth factors.

Effects of Ethanol. Production of B104 cells wascompromised by ethanol treatment. In fact, ethanolhalved the amount of cell production relative to theuntreated cells (Fig. 2). The depressive effect ethanolwas even more dramatic for cells co-treated with agrowth factor. Cell production in these cultures wasone-third of that for cultures treated with a growthfactor alone. The amount of cell death was unaffectedby ethanol, whether or not the cells were additionallytreated with a growth factor. The rates of cell produc-tion were significantly (p � 0.05) reduced by ethanol.The mean daily rate of cell production in culturestreated with no growth factor, bFGF, or PDGF-BBwere equivalently reduced by ethanol (32.3–40.7%).This contrasts with the rates of cell death, which wereunaffected by ethanol. Thus, the inhibitory effects ofethanol on B104 cells were mostly transduced throughthe proliferating population.

The effects of growth factors and ethanol on thebalance of cell production and cell death among B104cells are described in Figure 2. Ethanol halved thisratio. This implies that ethanol has a more deleteriouseffect on cell production than on cell death. Thisalteration is consistent with the effects of ethanol oncultured cortical neurons (Fig. 1) and neurons in vivo,specifically neurons in the principal sensory nucleusof the trigeminal nerve (Miller, 1999). Ethanol alsosignificantly (p � 0.05) decreased the growth factor-elevated ratio of cell production to cell death (Fig. 3).

DISCUSSION

Utility of the Mathematical Model

The present mathematical model has four strengths.(1) The model is based on three obtainable measures,the daily change in cell number, TC, and GF. (2) Theactivity of cultures with widely divergent growthcharacteristics can be directly compared. (3) An ob-jective assessment of total amount of cell death can beascertained. (4) The balanced impact of cell prolifer-ation and death on the numbers of cells in a definedpopulation can be determined.

The model is based on two assumptions; one as-sumption relates to the determination of the TC and

the other to the GF. As noted in the original paperdescribing the cumulative labeling method (Nowa-kowski et al., 1989), measures of cell cycle kineticsdepend upon the composition of the proliferating pop-ulation. That is, do the cells behave as a single pop-ulation with common cell cycle kinetics or as twosubpopulations each of which having a distinct set ofcell cycle kinetics? Previous in vitro studies of neo-cortical cells (Jacobs and Miller, 2000, 2001) andneuroblastoma cells (Luo and Miller, 1997) unani-mously show that the cultured cells act as a singlepopulation. Furthermore, in vivo data (e.g., Miller andNowakowski, 1991; Miller and Kuhn, 1995; Taka-hashi et al., 1996) show that the data for proliferatingcells in the developing cerebral wall are also best fit toa simple linear model, i.e., the proliferating cellsbehave as a single population.

The GF depends upon the stability of the BrdUlabeling index over time. Empirical data show that theGF is stable for as much as 72 h post-BrdU adminis-tration (Jacobs and Miller, 2001). The cumulativelabeling method assumes that (labeled) post-mitoticcells leave the proliferative zone (Nowakowski et al.,1989). In vivo this removal occurs by the migration ofneurons from the proliferative zones and by the deathof cells in the proliferative zones. In vitro, however,the migration of postmitotic neurons from the prolif-erative zones does not occur. Therefore, only celldeath and detachment from the culture plate can leadto the removal of cells.

Cell Production

With the present model, the number of cells poten-tially generated by a cycling population and the actualnumber of cells produced can be calculated. Accurateestimates of the amounts of cell production, for ex-ample, via flow cytometry of individual cell DNAcontent and biochemical assays of [3H]thymidine in-corporation, can be compromised. Often it is pre-sumed that all cells in a culture dish are proliferativelyactive. One example is studies in which flow cyto-metric measures are used to determine cell cyclekinetics. Various in vitro studies (Luo and Miller,1997; Jacobs and Miller, 2000, 2001) show that thisassumption is invalid, even for homogeneous celllines. The GF for most populations is �90%. Further-more, such an assumption inaccurately applies to invivo systems as well (Nowakowski et al., 1989; Millerand Nowakowski, 1991; Miller and Kuhn, 1995; Ta-kahashi et al., 1996). Cell growth (the change in cellnumber over time) and its common yardstick are alsocrude measures. They are confounded insofar as theyas composite measures that rely on cell cycle kinetics,

Mathematical Model of Cell Proliferation and Death 179

GF, and the incidence of cell death. In contrast tothese traditional estimates of cell generation, thepresent model not only estimates the potential prolif-erative capability of a defined population, but alsodetermines the real amount of cell production.

Cell Death

The total amount of cell death can be determined withthe present mathematical model. This model does notrely upon direct documentation of cell death. In theethanol-treated cultures, the projected amount of cellsdeath and the measured amount of cell death is re-markably similar. In contrast, in the control cultures,the modeled amount of cell death is twice the mea-sured amount. Thus, the amount of cell death deter-mined with the model does not necessarily concurwith the empirically determined amount of cell death.What is the source of this discrepancy?

1. The difference may reflect the time it takes acell to die (i.e., the time during which a cellexhibits an anatomically discernible sign of celldeath). For example, ethanol may double thetime required for cells to degenerate. Thus, thenumber counted at any one snapshot in timemay be less than the total number that woulddegenerate over an extended period such as aday. Accordingly, the similar number in theethanol-treated cultures would imply that thedegenerative process accounted for by TUNELand nuclear condensation would transpire overa 24 h period. Likewise, these same eventswould take about 12 h in the control cultures.

2. Alternatively, death may be via one of variousmodes (one of which is not detected usingTUNEL and a nuclear stain), and their may bean exposure-associated proclivity for one mode.Accordingly, death among the control cells maygo by two modes, whereas ethanol-induced celldeath may be fully accounted for by TUNELand nuclear condensations. Cell death can in-volve a particular mode (e.g., apoptosis or ne-crosis) or depend upon a specific signal trans-duction pathway (e.g., p53-dependent orindependent). In fact, ethanol affects apoptoticdegeneration (Bhave and Hoffman, 1997; Cart-wright et al., 1998; Cheema et al., 2000; Jacobsand Miller, 2001), and alters p53 expressionduring the period of naturally occurring neuro-nal death (Kuhn and Miller, 1998).

Various anatomical and biochemical methods canidentify dying cells; however, none can determine the

total amount of cell loss. Thus, the difference ulti-mately results from the inadequacy of the anatomicaland biochemical methods to be all inclusive methodsfor identifying cells that die from a plethora of modesand mechanisms.

The Balance of Cell Production and CellDeath

Each piece of data, daily cell growth, the TC, and theGF is an incomplete index of development. For ex-ample, one cannot determine the amount of cell pro-duction or death based on any one of these indices. Bycombining these three basic data, however, uniqueinsights into the balance of cell production and deathcan be garnered. Possibly the greatest utility of themathematical model is the ability to assess the effectsof growth factors and toxins. These substances haveprofound effects upon cell production and survival(Luo and Miller, 1998; Mooney and Miller, 1999;White et al., 2003). As shown in the present examples,the effects of two mitogens (bFGF and PDGF-BB)and of the antimitogenic toxin ethanol can be readilyappreciated. Such discriminations can be made be-cause the fundamental data (daily cell growth, TC, andGF) are absolute measures.

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