Blackwell Synergy: Cell Prolif, Vol 39, Issue 1, pp. = 3-14: Mathematical model for the cancer stem cell hypothesis (Full = Text)

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Cell=20 Proliferation
Volume=20 39 Page 3 - February 2006
doi:10.1111/j.1365-2184.2006.00369.x

Volume 39 = Issue=20 1

Mathematical model = for the=20 cancer stem cell hypothesis

R. Ganguly = and I. K.=20 Puri

Abstract. Recent research on = the origin=20 of brain cancer has implicated a subpopulation of=20 self-renewing brain cancer stem cells for malignant = tumour=20 growth. Various genes that regulate self-renewal in = normal=20 stem cells are also found in cancer stem cells. This = implies=20 that cancers can occur because of mutations in normal = stem=20 cells and early progenitor cells. A predictive = mathematical=20 model based on the cell compartment method is = presented here=20 to pose and validate non-intuitive scenarios proposed = through=20 the neural cancer stem cell hypothesis. The growths of = abnormal (stem and early progenitor) cells from their = normal=20 counterparts are ascribed with separate mutation=20 probabilities. Stem cell mutations are found to be = more=20 significant for the development of cancer than a = similar=20 mutation in the early progenitor cells. The model also = predicts that, as previously hypothesized, repeated = insult to=20 mature cells increases the formation of abnormal = progeny, and=20 hence the risk of cancer.

INTRODUCTION

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Top of pageINTRODUCTION = <<THE MODELCONCLUSIONREFERENCES

Self-renewal is an intrinsic property of some = cancer cells.=20 Various genes that regulate self-renewal in normal = cells are=20 also found in cancer cells (Al-Hajj & Clarke = 2004;=20 Clarke = 2004).=20 This implies that cancers can occur because of = mutations in=20 normal stem and early progenitor cells or from = abnormal=20 self-renewing cancer cells. The cancer stem cell = hypothesis=20 states that just as mature cells are maintained by=20 self-renewing stem cells, cancerous tumours are the = abnormal=20 progeny of abnormal stem cells or of progenitor cells = whose=20 DNA carries mutation(s). There is recent evidence that = brain=20 cancer tumours are driven by a very small = subpopulation of=20 self-renewing brain cancer stem cells (Reya et al. = 2001; Pardal = et al.=20 2003; Singh et = al. 2003,=20 2004; Clarke=20 2004). Brain tumour growth bears similarities to = other=20 cancer pathways for blood or epithelial cells (Al-Hajj & Clarke = 2004).=20 It is likely that cancerous stem cells may often be=20 responsible for recurrences that occur after = treatment.

Transformation of stem cells into malignant cells = is=20 important because they already have an activated = self-renewal=20 mechanism. These cells therefore require fewer = mutations to=20 become abnormal cells than are required to ectopically = activate a more differentiated cell. Because stem = cells are=20 self-renewing, they often persist for longer periods = and are=20 thus more likely to accumulate mutations than a = short-lived=20 progenitor or differentiated cell. A progenitor cell = could=20 inherit mutations from a stem cell and itself could = undergo=20 continuing mutation to cause transformation. When the=20 signalling pathways that normally regulate stem cell=20 self-renewal are disrupted, newly arising cancer cells = appropriate the machinery for self-renewal that is = normally=20 only expressed in stem cells. Many pathways associated = with=20 normal stem cells are also be involved in regulating = the=20 development of cancer cells, for example, Bmi-1 (Lessard & = Sauvageau=20 2003; Molofsky=20 et al. 2003), sonic hedgehog (Taipale & Beachy = 2001),=20 and Wnt signalling (Polakis 2000; Taipale & Beachy = 2001),=20 which are all implicated in both oncogenesis and stem = cell=20 renewal.

Predictive models of stem cell systems provide the = means to=20 pose and validate non-intuitive hypotheses. Hence, = they serve=20 as important tools to distinguish the underlying = regulatory=20 mechanisms that govern stem cell fate decisions. A = model can=20 also provide a basis for estimating the potential risk = of=20 malignant tumour growth in the event of a detectable = oncogenic=20 event (for example, formation of a cancer cluster), or = physiological perturbation to the tissue (for example, = radiation injury).

The literature cites several techniques for = mathematical=20 modelling of haematopoietic, neural and embryonic stem = cells=20 (Viswanathan=20 & Zandstra 2003). However, none of these = models can=20 quantitatively describe the recently proposed neural = cancer=20 stem cell hypothesis (Reya et al. = 2001; Al-Hajj = & Clarke=20 2004; Clarke=20 2004; Singh=20 et al. 2004). Here we describe a = mathematical model=20 for the hypothesis that is based on the cell = compartments=20 method (Wichmann=20 & Loeffler 1984). The role of internal cell = signalling=20 during cell proliferation is discussed. Growths of = abnormal=20 (stem and early progeny) cells from their normal = counterparts=20 are ascribed with separate mutation probabilities. The = relative importance of mutation probabilities of stem = and=20 early progenitor cells on abnormal progeny's growth = rates are=20 discussed. It is also demonstrated how repeated insult = to=20 mature cells can lead to enhanced population growth = rate of=20 the abnormal cells.

THE MODEL

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Top of pageINTRODUCTIONCONCLUSIONREFERENCES

The mathematical model describes formation of = cancer cells=20 from stem cells. We have used it to analyse the = influence of a=20 stimulus, such as depletion of mature cells because of = injury,=20 on overall cancer growth. The model is deterministic = (that is,=20 not stochastic) so that for specified input = conditions, it=20 always provides the same prediction. However, this is = not=20 restrictive, since randomness is readily modelled by = changing=20 various parameters (such as those reported in Tables 1 = and=20 2).

Figure 1 is = a=20 schematic representation of how normal stem cells and = early=20 progenitor cells undergo mutation to produce abnormal = cells=20 that give rise to abnormal progeny which ultimately = form=20 tumours (Clarke=20 2004). It is an evolutionary perspective that = assumes that=20 cells regularly undergo mutation in their DNA (the = vast=20 majority of which are detrimental to their survival). = However,=20 some cells, particularly those that are capable of=20 self-renewal, infrequently survive these mutations and = subsequently drive the formation of pathological = tumours.

The incorporation of cell biological mechanistic=20 information into stem cell models requires that = signalling=20 thresholds be mathematically represented in the = context of=20 cell fate choices. It has been suggested that the = diffusion of=20 a soluble signal can establish a concentration = gradient and=20 that cells within the gradient can adopt alternate,=20 'all-or-none' fates at critical threshold levels of = the signal=20 (Jessel &=20 Lumsden 1997; O'Neill & = Schaffer=20 2004). Our model is consistent with this = principle, but we=20 assume that this concentration gradient becomes = negligible=20 when the mature cell population reaches a desired = sustainable=20 level (that is, there must be an upper signalling = threshold=20 that should be modelled).

Consistent with previous findings (Oliver & = Wechsler-Reya=20 2004), the growth mechanisms and factors = influencing stem=20 cell self-renewal are considered to be the same for = both=20 normal and abnormal stem cells. Consequently, in our = model,=20 the self-renewal probabilities for normal and abnormal = (DNA=20 mutated) stem cells are identical. The same assumption = applies=20 for normal and abnormal early progenitor cells. Stem = cells are=20 assumed to self-renew an unlimited number of times (Al-Hajj = & Clarke=20 2004) with a self-renewal probability=20 P_SC. Cells that do not self-renew=20 differentiate to form early progenitor cells. = Depending on the=20 stem cell and early and late progenitor cell = populations, the=20 self-renewal probability varies between a maximum=20 P_max,SC and a minimum=20 P_min,SC.

Unlike stem cells, early progenitor cells can = self-renew=20 only a limited number of times (Al-Hajj & Clarke = 2004),=20 assumed to be k_EP in our analysis. = Depending=20 on early and late progenitor cell populations, the = early=20 progenitor cells' self-renewal probability=20 P_EP lies between = P_max,EP=20 and P_min,_EP. Late = progenitor=20 cells are assumed to undergo n_LP = stages of=20 cell division. Here, depending on the mature cell = population,=20 cell division regulatory feedback signalling controls = the=20 number of cell division stages (between=20 n_LP,min and = n_LP,max). This=20 regulatory signal is responsible for maintaining a = steady=20 population of mature cells. The mature cell population = also=20 provides feedback to the stem cells and thus = influences their=20 mitotic fraction and self-renewal rate. In Fig. 1, the = regulatory=20 feedback signals are represented by dotted lines and = numbered=20

=
through

. Table 2 = summarizes the=20 mathematical representations for these signals.

We have considered cancer-initiating mutations both = in=20 normal stem cells and in early progenitor cells. = Abnormal=20 progeny are produced either through mutation of stem = cells=20 into abnormal stem cells and their subsequent = self-renewal and=20 differentiation, or from differentiated early = progenitor=20 cells. Occurrences of mutation are probabilistic = events. Each=20 cell division event for stem cells or for early = progenitor=20 cells has the associated probabilities, = M_SC=20 and M_EP, that represent the = occurrence of=20 oncogenic mutation during DNA transcription. Hence, = following=20 cell division, one daughter cell acquires the mutated = gene=20 while the other retains the original DNA. As the = self-renewal=20 process has a probability P_SC, the=20 probability that the mutated gene is inherited in the = abnormal=20 stem cell lineage is=20 = M_SC =D7 P_SC. The=20 same rationale applies to mutated early progenitor = cells.=20 Cancer arises if a cell undergoes a critical number of = consecutive mutations and acquires the ability to = proliferate=20 indefinitely. Thereafter, the associated cell cluster = produces=20 a malignant tumour through its uncontrolled = proliferation (Reya = et al.=20 2001).

Each cell type is considered as a separate = compartment.=20 Cell population growth is modelled by considering = individual=20 rate expressions for each compartment.

(1) Stem cell compartment. During each cell = division, there=20 is a probability, P_SC, that the two = daughter=20 stem cells retain their original features. Thus, the = stem cell=20 population increases by an amount (2PSC-1). These = self-renewed=20 cells add to the stem cell lineage, whereas the = differentiated=20 cells contribute to the early progenitor cell pool. = When a=20 mutation occurs during the DNA replication process = (which is=20 assumed to have a probability M_SC), = one of=20 the two daughter cells inherits the mutated gene. = Thus, for=20 each cell division event, the number of cells lost = from the=20 normal stem cell pool is M_SC. = Consequently,=20 the net rate of change of the stem cell population = is

(1)

where=20 = _SC =3D (α_SC/τ_SC)ln2 = denotes the cell division rate, α_SC = the mitotic=20 fraction and τ_SC the cell cycle time = (that is, the=20 time required to complete a full cycle of cell = division). The=20 self-renewal probability=20 = P_SC =3D P_SC(N_SC=,=20 N_EP, N_LP,=20 N_MC) 3DE=20 [P_max,SC,=20 P_min,SC]. The mitotic fraction is a = bounded=20 function of the stem and early progenitor cell = populations (Wichmann = &=20 Loeffler 1984), that is,=20 = α_SC =3D α_SC(N_{SC,=20
N_EP) [α_max,SC,=20
α_min,_SC].}

(2) Early progenitor (EP) cell compartment. The = model for=20 this compartment considers that EP cells undergo a = limited=20 number (k) of self-renewal steps. Thus, the = early=20 progenitor cells are split into k = subcompartments that=20 contain identical EP cells, which differ only with = respect to=20 the number of times they have undergone self-renewal = (and,=20 hence, the number of times they can further = self-renew). We=20 refer to these subgroups as EP₁, = EP₂=85=20 EP_k.

When a subgroup EP_i undergoes = cell=20 division, it self-renews (with a probability=20 P_EP that is assumed identical for=20 i =3D 1 through = k − 1) into a=20 subgroup EP_i+1 and also = differentiates into=20 LP. The first subgroup receives a flux of = differentiated stem=20 cells from the stem cell compartment, whereas the last = one=20 completely differentiates into LP (late progenitor) = (i.e.=20 P_EP =3D 0 for the=20 k^th compartment). Each subgroup has = a=20 mutation probability M_EP (assumed = identical=20 for i =3D 1 through k). The = relations=20 describing the population changes in each subgroup = are

(2a)

(2b)

where=20 = _EP =3D (α_EP/τ_EPC)ln2= .=20 This is a set of state equations for the state = variables

=
.

The output equation

can = be used to=20 determine the EP cell population. Because EP cells do = not=20 differ in cell characteristics (other than their = ability to=20 further self-renew), the same values of _EP, = P_EP and M_EP are = considered=20 for all subgroups 1 through k_EP = (except that=20 the value of P_EP for the=20 k^th subgroup is zero). The total = efflux of=20 differentiated EP cells that enters the LP compartment = is

(3)

Thus, the resulting efflux of mutated EP cells is =

=
.

(3) Late progenitor cell compartment. The LP = compartment=20 receives an influx

as = described=20 previously. The cells contained in it undergo=20 n_LP successive stages of cell = division=20 before transforming into mature cells, where=20 = n_LP =3D n_LP(N_MC)= [n_LP,max,=20 n_LP,min]. Although a state=96space = model for=20 the n stages of cell division can be explicitly = written=20 (as for the EP compartment), its solution would be = needlessly=20 complicated, as the dimension of the state=96space = matrix is=20 itself a variable in this case. Instead, we follow an = approach=20 (Wichmann &=20 Loeffler 1984) that assumes that the efflux from = the EP=20 compartment is amplified by a factor = Zⁱⁿ as=20 soon as the cells enter the LP compartment. = Immediately before=20 leaving the LP compartment, they are further amplified = by a=20 factor Z^out. These two factors are = selected=20 such that

=
.

The total time τ_LP spent by a cell = in the LP=20 compartment is the sum of the cell generation time=20 = (τ_g =3D α_LP ln2/ = 64;_LP)=20 and the cell maturation time = τ_m. The factor=20 = Zⁱⁿ =3D (2ⁿ −&nb= sp;1)=20 = τ_g/τ_LP+2ⁿτ<= SUB>m/τ_LP,=20 and the LP population change is

(4)

and the efflux from the LP compartment (that is, = the=20 generation rate of mature cells)

. As = before,=20 = _LP =3D (α_LP/τ_LP)ln2.=

(4) Mature cell compartment. The mature cell = compartment=20 receives an efflux of

from = the LP=20 compartment. Considering a cell death (or apoptotic) = rate of=20 _0,MC, the rate of change in the mature cell = population

(5)

(5) Abnormal stem cell compartment. As mutated stem = cells=20 behave in the same manner as normal cells, the = relation=20 representing the growth rate of the abnormal stem cell = population is analogous to (1) except that the = mutating=20 fraction must be added to the cell population in this = case.=20 Thus,

(6)

(6) Abnormal early progenitor cell compartment. The = equations for the abnormal EP cell population also = resemble=20 Eq. (2a) and (2b), except that here too the mutating = cells add=20 to the population. Hence,

(7a)

(7b)

The overall population of abnormal EP cells can be = found by=20 solving the above state equations and substituting the = state=20 variables into the output equation

. = The outflux of=20 abnormal EP cells

(8)

(7) Abnormal progeny compartment. If they originate = from=20 cancerous abnormal stem cells or from abnormal early=20 progenitor cells, the abnormal progeny undergo = uncontrolled=20 cell division leading to a malignant tumour. The = growth of the=20 abnormal progeny

(9)

where _0,AP denotes the abnormal progeny = cell=20 death rate (or apoptosis).

RESULTS AND = DISCUSSION

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Top of pageINTRODUCTIONCONCLUSIONREFERENCES

Equations=20 1=969 are numerically solved through time marching = using the=20 parameters specified in Table 1 (Wichmann & = Loeffler=20 1984). The model is first tested for its stability = and=20 steady state response by selecting an initial = normalized=20 condition N_sc =3D 1.0, = with all=20 other normalized cell populations set to zero. The = response of=20 the SC, EP, LP and MC populations is then examined for = a=20 mature cell death rate of 0.01/h. For this base case, = the=20 mutation probabilities are assumed to be zero.

Figure 2a = presents the=20 growth of the EP, LP and MC populations from stem = cells in=20 response to these conditions. Initially, when these=20 populations are very small, feedback signals

,

and =

from = the EP, LP and=20 MC compartments increase the values of = α_SC and=20 α_EP so as to quickly replenish the = respective cell=20 populations (Fig. 2b). = Because=20 = P_SC =3D P_SC,min=20 (that is, 0.4) initially, rapid cell division leads to = an=20 early depletion in the SC pool. Subsequently, as the = value of=20 P_SC slowly increases, so does the SC = population stabilizes at=20 N_SC =3D 0.77. The values = of=20 α_SC and α_EP = concurrently decrease to=20 steady values of 0.28 and 0.34, and the EP, LP and MC = cell=20 populations likewise stabilize at various steady = values (3.19,=20 96.85 and 678.1, respectively) within the first = 400 h. Figure 2=20 confirms that the modelled process is self-regulating, = that=20 is, proper cell signalling limits cell proliferation = to a=20 steady value that is sufficient to replenish the = steady death=20 (apoptosis) of mature cells.

Next, we consider the occurrence of an oncogenic = event in=20 either of the SC or of the EP populations. First, we = assume=20 that there is a 10³ probability that a mutation = occurs=20 during stem cell self-renewal with=20 M_SC>>M_EP (or = M_EP≈ 0). Next, assuming that = the abnormal=20 progeny have an apoptotic rate (1%/h) equals the = mature cell=20 death rate, the growth rates of the SC_A,=20 EP_A and AP populations are determined as = shown in=20 Fig. 3a.=20 Since the mutation rate is very small, it has a = negligible=20 effect on the SC population. The abnormal cell = populations are=20 much smaller than those of healthy cells. Following an = initial=20 rapid growth period, the AP population grows steadily = and=20 nearly linearly to a normalized value of 0.15 at = 700 h.=20 This indicates that loss of self-regulation (that is = otherwise=20 observed in healthy cells) is successfully modelled. = When EP=20 cells undergo mutation (with=20 M_EP =3D 10³, and=20 M_EP>>M_SC), = the AP=20 population growth rate shown in Fig. 3b is = far smaller=20 than when the SC population is allowed to mutate. This = is to=20 be expected, since while stem cells can self-renew=20 indefinitely, EP cells have limited capability to do = so.=20 Hence, SC mutations have a much larger proliferation = potential=20 than mutations in EP cells. The model confirms that = mutations=20 leading to cancer are more significant when they occur = in stem=20 cells than when they occur in early progenitor = cells.

Regulation of healthy cells occurs through feedback = signals=20 that alter the values of P_SC, = α_SC=20 and α_EP (Fig. 1). = (This=20 mechanism is also responsible for tissue healing; Craig et al. = 1996;=20 Calza = et=20 al. 2004.) In order to investigate the = influence of=20 these feedback loops on the onset of cancer, we = simulated=20 repeated insult to the mature cell population by = periodically=20 depleting its population by a specified amount. Figure 4a = shows the=20 response of the SC, EP, LP and MC populations when 50% = of=20 mature cells are removed at 200 h intervals. = (Repetitive=20 radiation therapy used to treat cancer could lead to a = similar=20 condition, that is, when some mature cells also = periodically=20 die because of treatment; Fujimaki = 2005.)

Upon loss of mature cells, the feedback loop

=
immediately=20 increases proliferation of LP cells, and feedback = signal

=
contributes to an=20 immediate rise in the proliferative fraction of stem = cells.=20 This is reflected in Fig. 4b, = which shows=20 that the value of α_SC temporarily = increases to 0.34=20 from a steady value of 0.28. Therefore, repeated = insult=20 increases the proportion of the stem cell population = that=20 participates in cell division. Even assuming that the=20 probability of cell mutation remains unchanged, the = growth=20 rate of the abnormal progeny is consequently = enhanced.

A comparison of Figs 3b and = 4b=20 indicates that the abnormal progeny population becomes = larger=20 with repeated injury (when=20 M_SC =3D 10³ and=20 M_EP =3D 0, the AP = population=20 increases to 0.16 in 700 h for repeated insult, = whereas=20 it would have been 0.15 in the absence of any insult). = This is=20 in conformity with the earlier findings (Beachy et al. = 2004), which showed that repeated tissue invasion = leads to=20 an increased cancer risk. In comparison, the increase = in the=20 value of α_EP following insult is = negligible (Fig. 4c).=20 Hence, when only EP cells are assumed to undergo = mutation (for=20 M_EP =3D 10³ and=20 M_SC =3D 0), the AP cell = population=20 is relatively unaffected by repeated insult to the MC=20 population. The model indicates that repeated insult = increases=20 the cancer risk by influencing the mitotic fraction in = the=20 stem cell population.

When repeated MC insult occurs every 200 h as = shown in=20 Fig. 4b,=20 the value of α_SC first rapidly = increases to a=20 maximum but then decreases to its initial value within = 200 h. However, when the frequency of insult is = increased=20 to once every 10 h, α_SC does not = decrease to=20 that initial value at the end of each cycle. Here, Fig. 5 shows=20 that the actively proliferating stem cell pool keeps=20 increasing with time and consequently the AP = population grows=20 to an even larger value (N_AP = increases from=20 0.16 to 0.18 as the time between two consecutive = insults is=20 decreased from 200 to 100 h). This indicates that = the=20 cancer risk increases with increasing frequency of = injury to=20 the mature cell population.

The trend becomes obvious when the AP populations = after=20 700 h are plotted as a function of the mean time = between=20 two consecutive insults (Fig. 6). = Here, four=20 different combinations of the SC and EP cell mutation=20 probabilities are considered=20 (M_SC =3D 0.002 and 0.001 = with=20 M_EP =3D 0; and=20 M_EP =3D 0.002 and 0.001 = with=20 M_SC =3D 0) while all the = other=20 parameters were kept unchanged. As before, 50% insult = of the=20 MC population is considered. It is evident from Fig. 6 that = the AP=20 population increases as the time between two = subsequent=20 (regularly imposed) insults is reduced. The effect of = changing=20 the frequency of repeated insult is more profound when = the=20 mutations occur in SC rather than in EP cells. Also, = the=20 effect of changing the insult frequency is more = pronounced at=20 higher insult frequencies (that is, at smaller mean = time=20 between the insults =96 less than 100 h for the = case shown=20 here). With smaller time between two insults, MC = population=20 fails to rise to its normal value =96 leading to a = high level of=20 actively proliferating SC and EP cells at the end of = each=20 cycle. This means that the SC and EP cells undergo = more=20 frequent cell division. For a given probability of = mutation,=20 this essentially means a higher growth rate of the AP = cells.=20 When the mean time between two insults is very large = (e.g.=20 lager than 250 h for the cases presented here), = nearly=20 the steady normal MC population is re-established = after each=20 insult. Each consecutive insult therefore can be = treated as=20 nearly isolated events, and the AP growth rate becomes = almost=20 insensitive to the insult frequency.

CONCLUSION

Go to:

Top of pageINTRODUCTIONREFERENCES

A predictive model based on a cell compartments = method is=20 presented to pose and validate non-intuitive scenarios = arising=20 from the cancer stem cell hypothesis.

1	Under normal circumstances, the model = converges to a=20 steady population of the stem, early and late = progenitor=20 cells that maintain a constant mature cell = population=20 for a specified cell apoptotic = rate.
2	An oncogenic event in stem cells or EP cells = produces=20 abnormal stem or EP cells and eventually the = abnormal=20 progeny. For the same occurrence probability, = mutations=20 in stem cells (rather than in EP cells) lead to = faster=20 growth of the abnormal progeny.
3	When the mature cell population undergoes a = sudden=20 depletion because of an insult (for example, = radiation=20 damage to the mature tissue), regulatory = feedback=20 signals (that are responsible for tissue = healing)=20 increase the stem and EP cell proliferative = fractions.=20 This in turn leads to an increased growth rate = of the=20 abnormal progeny cells.
4	The abnormal progeny cell population growth = rate=20 increases as the mean time between two = consecutive=20 insults is reduced. This trend is more = pronounced when=20 the insult frequency is higher. However, when = this=20 frequency is small enough that the stable cell=20 populations are restored between two consecutive = insults, the AP growth rate is relatively=20 = unaffected.

REFERENCES

Go to:

Top of pageINTRODUCTIONCONCLUSIONREFERENCES = <<

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Al-hajj M, Clarke MF = (2004)=20 Self-renewal and solid tumor cells. = Oncogene=20 23, 7274.

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Calza L, Fernandez M, = Giuliani A,=20 Pirondi S, D'intino G, Manservigi M, De Sordi N, = Giardino L (2004) Stem cells and nervous tissue = repair:=20 from in vitro to in vivo. Prog. Brain = Res.=20 146, 75.

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Craig CG, Tropepe V, = Morshead CM,=20 Reynolds BA, Weiss S, van der Kooy D (1996) = In=20 vivo growth factor expansion of endogenous=20 subependymal neural precursor cell populations = in the=20 adult mouse brain. J. Neurosc. 16, = 2649.

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Fujimaki T (2005) = Surgical=20 treatment of brain metastasis. Int. J. Clin.=20 Oncol. 10, 74.

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Jessell TM, Lumsden A = (1997) In:=20 Cowan WM, Jessell TM, Zipursky SL, eds. = Molecular and=20 Cellular Approaches to Neural Development, = p. 290.=20 Oxford: Oxford = University=20 Press.

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Lessard J, Sauvageau G = (2003)=20 Bmi-1 determines the proliferative capacity of = normal=20 and leukaemic stem cells. Nature = 15,=20 255.

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Molofsky A, Pardal R, = Iwashita T,=20 Park I-K, Clarke MF, Morrison SJ (2003) Bmi-1 = dependence=20 distinguishes neural stem cell self-renewal from = progenitor proliferation. Nature = 425,=20 962.

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Singh SK, Hawkins C, = Clarke ID,=20 Squire JA, Bayani J, Hide T, Henkelman RM, = Cusimano MD,=20 Dirks PB (2004) Identification of human brain = tumor=20 initiating cells. Nature 432, = 396.

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Taipale J, Beachy PA = (2001) The=20 hedgehog and Wnt signaling pathways in cancer.=20 Nature 411, 349.

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Cell=20 Proliferation
Volume=20 39 Page 3 - February=20 2006

Received 22 July 2005; revision = accepted=20 7 November 2005

Affiliations

Department of Engineering = Science and=20 Mechanics, Virginia Polytechnic Institute and State=20 University, Blacksburg, Virginia, USA

Correspondence

I. K. Puri, Department of = Engineering=20 Science and Mechanics, Virginia Polytechnic Institute = and=20 State University, Blacksburg, Virginia 24061, USA. = Tel.:=20 (540)231 3243; Fax: (540)231 4574; E-mail: = ikpuri@vt.edu

Image Previews

[Full Size]

Figure 1. Schematic=20 representation of the model. SC, stem cells; EP, = early=20 progenitor cells; LP, late = ...

[Full Size]

Figure 2. Transient=20 response of the model beginning with=20 N_SC =3D 1.0 and with all = other cell=20 populations s...

[Full Size]

Figure 3. Growth rate of=20 populations of abnormal stem cells, EP cells and = abnormal=20 progeny for a speci...

[Full Size]

Figure 4. (a)=20 Transient response of the SC, EP, LP and MC = populations to=20 periodic insult that=20 depletes...

[Full Size]

Figure 5. Response of=20 the in SC_A, EP_A and AP = populations when=20 50% of the MC population is depleted=20 eve...

[Full Size]

Figure 6. AP=20 cell growth as a function of the time between = consecutive=20 insults. The AP cell=20 population...

[Full = Size]

Table 1. Parameters used in=20 the model

[Full = Size]

Table 2. Mathematical=20 representations of the regulatory signals (Wichmann &=20 Loeffler 1984)

To cite = this=20 article
Ganguly, R. = & Puri,=20 I. K. (2006)
Mathematical model for the cancer stem = cell=20 hypothesis.
Cell=20 = Proliferation 39 (1), 3-14.
doi:=20 = 10.1111/
j.1365-2184.2006.00369.x

Blackwell Synergy=AE is a Blackwell = Publishing, Inc.=20 registered trademark
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getObjectByName(nameOfObject){=0A= Obj =3D null;=0A= if (browser.ie) Obj =3D document.all[nameOfObject]=0A= else if (browser.dom) Obj =3D findDOMObject(nameOfObject)=0A= else if (browser.ns4) findLayer(window,nameOfObject);=0A= return (!Obj || ( browser.ns4 && Obj =3D=3D window ) ) ? "Object not = found" : Obj;=0A= }=0A= // helper function=0A= function findDOMObject(nameOfObject) {=0A= for (var i =3D 0; i < document.images.length; i++) {=0A= if (document.images[i].name=3D=3DnameOfObject) return = document.images[i]=0A= }=0A= return document.getElementById(nameOfObject)=0A= }=0A= // helper function=0A= function findLayer(node,nameOfObject) {=0A= if ( node.name =3D=3D nameOfObject ) Obj =3D node;=0A= for ( var counter =3D 0; counter < node.document.images.length; = counter++ ) {=0A= if (node.document.images[counter].name=3D=3DnameOfObject) Obj =3D = node.document.images[counter];=0A= }=0A= for ( var i =3D 0; i < node.document.layers.length; i++ ) {=0A= findLayer(node.document.layers[i],nameOfObject);=0A= }=0A= }=0A= =0A= function toggleVisibility(id) {=0A= obj =3D getObjectByName(id);=0A= if (obj) {=0A= v =3D obj.style.display;=0A= obj.style.display =3D (v =3D=3D 'none') ? 'block' : 'none';=0A= i =3D getObjectByName('hiddenImage');=0A= if(i) i.src =3D '/page/imageHack.jsp?object=3D' + id + = '&display=3D' + obj.style.display + '&dummy=3D' + (new Date()).getTime();=0A= }=0A= }=0A= =0A= /** CSS class support */=0A= /** return true if removed, false if not found */=0A= function removeClass(el, className) {=0A= if (!(el && el.className)) return false;=0A= var cls =3D el.className.split(" ");=0A= var ar =3D new Array();=0A= for (var i =3D cls.length; i > 0;) if (cls[--i] !=3D className) = ar[ar.length] =3D cls[i];=0A= el.className =3D ar.join(" ");=0A= return cls.length !=3D ar.length;=0A= }=0A= /** return true if element has that class */=0A= function hasClass(el, className) {=0A= if (!(el && el.className)) return false;=0A= var cls =3D el.className.split(" ");=0A= for (var i =3D cls.length; i > 0;) if (cls[--i] =3D=3D className) = return true;=0A= return false;=0A= }=0A= /** add class to element, if element already had that class - does = nothink */=0A= function addClass(el, className) {=0A= if (!el || hasClass(el, className)) return;=0A= if (el.className) el.className +=3D " " + className;=0A= else el.className =3D className;=0A= }=0A= /** Add CSS rule as last rule in last stylesheet - override all previous = definitions=0A= removing is harder - not implemented, use it with caution */=0A= function addCssRule(selector, rule) {=0A= if (document.styleSheets) {=0A= var css =3D document.styleSheets[document.styleSheets.length-1]; = // get last css=0A= if (css.addRule) css.addRule(selector, rule); // IE=0A= else if (css.insertRule) css.insertRule(selector + "{" + rule + = "}", css.cssRules.length); // W3C=0A= }=0A= }=0A= /** get CSS style of the document */=0A= function getCssText() {=0A= var cssText =3D "\n";=0A= var styleSheets =3D document.styleSheets;=0A= for (var i =3D 0; i < styleSheets.length; i++) {=0A= var css =3D styleSheets.item(i);=0A= if (css.cssText) cssText +=3D css.cssText;=0A= else {=0A= var cssRules =3D css.rules ? css.rules : css.cssRules;=0A= for (var j =3D 0; j < cssRules.length; j++) {=0A= cssText +=3D cssRules.item(j).cssText + "\n";=0A= }=0A= }=0A= }=0A= return cssText;=0A= }=0A= =0A= /** DOM Event support */=0A= /** addEventListener method may already exist, it's not trivial = implement it and not break it, use other mame 'addListener' is simple */=0A= function addListener(el, evname, func) {=0A= if (el.attachEvent) el.attachEvent("on" + evname, func);=0A= else if(el.addEventListener) el.addEventListener(evname, func, true);=0A= }=0A= /** removeEventListener */=0A= function removeListener(el, evname, func) {=0A= if (el.detachEvent) el.detachEvent("on" + evname, func);=0A= else if (el.removeEventListener) el.removeEventListener(evname, func, = true);=0A= }=0A= =0A= /** Server session access */=0A= /** helper method to access session object */=0A= function _getSession() {=0A= return window.session ? window.session : (window.session =3D new = Array());=0A= }=0A= /** send request to server to preserve some value between different = requests, value is accessible using getSessionAttribute */=0A= function setSessionAttribute(name, value) {=0A= /* just to test callback */=0A= function test(doc) {=0A= // alert("Server returned: " + doc.body.innerHTML);=0A= }=0A= = sendServerMsg("/action/sessionAccess?action=3DsetJavaScriptAttribute&name= =3D" + name + "&value=3D" + value, test);=0A= _getSession()[name] =3D value;=0A= }=0A= /** see setSessionAttribute */=0A= function getSessionAttribute(name) {=0A= return _getSession()[name];=0A= }=0A= =0A= /** send request - url - to the server, response is accessible in = document passed to callback function=0A= DO NOT Send anything before page is loaded!=0A= */=0A= function sendServerMsg(url, callBack) {=0A= /* called when frame is loaded */=0A= function _receiveServerResponse(ev) {=0A= var iFrame =3D window.event ? window.event.srcElement : = ev.currentTarget; // IE : W3C=0A= var doc =3D iFrame.contentDocument ? iFrame.contentDocument : = document.frames(iFrame.id).document; // W3C : IE=0A= if (iFrame.callBack) iFrame.callBack(doc);=0A= document.body.removeChild(document.getElementById(iFrame.id));=0A= }=0A= =0A= var iFrame =3D document.createElement('iframe');=0A= addListener(iFrame, "load", _receiveServerResponse);=0A= iFrame.setAttribute("id", "msgFrame" + new Date().getTime()); // = so IE can retrive document=0A= iFrame.setAttribute("src", url);=0A= iFrame.setAttribute("style", "display:none");=0A= if (callBack) iFrame.callBack =3D callBack;=0A= document.body.appendChild(iFrame);=0A= }=0A= =0A= /**=0A= copy innerHTML of src element to innerHTML of target element=0A= */=0A= function copyInnerHtml(src, target) {=0A= var domSupported =3D document.getElementById ? true : false;=0A= var se =3D (domSupported && typeof src =3D=3D "string") ? = document.getElementById(src) : src;=0A= var te =3D (domSupported && typeof target =3D=3D "string") ? = document.getElementById(target) : target;=0A= if (se.innerHTML && te.innerHTML) te.innerHTML =3D se.innerHTML;=0A= }=0A= /**=0A= filter select options, options that do not contain filter text are = removed=0A= in first use, original select options are backed up as attribute = optionsCopy=0A= */=0A= function filterSelect(selectId, value) {=0A= var select;=0A= if (document.getElementById && (select =3D = document.getElementById(selectId)) && select.options) {=0A= if (!select.optionsCopy) {=0A= select.optionsCopy =3D new Array();=0A= select.optionTexts =3D new Array();=0A= for (var i=3D0; i'+document.title+'');=0A= tmp.writeln(el.innerHTML);=0A= tmp.writeln('');=0A= tmp.close();=0A= }=0A= if (anchor) {=0A= var target =3D (tmp.getElementById) ? = tmp.getElementById(anchor) :=0A= (tmp.all) ? tmp.all[anchor] : false;=0A= if (target) {=0A= if (target.scrollIntoView) target.scrollIntoView();=0A= else if (window.scroll && target.offsetTop) = el.popupWindow.scroll(0, target.offsetTop);=0A= }=0A= }=0A= }=0A= }=0A= =0A= /** marks all check box */=0A= function markAllCheckboxes(aForm, aNamePrefix, aChecked)=0A= {=0A= var elmts =3D aForm.elements;=0A= for (var i=3D0; i 0);=0A= if (addElmt)=0A= url +=3D "&" + elmts[i].name + "=3D" + elmts[i].value;=0A= }=0A= window.location =3D url;=0A= }=0A= }=0A= =0A= /**=0A= * @param aAuthorArray array stores all authors=0A= * @param aKeywordArray array stores all keywords=0A= */=0A= function getAuthorsAndKeywords(aForm, aAuthorArray, aKeywordArray, = aForceAdd)=0A= {=0A= setCheckWhenDefine(aForm.sauthusercbx, true);=0A= setCheckWhenDefine(aForm.keyusercbx, true);=0A= =0A= var elements =3D aForm.elements;=0A= var cntAuthor =3D -1;=0A= var cntKeyword =3D -1;=0A= for (var i=3D0; i0) || (aKeywords.length>0))=0A= {=0A= pmurl =3D = "http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?dopt=3DDocSum&cmd=3DSearch= &db=3DPubMed&term=3D";=0A= =0A= for (var i=3D0; i0) && (aKeywords.length>0)) {=0A= pmurl +=3D "+OR+";=0A= }=0A= =0A= for (var i=3D0; i0) || (aKeywords.length>0))=0A= {=0A= adsurl =3D=0A= "http://adsabs.harvard.edu/cgi-bin/nph-abs_connect?db_key=3DAST" +=0A= "&sort=3DSCORE&ttl_syn=3DYES&version=3D1" +=0A= "&txt_syn=3DYES&txt_logic=3DAND" +=0A= "&aut_syn=3DYES&aut_logic=3DAND";=0A= =0A= if (aAuthors.length > 0) {=0A= adsurl +=3D "&author=3D";=0A= for (var i=3D0; i 0) {=0A= adsurl +=3D "&text=3D";=0A= for (var i=3D0; i0) || (aKeywords.length>0))=0A= {=0A= var array =3D [aAuthors, aKeywords];=0A= for (var i=3D0; i0&&parent.frames.length) {=0A= d=3Dparent.frames[n.substring(p+1)].document; n=3Dn.substring(0,p);}=0A= if(!(x=3Dd[n])&&d.all) x=3Dd.all[n]; for = (i=3D0;!x&&i 0) ? "&" : "?");=0A= x +=3D (aButton + ".x=3D1");=0A= aForm.action =3D x;=0A= }=0A= }=0A= else=0A= {=0A= var sltd =3D x.options[x.selectedIndex];=0A= var pfxJSBTN =3D "jsbtn:";=0A= if (sltd.value.indexOf(pfxJSBTN) =3D=3D -1)=0A= {=0A= var value =3D (pfxJSBTN + aButton + ";" + sltd.value);=0A= //x.options[x.selectedIndex] =3D new Option(sltd.text, value, true, = true);=0A= x.options[x.selectedIndex].value =3D value;=0A= }=0A= }=0A= aForm.submit();=0A= }=0A= }=0A= =0A= =0A= function onChangeClick(aSelect, button)=0A= {=0A= var frm =3D aSelect.form;=0A= submitFormWithButtonClicked(frm, button);=0A= }=0A= =0A= function onChangeClick2(aSelect, button)=0A= {=0A= var frm =3D document.forms[0];=0A= var x =3D frm.action;=0A= x +=3D "?"+button+".x=3D1";=0A= frm.action =3D x;=0A= frm.submit();=0A= }=0A= =0A= function submitArticles(aForm, action) {=0A= submitMultiArticles(aForm, action, true);=0A= }=0A= =0A= /* Function to show next search page */=0A= function searchNextResultPage(aForm, aOffset)=0A= {=0A= var x =3D eval(aForm.startPage.value) + aOffset;=0A= aForm.startPage.value =3D x;=0A= aForm.submit();=0A= }=0A= =0A= /* Function to show help page */=0A= function popupHelp(url)=0A= {=0A= popupHelp550(url);=0A= }=0A= =0A= /* Function to open up a new browser window, without a navigation bar */=0A= function newWindow(url)=0A= {=0A= var new_window;=0A= var windowProperties;=0A= windowProperties =3D = "width=3D750,height=3D700,top=3D30,left=3D230,toolbar=3D0,menubar=3D0,res= izable=3D1,scrollbars=3Dyes";=0A= if(new_window=3D=3Dnull) {=0A= new_window =3D window.open(url,null,windowProperties);=0A= } else {=0A= new_window.document.replace(url);=0A= }=0A= }=0A= =0A= /* Function to open up a new browser window, without a navigation bar */=0A= function newInstitutionWindow(url)=0A= {=0A= var new_window;=0A= var windowProperties;=0A= windowProperties =3D = "width=3D750,height=3D700,top=3D30,left=3D230,toolbar=3D1,menubar=3D1,res= izable=3D1,scrollbars=3Dyes";=0A= if(new_window=3D=3Dnull) {=0A= new_window =3D window.open(url,null,windowProperties);=0A= } else {=0A= new_window.document.replace(url);=0A= }=0A= }=0A= =0A= =0A= /**=0A= * Function to close the current window=0A= */=0A= function close_window() {=0A= window.close();=0A= }=0A= =0A= /**=0A= * get all the form elements and check if the "dbname"=0A= * maps to specifid value.=0A= */=0A= function checkDbName(aForm, aValue)=0A= {=0A= var elements =3D aForm.elements;=0A= for (var i=3D0; i0)) {=0A= alert("Sorry, email alert for journals with \n" +=0A= "full access rights is not available.");=0A= return;=0A= }=0A= =0A= var elmts =3D aForm.elements;=0A= for (var i=3D0; i