Cancer stem cells represent a novel therapeutic target. The major challenge in targeting leukemic stem cells (LSC) is finding therapies that largely spare normal hematopoietic stem cells (HSC) while eradicating quiescent LSCs. We present a mathematical model to predict how selective a therapy must be to ensure that enough HSCs survive when LSCs have been eradicated. Stem cell population size is modeled as a birth-death process. This permits comparison of LSC and HSC eradication times under therapy and calculation of the number of HSCs at the time of LSC eradication for varied initial population sizes and stem cell death rates. We further investigate the effects of LSC quiescence and resistance mutations on our predictions. From a clinical point of view, our models suggest criteria by which cancer stem cell therapy safety can be assessed. We anticipate that in conjunction with experimental observation of cancer stem cell killing rates, our results will be useful in screening targeted therapies for both hematologic and solid tumor malignancies. [Cancer Res 2009;69(24):9481–9]

Major Findings

We apply a birth-death process model to investigate how selective a therapy must be to ensure that an adequate number of normal hematopoietic stem cells survive when leukemic stem cells (LSC) have been driven extinct. Our models take into account quiescence and genetic heterogeneity and make specific predictions in terms of two interpretable quantities, therapeutic selectivity and killing efficiency, which are readily calculated for a proposed therapy using in vitro death rates. Major clinical implications include our finding that the number of LSCs at therapy initiation is less important than therapy selectivity and the necessity of directly targeting quiescent LSCs. Finally, we consider the effects of combining targeting LSC therapy with niche signaling inhibitors and conclude that modest increases in apoptosis rates can be augmented by inhibiting stem cell renewal, leading to cure.

Quick Guide to Equations and Assumptions

In studying stem cell population dynamics as a linear birth-death process Xt, it is helpful to clearly lay out all assumptions. Our simplest model is diagrammed in Fig. 1A.

$E(Xt)=ne(β−δ)t$
Equation A:

Several well-known formulas are required for understanding the implications of the birth-death models. The most important formulas cover the average (Eq. A) and SD $[n(β+δβ−δ)(e2(β−δ)t−e(β−δ)t)]12$ of the number of stem cells at time t, where n is the initial number of cells at the beginning time 0 (34). When the death rate exceeds the birth rate (δ > β), the most crucial feature of these formulas is the exponential decline in both the average and SD with time.

Major Assumptions

All LSCs divide at the same rate β and die at the same rate δ per cell. Thus, in the small time interval from t to t + Δt, a particle dies with approximate probability δΔt and symmetrically divides with approximate probability βΔt. Each stem cell behaves independently and initiates a clan of descendants, which individually behave like the ancestral cell. When the death rate exceeds the birth rate (δ > β), clan extinction is certain. The extinction time for the entire process is the maximum extinction time for the different clans issuing from the various ancestral stem cells. We ignore asymmetrical division that leaves a stem cell intact and produces one partially differentiated progenitor cell. Although such events are relevant to the entire organism's health, they are irrelevant to the ultimate fate of a stem cell population.

The probability that all cells derived from n clans are extinct by time t

$Pr(Xt=0)=[δ−δe(β−δ)tδ−βe(β−δ)t]n$
Equation B:

is also important.6

Major Assumptions

The independent behaviors of the different clans allow us to express the extinction probability as the nth power of the extinction probability of a single clan. Because δ > β, this probability tends to 1 as t tends to ∞.

Birth-death process theory permits us to answer questions such as what is the probability that at least one normal stem cell survives after all LSCs have been eradicated? The answer to this and more nuanced questions depends on the initial numbers of the two cell types as well as on the killing differentials. The extinction time Mn over all n clans initiated by the original population of stem cells, leukemic or normal, equals the maximum extinction time for the separate clans. Based on Eq. B, one can exploit the asymptotic theory of extreme value statistics (35) to show that the standardized random variables (δ − β) (Mnan) stabilize6 for the carefully chosen sequence

$an=1δ−β[lnn+ln(δ−β+βnδ)]≈1δ−β[lnn+ln(1−βδ)]$

In fact, the sequence of random variables (δβ) (Mn − an) tends to the Gumbel distribution with mean γ ≈ 0.57722 and variance π2/6 (36).

These considerations produce the accurate mean and variance approximations

$E(Mn)≈an+γδ−β,Var(Mn)≈1(δ−β)2π26$
Equation C:

The sequence an is a linear function of ln n. This weak dependence on n suggests the possibility of safely eradicating LSCs even when they sharply outnumber HSCs. In this regard, it is helpful to define therapy selectivity σ as the ratio of the differences between death and birth rates for LSCs versus HSCs, namely

$σ=δL−βLδH−βH$

Here, rates are subscripted by L for leukemia and H for healthy. For safe eradication, one needs σ to be substantially greater than ln nL/ln nH.

We also solve for the time t that renders the extinction probability (Eq. B) equal to a specific value p > 0. To good approximation6

$t=−1δ−βlnδ−δp1/nδ−βp1/n≈−1δ−βln(−lnp)+an$
Equation D:

Alternating between Eq. B and Eq. D allows us to determine the extinction probability for the healthy stem cells at the time when the extinction probability for the LSCs reaches a predetermined level p. When birth rates are unaffected by therapy, the survival probability of the HSC population increases quickly as the difference between δH and δL grows. For example, take δH = 0.08 week-1 and βL = βH = 0.024 week-1. To be 80% certain that at least one ordinary stem cell remains when we are 99.9% certain no LSC remains, δL must be 0.15 week−1 or larger, corresponding to therapy selectivity σ of at least 2.1. We provide more definitive estimates in Results.

Perhaps the most useful measure of success in a targeted therapy is the number of HSCs NH left at the precise moment of LSC extinction. We define the killing efficiency κ of a therapy as the ratio δLL of the death rate to the birth rate of LSCs. The average (expected) number of healthy stem cells present at the time when all the LSCs are eradicated6 depends on the selectivity σ and killing efficiency κ of the therapy through the formula

$E(NH)≈nHΓ(1+σ−1)[nL(1−κ−1)+κ−1]1σ≈nH[nL(1−κ−1)+κ−1]1σ$
Equation E:

where Γ(·) is Euler's γ function. For an effective therapy, δL > βL, κ > 1, and σ > 1. The presence of the σth root in the denominator of Eq. E implies that the expected number of HSC is driven more by therapy selectivity and the initial numbers of HSCs and LSCs than by killing efficiency.

Cancer stem cells play roles in both solid tumors (110) and hematopoietic malignancies (1115). Cancer stem cells are involved in disease pathogenesis and progression as well as in relapse and resistance to therapy (8, 9, 14, 15). Therapies proposed to target cancer stem cells include monoclonal antibodies (16); vaccines (17); apoptosis induction via oxidative stress responses, NF-κB inhibition, and p53 activation (18); drugs targeting the Notch and Hedgehog (Hh) signaling pathway (19, 20); and other targeted therapies (2123).

One major concern in targeting leukemic stem cells (LSC) is the risk of killing large numbers of nonmalignant hematopoietic stem cells (HSC). Therapy designed to completely eradicate cancer stem cells could lead to dangerously low HSC populations, threatening their ability to maintain homeostasis and tissue repair. Recognizing this threat, therapies are being designed to selectively eliminate cancer stem cells in acute myelogenous leukemia (18, 2426), chronic myelogenous leukemia (CML; ref. 27), and solid malignancies (23). However, it is unclear how selective a therapy must be to ensure that an adequate population of HSCs is present. Other major therapeutic complications include quiescence and heterogeneity in genetic expression and phenotype of LSCs both across patients with a given leukemia and within a patient over disease course (24). Quiescence is a challenge because quiescent LSCs may be resistant to targeted therapy (28). Finally, one must consider the role of the stem cell niche, the microenvironment that houses stem cells and regulates their proliferation, quiescence, and differentiation. The HSC niche is partially driven by the Wnt-Hh signaling pathway, a complex protein network with roles in both embryogenesis and carcinogenesis. Therapy inhibiting this pathway can impair CML stem cell self-renewal (20). Combining such therapy with a dasatinib-like drug, which induces apoptosis by binding to and inhibiting the constitutively active bcr-abl tyrosine kinase in CML cells, may lead to LSC eradication.

Mathematical models of both healthy and cancer stem cells provide insights into their biology (2933). This article discusses the clinical implications of a quantitative model developed in full mathematical detail elsewhere.6

6M.E. Sehl et al. Extinction models for cancer stem cell therapy, submitted for publication.

As an example, we apply our model to CML. The primary purpose of the model is to determine the differential in death rates a therapy must possess to eradicate a LSC population while maintaining an adequate HSC population. This differential is succinctly expressed by selectivity and efficiency indices introduced later. A branching process elaboration of the model highlights the importance of following quiescent as well as active LSCs during therapy. When the model incorporates features such as drug-resistant mutations, it becomes difficult to derive explicit analytic results. We therefore briefly describe numerical tools for fast simulation of complicated versions of the model. These tools allow us to explore combination therapies that target both LSCs and the microenvironmental niche signaling supporting their expansion.

### Quiescence

Quiescence is a reversible state of not dividing; the analogous state of senescence is not. Quiescent LSCs have been isolated in CML (37). Although some therapies are effective against both active and quiescent cells, other therapies such as dasatinib are effective in inducing apoptosis in active CML stem cells but do not eliminate quiescent ones (27). In this setting, we predict that the active LSC population will be eradicated first and leave behind a quiescent LSC population, which on awakening causes cancer recurrence.

Figure 1B presents a variant of our model that captures both HSC and LSC quiescence and awakening. Here, a quiescent stem cell dies with rate ν and awakens to an actively dividing state with rate α. Active stem cells divide or die. We consider the case where a therapy is less effective at killing quiescent LSCs than active LSCs so that ν < δ. For simplicity, we initially ignore transitions to quiescence by taking φ = 0 in Fig. 1. These transitions to quiescence only slow LSC eradication and do not detract from the thrust of our arguments.

Figure 1.

Modeling stem cell dynamics as a birth-death process. A, basic model: linear birth-death process. B, extension of model to include quiescence of LSCs. C, model including heterogeneity of LSCs.

Figure 1.

Modeling stem cell dynamics as a birth-death process. A, basic model: linear birth-death process. B, extension of model to include quiescence of LSCs. C, model including heterogeneity of LSCs.

Close modal

### Modeling LSC heterogeneity

Our simple model and its elaboration are both multitype branching processes (34). This general framework is compatible with even more complicated models, but mathematical analysis is more formidable than for single-type processes such as birth-death processes. If one is satisfied with population means, then considerable progress can be made. Multitype branching particles behave independently according to prescribed stochastic rules. Each particle of type i has death intensity λi. At the end of its life, a type i particle reproduces particles of its own type and particles of other types. Let fij be its expected number of type j daughter particles. The mean behavior of the system of particles is summarized by a matrix M(t) = [mij(t)] whose entry mij(t) equals the mean number of type j particles at time t starting with a single type i particle at time 0. When the process starts with ni type i particles, there are on average Σinimij(t) type j particles at time t. M(t) satisfies the matrix differential equation $ddtM(t)=M(t)Ω$, where Ω = (ωij) has diagonal entries ωii = λi(fii − 1) and off-diagonal entries ωij = λifij. Because the process starts with a single type i particle, we have initial conditions mii(0) = 1 and mij(0) = 0 for ij. The matrix exponential $etΩ=∑k=0∞tkk!Ωk$ solves the differential equation for M(t) subject to the initial conditions.

As an illustration, we model heterogeneity in gene expression of LSCs occurring across patients or within a patient over time. Let a stem cell acquire progressive decreases in susceptibility to therapy through a series of resistance mutations, each occurring at rate μ. For simplicity, we neglect other heterogeneity mechanisms such as epigenetics and niche regulatory signaling.

Figure 1C describes four LSC types, each with varying therapy susceptibility manifested through different death rates δ1,…,δ4. Each LSC type has the same symmetrical division rate β, and any cell of type 0 through 3 can acquire a mutation at rate μ rendering it less susceptible to therapy. As the figure indicates, more susceptible cell types gradually progress to less susceptible types. The model excludes back mutation to a more susceptible state. The HSC population constitutes a separate type, type 0, which mutates to a type 1 LSC with rate μ. As in our original model, HSCs divide with rate βH and die with rate δH per cell. Mutation is rare. As a guess, consider CML with ∼40 known mutations. If each mutation occurs at a rate of 10−8 per cell division, then μ = 4 × 10−7 (31).

In this extended model, the death intensities are

$λ0=δH+βH+μλi=δi+β+μ,1≤i≤3λ4=δ4+β.$

The matrix F = (fij) has trivial entries except for

$f00=2βHλ0,f01=μλ0fii=2βλi,fi,i+1=μλi,1≤i≤3f44=2βλ4.$

Thus, the matrix Ω is

$Ω=(β−δH−μμ0000β−δ1−μμ0000β−δ2−μμ0000β−δ3−μμ0000β−δ4)$

Because no back mutation occurs, Ω is upper triangular, and its eigenvalues appear along its diagonal. Along with their corresponding eigenvectors, these determine the asymptotic mean behavior of the system.

### Stochastic simulation

Stochastic simulation is useful in modeling stochastic processes where analytic results are not easily obtained. It can also verify complicated analytic results. We apply stochastic simulation to extend our birth-death process model to account for both LSC and HSC quiescence. This permits backflow into quiescence at rate φ per active cell (see Fig. 1). We also compare the matrix exponentiation and stochastic simulation results for our LSC heterogeneity model.

Stochastic simulation algorithms range from fine-grained exact algorithms in which every reaction is simulated (SSA; ref. 38) to coarse-grained approximation methods (39). The τ-leaping method is intermediate (40). Over time interval (t, t + τ), the number of times a reaction fires is approximated by a Poisson random variable with mean equal to τ times the propensity of the reaction. We apply a variant of the τ-leaping method, the SAL algorithm (41), which includes linear and quadratic changes in reaction propensities. These improvements increase accuracy for larger systems without compromising speed.

### Parameter choices

Supplementary Table S1 lists our model parameter choices. Murine experimentation reveals that 0.007% of bone marrow cells are long-term HSCs with life-long myelolymphoid reconstruction ability (42, 43). Extrapolating murine and feline data to humans predicts a total HSC population of 22,000 stem cells (44). We consider two coexisting stem cell populations, LSCs and HSCs, with separate rates of birth, death, quiescence and awakening, and distinct initial numbers. We consider advanced leukemia in which the LSCs outnumber the HSCs at therapy initiation.

In CML, birth and death rates per stem cell occur on a time scale of months. These assumptions are based on a 42-wk replication time for human HSCs, extrapolated from feline HSC replication (44), and a death rate of 1 per 71 wk, extrapolated from CML patients on imatinib therapy (31, 32). Because CML LSCs are insensitive to imatinib therapy (27), these death rates are likely to be close to natural death rates for LSCs. Under therapy directly targeting LSCs, such as dasatinib or farnesyl transferase inhibitors, the death rate should be much higher and we consider effects of varying death rate in our calculations and simulations. Death rates νL and νH for quiescent LSCs and quiescent HSCs, respectively, also vary under therapy and we consider the case where νL << δL and νH << δH.

HSC quiescence and awakening rates are unknown. However, ∼90% of LSCs are reproductively active (45), whereas ∼75% of HSCs are quiescent (46). We fix an awakening rate of α = 0.07 week−1 and estimate the backflow to quiescence by assuming equilibrium for the stem cell population. In Results, we consider two scenarios: the case in which the HSC and LSC quiescence rates are equal φH = φL and low, and the case in which φL << φH. In the first case, both the LSC and HSC populations respond to aberrant microenvironmental niche signaling. In the second case, LSCs fail to respond to normal niche signaling.

### Safety depends mainly on selectivity

First, we ignore quiescence. Figure 2 plots the trajectories of the average number of HSCs and LSCs, calculated using exact analytic results, and their extinction probabilities, calculated using the Gumbel approximation (3). Although LSCs initially outnumber healthy stem cells, the LSCs are much more likely to go extinct first. At the time when the LSC extinction probability approaches 1 (18–20 weeks), ∼1,000 HSCs remain.

Figure 2.

Mean population size and extinction probability of LSCs and HSCs during therapy. Here, δL = 0.59, δH = 0.08, and βL = βH = 0.024.

Figure 2.

Mean population size and extinction probability of LSCs and HSCs during therapy. Here, δL = 0.59, δH = 0.08, and βL = βH = 0.024.

Close modal

Figure 3A plots as a function of therapy selectivity σ the average number $E(NH)$ of HSCs left at the moment when LSCs reach extinction. As discussed, this average number is insensitive to changes in the killing efficiency κ (see Materials and Methods for definitions of σ and κ). Clearly, higher selectivity entails higher average numbers of HSCs at LSC eradication. For example, with κ = 25 and σ ≥ 10, we expect >1,000 HSCs to survive. The selectivity required to ensure that the mean number of surviving HSCs is over 1,000 at the time of LSC extinction (σ = 10) is much higher than the selectivity required to be 80% sure that at least one healthy stem cell survives (σ = 2.2). Although both results assess safety in theory, predicting $E(NH)$ is more valuable in practice because a handful of HSCs is unlikely to be sufficient for tissue repair and maintenance.

Figure 3.

Dependence of the mean (A) and distribution (B and C) of the number of HSCs at LSC extinction time on the selectivity (σ) of therapy. A, the mean is shown (solid line) ± 1 SD (dashed lines). δH = 0.08. B, δH = 0.08 and σ = 10. C, δH = 0.48 and σ = 1.2. In B and C, solid bars represent the distribution obtained using stochastic simulation, whereas lines represent the distribution obtained from analytic results.

Figure 3.

Dependence of the mean (A) and distribution (B and C) of the number of HSCs at LSC extinction time on the selectivity (σ) of therapy. A, the mean is shown (solid line) ± 1 SD (dashed lines). δH = 0.08. B, δH = 0.08 and σ = 10. C, δH = 0.48 and σ = 1.2. In B and C, solid bars represent the distribution obtained using stochastic simulation, whereas lines represent the distribution obtained from analytic results.

Close modal

We also examine the full distribution of HSCs at the precise moment when LSCs become extinct. This distribution is more helpful than the average in predicting the rarity of the event that HSC numbers become dangerously low under targeted therapy. Results obtained using analytic methods6 and stochastic simulation show nice agreement. Figure 3B and C depicts distributions obtained for varying selectivities. With decreasing selectivity, HSCs will very likely be wiped out by the time of LSC eradication.

It is interesting to compare the dependence of the mean number $E(NH)$ of HSCs at the time of LSC extinction on the selectivity, killing efficiency, and the initial proportion nL/(nL + nH) of LSCs. We start with σ = 10, κ = 25, and nL/(nL + nH) = 0.8 and change one variable at a time, holding the others constant. Based on Eq. E, Table 1 shows that $E(NH)$ is more sensitive to selectivity than to killing efficiency or the initial fraction of HSCs. When σ = 1.25, $E(NH)$ approaches dangerously low numbers. $E(NH)$ also falls as the initial proportion of nL increases but less dramatically than with σ. When the proportion of LSCs is 99%, there are still 77 HSCs at the time of LSC eradication.

Table 1.

The average number of HSCs remaining at LSC eradication depends on σ, but the mean time to LSC extinction depends on κ

VariableValue$E(NH)$$E(Mn)$ (wk)
σ 10.00 1,581 *
5.00 577 *
2.50 80 *
1.25 *
κ 25.00 1,581 18.05
12.50 1,588 37.50
6.25 1,603 81.43
3.13 1,637 196.55
nL/(nL + nH0.80 1,581 18.05
0.90 781 18.25
0.95 389 18.35
0.99 77 18.42
VariableValue$E(NH)$$E(Mn)$ (wk)
σ 10.00 1,581 *
5.00 577 *
2.50 80 *
1.25 *
κ 25.00 1,581 18.05
12.50 1,588 37.50
6.25 1,603 81.43
3.13 1,637 196.55
nL/(nL + nH0.80 1,581 18.05
0.90 781 18.25
0.95 389 18.35
0.99 77 18.42

NOTE: We start with σ = 10, κ = 25, and nL/(nL + nH) = 0.8, varying each parameter in turn while holding the others constant at their initial values.

*When βL and κ are held constant, $E(Mn)$ is not a function of σ.

Although κ has little effect on $E(NH)$, it determines the LSC mean extinction time. This fact becomes evident when we rewrite $E(Mn)$ (Eq. C) in terms of κ as

$E(Mn)=1βL(κ−1)[lnnL+ln(1−1κ+1nLκ)+γ].$

For example, κ = 25 leads to LSC eradication in 18 weeks on average, whereas κ = 3.125 increases mean time to eradication to ∼4 years (Table 1). Higher killing efficiency should lead to faster cures, which are important in reducing morbidity and mortality from side effects.

### Safe therapy must target quiescent LSCs

When the active LSC elimination rate, δLβL, is greater than the quiescent LSC elimination rate α + ν, recurrence is likely. To highlight the danger in targeting only active LSCs in the parameter regime α + ν < δLβL, we derived6 the approximation

$G(t)≈1−e−(α+ν)t[1+c+o(1)]$
(F)

to the probability that a quiescent LSC and all of its descendants are extinct by time t. Here, o(1) is an error term that tends to 0 exponentially fast as t tends to ∞, and c is a positive constant that is well approximated by

$c≈(δL−βL)αδL[(δL−βL)−(α+ν)],$

when δL >> βL. Thus, the behavior of G(t) is determined by the effective elimination rate α + ν of quiescent LSCs and to a lesser extent by the difference (δβ) − (α + ν).

If we start with nQL quiescent LSCs, then asymptotic extreme value theory entails a mean extinction time of

$1α+ν[lnnQ+ln(1+c)]+γα+ν.$
(G)

for the quiescent LSCs and their descendants. In contrast, the mean extinction time for the nL active LSCs is

$1δL−βL[lnnL+ln(δL−βL+βLnLδL)]+γδL−βL.$
(H)

Because α + ν < δLβL, the multipliers of ln nQ and ln nL in these formulas satisfy (δL − βL)−1 < (α + ν)−1. Thus, our analytic results show that when α + ν < δLβL, the mean extinction time of the nQL quiescent LSCs and their descendents exceeds the mean extinction time of the nL active LSCs unless nL is much larger than nQL. Therefore, oncologists potentially face two dangers: (a) driving the active healthy stem cell population to a low level or (b) prematurely stopping therapy shortly after actively dividing LSCs are eradicated and risking quiescent LSC awakening to revive the cancer.

When we consider quiescence in the HSC population, the relevant population to preserve includes both the active and quiescent HSCs. Stochastic simulation helps us understand the effects of incorporating backflow to quiescence and the effects of letting the quiescent LSC death rate νL approach the active LSC death rate δL. Figure 4A, to C shows the results of stochastic simulation when we incorporate both quiescence and awakening of LSCs and HSCs. Simulations were predominantly implemented by SAL with SSA steps when needed and run for fixed time periods with increments of 8.4 weeks. Mean population counts over 10,000 trials are plotted for each run. Negative population counts were avoided by rejecting any leap taking a population below zero. We considered three scenarios by varying death rates of quiescent LSCs and quiescence rates of LSCs. For σ = 10 and a lower quiescence rate for LSCs than for HSCs (Fig. 4A), there are adequate HSC population counts (quiescent and active) at the time of LSC eradication. The LSC extinction time is far longer when quiescence is involved (around 100 weeks versus 18 weeks).

Figure 4.

Effects of quiescence and heterogeneity of LSCs on population sizes during therapy. A, φL = 0.007 << φH = 0.21 and νL = νH = 0.00024 week−1. The coefficient of variation of population size increases as stem cell counts approach very small numbers, with ranges from 0.17 to 1.2 for LSCs, 0.03 to 0.11 for HSCs, 0.04 to 0.44 for quiescent LSCs (qLSCs), and 0.03 to 0.17 for quiescent HSCs (qHSCs). B, φL = φH = 0.007 and νL = νH = 0.00024 week−1. C, φL = φH = 0.007, νL = 0.25, and νH = 0.00024 week−1. In A to C, δL = 0.59, δH = 0.08, βL = βH = 0.024, and αL = αH = 0.07 week−1. D, arrow, increasing δH. In all cases, δ4 = 0.52, δ3 = 0.57, δ2 = 0.62, and δ1 = 0.67. Model 1: δH = 0.08 week−1. Model 2: δH = 0.31 week−1. Model 3: δH = 0.48 week−1.

Figure 4.

Effects of quiescence and heterogeneity of LSCs on population sizes during therapy. A, φL = 0.007 << φH = 0.21 and νL = νH = 0.00024 week−1. The coefficient of variation of population size increases as stem cell counts approach very small numbers, with ranges from 0.17 to 1.2 for LSCs, 0.03 to 0.11 for HSCs, 0.04 to 0.44 for quiescent LSCs (qLSCs), and 0.03 to 0.17 for quiescent HSCs (qHSCs). B, φL = φH = 0.007 and νL = νH = 0.00024 week−1. C, φL = φH = 0.007, νL = 0.25, and νH = 0.00024 week−1. In A to C, δL = 0.59, δH = 0.08, βL = βH = 0.024, and αL = αH = 0.07 week−1. D, arrow, increasing δH. In all cases, δ4 = 0.52, δ3 = 0.57, δ2 = 0.62, and δ1 = 0.67. Model 1: δH = 0.08 week−1. Model 2: δH = 0.31 week−1. Model 3: δH = 0.48 week−1.

Close modal

When LSC and HSC quiescence rates are equal, it is difficult to eradicate LSCs before active HSCs reach a critically low number at ∼100 weeks (Fig. 4B). Finally, when we increase the death rate of quiescent LSCs to about half the death rate of active LSCs, there are still adequate numbers of HSCs at the time of LSC eradication (Fig. 4C) and a faster time to cure is observed (around 30 weeks). We conclude that targeted LSC therapy must kill quiescent LSCs to be safe. Our results highlight the need for experimental determination of quiescence and awakening rates for both LSCs and HSCs.

### LSC heterogeneity

Figure 4D plots the mean trajectories of HSCs and a heterogeneous populations of LSCs using matrix exponentiation. Results from stochastic simulation closely match our analytic results. The arrow in Fig. 4 denotes gradually increasing HSC death rates. Not surprisingly, the least susceptible LSC population drives the overall LSC eradication dynamic. Here, we made the simplifying assumption of equal initial numbers (4,400) of each LSC type. Changing the initial numbers of each LSC type may shorten the overall extinction time. For example, less susceptible LSCs may go extinct before more susceptible LSCs if there are initially only a few less susceptible LSCs.

As we decrease the difference between δH and the most susceptible death rate δ1, there are fewer expected HSCs at the time of LSC eradication. With δH = 0.48, Fig. 4D shows that HSCs are nearly extinct by the time all LSCs are eradicated. Therapy selectivity ranges from 1.1 for the least susceptible LSC to 1.4 for the most susceptible LSC. Based on these results, it seems reasonable to define the selectivity of a heterogeneous LSC population using the death rate of the least susceptible population.

### Role of the stem cell niche

Stem cell microenvironmental signaling may play a large role in cancer stem cell dynamics. Hh signaling inhibition slows LSC self-renewal, suggesting a potential benefit of combining this therapy with standard tyrosine kinase inhibitors such as imatinib or dasatinib. Estimated death rates of LSCs on imatinib therapy are low (31). On imatinib therapy alone, we predict using Eq. C that extinction would take years. However, if Hh inhibition is added to imatinib therapy, thus decreasing the birth rate βL relative to the death rate δL, the killing efficiency may become sufficient to eradicate a LSC population in a reasonable amount of time.

Figure 5.

Effects of combining therapy inhibiting self-renewal with tyrosine kinase inhibition. A, effects of varying δL on expected time to eradication. Here, we hold the initial proportion of LSCs constant at 80%. B, effects of varying initial proportion of LSCs on expected time to eradication. Here, δL is held constant at 0.10.

Figure 5.

Effects of combining therapy inhibiting self-renewal with tyrosine kinase inhibition. A, effects of varying δL on expected time to eradication. Here, we hold the initial proportion of LSCs constant at 80%. B, effects of varying initial proportion of LSCs on expected time to eradication. Here, δL is held constant at 0.10.

Close modal

In conclusion, rational therapy is informed by mathematical modeling. The birth-death model gives us hope that attacking LSCs can cure CML. To be successful, targeted therapies must be tuned to spare HSCs and eradicate quiescent as well as active LSCs. This second criterion eliminates therapies that only attack dividing cells.

Highlighting $E(NH)$ dependence on selectivity is helpful in considering therapies that might differentially affect birth rates rather than death rates. For example, if we start with different birth rates for HSC and LSC and apply a therapy that decreases βL without affecting βH, we can calculate selectivity and $E(NH)$ in a similar fashion. Thus, our model is equally useful in predicting safety in this setting.

Our mathematical models may prove helpful in predicting therapeutic safety and efficacy as well as time to cure. Predictions will require reasonable estimates of LSC and HSC birth and death rates. Recently developed high-throughput techniques allow the identification of agents that selectively target cancer stem cells and spare ordinary cells (23). Of course, in vivo rates may be less than in vitro rates, so these predictions must be confirmed in animal studies. Some therapies will demand a delicate balance between the time to cure and safety. For example, a therapy may have a high killing efficiency with a time to cure of 3 weeks but may have low selectivity and cause dangerously low levels of HSCs for a transient time period. Another therapy may have high selectivity, ensuring an adequate HSC number when all LSCs have been eradicated, but still have a reasonable expected time to cure, say <1 year. From our perspective, slower cures are likely safer because they permit better monitoring of both HSC and LSC levels. On the other hand, the adverse side effects could tip the treatment balance toward quicker cures.

Our results may generalize to other leukemias and solid tumors in which cancer stem cells are being targeted. Their relevance depends on how well the modeling assumptions fit these applications. A limiting assumption in our model is that cancer stem cells behave independently. Although LSC populations occupying separate bone marrow niches may not influence each other, spatial considerations may be important in solid tumors. Stem cells are influenced by their microenvironment, and their fate is regulated by interactions with cells of the niche, as well as other stem cells, adhesion molecules, extracellular matrix, and gradients in oxygen tension, pH, calcium concentration, growth factors, and cytokines. Furthermore, gradients in blood supply and hence drug delivery to solid tumors may render some stem cells within a tumor more susceptible to therapy. Other mathematical models address vasculature and spatial effects on drug delivery (47). Incorporation of stem cell spatial dependence in our models would help generalize our results to solid tumors.

No one should underestimate the difficulty in developing targeted drugs. For the full spectrum of cancers, this may take decades. Although great improvements are under way (48), the ability to monitor and detect stem cells is currently primitive. The mean extinction time and expected number of remaining HSCs at the time of LSC eradication are predicted by the killing efficiency and selectivity, which in turn are based on our chosen death rates and initial LSC population size. The minimum HSC numbers adequate for tissue survival and maintenance are not known and may vary by tissue type. Therapy dosing and duration must be determined by observations of toxicity in preclinical and early clinical trials. Such trials should also address whether stem cell therapies should be combined with standard chemotherapy regimens targeting the differentiated cells comprising the bulk of the tumor. If available in the future, direct measurement of both the actively dividing and quiescent cancer stem cell populations, used in conjunction with our modeling predictions, may be ideal for making decisions about discontinuation of therapy.

Therapy successes are apt to come first for well-characterized cancers such as CML. Our optimism is not entirely predicated on the stem cell hypothesis of cancer. Recent evidence suggests that the number of cells with tumorigenic potential under various conditions may be several orders of magnitude higher than the number of cancer stem cells (49). Our models are flexible enough to include a wider class of cells. Provided the selectivity and killing efficiency are sufficiently high, nothing in principle prevents a cure. Cancer stem cell targeted therapy is more radical and risky than management of cancer as a chronic disease, and it must be performed with extraordinary care. Finally, it is helpful to keep in mind the potential of targeted therapies for diseases such as HIV. If a therapy could be devised to preferentially kill infected immune cells, it might have a chance of succeeding.

No potential conflicts of interest were disclosed.

Grant support: USPHS grants GM53275, MH59490, and GM008185 and American Society of Clinical Oncology Young Investigator Award.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

We thank Drs. Richard Gatti, Elliot Landaw, Julian Martinez, Van Savage, and Desmond Smith and Lindsay Riley for critical discussions and manuscript review.

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