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The Fuzzy Math of Solid Tumor Stem Cells: A Perspective

¹ The Sidney Kimmel Comprehensive Cancer Center at Johns Hopkins University, Baltimore, Maryland and ² The Norris Comprehensive Cancer Center at The University of Southern California Keck School of Medicine, Los Angeles, California

Requests for reprints: Scott E. Kern, Department of Oncology, Johns Hopkins University, 1650 Orleans Street, Baltimore, MD 21231. Phone: 410-614-3314; Fax: 443-287-4653; E-mail: sk{at}jhmi.edu.

Abstract

Apparently effective therapeutic agents very often fail to cure cancer patients. It is therefore attractive to wonder whether a specific resistant cell subset should be recognized and separately targeted. In solid tumors, such as carcinomas, a minor population of "cancer stem cells" has been proposed and sought experimentally in human tumors and isolated cell populations. It is often overlooked that the rationale and supportive data are essentially numerical and can be evaluated as such. A reevaluation of the published studies and related claims within awarded U.S. patents suggests that the mathematical support for the concept of therapeutically useful stem cells is weak and may even invalidate the foundations of these publications and patent claims. Mathematical arguments should be used more consistently, because they can serve as a guide for interpreting studies into cancer stem cells of solid tumors. [Cancer Res 2007;67(19):8985–8]

"When you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science." (William Thomson, Lord Kelvin).

The concept of cancer stem cells is intuitively suggested by clinical experience with leukemia, wherein cytologic features, cell cycle dynamics, and response to therapy of circulating leukemic cells can differ greatly from the proliferating leukemic blasts in the bone marrow. The concept is also evident in teratocarcinomas (1). It has become recently popular to extend the concept to other solid tumors, such as adenocarcinomas and brain tumors. There were more published original research articles on carcinoma stem cells within the first 4 months of 2007 than in the preceding 4 years. The U.S. Patent and Trademark Office has issued patents covering cancer stem cells, of which two have proved especially influential and served as a potential basis of biopharmaceutical business development.

To date, there is little published commentary examining either the publications or the patent documents, although a recent review by Hill offered a thoughtful and skeptical perspective (2). It would be appropriate to extend such an analytic examination of the solid tumor stem cell concept. We agree with the authors of the proponent publications and the patent inventors, as well as with Hill, that the stem cell concept is rooted in the analysis of cell numbers of the original unfractionated tumor cell populations and in sorted subfractions. Readers, however, may have not recognized yet the full measure of what the numbers tell us.

The Mathematical Genesis of the Solid Tumor Stem Cell Concept

Briefly, stem cells are proposed to be a minor and phenotypically distinguishable population of cells that is fully responsible for the numerical growth in the cancer cell population due to their long-term ability to produce an exponentially expanding reproductive lineage. The nonstem cells are roughly conceived as lacking this ability.

For example is a tumor successfully treated so that only 0.5% of the original cells remains. If the original tumor had a stem cell population constituting 0.5% of its total cells and these cells preferentially escape therapy, numerically this once 0.5% resistant population (now 100% of residual cells) would be fully explained. One would then wish to isolate such cells, study them, and identify a feature (such as a specific surface marker) that would allow new therapies to kill this special stem population. One would need to target all or the vast majority of the stem cells because simply killing some of them would not significantly slow tumor recurrence.

But let us now look at a different scenario, perhaps more common in the clinic: a tumor having a partial response to a treatment such that the diameter is reduced by 50% and is accompanied by a proportional loss of tumor cell number (Fig. 1 ). The volume becomes reduced by 87.5% (1–1/2³), leaving 12.5% of the original tumor cells. A minor population of stem cells in this setting cannot be responsible for treatment failure, as noted by Blagosklonny (3). Nonetheless, mathematical expectation is that any resistant stem cells, if they are fully resistant, might become enriched in the remaining tumor mass, being enriched from 0.5% in the untreated tumor to 4% (0.5% times 2³) in the treated mass. In this case, even killing 87.5% of the originally resistant stem cells would only return the stem cells to their original value of a 0.5% subpopulation. It is also possible that the stem cells have no special resistance to the original therapy and would remain always as 0.5% of the total tumor cells (Fig. 1).

View larger version (24K): [in this window] [in a new window] [Download PPT slide]   Figure 1. Stem cells may or may not be enriched by therapy. A partial clinical response is illustrated. Numerical data are not yet available from studies of treated human solid tumors to discern whether stem cells are enriched after partially successful therapy.

Thus, the identification of the putative stem cells for therapeutic targeting would be most important in instances of minimal residual disease, wherein most nonstem cells are already eliminated. The concept would be bolstered by a demonstration that stem-like cells are numerically enriched in effectively treated tumors. Treatments would need to effectively target >99% of the cancer stem cells under the known difficulties attending clinical settings. Unfortunately, as detailed below, cancer biologists cannot yet target cells with this level of accuracy even under well-controlled laboratory conditions.

The Numerical Inefficiencies of Isolating, Identifying, and Targeting Solid Tumor Stem Cells

A practical problem exists and impairs the utility of the solid tumor stem cell concept. No cellular fraction from human tumor tissues has yet been enriched, in which functionally measured stem cells were >5% of the isolated population, owing to the marker being promiscuously expressed outside the definable stem population (Fig. 2 ; refs. 6–8). For this very reason, the marker-defined subpopulation is often a sizable fraction of the originally isolated cell population. At this level of dilution, it is not possible to distill out the markers or the therapeutic targets that would be highly specific for stem cells (2).

View larger version (26K): [in this window] [in a new window] [Download PPT slide]   Figure 2. Tumorigenic cells are better detected after cell fractionation techniques, but neither population is demonstrably pure. In published reports, the calculated sum of tumorigenic cells in the "stem" and "nonstem" populations often exceeds that of the original unfractionated population, indicative of uncharacterized inhibitors or other unanticipated artifacts.

Li et al. reported that whereas a fraction from pancreatic cancers containing a defined marker phenotype (CD44⁺CD24⁺ESA⁺) had an improved rate of generating xenografts, all other mutually exclusive fractions also retained the ability to give rise to xenografts (8). Li et al. explained that there is "1% to 3% of tumorigenic cells that invariably contaminate the nontumorigenic cells" due to the imprecision of cell sorting (8). Thus, Li et al. suggested that a nontumorigenic population cannot be obtained in practice. It is challenging to consider published claims that "1% to 3% tumorigenic cells" were in the "nontumorigenic cells" fraction after sorting, alongside the evidence that the unsorted original population already had <1% tumorigenic cells. Part of the problem is the inconsistent use of the term "tumorigenic" to mean alternately "marker-defined" cells or "tumor-producing" cells.

This numerical measure of impurity has dire implications for therapeutic targeting. Very many of the "xenograftable" cells remain in the so-called nontumorigenic fractions. Li et al. reported that the optimal fraction (CD44⁺CD24⁺ESA⁺) "had at least a 100-fold increased tumorigenic potential compared with nontumorigenic cells" and that <1% (0.2–0.8%) expressed this phenotype in the original unfractionated tumor. Thus, numerically, many or even the majority of the tumorigenic cells apparently were retained within the nontumorigenic fractions (Fig. 2; ref. 8). The mathematical analysis is as follows: the fraction having the stem phenotype times the fold increase in activity (0.2% x 100 = 20) is added to the residual nonstem phenotype times the unity activity (99.8% x 1 = 99.8) to generate the denominator for total tumorigenic activity of the combined populations (119.8). The stem cell activity is solely represented by the value 20. The "stem cell" fraction is thus calculated to contain <17% of the tumorigenic cells (20 divided by 119.8). More than 83% of the tumorigenic cells lie in the nontumorigenic fractions. Presuming even that stem-cell targeting could be made as efficient in the patients under ideal laboratory sorting conditions, antistem therapy would fail to target significant numbers of the stem cells. Patients are not clamoring for yet another therapy that reduces tumor growth by 17%.

Collins et al. reported numbers supporting an even more severe inefficiency of stem cell identification (5). They used the CD44⁺2ß1⁺CD133⁺ population to isolate stem cells from prostate cancer, estimating that 0.1% of cells had the phenotype. The fraction was only 3-fold to 4-fold enriched in clonogenicity over the unselected population (their Fig. 1), allowing readers to derive a mathematical estimate that >99.6% of the stem cells lacked the phenotype and resided in the "stem-depleted" fraction. If stem-directed therapy could deplete 0.4% of the cancer stem cells of patients, fuzzy math will become reflected in fuzzy therapy.

Patent Claims Depending on Numerical Criteria

Ironically, the governmental patent process often requires stricter definitions than found in scientific publications. Patents are, thus, a rich source of public access scientific literature. The numerical problems surface again when considering the claims awarded by the U.S. Patent and Trademark Office. In U.S. patent 6984522,³ all three independent aims involve examining cells for CD44 and CD24 surface markers. Two of the independent claims include a functional test as a requisite defining term for the claimed entity, a population of cells. Specifically, claims 1 and 20 define "solid tumor stem cells" as distinct from the remaining cells, termed simply "solid tumor cells." "Solid tumor stem cells" are stated to satisfy the criterion "are tumorigenic," a nonzero number. "Solid tumor cells" are, by contrast, stated to be "nontumorigenic," referring to the unambiguous number "zero".

The requirement for a functional test to define the subject covered is uncommon and is usually avoided in patent claims. In this instance, the functional test seems unacceptable for a number of reasons: (a) The claims do not specify the tumorigenicity test to be used; many methods exist. (b) Claim 1 requires that at least 75% of the "stem cell" population be tumorigenic, but current assays for tumorigenicity of tumor-isolated cells do not test individual cells, and thus a quantitation of the fraction of cells, as required by the claim language, may be impractical. For example, if 50% of cells were tumorigenic and thus violated the criterion of claim 1 and if the assay used 100 cells per injection site in a mouse, then only in 0.5¹⁰⁰ of the tests (or virtually never) would the assay fail to give rise to a tumor. (The estimated number should be adjusted based on the number of cells needed for minimal acceptable xenograft success. The patent lists 100 cells as the minimal example.) Obviously, the numerical efficiency of xenografting is less than this level of performance, and thus no current assay could establish a population to have survived the test of this criterion specified within the claim. Claim 1 thus seems to be unworkable — an impractical claim. (c) Prior studies have not established whether a cell population can be isolated from a solid tumor that is inherently and stably "nontumorigenic" (Fig. 2). Presumably, if the population can be treated in such a way as to improve the efficiency of xenografting for this same population, the population would then violate the very term used to define it (more on this subject below). (d) It is virtually impossible to satisfactorily define a cell population as "tumorigenic" or as "nontumorigenic." The definition is thus a fallacy. As noted above, the inventors themselves lamented the inevitability of contaminated cell populations in Li et al. (8).

Variables Affecting Xenograft Rates

A tumorigenic population may become damaged or inhibited by mechanical, chemical, or biological means so that it seems "nontumorigenic." Such an inhibitor was described explicitly by O'Brien et al. (6). In their study, the addition of CD133^– cells reduced the tumorigenicity of the CD133⁺ cells by 95% (from 20 to 1 tumor-initiating cells per 57,000 tested cells). The investigators proposed that the marker-negative cells "are negatively regulating the growth of the CD133⁺" tumor-initiating cells. Such a finding seems to fully invalidate the xenograft assay to measure stem cells (Fig. 2).

Conventionally, when the purity rather than the quantity of a subject dominates an assay's results, an assay inhibitor is suspected. As another study of colorectal cancer remarked, "despite the higher number of CD133⁺ cells present in 10⁶ unseparated cells, tumor formation after the injection of the total colon cancer population was slower and less efficient than that obtained with purified CD133⁺ cells, in line with results reported for breast cancer stem cells" (12).

The inventors on the patents saw similar evidence in a numerical discrepancy pointed out by Hill (2). If 1% of the sorted CD44⁺CD24^– cells are tumorigenic stem cells and 11% to 35% of the cells have this marker profile, as reported in the study by Al-Hajj et al. (7), the extreme requirement for >10,000 of the original unfractionated breast cancer–derived cells to produce a xenograft is a numerical discrepancy.

A study of brain tumors suggested the same problem (13), although the xenograft efficiency of the original unfractionated cells unfortunately was not reported. CD133⁺ cells constituted 6% to 29% of the cells from glioblastomas and medulloblastomas; as few as 100 fractionated CD133⁺ cells gave rise to xenografts. By mathematical extension, 333 to 3,333 of the original unfractionated cells in the absence of an inhibitor would have also produced xenografts. Yet the investigators in the same paragraph noted that xenografts of human tumor cells usually require 10⁵ to 10⁶ injected cells.

A related numerical discrepancy exists in the study by Al-Hajj et al. (7). The authors report that 11% to 35% of human breast cancer tumor or ascites cells (depleted of lineage markers of normal cells) were of the CD44⁺CD24^–/low phenotype and that this fraction was 10-fold to 50-fold enriched for tumor-forming activity compared with the original unfractionated tumor cells. The fractionated cells thus contained anywhere from 110% (11% times 10-fold) to 1,750% (35% times 50-fold) of the activity of the unfractionated original samples.

Thus, we should infer that a strong inhibitor of grafting was probably present in at least some of these experiments, if not frankly dominating most of them. Something had been created from nearly nothing, which in the course of laboratory investigation is a highly suggestive indication of an inhibitor, one having been expunged by mechanical fractionation. Interested investigators should assay for inhibitors using a number of methods, including the extended dilution of marker-negative cells to dilute any inhibitors and by spiking marker-positive cells into marker-depleted fractions to determine whether the xenograft-supporting qualities of the marker-positive population become impaired in the presence of the marker-negative sample.

Aside from the issue of inhibitors, the inventors in U.S. patents 6984522 and 7115360³ used a number of improvements to aid xenografting. Implicitly then, stemness could not be the sole variable affecting the numerical results produced by these so-called stem cell xenograft assays. These additional listed variables include the varying use or omission of Matrigel, use of the mammary fat pad but not other sites, pretreatment of the host with estrogen pellets, inclusion in the "culture medium capable of supporting cell growth" of calf serum, hormones, insulin, transferrin, selenium, or B27 supplement, use of serum-free media containing "growth factors effective for inducing stem cell proliferation," and physiologic O₂ concentrations.

U.S. patent 7115360 adds another variable, noting that etoposide (VP-16) is given to some SCID mice. The reason given in the patent that in this manner the "mice can be further immunosuppressed," seems to contradict the statement in the inventors' U.S. patent 6984522 that "immunodeficient mice do not reject human tissues." The inventors comment on the discrepancy in Li et al., where they reveal that in their prior experiments concerning breast cancers, there was "an improvement in the rate of engraftment with pretreatment of mice with VP-16" (8).

Summary

Readers should realize that the above arguments are made from the numbers, terms, and conclusions provided by proponents, authors, and inventors of solid tumor stem cell reports. We have not applied a priori objections to the stem cell concept, questioned whether the markers were transient phenotypes or were altered by the experimental manipulations, or challenged the relative teleologic attractiveness of a theory purporting that tumor evolution selects consistently for clones of cells that will self-regulate so as to remain a small minority of the overall cell population. Instead, we provide multiple examples of strategies arising from a respect for numerical validity that can be used to guide and critique interpretations of stem cell data.

Personally, we suspect that tumorigenic behavior might be a varying probabilistic potential for all tumor cells rather than quantal and deterministic feature of a minority of tumor cells (2). A definition of "solid tumor stem cells" may evade us for some time. In the meantime, numbers, as well as simple schematic tree diagrams reflecting the clonal evolution of examined tumors (14), provide constraints on any definition of "stemness." We encourage authors, peer reviewers, patent examiners, and journals to systematically use such analytic aids to examine the data in front of them and to better convey conceptual models of solid tumor stem cells. The fuzzy alternative "is of a meager and unsatisfactory kind."

Acknowledgments

Grant support: NIH grants CA62924 and CA111940.

Footnotes

³ U.S. patents are publicly available online at http://www.uspto.gov.

Received 5/30/07. Revised 6/19/07. Accepted 7/ 6/07.

References

1. Sell S, Pierce GB. Maturation arrest of stem cell differentiation is a common pathway for the cellular origin of teratocarcinomas and epithelial cancers. Lab Invest 1994;70:6–22.[Medline]
2. Hill RP. Identifying cancer stem cells in solid tumors: case not proven. Cancer Res 2006;66:1891–5.[Abstract/Free Full Text]
3. Blagosklonny MV. Target for cancer therapy: proliferating cells or stem cells. Leukemia 2006;20:385–91.[CrossRef][Medline]
4. Potten CS, Loeffler M. Epidermal cell proliferation. I. Changes with time in the proportion of isolated, paired and clustered labelled cells in sheets of murine epidermis. Virchows Arch 1987;53:279–85.
5. Collins AT, Berry PA, Hyde C, Stower MJ, Maitland NJ. Prospective identification of tumorigenic prostate cancer stem cells. Cancer Res 2005;65:10946–51.[Abstract/Free Full Text]
6. O'Brien CA, Pollett A, Gallinger S, Dick JE. A human colon cancer cell capable of initiating tumour growth in immunodeficient mice. Nature 2007;445:106–10.[CrossRef][Medline]
7. Al-Hajj M, Wicha MS, Benito-Hernandez A, Morrison SJ, Clarke MF. Prospective identification of tumorigenic breast cancer cells. Proc Natl Acad Sci U S A 2003;100:3983–8.[Abstract/Free Full Text]
8. Li C, Heidt DG, Dalerba P, et al. Identification of pancreatic cancer stem cells. Cancer Res 2007;67:1030–7.[Abstract/Free Full Text]
9. Zheng X, Shen G, Yang X, Liu W. Most C6 cells are cancer stem cells: evidence from clonal and population analyses. Cancer Res 2007;67:3691–7.[Abstract/Free Full Text]
10. Patrawala L, Calhoun T, Schneider-Broussard R, Zhou J, Claypool K, Tang DG. Side population is enriched in tumorigenic, stem-like cancer cells, whereas ABCG2+ and ABCG2- cancer cells are similarly tumorigenic. Cancer Res 2005;65:6207–19.[Abstract/Free Full Text]
11. Szotek PP, Pieretti-Vanmarcke R, Masiakos PT, et al. Ovarian cancer side population defines cells with stem cell-like characteristics and Mullerian Inhibiting Substance responsiveness. Proc Natl Acad Sci U S A 2006;103:11154–9.[Abstract/Free Full Text]
12. Ricci-Vitiani L, Lombardi DG, Pilozzi E, et al. Identification and expansion of human colon cancer–initiating cells. Nature 2007;445:111–5.[CrossRef][Medline]
13. Singh SK, Hawkins C, Clarke ID, et al. Identification of human brain tumour initiating cells. Nature 2004;432:396–401.[CrossRef][Medline]
14. Tsao J, Tavaré S, Salovaara R, Jass J, Aaltonen L, Shibata D. Colorectal adenoma and cancer divergence: Evidence of multilineage progression. Am J Pathol 1999;154:1815–24.[Abstract/Free Full Text]

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