In terms of the mechanism of metastasis, the rate of metastasis/migration surely depends upon differentiation status among individual cells (clinically undifferentiated cancer cells are commonly metastatic) and it also surely depends upon molecular, inter-cellular, intra-tumor, and intra-organ interactions that are both conceptually distinct and not fully understood. We expect that these interactions will arise naturally from extensions of this model when all variables, such as local hormone levels and growth factors, are available for a patient. At this time, the establishment of metastatic lesions in ectopic sites is expressed in our model as the function of a cell's differentiation status (differentiation coefficient), which is impacted by random mutations per cell division and the summary value of all external stimulations (including unfavorable growth conditions during migration and in the ectopic sites). How a metastatic cancer cell survives and proliferates to form a metastatic lesion is expressed in Eq. 9 and simulated in Fig. 5. Eq. 9 is the same equation used for primary tumor growth. However, the growth condition (β) is remarkably different for a cell in ectopic sites (soil effect).

“Neoplasm” is currently described in our model as a disease of altered proliferation and differentiation. Differentiation status, represented by the differentiation coefficient (Eq. 4), was found in this model to be the most important factor to describe a single cancer cell (whereas a tumor is a mass equivalent to a population resulting from proliferation of individual cells). Proliferation, driven by the proliferation potential of individual cells, is an important force to drive clonal expansion and formation of a clinically detectable mass. As in the case of diagnosis of cancer, cellular differentiation is traditionally described based on morphological features. Despite acceptance of the importance of differentiation status in characterizing cancer, research has yet to establish a molecular measurement to quantitatively determine how differentiated a cell is, covering cells in all developmental stages from stem cell to proliferating cell to terminally differentiated cell to senescent cell. As you point out, our current understanding of differentiation is vague, descriptive, and arbitrarily and largely based on visual inspection. The complete reliance on the visual inspection by a pathologist (pictorial description) in cancer diagnosis, and by extension in the description of tumor cell differentiation, has hampered the development of independent mathematical cancer models (schematic description) because of the following hurdles:
1. There is no tumor if a pathologist cannot see a mass, which makes it almost impossible to study subclinical and transient/dormant tumors in humans;
2. Since pathological diagnosis as a morphological inspection remains the standard, any molecular feature not consistent with the morphological diagnosis has to be discarded. There is no malignant feature if a pathologist asserts that the mass or cell is non-cancerous. Molecular study and analysis is completely restrained by the traditional morphological diagnosis.
3. Because the morphological diagnosis is so descriptive and rough in the categorization from benign to extremely malignant tumors, it lacks the distinguishing power to reveal the detailed involvement of multiple genetic factors and their dynamic interaction with changing environmental influence over a patient's lifetime. Internal insults (genetic alterations, mi in Eq. 7), external influence (environmental factors, β in Eq. 8) and patient age (chronological time, t) are the three basic parameters in our model. Their interactions and relative contributions could not be resolved by grouping tumors into 3-4 groups (tumor grades or clinical stages).

Despite many decades of painstaking efforts to establish molecular features of cancer, there is no molecular diagnosis which is independent and has the potential to replace the traditional pathological diagnosis, which remains the standard of patient management. The most prominent summary of cancer hallmarks has been challenged as it does not effectively distinguish between a malignant and benign tumor (Lazebnik, Y 2010 What are the hallmarks of cancer? Nat Rev Cancer. 10:232).

Differentiation is described in our model as a cell's capability to maintain homeostasis. Every cell has its inherent developmental program. The process of clonal expansion and cell multiplication is remarkably different in various tissues. For instance, certain epithelium has a turnover time of a few days, while other tissues, such as neurons and cardiac muscle may never regenerate. Many epithelial tissues have a high regeneration rate and happen to be the tissues/organs with high cancer incidence. A cell's capability to cause its own death is as important as a cell's capability to proliferate and is necessary for tissue regeneration and remodeling. This feature is expressed in our model as the capability to maintain homeostasis in order to follow the inherent developmental program (Eq. 3-5). Any disruption to such program may lead to a variety of pathological changes in addition to cancer.

Differentiation coefficient, k, is used in our model (Eq. 4) as a continuous and quantitative measurement of a cell's capability to maintain homeostasis. It has a low value in proliferating cells in the early stage of development, and is given a high value in a terminally differentiated cell as demonstrated in Eq. 4. A high k value can translate into a large resistance potential (Eq. 6) which pulls a cell back toward the inherent rate of cell proliferation if it deviates from the program (Eq. 8). A cell with proliferation over the programmed level will be affected negatively in order to slow down. A terminally differentiated cell at a senescent stage (with high differentiation coefficient) can only proliferate under strong external hormone stimulation which has to overcome an increasingly high and negative resistance potential. A sudden withdrawal of external hormone will result in massive cell death (simulated in Fig. 2B).

Thus, our model provides the concept of malignancy in a single cell. A single cancer cell poses minimal threat. The morbidity and mortality of cancer arises from the mass we could not control. The malignancy of a cell is described by differentiation coefficient with a continuous spectrum. The probability of a cell to spawn a tumor has to be calculated with quantitative values of hormone and other environmental factors (Eq. 9). The clinical outcome of an established tumor should be evaluated with analysis of its potential to further expand locally (Fig. 3) and the capability of its member cells to migrate and establish metastatic lesions (Figs. 4 and 5).

Additionally, the growth condition (soil) in an ectopic site is very important for the establishment of metastatic lesions. In an extreme case, benign endometrial cells can grow and form a mass in ectopic sites when the monthly cyclic estrogen is high and stimulatory. However, at the individual cell level, these metastatic (benign) cells are not different from those cells in the primary site (uterine endometrium) and have a high k value. The inherent feature of a cancer cell, the differentiation status expressed by k value, has a critical role to form metastatic lesions. Only cancer cells (with k ≈ 0) can form metastatic lesions without hormone stimulation (Fig. 5A). Thus, a continuous and quantitative valuation of differentiation status in the differentiation coefficient can be used to calculate the probability of a cancer cell to spawn a metastatic lesion under varying growth conditions in different ectopic places. The probability of a metastatic lesion is a function of differentiation coefficient, with many other variables.

In order to understand the metastatic process, the malignancy (including metastatic potential) of individual cells is critically important since hematogenous and lymphagenous metastasis are generally through migration of individual cancer cells from the primary tumor. More specifically (regarding the definition of cancer), if every malignant cell has the power of self-sufficiency in growth signals, limitless replicative potential, insensitivity to anti-growth signals, evading apoptosis, sustained angiogenesis and tissue invasion & metastasis, cancer cells should be able to colonize anywhere and grow in any condition. But as we already know, cancer cell metastasis is an extremely low success event even if numerous cancer cells are detached and migrate into circulation. How can we understand these seeming contradictions? In our view, these contradictions arise because of the indirect relation of inherent cellular properties to pathological diagnosis and clinical outcome.

Patient survival (clinical outcome), often expressed in terms of the 5-year survival rate, is the combined effect of numerous factors including early detection and effectiveness of treatment. Compared to the inherent malignancy of a tumor, the efficacy of intervention may be a more important factor in clinical outcome, and clinical outcome, in turn, serves as the basis for the commonly used staging and tumor grade assessments. Therefore, it is unclear what malignancy means as a factor in patient survival since patient survival, in part, defines it.

The relation is even more ambiguous between patient survival and the inherent properties of individual tumor cells. Patient outcome has been used not only to define the malignancy of a specific type of cancer, but also to discretize tumor stage and grade (or differentiation) of the same type of cancer. This discretization is largely retrospective and based on the correlation between patient outcome and the features presented in a tumor, which is a population of many cancer cells. Since a patient has only one discrete outcome, that outcome cannot be directly used to describe the quasi-continuous malignancy of the many individual cells in a tumor. An individualized description of malignancy for individual cancer cells in a tumor may have to include distinct and varying inherent features, such as genetic alterations, in order to address intra-tumor heterogeneity. Unfortunately, there are currently no diagnostic molecular and cellular criteria, especially in solid tumors. Pathologists still make their diagnoses without much regard for information about genetic alterations in individual cells. Instead of relying upon distinct and varying molecular and genetic features, traditional pathological diagnosis of cancer relies on clinical information, including patient history, the organs involved, morphological features and the relative location of cancer cells in tissue architecture.

There seems to be a gap in the link between the features in individual cancer cells and the pathological diagnosis of a tumor. This gap may be negligible if all cancer cells are homogenous in the tumor. But it poses a tremendous problem in cancer research if cancer cells are, in fact, quite heterogeneous since most cancer cell lines are established through clonal expansion of individual cancer cells from a tumor. As reported recently in the Wall Street Journal (Front page on 4/21/12), cancer cell line repositories commonly have a significant proportion of misidentified cancer cell lines. However, only recently, with routine genetic profiling of these cell lines, have researchers begun to notice the extent of the confusion. This is because the identity of a cancer cell line is defined by its direct descent from the original tumor and there are few cancer cell features that could be diagnostic independent of history and relationship with other cell lines. Disturbingly, this suggests that our current definition/diagnosis of cancer is phenomenalogical and does not rely on either of the two most prominent factors in cancer development: inherent genetic features and environmental factors. A clinician standing by the pathologist who is inspecting cells from a cancer cell line may ask a more relevant question: what might the potential outcome be if some of these cells were to be put back into a patient (hypothetically)? While these cells may hypothetically grow to become a tumor if they are placed at the same site in the original patient, it is also possible that the cells may not even survive in any patient at all. For a cancer biologist who uses cancer cell lines to establish so many oncogenic pathways, a cancer cell line represents but one of billions of cells in a tumor not a entire tumor and certainly not a class of tumors. A cancer cell line that has been cultured for many years may actually represent nothing or something alien because of the enormous number of genetic alterations that can occur with many passages and deprivation of patient specific environment. If we can't be sure if a cancer cell line would become a cancer in humans, how should we use the research findings in cancer cell lines accumulated over several decades? Some findings may be true and others could be misleading. Even worse is that we may not know how to tell them apart. What are the inherent molecular and morphological features which can determine the malignancy of an individual cancer cell?

Currently, malignancy of an individual cancer cell, however that is defined, is indirectly related to the malignancy of a tumor. The later is vaguely defined based on the patient outcome and has not been established with its own inherent features. This vagueness presents a challenge for us to understand the dynamic process of tumor development (including metastasis) from a cancer cell and the resulting heterogeneity inside the tumor. While metastasis is recognized as the most important and lethal of cancer phenotypes, lack of description of metastatic potential in individual cancer cells has hampered our capability to more accurately predict metastasis through examination of the primary tumor. These issues are attributable to a lack of prospective studies of human cancers (observations of the natural course of a cancer). While cancer cell lines and animal models are used to model human cancer, a phenomenon which we do not comprehend and cannot replicate, the validity of these models remains uncertain unless and until we actually understand the disease they are meant to model. Empirical cancer research has yet to break out of this deadly trap and fulfill its promise to substantially reduce cancer mortality.