Staging of cancer is the main objective of cancer screening, and considered the most important predictor of survival, where cancer treatment is primarily determined by staging which can be changed according to progression of the disease. There are two types for cancer staging, clinical and pathologic, where both are considered a supplement for each other (Lucas 2006), each of them describes the anatomic extent of the cancer at the time of diagnosis before the application of definitive treatment to develop classification into stages, which serves for treatment guidance and for comparing with the end results of the applied treatment (Wilson and Jungner 1968). Clinical stage is based on all of the available information obtained before a surgery to remove the tumor, while pathologic stage adds additional information gained by examination of the tumor microscopically after surgery expressing the stage before therapy only (Lucas 2006). Usually, cancer staging by either type expresses the extent that cancer has spread, and is usually described by numbers I to IV with IV having more progression. Such classification, which is well known by the TNM system, involves the tumor size (T) and the regional lymph node involvement (N) and/or distant metastasis (M), is based on the premise that cancers of the same histology and anatomic sites share similar patterns of growth and metastasis (AJCC 1998). Thereby, staging by those numbers which reflect the histologic grade is considered very wide and unspecified to describe the extent and aggressiveness of the disease in patients, allowing to either of over or lower estimation to the administered dose that contributes to risks of tumor regrowth and metastasis (Moawad 2011). E. Moawad has introduced a clinical staging by imaging techniques that allow accurate cancer staging that helps to administer the appropriate low-waste dose and modify it by monitoring an earlier response to therapy which contributes besides developing dose-delivery skills to the success of all types of cancer treatments (Moawad 2010, 2011). In the same time, it was possible to correlate the in vitro data with patient response to therapy and the incorporation of [^{3}H] tritiated thymidine and the [^{14}C] thymidine by slices of their tumors (Baserga and Lisco 1963; Wimber and Quastler 1963). No responses to therapy were found in those patients whose tumors when incubated in vitro with thymidine showed poor nucleoside incorporation. In contrast, patients receiving benefit from cancer therapy were those whose tumors when incubated in vitro with thymidine showed good nucleoside incorporation (Wolberg and Brown 1962). Such great variability reflects their variability in growth rate propensity for metastasis, or in other words, in their stages and consequently their corresponding grades. Thus, pathological staging has been assessed on the basis of measurements of the incorporation of [^{3}H] tritiated thymidine and the [^{14}C] thymidine in vitro in the tumor slices. Current approach aims to present clinical and pathological staging models of the cancer at the nanoscale to obtain more accurate assessment for the main factors of the prognostic determinants in the classification for cancer staging.

### Clinical staging model at the nanoscale (CSMN)

E. Moawad has presented a clinical model for cancer grading in which the patient-specific histologic grade has been identified by evaluating the tumor energy (*E*_{Tumor}) based on measuring tumor doubling time (*t*_{D}) which expresses the rate of growth by imaging techniques (Moawad 2010, 2011), and estimating the percentage of the hypoxic cells (*H*%) which expresses the tumor histologic classification, and inversely proportional to the total number of the malignant cells of the tumor whose value is ranging between 8 and 20 % (Moawad 2010, 2011). Thus, the energy of the hypoxic cell (*E*_{Hypoxic.cell}) which expresses the histologic grade at the cellular level can be estimated from Emad formula after measuring the tumor *t*_{D} as follows:

Thus, provided that patient-specific histologic grade (*E*_{Tumor}) is the summation of energies of all the tumor hypoxic cells (Σ*E*_{Hypoxic.cell}) then:

where *M*% is the percentage of the tumor malignant fraction, and *C*_{0} is the total number of tumor cells (Moawad 2010, 2011). Knowing that a tumor of 1 g converted into 10^{9} ng contains 10^{9} cells, it would be more convenient to express the tumor histologic grade by nanoscale as equivalent to the average growth energy of a tumor of 1 ng or one nanoparticle (*E*_{ng}) investigating whether we can directly control matter on the molecular scale. Hereby, in all sections of the current approach the tumor cell will be expressed by the tumor nanoparticle, i.e., *E*_{Cell} = *E*_{ng}. Thus, from Eq. 3, it can be deduced that

and consequently the average doubling time of the tumor nanoparticle would be equivalent to:

Thus, from Emad formula the histologic grade of the tumor nanoparticle would be:

### Pathological staging model by [^{3}H] tritiated thymidine at the nanoscale

Based on the in vitro measuring of cell proliferating of tumor slices by [^{3}H] tritiated **t**hymidine incorporation, current approach introduces another pathological model for cancer grading at the nanoscale (pathological staging model by [^{3}H] tritiated thymidine at the nanoscale (PSM [^{3}H] N)); as labeling of cells by [^{3}H] tritiated thymidine has been commonly used as an indicator of the proliferative capacity of tumor cells (Lieb and Lisco 1966), conversely the unlabeled tumor fraction (*U*%) has been hypothesized by this model as an indicator for the inhibition to cell-proliferating rate. As inhibition to cell-proliferating rate accompanied by cell cycle arrest distinguishes the malignant cells from the normal ones, then the unlabeled tumor fraction (*U*%) can be considered the malignant tumor fraction (*M*%) that previously presented as one of the factors of the clinical staging shown in Eq. 3, i.e., *U*% = *M*%. Thus, such inhibition to cell-proliferating rate can be monitored by the deficit of [^{3}H] tritiated thymidine incorporation in the tumor malignant cells, provided that energy of such deficit part (*U*%) of the thymidine dose is equivalent to the energy of the malignant fraction (*M*%) of that tumor and denoted by *E*_{Tumor} which expresses the patient-specific histologic grade. Accordingly, by knowing the percentage of the unlabeled cells (*U*%) of the tumor then:

on condition that *U*% < 1. Consequently, the patient-specific histologic grade at the nanoscale would be:

### Pathological staging model by [^{14}C] thymidine at the nanoscale

It is well known that in all cancer stages from early to advanced disease, cancer cells are known to have alterations in multiple cellular signaling pathways drives normal cell to carcinoma (Wu et al. 2009). One of the most important signals is the continuous deficit in cell proliferation rate accompanied by a progressive cell cycle arrest along those stages (Reiskin and Mendelsohn 1964). Significant efforts have been made to understand the kinetic analysis of cell proliferation that drives cancer development and progression (Allard et al. 2004). There is as yet little to distinguish the cancerous cell from a variety of normal cells which have also been analyzed (Reiskin and Mendelsohn 1964) as basis of such efforts. Since labeling of cells by [^{14}C] thymidine is commonly used as well in vitro in measuring cell proliferation rate, where the incorporation of the [^{14}C] thymidine into the dividing cells and the level of this incorporation is proportional to the amount of cell proliferation (Reszka et al. 2001). Thus, the second pathological staging model presented by current thesis (pathological staging model by [^{14}C] thymidine at the nanoscale (PSM [^{14}C] N)) posits that the inhibition to cell-proliferating rate due to all the genetic variations and the aberrant activations accompanied by cell cycle arrest that drive the normal cell to carcinoma induces a deficit of [^{14}C] thymidine incorporation in the detected samples compared to the control one. Consequently, percentage of the deficit of [^{14}C] thymidine incorporation (*D*%) in those samples compared to the control one is equivalent to the increase of the tumor nanoparticle *t*_{D}, i.e., *t*_{D.ng} compared also to the cell doubling time at the natural background radiation (*t*_{D.NBR}) which is equal to 1.884220083 s (AJCC 1998), i.e.

on condition that *D*% < 1. And thus PSM [^{14}C] N is valid for samples of nanoparticle doubling time less than twice that of normal tissue nanoparticles at the NBR, i.e., *t*_{D.ng} < 2*t*_{D.NBR}. Accordingly, the histologic grade of the detected samples at the nanoscale can be derived by Emad formula as follows:

Since the identified histologic grade (*E*_{ng}) by those models shown in Eqs. 6, 8, and 10 should be identical, and as factors concerned in those models should confirm basis of the main factors assessment of the prognostic determinants in the classification for cancer staging by the TNM system. The objective of the current approach is to investigate the consistency of the results of those models to provide a clear-cut criterion for accepting or rejecting the hypotheses of those models.