### 2.1 Mathematical model

A mathematical investigation of the multiple pathways to recurrent cancer is covered: there are two reasons for tumor regrowth; these reasons are either underestimation or overestimation of the administered dose. The curves of energies of treated tumors by either of Lower Irradiated Dose Treatments LIDT or Over Irradiated Dose Treatment OIDT would have different attitudes for each case. An important aspect of the model is that tumor size may vary during treatment; if rate of growth is faster than that of cell killing, the tumor volume will increase. Conversely, if rate of cell killing is faster than that of the growth, the tumor volume decreases. The tested hypothesis of the current mathematical model is that summation of tumor growth energy along the studied duration results from the balance between initial tumor energy E_{0.Tumor}, initial drug energy E_{0.Doses}, and, finally, amount of energy that the whole body disposed of by rate of radionuclide decay constant within the same duration, which is known by summation of Whole Body Cell Energy Burden ∑WBCEB, such that:

Provided that the way of whole body self adaptation, according to the whole-culture measurement approach point of view, is that radiation effects propagate gradually by radionuclide decay constant circularly from internal nanoparticles (cells) to be released in the neighborhood and so on till the outer nanoparticles (cells) and then to the surrounding environment. This recovery operation lasts till Whole Body Cell Energy Burden (WBCEB) for all body nanoparticles (cells) reaches the Natural Background Radiation (NBR) level settled by the Committee on the Biological Effects of Ionizing Radiations (BEIR) of the National Research Council, and shown by Emad Moawad that it is equivalent to (E_{NBR} = 0.0000538132 Emad) (Moawad 2011). Thus, all body nanoparticles (cells) were involved in recovery burden, and then ∑WBCEB = the Whole Body Cell Energy Burden (WBCEB) gained due to radiotherapy × C_{0} (the total number of the body nanoparticles (cells)). Negative or positive sign (∓) to cover all types of treatments with respect to dose energy, negative for either of the Over Irradiated Dose Treatment OIDT or treatments that follow work–energy principle WEPT (Moawad 2010), and positive for the Lower Irradiated Dose Treatment LIDT. The main features and assumptions of the mathematical model describing the response of the tumor to radiotherapy are as follows: the tumor is viewed as a densely packed, radially symmetric sphere. Cell movement is produced by the local volume changes that accompany cell proliferation and death. The spheroid expands or shrinks at a rate that depends on the balance between cell growth and division and cell death within the tumor volume (O’Donoghue 1997). Controlled tumors follow a growth curve by an exponential function of growth constant equivalent to ln2/*t*_{D}, where t_{D} is the tumor doubling time; the initial tumor cell energy can be determined by Emad’s formula (Emad 2010):

According to the work–energy principle, the accuracy of estimating the initial effective radioactive dose depends on the equivalence of the initial growth energy of the tumor and the initial decay energy of the effective radioactive dose (Moawad 2010), i.e.,

Such Work–Energy Principle Treatment WEPT posits that the tumor will be cured such that the tumor shrinkage constant will be identical to the decay constant of the used radionuclide. This means that in such a case, the value of the used radionuclide half-life time will be an approximate value for the half-time of doomed cell loss:

i.e., the treated tumor according to WEP will be cured and shrunken exponentially by the decay constant of the used radioactive dose. This provides a hypothesis that curves of energies of each of the treated tumor according to WEP, and the radioactive dose as functions of time are congruent along the whole treatment. This means that energy of the tumor *E*_{
Tumor
} along the WEPT is

Accordingly, the ratio of the dose released energy during a certain time to summation of tumor energy along the shrinkage stage in the same duration is equivalent to the radionuclide decay constant, i.e.,

Such a hypothesis can be tested by integrating the function of the tumor energy pathway along the studied duration. In addition, this approach posits that in case of WEPT treatments, the time passed for the WBCEB under a successful cancer therapy, without tumor regrowth to reach the E_{NBR} is the curing duration. Then, in such a case, from Eq. (2.1.1), summation of Whole Body Cell Energy Burden is equivalent to the sum of energies of each of the dose and the tumor, i.e.,

since E_{0.Tumor} = E_{0.Dose} in WEPT as shown in Eq. (2.1.3), then . Such curing time should be minimized as much as possible to reduce serious normal tissues toxicities (Schneider and Besserer 2010). Therefore, radionuclides with short half-lives offer advantages over those with longer lives; advantages over existing techniques include extremely low radiation dose because of the short half-life of the isotope ease. After passing *n* radionuclide half-life times the WBCEB will be decreased to the *E*_{
NBR
}. Accordingly, , and then curing time of the radiotherapy treatments is:

The significance of the above relation shows the possibility to decrease time of both of phase II and phase III in which the treated body disposes dose and tumor energies and comes back to ENBR or in other words: disposes tumor and drug toxicities to decrease the risks for inducing second cancer (Schneider and Besserer 2010). The remission duration is taken as the time between the start of treatment and tumor regrowth to some size or cell number threshold (O’Donoghue 1997). In either of LIDT or OIDT, summation of WBCEB will be decreased by energy consumed for regrowth, and then from Eqs. (2.1.7) and (2.1.1)

Also, for time of tumor regrowth from Eq. (2.1.9)

This equation is not applicable for tumor regrow time prediction as ∑E_{Tumor.Growth} must be known first, but it contributes to prove that the energy balances during radiotherapies for all types of tumor responses in accordance to the given experimental data. In addition, the physical quantity, ∑WBCEB, introduced in the presented mathematical model can be calculated according to the whole body measurement approach point of view, by considering that whole body cells gain energy after exposure to radiation, which leads to the increase of their growth energy exponentially by the growth constant of used radionuclide. Accordingly, if a healthy subject has been exposed to radiation dose for a certain duration (*T*), then

from Eq. (2.1),

In LIDT, in the absence of energy equilibrium between initial tumor energy, E_{0.Tumor}, and that of administered dose, E_{LIDT}, tumor growth will be the resultant of the activated nuclear transmutations, as shown by Emad Moawad (2010); LIDT curve would be grown to a level of tumor energy equivalent to

where ΔE_{Doses} is the difference between the initial tumor energy and that of insufficient dose administered in LIDT, i.e., . In addition, summation of tumor response growth energy after dose delivery would be: as postulated in Eq. (2.1.1). Then, from Eq. (2.1.12)

and can be checked through integrating the area shown in Fig. 1 between tumor energy curve of LIDT and that of the initial energy level. While for OIDT compared to WEPT in case of tumor shrinkage, the difference between their released energies would be equivalent to the summation of the difference of their tumor energies along shrinkage duration , i.e.,

To test the previous hypothesis, the tumor energy progression should be determined along the whole treatment for OIDT and WEPT. The accumulated difference of the tumor energies along both treatments, which is the sum of difference of energies of the faster shrunken than the slower one, should be equivalent to the accumulated difference of energy of the administered doses in both treatments, i.e.,

The different tumor responses along treatments of different dose energies with respect to that of the tumor are represented graphically in Figs. 1 and 2.

For OIDT, curve of tumor response energy would be compressed, following a higher decay constant than that of the used radionuclide, of half-life time t_{1/2.Shrinkage} > t_{1/2.Isotope}, leading to faster shrinking than that of the WEPT as shown in Figs. 1 and 2, i.e.,

At the same time, this model enables to predict the shrinkage half-life time during a certain time *T* in either WEPT or OIDT according to the following equation:

where E_{0.OIDT} ≥ E_{0.Tumor}. Hence, by trial and error method, assuming values of *n* satisfies

since

then Eq. (2.1.17) can be simplified for long-term radiotherapy effects to . Moreover, Eq. (2.1.17) shows that for WEPT: as postulated for the presented mathematical model. This fast shrinkage will last, under the condition that the difference of the total decay energies that are released from both treatments, OIDT and WEPT, is higher than the accumulated difference of the tumor energies along both treatments, i.e., , as shown in Fig. 2. Once these differences become close to each other, as the shrinkage slows gradually to the minimum tumor size of OIDT, and then reverses its response, regrows to achieve the balance with that of WEPT according to its time course when . Afterwards, the tumor energy curve of the OIDT will continue to regrow negatively above that of the WEPT as shown in Fig. 2 to a level of energy such that this accumulated tumor response energy would be equivalent to as previously postulated for the mathematical model in Eq. (2.1.1); from Eq. (2.1.12)

which can be checked through integrating the area shown in Fig. 2 between the tumor energy curve of OIDT and that of WEPT starting from their intersection.

### 2.2 Lower irradiated dose treatment

This application shows that tumors under LIDT will grow or gain energy equivalent to the difference in energy of WEPT from that of LIDT, i.e., , as shown in Eq. (2.1.13). Furthermore, it tests the hypothesis of the LIDT mathematical model as shown in Eq. (2.1.1). Thakur et al. (2003) showed methods and materials for experiments in nude mice bearing human tumors: approximately 5 × 10^{6} viable human prostate (DU145), breast (T47D), or colorectal cancer (LS174T) cells were implanted into nude mice in groups of ten mice each, and tumors were allowed to grow to (0.61 cm in diameter) 5 × 10^{8} ng and treated with 16.7 MBq (450 μCi). 111In-oxine grew, on the average, only 17% irrespective of their type—breast, prostate, or colorectal, within 28 days after injection (Thakur and Ron Coss 2003), while those treated by 18.5 MBq (500 μCi) did not grow within the same duration as shown in Figs. 3 and 4.

To test whether the lower administered dose (16.7 MBq of 111In) was appropriate or not, the tumor cell growth energy, E_{0.Cell}, and its doubling time, t_{D}, which is adequate for such a dose can be derived from the equations

(Moawad 2010), which show that this dose was appropriate for tumor cell (nanoparticle of 1 ng) growth energy equivalent to E_{o.Cell} = E_{o.ng} = 4.385 Emad, corresponding to tumor doubling time, t_{D} = 5384.51 s = 0.06 days, while the presented data shows that t_{D} was equivalent to 28 days. This great difference in dose energy supply from that of the tumor allows tumor growth through the phenomenon of transmutation that permits transformation of elements in live organisms [9 s]. For growth calculations, Emad Moawad explained (2010) that those little doses were not sufficient. The growth energy of the untreated (controlled) tumor was 1.0091 J, while the decay energy of the insufficient dose from In-111 = 0.816 J only. Tumor growth energy (17%) + dose decay energy (insufficient dose) = 0.1717 + 0.8161 = 0.988 J, which achieves an accuracy of 98% of the growth energy of the untreated (control) tumor (1.0091 J). In addition, . At the same time, the regrowth energy, ΔE_{Regrowth}, was , following a doubling time of 123.6156238 days. This indicates that the tumor regrowth energy due to LIDT is less than the difference between WEPT and LIDT dose energy. To check the hypothesis of the mathematical model: summation of tumor growth energy, , can be calculated along time of growth 28 days as presented in the experimental data as follows: from Eq. (2.1.1 & 2.1.14)

Tshis value can be reached by integrating the presented experimental observations by Thakur et al. as follows: as shown in Fig. 3 (Fu et al. 2004). This is nearly 100% identical to the experimental data that presented by Thakur et al.’s measurement (2003). In addition, from Eq. (2.1.17), if the administered dose would be 20.66 MBq (1.0091 J), as shown by Emad Moawad, instead of the applied ones (16.7 MBq, 18.5 MBq) by Thakur et al. to satisfy the WEPT, the shrinkage half-life time is supposed to be

by trial and error method. The value of *n* that satisfies Eq. (2.1.18) is *n* = 9.9 as it gives t_{1/2.Shrinkage} equivalent to 2.83 days; this rate corresponds to the summation of tumor energy

along 28 days as shown in Fig. 4 (Brown 1999); this is also nearly 100% identical to the ratio of the dose released energy during the same period by the decay constant of the radionuclide, as previously postulated for the features of the mathematical model in Eq. (2.1.6) for WEPT, where .

### 2.3 Over irradiated dose treatment

O’Donoghue et al. (2000) showed the temporal behavior of a surviving fraction for a tumor of (1 × 10^{11} ng) initial mass with the baseline response parameters. The t_{D} of the tumor cells was taken as 4 days. This represents a central estimate of values measured by bromodeoxyuridine labeling in human tumors (Terry et al. 1995; Tsang et al. 1995; Bolger et al. 1996; Bourhis et al. 1996). The single, large administrations of LSA treatment consists of an administration of 8.25 GBq (223 mCi). A value of 3 days was used as an approximate value for the half-time of doomed cell loss (Ts). The time courses of tumor regression and recurrence for the treatment showed that the minimum tumor size reached was (7.2 × 10^{8} ng) at 27.6 days for LSA. If remission duration is defined as the time to regrow to a tumor mass of (5 × 10^{9} ng), then this was 53.2 days counted from the start of dose delivery (O’Donoghue et al. 2000); these experimental data are shown in Fig. 5.

Checking the thesis of this approach and mathematical model accuracy:knowing that t_{D} = 4 days, from Eq. (2.1.13), the growth energy of tumor nanoparticle (cell), E_{ng}, can be determined by Emad’s formula from Eq. (2.1.2) as follows:

Then, the total tumor growth energy

While the administered dose was 8.25 GBq, of decay energy.

which is obvious much more than *E*_{0.Tumor} representing an OIDT, as from the point of view of this approach the initial decay energy of the administered dose was supposed to be 192.065 J, instead of 1284.837 J, and all such energy difference (1092.772 J) is considered an over irradiated dose, responsible for the consequent tumor regrowth. During this treatment, the initial tumor size shrunk, and the time courses of tumor regression and recurrence for the treatment showed that the minimum tumor size reached was 7.2 × 10^{8} ng at 27.6 days. The corresponding half-life time of tumor decay was 3.87761 days. The first hypothesis of the OIDT mathematical model in which the accumulated difference of the tumor energy along OIDT from that of WEPT will be equivalent to the accumulated difference of energy of the administered doses in both treatments, i.e.,

as shown in Eq. (2.1.16) can be tested as follows: the tumor energy progression should be determined along the whole treatment for both administrations, OIDT and WEPT. The accumulated difference of the tumor energies along both treatments, which is the sum of the difference of energies of the occasional faster shrinkage than the slower one, and the consequent regrow, should be equivalent to the accumulated difference of the energy of the administered doses in the same interval from treatment start until equivalence of tumor energies in either treatment, i.e., until their curve intersection (balancing point). Firstly, from *t* = 0 to *t* = 27.6 days, the stage of the fast shrinkage, size of the over irradiated tumor decreased faster than the size of the one irradiated according to WEP, due to the over irradiated dose, the difference between its decay energy, and the decay energy administered by WEP released within this interval, Δ*E*_{
Doses
}, where

While the accumulated energy differences between tumor energies along the same period is , then

Secondly, from *t* = 27.6 days to *t* = 53.2 days, the time courses of the over irradiated treatment showed that the tumor size regrow from 7.2 × 10^{8} ng and reached 5 × 10^{9} ng. This shows that the regrowth doubling time was 9.1564 days. The WEP treatment showed that the tumor size decayed exponentially following the decay constant of the used radionuclide. Consequently, the time courses of both treatments showed that tumor energy would get the same energy in both treatments after their start by 43.419 days. At this balancing point, the accumulated differences between tumor energies along the same period [from start till balancing point] is , where

From Eqs. (3.2.1) and (3.2.2),

as shown in Fig. 5. While the difference of the released energy between the over irradiated dose and the WEP one is

From Eqs. (2.3.3) and (2.3.4), it can be deduced that in the same period, from *t* = 0 to *t* = 43.419 days, the difference between the decay energy of the over irradiated dose and that administered by the WEP that was released within this interval, ΔE_{Doses}, is equivalent to the accumulated differences between tumor energies, , i.e., , along the same period [from start till balancing point] as previously postulated by Eq. (2.1.15). To check the second hypothesis of the OIDT mathematical model: summation of tumor growth energy, , can be calculated along the time of growth, 53.2 days, as shown by O’Donoghue et al. as follows: from Eqs. (2.1.1) and (2.1.19)

This value can be reached by integrating the presented experimental observations (O’Donoghue et al. 2000) as follows:

as shown in Fig. 5. This is nearly 100% identical to the experimental data that was presented by O’Donoghue et al.’s measurement (2000). This test can be executed conversely; by knowing , duration of such tumor response can be determined from Eq. (2.1.11) as follows:

as previously presented in the experimental data. This success in determining enables us to predict whether tumor will regrow or be cured after a certain time due to cancer treatment. In addition, to check the third hypothesis of OIDT mathematical model for fast shrinkage rate, the shrinkage half-life time can be predicted from Eq. (2.1.17) as follows:

by trial and error method. The value of *n* that satisfies Eq. (2.17) is *n* = 4.6 as it gives t_{1/2.Shrinkage} equivalent to 3.877 days, which is also 100% identical to O’Donoghue et al.’s presentation (2000). As the goal of our model development is second cancer risk prevention, this approach, hereby suggests that WBCEB should be less than the Low Dose Radiation (LDR) effect that was settled by BEIR and that was shown by Emad Moawad to be equivalent to E_{LDR} = 0.000538132 Emad or 12.503 MeV or 2.0030088 × 10^{−12} J (Moawad 2011). In application 2.3-, ∑WBCEB has been increased from (2 × 192.065) × C_{0} J in WEPT to in OIDT; this led to prolongation of the curing time shown in Eqs. (2.1.8) and (2.1.9) from 38.39 days in WEPT to 54.3 days in OIDT that could lead to serious normal tissue toxicities and contribute in increasing second cancer risks. Therefore, OIDT is also considered one cause of second cancer.

### 2.4 Estimating the WEPT from the tumor response through the LIDT or the OIDT

This application for checking the efficacy of radiotherapies after their execution helps in preserving patients’ rights against the randomized statistical dose assessment that ignores patient-specific factors. It shows that as the tumor sizes and their doubling time in patients varied widely, as these differences produced significant differences between doses assessed physically even for the same sizes (Rajendran et al. 2004). Barendswaard et al. (Barendswaard et al. 2001) showed that 4- to 6-week-old athymic female Swiss (*nu*/*nu*) mice, body weight , from their in-house nude mouse facility were injected with 10 × 10^{6} SW1222 cells in the left thigh muscle. After 5–7 days, mice bearing tumors of 1.40–9.0 × 10^{8} ng were selected. A total of 169 mice were divided into groups of 4–9 mice. Fourteen groups were administered varying amounts of mAb A33 labeled with 131I. The activities of 131I-A33 ranged from 0.925 to 18.5 MBq (0.025–0.5 mCi). Tumor size was measured bidimensionally with calipers, and the volume was calculated assuming elliptic geometry. Initial tumor sizes were between 0.14 and 0.90 cm^{3} (mean, 0.44 cm^{3}), i.e., initial tumor masses were between 1.4 and 9.0 × 10^{8} ng, mean 4.4 × 10^{8} ng. Mice with tumors of differing sizes were divided into groups such that the tumor size spectrum for each group was similar. The tumors were measured every 3 or 4 days for 100 days or until the death of the animal. Mice were killed when the tumor caused apparent discomfort in walking or when its volume exceeded 2 cm^{3}, i.e., when tumor mass exceeded 2 × 10^{9} ng. Observations showed that tumor growth was retarded after treatment to an extent that was dependent on the amount of activity administered. Barendswaard et al. showed that “tumors were considered cured if they failed to regrow over the period of observation (100 d after treatment), while occasional tumor cures were seen at intermediate administered activities of 131I (3.7–11.1 MBq), but a higher value (14.8 sMBq) did not produce any cures. Four of five tumors in this group became temporarily undetectable but subsequently recurred between day 40 and day 80. The highest activity of 131I administered (18.5 MBq) resulted in tumor cures in all four animals in that group.” (Barendswaard et al. 2001). Barendswaard et al. showed that the maximum tolerated activities of 131I were 18.5 MBq (0.5 mCi) in this model system. Activities of 18.5 MBq 131I caused petechiae, which became apparent after 2 days and confluent after 4 days; these activities also caused progressive weight loss. Median tumor volume, normalized to initial volume, as a function of time in nude mice bore SW1222 xenografts when treated and shown in Fig. 6 as presented by Barendswaard et al. (2001).

Checking the postulates of this approach:

First: from tumor response of the controlled group, the tumor doubling time was 4 days; consequently, from Eq. (2.1.2) the cell (nanogram or nanoparticle) growth energy was

Second: as the initial tumor size was not provided by Barendswaard et al., numerical simulations of Eq. (2.1.16) can be applied to investigate the tumor’s initial size, which can be estimated from the thesis of the equivalence of the difference between areas under the curves of tumor response during WEPT and OIDT and the difference between drug released energy and that of the tumor, i.e.,

as shown in Eq. (2.1.16). Then, by substituting data of this experiment, shown by Barendswaard et al. (2001) (shrinkage half-life time was 6.5 days along 20 days, for 18.5 MBq initial dose activity of 131I that corresponds to 2.881 J) in Eq. (2.1.16), calculation of the initial tumor energy shows that

that corresponds to 8.168 MBq only. This means that the administered dose of WEPT was supposed to be 8.168 MBq; consequently, the doses of 14.8 MBq and 18.5 MBq are considered OIDT.

which is accepted as a median value for the set of treated tumors ranged (1.4–9.0) × 10^{8} ng, with a mean of 4.4 × 10^{8} ng, as given by Barendswaard et al. (2001).

Forth: a dose–response relationship has been studied for the different administered doses. The highest activity of 131I administered (18.5 MBq) resulted in tumor cures in all animals, but the value of 14.8 MBq did not produce any cures and tumor regrowth was between day 40 and day 80. The tumor response after dose delivery followed an exponential shrinkage of half-life time of 6.5 days, continued only for the tumor that was assigned to the 18.5 MBq of 131I dose, and cured. On the contrary, the tumor that has been assigned to a dose of 14.8 MBq reached its minimum size after 40 days; afterwards, it relapsed, following an exponential growth of 14.46 days doubling time, and reached a relative tumor size of 6.8 with respect to the minimum size after 80 days from the dose delivery. To check the hypothesis of OIDT mathematical model for fast shrinkage rate of the 14.8 MBq (2.305 J) dose, the shrinkage half-life time can be predicted from Eq. (2.1.17) as follows:

by trial and error method. The value of *n* that satisfies Eq. (2.1.18) is *n* = 5.5 as it gives *t*_{1/2.Shrinkage} equivalent to 7.4 days. This rate causes the tumor energy to decrease from 1.2721 J to 0.028 J after 40 days as

Consequently, summation of tumor growth energy, , can be reached by integrating the presented experimental observations by Barendswaard et al. as follows:

as shown in Fig. 6. The postulate of the provided mathematical model summation of tumor growth energy, , can be calculated by determining the required time of regrowth as follows: from Eq. (2.1.1 & 2.1.19)

This is 100% identical to Barendswaard et al.’s presentation, and shows that is the resultant of tumor and dose energies against ∑*WBCEB*.

Fifth: to check the hypothesis of OIDT mathematical model for fast shrinkage rate, also for the 18.5 MBq (2.88 J) dose, the shrinkage half-life time can be predicted from Eq. (2.1.17) as follows:

by trial and error method. The value of *n* that satisfies Eq. (2.1.18) is *n* = 3 as it gives t_{1/2.Shrinkage} equivalent to 6.72 days, which is also 97% identical to Barendswaard et al.’s presentation (6.5 days) (2001).