A Review of Mathematical Concepts for
Calculation of Cancer Parameters
Saganuwan Alhaji Saganuwan*
Department of Veterinary Physiology, Pharmacology and Biochemistry, College of Veterinary Medicine,
University of Agriculture, Nigeria
Submission: November 02, 2018; Published: January 11, 2019
*Corresponding Address: Saganuwan Alhaji Saganuwan, Department of Veterinary Physiology, Pharmacology and Biochemistry, College of Veterinary Medicine, University of Agriculture, P.M.B. 2373, Makurdi, Benue State, Nigeria
How to cite this article: Saganuwan Alhaji Saganuwan. Saganuwan Alhaji Saganuwan. Canc Therapy & Oncol Int J. 2019; 12(5): 555849. DOI:10.19080/CTOIJ.2019.12.555848
Various mathematical models such as Gompertzian, logistic, exponential and other immunogenic models were integrated for calculation of various cancer parameters. Pharmacokinetic and pharmacodynamic models likened to other models especially, absorption and elimination are considered similar to proliferation and elimination phase of cancerous cells respectively.
Doubling time (TD) is equal to whereas reduction time
(RT) is and mean proliferation time (MPT) of cancer cells is . Some revised formulas fit cancer chemotherapy, immunotherapy and chemotherapy/immunotherapy. Some parameters were also calculated to validate the established and newly derived formulas for relevance, construct validity, prediction and reliability. Conclusively, none of the formulas is reliable. However, the formula that considers body surface area may be more relevant to monogastric animals especially dogs and humans. BSA with constant (k), height (0.528) and weight (0.528) functions better for human and dogs. Many other formulas highlighted are also useful.
Keywords: Cancer cell; Gompertzian formula; Chemotherapy; Immunotherapy
The first central nervous system (CNS) Anticancer Drug Discovery and Development Conference (ADDDC) was organized and convened out of frustration by dearth of effective anticancer drugs . About 169.3 million years were lost due to cancers in 2008, with colorectal, lung, breast and prostate cancers, respectively  having 5-year survival rate as highest for breast cancer, followed by colorectal and prostate cancer, respectively  with African countries recording 541,800 deaths . In 2007, over 12-million people were diagnosed with cancer. At least one-third of these individuals are not expected to survive the disease, making cancer the second most prevalent cause of death worldwide. Systemic chemotherapy forms the mainstay of cancer treatment and antimitotics are commonly used to treat a wide variety of cancers .
The strategy of chemically targeting cancerous cells at their most vulnerable state during mitosis has instigated numerous studies into the cell death , indicating that, there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optional therapies into medical practice due to patients, cancer and therapy specific components . Therefore,
the development of optional vaccine-chemotherapy protocols for removing tumor cells would be another appropriate strategy in cancer treatment . Since polymorphism can be maintained in a finite population by adaptively turning selection, there is need for a model of resistance in a stochastically evolving cancer cell population , with intent to reducing adaptive therapy. However, the growth rate of healthy and tumor cells approach the carrying capacities K1 and K2 respectively .
The effect of immune system is to kill the mutated and cancer cells at proportional rates d1 and d2, through apoptosis . The coefficient c represents the portion of the healthy cells, whose genome is disordered by the external esterase. These cells start the neoplastic transformation and are added to the tumor cells . The tumor competes with healthy cells for resources such as blood, nutrients and space . However, an optional control problem for combination of cancer chemotherapy with immune-therapy in form of a boost to the immune system is considered as a multi-input optional control problem .
Various literatures were searched for mathematical models used in calculation of doubling time of tumor cells, proliferation and loss, tumor volume, immuno-competent cell density,
colonization rate and other tumor parameters. Gompertzian,
logistic, exponential and other methods were re-viewed for
optimization of cancer chemotherapy and immunotherapy [5-
48]. All the formulas derived from various sources are given in
equations 1-25. New formulas for calculation of parameters of
tumor growth and cytotoxic drugs, cancer immunotherapeutic
and cytotoxic/immunotherapeutic were independently and
combinedly derived. The reported parameters are given in Table
1. Whereas analysis of doubling times in cancers of selected
origin are given in Table 2. However, various parameters are
recalculated for some cancer cell types and their cytotoxic drugs
are recalculated for construct validity, reliability and prediction of
the new formulas.
Cancer cells grow over time. But the rate of growth decreases
as the cancer mass increases. Therefore, cell tumor size and tumor
volume are proportional
dv/dt= The rate of change in tumor volume per unit time
kp = the rate constant for cell production
kl = The rate constant for cell loss
V = Vo exp [(kp – kl)/ (t2 – t1)]
Vo = represents the volume of the tumor at time zero
V = represent the volume of the tumor after the time interval
has elapsed (t2 – t1)
Td = TD = Doubling time of the tumor
Absorption half-life (T½α) = ln2/α
By the time a tumor becomes chemically detectable, it has
achieved a mass of approximately lg or 109 tumor cells.
1g of tumor mass = 109 cells = 30 doublings and its growth are
no longer exponential. The additional 10 doublings is required to
produce 1012 cells or 1kg lesion – a tumor burden at which most
patients succumb-occur much more slowly than do the previous
30 doubling and represents a fraction of the tumor’s growth.
Tumor growth delay for tumor in vitro and in vivo cell lines are
presented in Table 2.
The number of cells surviving at a given dose of a drug (dN)
is proportional to both the drug dose and the number of cells at
risk for exposure to the drug (NdD), where N = number of cells in
tumor, D = drug dose, dN = - KndD, where proportionality constant
– K is introduced with a negative sign. Because the number of cells
is expected to decrease with increasing drug dose, the formula is
rearranged as follows:
N = No exp – K (D – Do), where the subscript (o) indicates the
initial dose and cell number.
Imaging at the beginning of treatment, a tumor contains 10
cells, if each course of treatment results in death of 99.9% of these
cells, if no log of cell growth occurs between courses of treatment,
five courses of treatment are required to dominate the last cell.
The exponential relationship between drug dose and tumor
survival dictates that a constant proportion, not number of tumor
cells is killed with each treatment cycle. In this example, each
cycle of drug administration results in 99.9% (3log) of cell kill,
and log of cell growth occurs between cycles.
Assume a tumor contains 1011 cells and the proportionality
constant (-K) = -5 for cyclophosphamide (an alkylating agent). If
1.5 of cyclophosphamide is delivered, the tumor will be left with
5.5 x 107 Cells.
N = No exp – K (D – Do); No = 1011 when Do = O
∴N = 1011 exp – 5 (1.5 – O) = 5.5 x 107 cells. If the oncologist
chooses to administer 0.75g of cyclophosphamide instead of 1.5g,
N = 1011 exp – 5 (0.75 – O) = 2.4 x 109 cells.
The result is that a 50% decrease in dose has translated into
a 98% increase in cell survival. Therefore, let liken tumor growth
with compartment model of drug disposition in pharmacokinetic.
Alkylating agents with antitumor and myelo-suppressive effects
are directly proportionate to dose and to the total area under
the concentration versus time curve (AUC) rather than to
instantaneous plasma drug concentration.
MAT = Mean absorption time
Therefore, substitute for T½ α in equation 1
whereas TD is tumor doubling
MAT = MPT = Mean production time
Therefore, the derived formulas are presented as follows:
x = healthy cells; y = cancer cells
Their values are non-negative i.e. x ≥ 0; y ≥ 0
The coefficient a, growth rate of healthy cells, a1 = growth rate
of cancer cells
K1 = carrying capacity of healthy cells, K2 = carrying capacity
of tumor cells, immune system should kill mutated cells at (d1)
and cancers cells at (d2)
The tumor competes with healthy cells for resources; blood,
nutrients and space. The competition coefficients are b1, b2, and g.
6: The effect of anticancer chemotherapy:
dm/dt = μm +Vm (t)
1.2m/0.8+m for chemotherapy fraction cells kill.
Optional vaccine-chemotherapy protocols for removing tumor
cells maybe an appropriate strategy in cancer chemotherapy. A
proper treatment method would reduce the population of cancer
cells and changes the dynamics of cancer .
7: N = ns + nr (N = Inner tumor composition; ns = drug sensitive
cells; nr = drug resistant cells)
The capacity to involve and adapt makes successful treatment
of cancer difficult. Therefore, high-resolution monitoring of the
target population is important .
8: Y = μ1 (x − β x2 ) α − dδ y + x + ky yv
An optional control problem for combination of cancer
chemotherapy with immunotherapy in form of a boost to immune
system is considered as a multi-input optional control problem.
Simplified mathematical model may be useful to give some
guidance . The exponent 0.67 is needed since anticancer has
to be released through the surface of the tumor  but 0.528
correlates very well with both human and dogs . However,
various body surface area formulas have various exponents which
can grossly affect the results .
9: I (Inhibitor) = dp 2/3 q
Resistance factor should be responsible for the effect of drug
resistance of tumor cells on the dynamical growth for the tumor.
Optional control problems have common point wise both different
integral constraints on the control. Bang-bang control is optional
if the resistance is sufficiently strong .
10: Mt = – M (L) +V (t)
The drug level function m=m(t) obeys linear differential
equation with a positive drug decay rate where v(t) denotes the
drug dose administered per unit time. The fate of anticancer
drugs from introduction into the body to intracellular targets can
be represented by pharmacokinetic (pk) compartmental ordinary
differential equations (ODEs) for their concentration. This fate
is theoretically representable by partial differential equations
(PDFs) with boundary conditions instead of exchange rule. But in
cell medium, pharmaceutical differential equation must be used
to relate local drug concentration with molecular effects on their
targets, delte Billy et al. and , also .
The most common models of tumor growth are the exponential
model DN/DTand the logistics(DN/dt=λ N,(1-N/K), Where K is the maximum tumor size, or carrying capacity of
the environment and the Gomperz (DN/dt=λ Nln(K/N))where
again K is the carrying capacity. Contrary to the exponential and
logistic models the Gompertz model was initially developed in the
context of insurance  and was just used in the nineties to fit
exponential data of tumor growth . Murray  considered
two – population Gompertz growth model with a loss term to
model the effect of cytotoxic drug.
Where b is the rate of tumor-induced vascular formation, K
+ dN2/3 represents the rate of spontaneous and tumor-induced
vascular loss, g (t) ≥, 0 represents the antiangiogenic drug
μ = dose of antiangiogenic drug; V = dose of cytotoxic drugs;
Q1Y1, n = their effects in tumor cells and on vasculature.
Where m denotes the density of endothelial cells; Dm=
diffusion rate; αm = proliferation rate; Xm = chemotaxis rate; δm
= death rate; w = the concentration of chemoattractant substance;
Dw = diffusion rate; δw = production rate that depends on the
density of quiescent tumor cells q; δw = degradation rate.
A mathematic model for time used to theoretically investigate
anticancer therapy such as surgery and chemical treatments has
been established. Theoretically optional schedules are derived
which show superiority of a metronomic administration sequence
on a classical maximum tolerated dose scheme for the total
metastatic burden in the organ, .
Tumors have two phenotypical traits: volume denoted by V,
also to as size expressed in mm3 and caring capacity denoted by
K, expressed also in mm3. Hence, physiological domain where
metastases live is the square Ω = [Vo −V max]*[Vo,V max] whose
boundary is devoted by ∂Ω with external normal vector V(δ).
a = parameter controlling the cancer cells proliferation
a = parameter controlling the cancer cells proliferation
d = Parameter for production and effect of angiogenesis
The main assumption underlying the model is that, clearance
rate of inhibitors (e.g. endostatin, angiostatin, thrombospondin
1) is much smaller than clearance stimulators (e.g. vascular
endothelial growth factor based on fibroblast growth factor. The
concentration of inhibitors should be proportional to the surface
of the tumor giving rise to 0.67 power in the inhibition term. The
number of metastases emitted by a tumor with volume V for unit
of time is given by
15: Β (V) = mVα where α = 0.67 or fully penetrating. The tumor
(α = 1) and 0.75 or even having any fractal dimension between 2
and 3 or 4.
16:A(t)=DNI-1exp(-Ur(t-ti)lt≥ti where D is the
administered dose ti’s are the administration times and is a
Heaviside function having value 1 if and only it t ≥ tj .
A total amount (Amax) has to be given at a constant rate
during administration time (t), followed by a rest period from (t)
to an arbitrary end time (t).
Modeling and prediction of the effect of chemotherapy was
developed using fractional diffusion equation. The methodology
is useful for analysis of the effect of special drug and cancer .
Also, a mathematical model for the scheduling of angiogenic
inhibitor in combination with a killing agent was considered as
an optional control problem, . The initial condition well posed
for the optional control problem is not difficult to determine,
because the first order necessary conditions for optimality of the
controls U and V given by pontryagin maximum principle states
that there exists a constant (λo≥O) and an absolutely continued
co-vector λ satisfying the equation transversally. Cancer cells can
be eradicated in a very short time with a small amount of drug
using an optional administration therapy . The best way of
reducing the tumor burden after a fixed period of treatment is
to keep the tumor size to minimum initially and then fire high
intensity treatment towards the end of the treatment period
. But stochastic model provides a description of the optimal
therapeutic regimen  Endothelial birth (b) and death(d) rate
depend mainly on the type of tumor and the patient tumor cannot
increase over the maximum volume: C∞ = e∞ = (b/d)³/² nd then
does not consider evolution to metastasis .
18: However, tumor width (mg) =a*b2/2
a & b, the tumor length and weight in mg must be considered
in the calculation of cancer parameters.
T - C; T is the median time (days) required for tumors to reach
a predetermined size (e.g. 1000mg), and C is the median time
(days) for the control tumors to reach the same size.
T - C is the tumor growth delay; Td = the tumor volume
doubling time (days) estimated from the best fit straight line from
linear growth plot of the control tumors in exponential growth (100 – 800 range). The conversion of the T-C values to log10 cell kill
is possible, because the Td of tumors re-growing post treatment
(Rx) approximates the Td values of the tumors in untreated control
mice. The calculations for net log10 tumor cell kill is provided by
subtraction of the duration of the treatment period from the T-C
value and then divide by 3.32 x Td
Many solid tumors have shown empirically  to follow the
Gompertz growth law .
Where V = Volumetric size of the tumor of time, t and V0,
K0, and α are constants
21: But growth equation with growth constant
With V(o) = Vo, K(o) = Ko, show that an equivalent result is
obtained by the assumption 
22: Since all mitotic phases express ki – 67 antigens,
But tumors with less than 250 MIB-1 positive cells are
23: Cell cycle time and potential doubling time are calculated
for meningiomas and neurinomas as follows:
24: However, tumor potential can also be calculated as follows:
whereas Tpot is the tumor potential,
25: Tumor doubling time (Td) can also be calculated as
volume after t days 
(22 – 26) are very useful for calculation of cancer parameters
for meningiomas and neurinomas .
However, tumor inhibition rate % = (Mean tumor weight
of control group – Mean tumor weight of treated group) over
mean tumor weight of control group x 100 can be applied for
hepatocellular carcinoma H22 cell line in mice . The time it
takes for a tumor mass to double is known as the doubling time
which varies according to the size of tumor, but for most solid
tumors, it is about 2-3 months (60-90 days). Initially the growth
is exponential and then slows as the tumor increases in size
and age called Gompertzian growth. Generally, chemotherapy is
most successful when the number of tumor cells is low and the
growth fraction is high, which is the situation in the very early
stages of cancer. The larger the tumor mass, the more likely it has
metastasized to other sites . Cancer chemotherapy is goalspecific.
For the patient to be cancer-free, the treatment must be
total. But if the treatment is palliative, the quality of life may be
improved, and higher heart rate variability does not only predict
lower tumor burden but also improves survival in humans .
Therefore, cell kill hypothesis is a theoretical model that
predicts the ability of antineoplastic drugs to eliminate cancer
cells. A 1-cm breast tumor may already contain 109 cancer cells
before it can be detected during manual examination. After first
round of chemotherapy the cancer cells is reduced to 107(99%
kill). When second round of chemotherapy is applied the cancer
cells reduce to 106. At this point, T cells are removed remaining
cancer cells. It is likely that no antineoplastic drug or combination
of drugs will kill 100% of tumor cells. A relatively small number
of cancer cells may be removed after chemotherapy suggesting
that early diagnosis and treatment may be the goal standard
. But the reviewed formulas can be used to determine cancer
parameters whose values may indicate whether one or more of
therapeutic interventions can be successful [38-48].