The Gompertz model utilized to represent the growth of tumors. The Gompertz function models the development of a tumor during a specific time length, where the growth is progressively getting slower at the end of a time span.
The variable N? represents the plateau cell number, when the values for “r” are big, and the parameter “b” is linked to the start of the tumor producing rate. The equation written out below is the same as the first one.
The variable K represents the carrying capacity, the largest the tumor can grow using the amount of food in the body. The variable “?” is a constant that represents how quickly the cancer cells will multiply.
Only one pair of growth parameters isn’t sufficient enough to represent the clinical information. Research shows that cancer cells most likely have varying growth rates depending on the patient.
Model based on metabolic considerations
In order to make sense of a mathematical function used to predict the growth of tumors, one has to first learn the system of the ontogenetic development, the origination and development of an organism, in an organism. Ontogenic development is driven by metabolic rate and it is similar to a certain path that is usually seen in cell division. All food that is used for an organism’s growth can go into two things, the development of new tissue or the upkeep of the existing tissue. This can be expressed as
The variable B equals the fuel that a living body consumes when stationary. The variable, Bc, is the metabolism of a single cell and the variable Nc is the number of cells contained in a certain organism. The term NcBc stands for the fuel needed to support the old tissue. The variable Ec stands for the energy required to develop new tissue from a single cell.
One is to presume that the variables Bc, mc and Ec stay the same during the lifetime of an organism and that they are specific to a certain type of organism. Therefore the mass of an organism, represented by the variable m, is found from the mass of a single cell and the amount of cells in the organism , the equation for this is m = mcNc. When we differentiate and substitute this into the equation above, it comes out to
We are given
B0 is dependent on a certain taxon ,a taxonomic group of any rank, such as a species, family, or class.
Mc and Ec remain the same, we can write the above equation as
so a ? B0mc/Ec and b ? Bc/Ec.
The 3/4 exponent is universally constant for all living bodies, regardless of what type they are. Therefore the 3/4 exponent includes the general allometry, the study of the relative change in proportion of an attribute compared to another one during organismal growth, of B from the start of life to the end. The exponent represents the proportion in the total amount, Nt, of capillaries. We said before that the amount of cells is directly proportional to the mass of the body. The 3/4 exponent provides the boundaries on an organism’s growth. So the moment the organism’s growth ceases , i.e. dm/dt = 0, which is represented by this equation
The variable M represents the largest size of the body. Therefore the diversification of the variable M amongst several species inside of a taxon is found completely dependent on the cell’s metabolic rate, Bc, which turns into M-1/4. The growth of tumors sticks to the same laws, like any other living organism nutrients and blood are needed. One hopes that the same laws will be exercised and by utilizing the Universal Law for ontogenetic growth we expect to infer a congruent universal law for tumors. The a variable isn’t dependent on M and the variable b = a/M1/4. We can rewrite the above equation as
When solved, the equation gives us
The variable m0 represents the mass of the organism at the start of its life (when t = 0). It is possible for a and b to be found from key frameworks of the cell. From the charts below, we can tell that the growth functions form very similar pathways. One can determine that if the same implications were applied to tumors, an almost identical growth curve will appear.