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Skipper’s Laws

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There are two laws in Skipper’s laws. The first law states that the doubling time of a tumor is constant. By using a semi-log graph, it plots the growth of cells in a tumor over time to form a straight line. Researchers studied tumor growth in animals, specifically mice. The research showed that the mice would die when the tumor grew to a certain size that was a percentage of the mouse’s bodyweight. Skipper’s second law states that chemotherapy, treatment for cancer, works by first order kinetics. Going by first order kinetics, a specific amount of the tumor cells is eliminated regardless of the size of the tumor.

Log-Kill Hypothesis

The log-kill hypothesis was developed from skippers law. If tumor doubling time is perpetual; the time (represented by variable x) the tumor takes to grow from one thousand cells to ten thousand cells takes the same length of time to grow from ten million to one hundred million cells. The log-kill hypothesis analyzes the opposite view of the growth. The percentage of the malignant cells that are eliminated by chemotherapy treatment is always a fixed amount, no matter the amount of cells in the tumor. The log kill number is the base ten logarithm of the ratio of cell number reduction. Therefore two log kill equates to 99% reduction, six log kill equates to 99.9999% reduction.

Norton-Simon Hypothesis

This hypothesis claims that tumor cells are eliminated in response to chemotherapy at a rate correlative to the tumor growth rate at the beginning of the treatment.

Skipper-Schabel-Wilcox Model

The premise that “a given dose of a given drug will kill approximately the same fraction, not the same number, of widely different-sized tumor cell populations —so long as they are similarly exposed and both the growth fraction and the proportion of drug-resistant phenotypes are the same.” This model developed the log kill hypothesis.

Goldie-Coldman Hypothesis

The Goldie-Coldman hypothesis anticipates that the tumor cells grow opposing phenotypic adaptations towards chemotherapy. This resistance is uncontrolled by chemotherapy and reliant on the amount of cell divisions that happens after treatment starts. The bigger the tumor is or the longer the wait for beginning chemotherapy, the more the cells are less likely to be killed. The hypothesis is that a system of changing cycles of 2 different distinctive types of chemotherapy medicines produces a higher chance of tumor elimination.

Fractional Cell Kill Hypothesis

The fractional cell kill hypothesis claims that when administering a certain concentration of chemotherapy medicine for a length of time to a group of tumor cells, a fixed percentage of the tumor cells will be eradicated. It is because only a percentage of the tumor is eradicated with one prescription of the medicine, repeated doses have to be administered to shrink the tumor. Chemotherapy schedules repeat dosage of the medicine in loops for a period of time that is confined by toxicity to the patient receiving it. Nevertheless, the tumor can grow again during the tine in between each chemotherapy treatment done which can slow the killing of the tumor.

Exponential Growth Model

A model that has been used by scientists to represent tumor growth is the exponential growth model1. The exponential doesn’t characterize the growth rate in vivo which slows down as the tumor gets bigger.

Gompertz Model

Another model utilized to represent tumor dynamics is a Gompertz function or curve. It is a sort of mathematical model for a time period, where growth is progressively getting slower at the end of a time span. The variable N8 is the plateau cell number, when the values of r are big, and the parameter “b” is linked to the start of the tumor growth rate. This is the same equation again. The variable K is the carrying capacity, the maximum size of the tumor that can be achieved with the availability of nutrients in the body, and a is a constant that represents how quickly the cancer cells multiply. Only one set of growth parameters isn’t sufficient enough to represent the clinical information. Research shows that tumor cells most likely have different growth rates in different people.

Model based on metabolic considerations

To be able to understand a mathematical model for the growth of tumors, we have to first learn the process of the ontogenetic development, the origination and development of an organism, in an organism. Ontogenic development is driven by metabolism and it follows a certain path which is mainly cell division. The total fuel that goes towards an organism’s growth either goes into the development of new tissue or the upkeep of the existing tissue. The variable B equals the energy that an organism consumes when at rest. The variables, Bc and Nc, stand for the metabolism of a single cell and the number of cells in a certain organism. The term NcBc stands for the energy needed to support the existing tissue. The variable Ec stands for the energy required to develop new tissue from a single cell.

We are to assume that the variables Ec, Bc and mc are all constant during the time of an organism’s growth and that it is specific to a certain type of organism. Therefore the total mass of an organism, m, can be found from the mass of a single cell and the amount of cells in the organism, m = mcNc.

We are given B0 is dependent on a certain taxon,a taxonomic group of any rank, such as a species, family, or class. Like the B0, Ec and mc terms are both constant, we can write the above equation as so a = B0mc/Ec and b = Bc/Ec.

The 3/4 exponent is approximately the same for all organisms, regardless of what type they are. Therefore the 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. This exponent represents the scaling in the total amount, Nt, of capillaries. We said before that the total number of cells is directly proportional to the organism’s mass. The 3/4 exponent provides the boundaries on the growth of an organism. So the moment an organism stops growing, i.e. dm/dt = 0, we can see that through this equation The variable M represents the evidential largest body size. Therefore the diversification on the variable M amongst several species inside of a taxon is found completely by the cell’s metabolic rate, Bc, which turns into to M-1/4. The growth of tumors draws from the same laws, nutrients and blood, enter and nourish the tumor, we hope that the same scaling laws will apply and by utilizing the Universal Law for ontogenetic growth we expect to infer a congruent universal law for tumor growth. Like B0, mc and Ec are almost constant variables, the a variable is independent of M and the variable b = a/M1/4. We can rewrite the above equation as

Solving this equation gives us

The variable m0 represents the mass of the organism at the start of its life (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 all growth curves form very similar paths. We can determine that if almost identical contemplations are made for these cancerous cells, an almost identical growth curve will appear.

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