Mathematical Models for Cancer Growth
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The first of Skipper’s laws is that the doubling time of a tumor is constant. A plot of the number of cells in the tumor over time on a semi-log graph forms a straight line. Working with animals, Skipper showed that organism death happens when the tumor size reaches a certain fraction of the mouse weight, which means survival is associated with the number of tumor cells.
Skipper’s second law states that chemotherapeutic agents work by first order kinetics; a fixed fraction of the tumor cells is killed regardless of the tumor size.
This hypothesis was developed from Skipper’s laws. If tumor doubling time is constant; the time (x) the tumor takes to grow from 1000 cells to 10,000 cells is the same amount of time to grow from 10 million to 100 million cells. The log-kill hypothesis looks at the opposite direction, The fraction of the tumor’s malignant cells that are killed by a chemotherapy application is always the same – no matter how many cells are in the tumor. The log kill number is the base 10 logarithm of the ratio of cell number reduction. Thus 2 log kill means 99% reduction, 6 log kill means 99.9999% reduction.)
This hypothesis states that tumor cells are killed in response to a chemotherapeutic agent at a rate directly proportional to the tumor growth rate at the start of treatment. This is different from the log kill hypothesis.
Postulate 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 resulted in development of the log kill hypothesis.
This hypothesis predicts that the tumor cells develop resistant phenotypic variations towards chemotherapeutic agents. This resistance is independent of the chemotherapeutic agent and dependent on the number of cell divisions that occur after treatment begins. The larger the tumor size or the longer delay in initiating chemotherapy, the more resistant cells. The implication is that a regimen of alternating cycles of two different non–cross-resistant chemotherapy medicines yields a better chance of tumor eradication.
Fractional Cell Kill Hypothesis
This hypothesis states that when applying a certain concentration of a chemotherapeutic agent for a period of time to a mass of tumor cells, a constant fraction of the tumor cells will be killed. Because only a fraction of the tumor is killed with one dose of the agent, repeated doses must be given to reduce the tumor size. Chemotherapeutic regimens repeat administration of the medicine in cycles for a duration that is limited by toxicity to the patient. However, the tumor can re-grow during the intervals between the treatment cycles which limits the net reduction of each fractional kill.
Exponential Growth Model
One model that has been used to describe tumor growth is the exponential growth model1.
The exponential does not describe the growth rate in vivo which slows as the tumor size increases.
Another model use to describe tumor dynamics is a Gompertz curve or Gompertz function. It is a type of mathematical model for a time series, where growth is slowest at the end of a time period1.
where N∞ is the plateau cell number which is reached at large values of r and the parameter b is related to the initial tumor growth rate. This is the same equation recast (taken from Wikipedia.)
where K is the carrying capacity, i.e. the maximum size that can be reached with the available nutrients, and α is a constant related to how fast the cancer cells multiply.
Even with Gompertzian growth, a single set of growth parameters is insufficient to model the clinical data. Tumor cells almost certainly have different growth characteristics in different patients, and individual micrometastases within a single patient may also have different growth parameters.
Model based on metabolic considerations
To properly understand or derive a mathematical model for cancer growth, we must first understand the process of the ontogenetic development of an organism. This process is fueled by metabolism and follows a certain pattern which occurs primarily through cell division. In a recent paper, a mathematical model based on energy conservation was derived to model such growth and showed that regardless of the different masses and development times, all taxons share a common growth pattern2. It may be possible that cancer growth may be modeled in very much the same way. As the total energy that goes into an organism’s development either goes into the maintenance of existing tissue or the creation of new tissue, we can express this as
where, B is the energy that an organism uses while at rest. The variables, Bc and Nc are the metabolic rate for an individual cell and the number of cells in a particular organism respectively; the NcBc term represents the energy to maintain existing tissue. Ecis the energy needed to create new tissue from an individual cell.
We assume that variables Ec, Bc and mc all remain constant during an organism’s growth and is pertinent to a particular type of organism. Thus the total mass of an organism, m, can be determined from the mass of an individual cell and the number of cells, m = mcNc. By differentiating and substituting this into eq (3), we get
where B0 depends on a particular taxon,
As the B0, mc and Ec terms are constant, we can express the above equation succinctly as
where a ≡ B0mc/Ec and b ≡ Bc/Ec.
The 3/4 exponent is roughly the same for all organisms, whether they be mammals, birds, fish or plants. Thus the exponent describes the overall allometry of B from birth to maturity. As there is a tendency for natural selection to optimize energy transport, this has lead to the evolution of a fractal-like distribution network. This exponent is related to the scaling in the total number, Nt of capillaries3,4. As we already know, the total number of cells is related to the organism’s mass. This exponent has the profound implication in that it sets limits on the growth of an organism. Thus the point in which an organism stops growing, i.e. dm/dt = 0, we see that
where M is the symptotic maximum body size. Thus the variation on M among different species within a taxon is determined entirely by the cellular metabolic rate, Bc, which scales to M-1/4. As cancer growth follows the same principles, and as blood and nutrients enter into and feed a tumor, we expect the same scaling principles to apply and thus by using this Universal Law for ontogenetic growth we hope to derive a similar universal law for cancer growth. As B0, mc and Ec are approximately constant, a is independent of M and b = a/M1/4. We can thus rewrite eq (7) as
Solving this differential equation gives
where m0 is the mass of the organism at birth (t = 0). As a and b can be determined from fundamental parameters of a cell, a universal equation has been derived. We can see from the figures below that all growth curves follow the same path. We can infer that if similar considerations are made for cancerous cells, a similar growth curve will be obtained.
Empirical Tests of Mathematical Models for Tumor Growth
Exponential Growth Law
All tumors follow a standard growth pattern, growing fastest in the beginning and eventually reaching a maximum size. There are some weaknesses in the exponential growth model as it fails to model this behavior in vivo.
The Gompertz Model, while it does model the behavior as a tumor increases in size, it is not an empirical model. The model has too many variables to consider, such as types of cancers as well as environmental conditions. These may even vary considerably for patients with the same types of cancers1.
The Universal Law model proposed by West, adequately describes the growth rates of organisms by taking energy considerations into account. This model was tested against empirical data5. As powerful as West’s model for growth is, it only applies to organisms growing in unrestricted dietary conditions. Thus only fully replenished tumors will follow this universal growth curve. Differences in growth rates and saturation sizes of up to a factor of 500 were found in tumor spheroids cultured in media with different oxygen levels and glucose concentrations6. Thus deviations from West’s Law depend on particular environmental conditions.
As a tumor reaches a certain size or volume, metastatic dissemination will occur7. As the rate of cell growth is dependent on its development phase, this may serve as a possible explanation why recurrent cancers grow at much slower rates than the primary tumors2. In this case, the residual clonogenic cells of the primary tumor generate cells from an older development phase and thus multiply at a slower rate.
In general, the West model for ontogenetic growth of living organisms applies to the case of solid malignant tumors. The results fit a variety of in vitro and in vivo data5.
Growth Rates for Common Cancers
Human tumor growth rates are approximately exponential when in vivo which means that the rate of growth depends on developmental stage8. Tumor growth follows an exponential growth, as seen in the exponential, Gompertz and universal law models which means rate changes over time. To best describe the growth rate of a tumor, it is best to describe the growth rate in terms of doubling time, or the time it takes for the population of cells to reach twice its size9-11.
As there is a lack of clinical data at non-symptomatic stages, it can be assumed that two to three decades can elapse between the first carcinogenic stimulus and the emergence of the neoplasm12. Typical doubling times for a tumor are expected to be between 60 days for very aggressive tumors and 100 days for non-aggressive tumors.
Other Points of Interest
While there isn’t one specific model that can adequately describe any one tumor, each model does highlight certain aspects of tumor kinetics. The Gompertz model, for example, shows that a tumor’s rate of growth is greatest at the beginning stages; the point when there are no means to detect them clinically. The model also predicts that as the tumor grows, its growth rate slows down. Despite predicting these behaviors there is no real basis for explaining these attributes as this model is an empirical law.
The West law on the other hand is a fundamental law as it is based on basic physical laws; the conservation of energy. From this law we can intuit and explain certain observed behaviors, for one, that there would be a maximum size that a tumor will attain. This maximum size is due the ‘fractal’ pattern of the veins that feed and nourish the tumor; as a vein bifurcates there comes a point that the vein is too small to adequately supply nutrients to the cancerous cells. As the West Model also says that the growth rate changes with developmental stage, it explains why the growth rates of tumors that have metastasized don’t grow as rapidly as the tumorous cells from which they originated.
The National Library of Medicine of Medicine has an extensive page about cytokinetics which may be of interest to people who want to learn more:
The development of tumor models is important as they offer a way to better understand the growth kinetics of malignant tumors which may lead to the development of successful treatment strategies.13,14 15
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