Moreover, the function makes use of initial growth rate, which is commonly seen in populations of bacterial and cancer cells, which undergo the log phase and grow rapidly in numbers. Despite its popularity, the function initial rate of tumor growth is difficult to predetermine given the varying microcosms present with a patient, or varying environmental factors in the case of population biology. In cancer patients, factors such as age, diet, ethnicity, genetic pre-dispositions, metabolism , lifestyle and origin of metastasis play a role in determining the tumor growth rate. The carrying capacity is also expected to change based on these factors, and so describing such phenomena is difficult. Metabolic curve[ edit ] The metabolic function is particularly concerned with accounting for the rate of metabolism within an organism. This function can be applied to monitor tumor cells; metabolic rate is dynamic and is greatly flexible, making it more precise in detailing cancer growth.
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Brain-Cousens equation Polynomials Polynomials are the most flexible tool to describe biological processes. They are simple and, although curvilinear, they are linear in the parameters and can be fitted by using linear regression. One disadvantage is that they cannot describe asymptotic processes, which are very common in biology.
Furthermore, they are prone to overfitting, as we may be tempted to add terms to improve the fit, with little care for biological realism. Nowadays, thanks to the wide availability of nonlinear regression algorithms, the use of polynomials has sensibly decreased; linear or quadratic polynomials are mainly used when we want to approximate the observed response within a narrow range of a quantitative predictor.
On the other hand, higher order polynomials are very rarely seen, in practice. Linear equation Obviously, this is not a curve, although it deserves to be mentioned here. We will list the most used, here.
Therefore, Y increases by an amount that is proportional to its actual level. The exponential curve is used to describe the growth of a population in unlimiting environmental conditions, or to describe the degradation of xenobiotics in the environment first-order degradation kinetic. Error t-value p-value init: Intercept This equation is used in several different parameterisations and it is also known as Monomolecular Growth, Mitscherlich law or von Bertalanffy law.
This is used, e. Error t-value p-value a: Intercept 4. We show some simulated data as examples. Error t-value p-value a: Intercept 1. Error t-value p-value a: Intercept 2. Error t-value p-value d: Intercept Indeed, the initial slope of a Michaelis-Menten can be assumed as a measure of competition, that is the reduction in yield Y when the first weed is added to the system. We show an example relating to sunflower grown at increasing densities of the weed Sinapis arvensis.
Error t-value p-value i: Intercept 8. This is logical, but, it has the important consequence that the weed-free yield is constrained to be equal to the observed weed-free yield, which is not realistic. Therefore, we can reparameterise the yield-loss function, in order to use the observed yield as the dependent variable. Error t-value p-value YWF: Intercept They are parameterised in countless ways, which may be often confusing.
Therefore, we will show a common parameterisation, that is very useful in biological terms. The above function is known as four-parameter logistic. If necessary, contraints can be put on parameter values, i. Logistic functions are very useful, e. Error t-value p-value b: Intercept The difference is that this curve is not symmetric around the inflection point. Another type of asimmetry We have seen that, with respect to the logistic, the Gompertz shows a longer lag at the beginning, but raises steadily afterwards.
Also, I am not aware of a particular name for this function. The logistic function is black, the Gompertz function is red and the reparameterised Gompertz is blue. Therefore, using a function that is defined also for non-positive numbers may seem unrealistic. All the above described sygmoids may be based on the logarithm of X, which gives us more realistic models. Log-logistic curve In many applications, the sigmoidal response curve is symmetric on the logarithm of x, which requires a log-logistic curve a log-normal curve would be practically equivalent, but it is used far less often.
We show an example of a log-logistic fit, relating to a bioassay with Brassica rapa treated at increasing dosages of an herbicide. Error t-value p-value b: Intercept 1. Weibull curve type 2 The type 2 Weibull curve is for the Gompertz curve what the log-logistic curve is for the logistic curve.
Life[ edit ] Of the German Jewish family of Gompertz of Emmerich , he was born in London, where his father and grandfather had been successful diamond merchants. Debarred, as a Jew, from a university education, he studied on his own from an early age, in the writings of Isaac Newton , Colin Maclaurin , and William Emerson. He became a member of the Mathematical Society of Spitalfields , and served as its president when it was merged with the Astronomical Society of London. In he was elected a F.
Gompertz–Makeham law of mortality
Some useful equations for nonlinear regression in R
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