User:I9T1997/sandbox

Peer Review - Isabella De Marchena

I think that you did a good job at selecting what information to include. You also effectively utilized the paragraph formatting tool to create different levels within your article. What appears to be missing are your citations. Make sure your work is properly cited and that you add your citations using the appropriate tools to the reference page.

Gompertz Function

The Gompertz curve or Gompertz function, is a type of mathematical model for a time series and is named after Benjamin Gompertz (1779-1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-hand or future value asymptote of the function is approached much more gradually by the curve than the left-hand or lower valued asymptote. This is in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regards to detailing populations.

History

Benjamin Gompertz originally designed the function for the Royal Society in 1825 to detail his law of human mortality. The law rests upon a priori assumption that a person's resistance to death decreases as his years increase. The model can be written in this way:

$N(T)=N(0)\exp -c((\exp at)-1)$

where:

• N(0) is the initial number of cells/orgamisms when time is zero
• a is an asymptote
• b and c are positive numbers
• b denotes the displacement across the x-axis
• c denotes the rate of growth
• e is Euler's Number

N(t) represents the number of individuals in the given time period, t. The letters c and a are constants. This model is a modification of a demographic model of Robert Malthus. It was commonly used by insurance companies to calculate the cost of life insurance. This equation is known as a Gompertz function.

Applications

Gompertz Curve

Population biology is especially concerned with the Gompertz Function. This function is especially useful in describing the rapid growth of a certain population of organisms while also being able to account for the eventual horizontal asymptote, once the Carrying capacity is determined (plateau cell/population number).

It is modeled as follows:

$N(t)=N_{0}(\exp \ln(N_{I}/N_{0})(1-\exp(-bt))$

where:

• t is time
• N₀ is the initial amount of cells
• N_I is the plateau cell/population number
• b is the initial rate of tumor growth

This function consideration of the plateau cell number makes it useful in accurately mimicking real-life Population dynamics. The function also adheres to the Sigmoid function, which is the most widely accepted convention of generally detailing a population's growth. 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

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.The metabolic curve takes in to consideration the energy the body provides in maintaining and creating tissue. This energy can be considered as metabolism and follows a specific pattern in cellular division. Energy conservation can be used to model such growth, irrespective of differing masses and development times. All taxons (a group of one or more populations of an organism) share a similar growth pattern and this model, as a result, considers cellular division, the foundation of the development of a tumor.

$B=\sum _{C}(N_{C}B_{C})(E_{C}{\operatorname {d} \!N_{C} \over \operatorname {d} \!t})$

• B = energy organism uses at rest
• N_C = number of cells in the given organism
• B_C= metabolic rate of an individual cell
• N_CB_C= energy required to maintain the existing Tissue (biology)
• E_C= energy required to create new tissue from an individual cell

The differentiation between energy used at rest and metabolic rate work allows for the model to more precisely determine the rate of growth. The energy at rest is lower than the energy used to maintain a tissue, and together represent the energy required to maintain the existing tissue. The use of these two factors, alongside the energy required to create new tissue, comprehensively map the rate of growth, and moreover, lead in to an accurate representation of the Lag phase.

Gompertz function (link)

The information in this page is very relevant to the article topic. It is written very concisely and right to the point. In my opinion, the article does not have anything distracting on it. Perhaps, the images could be aligned in a better orientation, but even then, they are pretty relevant and it is nice to be able to visualize what you are reading. Being a page about a mathematical property, there is not much room for bias. The page most deals with how the function is derived and its applications. It does not delve in to reliability of function in different fields, so there is no real bias present. I certainly think that the mathematics in the wikipedia page could be better elaborated on. It makes an assumption that the reader has a very strong background in mathematics. Even though the information is well displayed, it could certainly show more steps or provide more information in to understanding how the formulas came to be.

The citations all work and are all highly relevant to the article at hand. The article has clearly been developed using the sources at hand. Most facts are linked to a source, and all the sources are of very reliable quality. These sources are from scientifically published journals and mathematical pages, which have been peer reviewed by experts in the field. Being a mathematical topic, there is limited bias and whatever information was used was based upon the mathematics. I do think the page could benefit from more discussion about the applications of the Gompertz function, as currently, it just listed in the article and not really discussed.

As i suspected, the article is rated "start class" and could definitely be expanded upon. It also a part of the WikiProjects Mathematics and WikiProjects Statistics. The conversations centering this article on the Talk page, seem to deal mostly with communication, and how to explain/better represent what certain mathematical variables mean.

Article expansion draft

Outline as well as talking points for my article expansion. Bibliography can be found at the bottom. It is non-structured for now, as I compile all my information. I am simultaneously using a Word document to draft the actual wikipedia article. Information for now may not be clearly understandable, especially which source is which. This will be sorted out soon as I am testing out whether I should write a new article or expand on the current one. Updates will follow shortly.

1. Gompertz function : expansion article

I think I could certainly contribute towards expanding knowledge in modeling tumors using the Gompertz curve. There are several published research works that I could draw from to discuss how to use and adapt the Gompertz curve to graph tumor models . The link (Gompertz function) states applications, but could certainly use more work to expand and explain how Gompertz curve is applied.

The following link explains how the Gompertz curve can be applied in terms of tumors, and how the formula can be adapted to consider other variables : https://hrcak.srce.hr/file/2874

This link talks about the gompertz curve in the context of tumors, looks at how relevant the curve is, adaptations that could be made, and how effective it is in modeling tumors. Moreover, it goes through the mathematics of the formula and its adaptations. I think, I could definitely add to other applications of the curve. I think I need to decide whether I want to create a new article discussing applications, or is it better to just improve the existing one. The following link: https://link.springer.com/article/10.14441/eier.3.239 discusses how the sigmoid nature of Gompertz' curve can be used to model economic growth, using 3 main equations. They go through the differentiation process and explain how modeling can be representative of economic shortcomings (leveling off).

This publication talks about the gompertz curve in the context of tumors, looks at how relevant the curve is, adaptations that could be made, and how effective it is in modeling tumors. Moreover, it goes through the mathematics of the formula and its adaptations. I think his could be one of main resources for my discussion on tumor modeling using the Gompertz Curve. The article is very well cited despite it being more than twenty years old. This means the mathematics is still relevant and well-accepted. The article is aimed at those in the field of biology and maths and thus, is very relevant to my wikipedia article. I think the mathematics is at a level that I could understand and relay in my article.

This source discusses how the sigmoid nature of Gompertz' curve can be used to model economic growth, using 3 main equations. They go through the differentiation process and explain how modeling can be representative of economic shortcomings (leveling off). This source offers me another example of how the Gompertz Curve can be applied. The source is pretty reliable given that it is published in a rather famous journal. This source is also from 2017, and given that economics trends and understandings change with consumerism, the mathematics is more likely to be relevant than a similar article from say, ten years ago. Again, like the previous source, there is a thorough explanation of how the original formula is derived and manipulated to model economic growth.

This source is perhaps the most complicated one to use. This publication discusses how the parameters of the original curve can be re-parameterized to model bacteria or cancer cells. One such re-parameterizations is the Gompertz-Laird the birth or hatching value is accounted for in place of the inflection-time parameter. It also examines the use of relative growth rate instead of growth coefficient. There are several modifications present, including the formulas and growth curves. Despite the plethora of information, I plan to use this source mostly as a means of broadening my knowledge. The information in this source is pretty hard to understand without a very strong background in mathematics. I think I will include only small bits of information and focus on showing how the original curve can be modified, and briefly explain why such modifications are important.

This paper provides an example of how the Gompertz Curve can be applied to mobile user growth, and is directly relevant to how the market growth of a product has a sigmoid growth curve. This model uses the least-squares principle, which I am vaguely familiar with. I think I will probably be able to use this paper as an example of how the Gompertz curve can be applied a product growth and probably won’t focus on the derivations/mathematics. I think I will use this source just as a means to show the sigmoid growth and explain very briefly (not in mathematical terms), how the general formula can be adapted to model this example. Again, the mathematics of this paper is not something I am very comfortable with, so I’ll talk about how the general principle can be applied in a wider context. Again, this paper is more for my general understanding, and because of how brief this paper is, I will try to incorporate the big picture ideas in to my wikipedia article.

This paper will serve as the basis of my third example, and is very relevant, given how many people are interested in stocks. This article uses the Gompertz curve to predict the time series of a stock. The data it bases its principles from is from the Composite Index of Shanghai Stock Exchange, making the mathematical models accurate as it draws from real data. It uses concepts such as local maxima and minima and averages to determine how the Gompertz curve can be fit to model stocks. The mathematics is definitely a bit of a struggle to understand given the brevity of the article. Again, the focus on this article will be more on the modifications that can be made to the curve and why it is used, than the mathematics side of it. The data itself is too specific to reference for my wikipedia article, so it makes more sense to talk about how the Gompertz Curve can be used in stocks as an example, and not focus much on the mathematical side.

UPDATE 1: I think I will include a bit of historical context

UPDATE 2: I have decided to contribute to original page, and details to the function, history, applications, and shortcomings.

UPDATE 3: Have moved information in to original page. Made several edits and made changes choice of words and expanded concepts as well as added my own

UPDATE 4: Images have been added to wikipedia page. Edits are being made on other page. Information is no longer being added here.

References

Gompertz, Benjamin (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society of London. 115: 513–585. doi:10.1098/rstl.1825.0026

Marusic, Miljenko. “Mathematical Models of Tumor Growth.” Mathematical Communications, vol. 1, no. 2, Jan. 1996, pp. 175–189.

Jarne, Gloria, et al. “S-Shaped Curves in Economic Growth. A Theoretical Contribution and an Application.” Evolutionary and Institutional Economics Review, vol. 3, no. 2, Sept. 2007, pp. 239–259., doi:10.14441/eier.3.239.

Tjørve KMC, Tjørve E (2017) “The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family.” PLoS ONE, vol. 12. no. 6, June 2017, pp. 17-23.

Francesco Benedetto, Domenico Izzo, "Performance improvements of power management in CDMA systems by adaptive modulation", Telecommunications and Signal Processing (TSP) 2013 36th International Conference on, pp. 149-153, 2013.

Chi-Chang Yang, et al. "Application of the Gompertz Growth Curve in the Time Series of Stock," 2013 Third International Conference on Intelligent System Design and Engineering Applications, Hong Kong, 2013, pp. 373-376

Gompertz, Benjamin (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society of London. 115: 513–585. doi:10.1098/rstl.1825.0026