Research
Abstract
The concept of mathematical modeling seeks to represent scenarios observed in the real world through a proven collection of equations, expressions, and variable because even though real-world data is complex, mathematics is able to approximate certain phenomena. Through the creation of mathematical relationships using an overall error function to describe the model’s overall error and its translation into mathematical code represented in MATLAB, a (sensitive) model was created to represent data related to injection dosages of lidocaine and the amount absorbed by the bloodstream and muscles after a certain period of time. There are other methods available to be reviewed and considered other than Newton’s method utilized for this study, but due to time constraints, what is presented here is as efficient as currently possible. The use of Newton's method for this research was chosen due to its ability to rapidly calculate the intended approximations during its iterative process; although only one study was used to verify the model, it should additionally have the ability to create an estimation of other data using close initial guesses.
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Rationale
As they exist in mathematics, differential equations are able to represent scenarios in the real-world that base themselves as functions of time. Some applications of them include how fast an object is moving based on another object, radioactive decay of an element, and multiple other scenarios. There are unique events, however, that may involve a system of differential equations being related to each other in terms of the same functions, but the coefficients pertaining to each of those two equations are generally different from each other. In order to actually represent two functions that model the entire systems of differential equations overall, a model can be created using the values of each coefficient that minimizes the error between the actual collected data points and the corresponding approximated data (the concept of “tolerance” used to judge this model is explained in the Research Map). At the beginning of the internship process when I met Dr. Croicu, I heard a rough outline of her ideas pertaining to this concept that she had wanted to investigate but did not have enough time to actually pursue this. I was extremely intrigued, but with only my experience in AP Calculus from last year, I knew that I would have to learn an incredible amount of linear algebra in a short period of time to do this. This did end up being the case as I had to learn about eigenvalues, eigenvectors, Newton's method, matrix properties, partial derivatives, and how MATLAB functions, but all of those topics culminated together to create a model that both my mentor and I can be proud of at this point in its creation process.
Results, Conclusions, & Implications
Data
The above error function describes the overall definition of the mathematical model for a system of differential equations in terms of a, b, c, and d; it is defined in terms of two functions of k, p, and lambda (defined with subscripts 1 and 2) that have their own lengthy definitions in terms of the same coefficients of the error, so even one partial derivative of the error requires six additional partial derivatives.
NOTE: The full mathematical process of this error function and all of its parts are described in Appendix I of the research paper (link at the bottom of this page).
From the error function above and following Newton's method to estimate the coefficient values that minimize this function, the graphs to the left show the two components generated by the model graphed against the actual data perturbed by about 30% (in order to simulate large real-world error).
NOTE: The final code created that generated these graphs and additional graphical representations of different perturbation results are shown in Appendices II & III of the research paper, respectively (link at the bottom of this page).
Conclusions, Implications, & Future Research
The mathematics described in Appendix I have been reviewed thoroughly and confirmed, so the more tasking part was being able to translate all of the mathematical concepts into mathematical code in MATLAB. The purpose is relatively simple: from inputted data, the coded subroutines should output certain coefficients representable in a system of differential equations from which exact solutions can be isolated and graphed. From reviewing the graph of 30% perturbation, the model appears to follow the overall shape of the graphed data points well. On account of larger amounts of error accumulated because of both rounding errors and numerical methods, a tolerance range of at least 50% must be considered when comparing the 30% perturbed values; all of the values from both components of the model’s system of differential equations were within this tolerance range.
Unfortunately, the limitation that currently prohibits this model to be applied to any real-world data to create a mostly valid model is Newton’s method itself is extremely dependent on the initial guesses. If they are too extreme to one side of the exact solutions or the other, it can quickly output values that are nowhere close to the desired values. In order for this model to be truly applicable in the future, other methods must instead be implemented and observed to not be as quickly skewed as Newton’s method. Continuation of this study would most likely consist of applying a global optimization method through MATLAB code, and the final goal is for the model to be completely applicable to any real-world study in terms of its effectiveness of modeling the data. At this point, a model has been completed that is able to effectively accept data with close enough initial guesses for the coefficient values and output a sufficient estimation for the actual data, allowing for additional interpretation for this lidocaine study specifically by other researchers and possibly other future models when utilized for additional modeling.
Legacy
When I was not working on the mathematical model at home during my research time, I was sitting in the seat I am standing behind below taking notes on Dr. Croicu's lesson and observing student involvement for nearly every lesson taught. The classroom was full of college students also taking notes on the College Algebra course I sat in on, and I was able to assist the students that asked for my help during review days in class and in a GroupMe chat. I hope that I have been able to help students with understanding the content in the course throughout the semester so they can do their best on their final test (which was given after my internship ended).
Related to my research accomplishments, my mentor is thrilled to see her idea come to life with our work on this mathematical model. The paper we hope to get peer-reviewed and published in the future is something that my mentor is excited about for both her and I, and I am simultaneously proud to be involved in this achievement. I have learned many additional mathematical topics throughout this research process I had never heard of, and now that I have developed this base for my knowledge of more advanced topics, I intend to continue building upon it. In the future, I hope to keep contributing to the mathematical world through additional research with others, and this internship experience has shown me just how to do so.
Research Paper
Click HERE to view my Final Research Paper