deletions | additions       diff --git a/Shot optimization.tex b/Shot optimization.tex  index 97578c7..d580b85 100644  --- a/Shot optimization.tex  +++ b/Shot optimization.tex 
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\section{Shot Optimization}  Shot optimization was performed using an asexual genetic algorithm. A genetic algorithm is If the distance from  a stategy for finding an optimal value out cell to the center  of all possible values. The idea is inspired heavily by biology; each iteration a shot was less than the radius  of the process produces shot, the cell was defined as killed. In this way, shot fitness could be assessed as  a new set of values, each with small modifications binary classification score to measure the overall accuracy  of the current set, while favoring shots. Calculating fitness in this manner allows for  the optimima. use of more advanced search strategies than simple brute-force searching.  Shot optimization was performed using an asexual genetic algorithm. A genetic algorithm is a stategy for finding an optimal value out of all possible values. The idea is inspired heavily by biology; each iteration of the process produces a new set of values, each with small modifications of the current set (i.e., radius could be mutated from 2 cells to 7 cells), while favoring the optimimum.  For the present study, the typically an  asexual operator, a mutation, genetic algorithm was used. That is, only mutations within genes were used,  was used: a single parent chrosome is altered randomly to produce a new one. With each iteration, or generation, members of a population, in this case the set of possible circle arrangements in the square lattice, are then assessed by a fitness function. The fitness determines which genes are passed on to the next generation, a kind of "selection" function. The selection function can be described as a fitness proportionate selection: let $p$ denote population size, let $m_i$ denote population members, for $1≤i≤p$, and let $f(m)$ denote the fitness function for a gene; the total fitness is set by \begin{equation}  F= \sideset{}{}\sum_{1}^{p}f(m_i) 
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The sum of probabilities of a single gene being chosen by the fitness function is equal to total fitness. The fitness level is used to associate a probability of selection with each individual geme (in this case, the genes for radius or position). In other words, genes are selected in proportion to their fitness.  The fitness function $f(m)$ was established as an $F_1$ score. It utilizes two terms: precision and recall. Precision is the number of correct results divided by the number of all returned results; recall is the number of correct results divided by the number of results that should have been returned. In context, therefore, precision was defined as the number of tumor cells covered in each shot divided by the total number of cells covered by the shot. Recall was defined as the number of cancer cells covered in a shot divided by the total number of cancer cells. The $F_{1}$ score was thus calculatedwith  as \begin{equation}   F_{1}=2\frac{(\text{Precision})(\text{Recall})}{(\text{Precision} + \text{Recall})}  \end{equation}  So essentially what we did was implement these two concepts in terms The result  of Python-written code that generated a 100x100 brain "grid" and a. generated the use of the $F1$ score as the fitness function is  a maximization of both  tumor b. implemented shots tissue covered, as well as a minimization  of 4, 8, 14, and 18 size radius for normal tissue covered. Additionally, as  the necessary amount proposed algorithm generalizes optimized shots for each arrangement  of times to achieve $k$ circles individually,  thehighest F_{1} score. Additionally, we simulated our algorithm multiple times to see indeed which shot  number will result with the greatest F_{1} score. of shots most appropriate for patients can be determined on a case-by-case basis.