General Information

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Project Description

Cancer is the second leading cause of death in adults, claiming the lives of more than half a million Americans every year. Tumors start from a single rogue cell in the body, which becomes defective, and the clones of that initial cell start to grow (or die out), and acquire new mutations depending on certain parameters stochastically; and by the time of detection the tumor has ~109 cells. During treatment, the patient initially responds and the tumor shrinks, but remaining cells inevitably develop mutations that give them drug resistance, and the tumor starts to grow again until the patient dies. It is becoming apparent that computational algorithms and mathematical frameworks can identify subtle patterns in patient profiles, which can help with early detection and personalized treatment, improving their outcome. For instance, my mentor and I will be developing mathematical frameworks to model tumor development and emergence of resistance against cancer drugs, using Galton Watson branching process - an area of stochastic processes, which also has applications in diverse areas (e.g. option pricing, predicting viral outbreak, news propagation on social networks). We are combining mathematical models with clinical data from cancer patients to ask some fundamental questions - (i) how long does it take a tumor to grow, and can we detect cancer early? (ii) can we identify emergence of resistance very early and accordingly change the treatment strategy? The project will aim to develop mathematical frameworks to model tumor growth using branching process.

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Weekly Log

Week 1:

This was the first week of my DIMACS experience. On the first day, I met with my mentor, Subhajyoti De. My mentor provided me with a basis for what the project would center around. To help contextualize these ideas, he asked me to read two articles, detailing numerous methods that had been researched detailing the heterogeneity of tumors and how it can be determined. The next day was orientation where I met with my peers and started familiarizing myself with the area and the resources I had available. After I had a few days to review and reflect on the assigned material, I met with my mentor again and we decided on our first plans of action for this project. I prepared the slides for my presentation, which I’ll be delivering on Monday. I also successfully managed to create and update my website. I look forward to the coming weeks of my DIMACS experience.

Week 2:

We began week 2 with our presentations, where I described the basic gist of my project and was able to see what my peers were working on. I also attended a lab meeting at CINJ, which was a very interesting and enriching experience. I was surprised to find I would like the group meeting environment, in which my coworkers described what they had learned at the seminar the week before. I look forward to giving my own informal presentation as I really start tackling my project. This week, though, focused on analyzing one of the papers [3.] my professor gave me. Prof. De expressed that his focus for the project was devising a code or mathematical model which might have a hope of accurately predicting the growth of cancer. I analyzed the paper in order to see what elements I would need to incorporate into my own research. From this paper, I was exposed to a lot of details about cancer which I did not know about before. I have learned that the major elements that needs to be considered when monitoring cancer’s growth are : population size, mutation rates, probabilities of these mutations, and spatial relationships between cells. Cancer is an evolutionary process, which stems from a single cell that mutates and continues to reproduce. The analyzed the amount of time at which a single cell with a mutation might enable other cells around it to mutate. For example : 
Normal cell : Type 0 cell Precancerous cell : Type 1 cell Cancerous cell : Type 2 cell If precancerous cells began with a mutation of type 0 to type 1, at what rate would this mutation occur, and would it obey the spatial limitations ie. cell number within a certain bisection. In addition, the paper modeled this growth and I was able to successfully replicate it in Java. The paper described at what rate would type 1 cells replicate to type 2 cells. These models, though, do not accurately predict the development of cancer, which is why I’ll be working on devising my own model to monitor and predict the growth of cancer. Although there is not too much research that deals directly with what my project will be focusing on, I’m excited to see how I can contribute to the research.

Week 3:

This week proved to be a very challenging week and I spent a large majority of it brainstorming. Although my topic and my goal are quite clear and concise, the methods I would need to use to arrive at such a goal are a bit more vague. I really wanted this research to be my own, so I focused on coming up with my own ideas for how to proceed with my project. I tried several approaches but was frequently frustrated when encountering a dilemma which I felt would hinder the project more than help it. Thankfully, I discovered the method I wanted to follow while I was typing up the formulas of a mathematical model [5] in one of the papers I read. For two of the papers I analyzed [3] and [5], I began to see the areas that their mathematical models overlapped and where each model fell short. I thought, instead of making my own model from scratch, I would focus on improving one model by using the another model. By using the strengths and weaknesses of the two models, I might have hope of making a better and more accurate model of my own. I discussed this idea with my mentor and he agreed that it was a great idea and basis for the rest of my search.

Week 4:

For my previous weeks at DIMACS, I have been working on mathematical models presented which two papers [3] and [5] my mentor provided. This week was a very daunting week as I was asked to give a presentation of my own within our CINJ lab meeting. The goal of my presentation was to explain the mathematical of within the papers I read to convey my thoughts and goals for this project. I was excited to discover that a lot of areas seemed a little murky, my fellow coworkers were able to shed light on the topics. Even more so, the questions they asked really helped to contextualize my understandings of the readings. The next day I met with my mentor again and he asked me to focus on the mathematical models within [5] . I spent the rest of the week coding the formulas and trying to replicate the diagrams provided. I found myself enjoying programming very much and I was able to learn more techniques for coding in Java. Although, I have hit several blocks in designing my code and debugging it, the results are extremely rewarding. Another great week at DIMACS!

Week 5:

This week, I devoted myself to designing and running my program which would work towards replicating the graphs found with the paper. There are a total of 6 different methods. - all monitoring tumor growth under different conditions. I’ve managed to successfully code 3 of the methods. The remaining 3 methods are much more complicated and intensive. I wrote to the author some of my questions and very luckily, he replied. His feedback greatly helped me progress my work. The week ended with a fun and very education field trip to IBM. I look forward to progressing my code further next week.

Week 6:

This week went by extremely fast, as Independence Day was on Tuesday. The festivities aside, I continued working on my code on the mathematical models. After running simulations and calculating the figures by hand, I can confidently say that my code is working and accurate. Unfortunately, I met delays when specific elements in the code, which heavily involves complex analysis and Laplace transformations. As I’ve never had taken a class on this, this dilemma proved to extremely challenging and exciting, as I was learning something new. Once I research and learn more on the subject, I’ll be able to finish my code and replicate the graphs. Overall, this was a very productive week !

Week 7:

Week 7 proved to be one of my busiest weeks. I began by refining my code in preparation for the end of the program. In addition, I spent a few days making my second presentation and practicing it. Although I produced my results throughout my weeks, I wanted to rerun my program and have a fresh excel sheet to work with. It was very fun to hear everyone's projects and progress. The presentation really helped me look back on my work and evaluate my own progress. I feel I have come quite a way away from how I was at the beginning of the program. Still, I cannot believe 7 weeks have flown by already.

Week 8:

Week 8 consisted of desiging my code to incorporate more variables. I examined the Mathematica notebook that accompained [6]. I devoted the whole week to designing my code and planning for my final report.

Week 9:

Week 9 has finally come to an end. I spent the beginning of the week planning and drafting my final report. Finally, my report is finalize and I am ready and packed to go home.

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Presentations

• First Presentation
• Second Presentation

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Additional Information

• My Mentor
  • Subhajyoti De

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References

[1.] Jiang, Yuchao, Yu Qiu, Andy J. Minn, and Nancy R. Zhang. "Assessing Intratumor Heterogeneity and Tracking Longitudinal and Spatial Clonal Evolutionary History by Next-generation Sequencing." Proceedings of the National Academy of Sciences. Ed. David O. Siegmund. PNAS, 29 Aug. 2016. Web. 30 May 2017.

[2.] El-Kebir, Mohammed, Layla Oesper, Hannah Acheson-Field, and Benjamin J. Raphael. "Reconstruction of Clonal Trees and Tumor Composition from Multi-sample Sequencing Data." Bioinformatics. Oxford University Press, 15 June 2015. Web. 30 May 2017.

[3.] Durrett, R., J. Foo, and K. Leder. "Spatial Moran Models, II: Cancer Initiation in Spatially Structured Tissue." Journal of Mathematical Biology. J Math Biology, 1 July 2015. Web. 11 June 2017.

[4.] Ryser, Marc D., Walter T. Lee, Neal E. Ready, Kevin Z. Leder, and Jasmine Foo. "Quantifying the Dynamics of Field Cancerization in Tobacco-related Head and Neck Cancer: A Multi-scale Modeling Approach." Cancer Research. American Association for Cancer Research, 20 Oct. 2016. Web. 03 June 2017.

[5.] Paterson, C. et al. An exactly solvable, spatial model of mutation accumulation in cancer. Sci. Rep. 6, 39511; doi: 10.1038/srep39511 (2016).

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