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## Comparative Dissertation: Resolving Tumor Growth Using Creative Mathematical Transformations
### **Abstract**
This dissertation explores the application of creative mathematical transformations—swap, invert, and mirror (s, i, n)—to tumor modeling methods commonly used in cancer research. We compare these playful methods with traditional mathematical modeling approaches, demonstrating both their interpretive potential and their limitations in scientific contexts.
---
### **1. Introduction**
Mathematical modeling is central to cancer research, allowing scientists to simulate tumor growth, invasion, and therapy outcomes without immediate recourse to clinical trials[1][3][4][5][7]. Standard models often use partial differential equations (PDEs), spatial geometry (frequently involving π), and computational algorithms to capture complex biological dynamics[4][7]. Here, we introduce an alternative, creative approach: applying swap, invert, and mirror transformations to mathematical representations and compare their interpretive value to established methods.
---
### **2. Standard Mathematical Method: Spheroid Tumor Growth Model**
A classic model for early-stage tumor growth is the spheroid model, which assumes a tumor grows as a sphere:
$$
V = \frac{4}{3}\pi r^3
$$
where $$V$$ is tumor volume and $$r$$ is radius. This model is widely used for its simplicity and its fit to in vitro tumor data[4].
---
### **3. Creative Transformations Applied**
#### **A. Swap (s)**
- Swap digits in π (3.14) → 1.43.
- Apply to the volume formula: substitute π with 1.43 for a "swapped" model:
$$
V_{swap} = \frac{4}{3} \times 1.43 \times r^3
$$
#### **B. Invert (i)**
- Invert π (3.14) visually or numerically (e.g., upside down on a calculator, or reciprocal 1/π).
- Using the reciprocal:
$$
V_{invert} = \frac{4}{3} \times \frac{1}{\pi} \times r^3
$$
#### **C. Mirror (n)**
- Mirror π (3.14) visually, which resembles "PIE"—a playful, non-numeric transformation.
- For a mathematical mirror, reverse the digits: 3.14 → 4.13.
$$
V_{mirror} = \frac{4}{3} \times 4.13 \times r^3
$$
---
### **4. Comparison Table**
| Method | π Value Used | Formula | Scientific Validity |
|----------------|-------------|------------------------------------------|---------------------------|
| Standard | 3.14 | $$V = \frac{4}{3}\pi r^3$$ | High (widely accepted) |
| Swap (s) | 1.43 | $$V_{swap} = \frac{4}{3} \times 1.43 r^3$$ | Playful, not scientific |
| Invert (i) | 1/3.14 | $$V_{invert} = \frac{4}{3} \times 0.318 r^3$$ | Not used in practice |
| Mirror (n) | 4.13 | $$V_{mirror} = \frac{4}{3} \times 4.13 r^3$$ | Playful, not scientific |
---
### **5. Discussion**
- **Standard models** (using π) are validated against experimental and clinical data, providing reliable predictions for tumor growth and therapy outcomes[4][7].
- **Creative transformations** (swap, invert, mirror) offer a playful lens for exploring mathematical symbolism but do not yield scientifically valid results for cancer modeling.
- The act of transforming constants like π highlights the importance of mathematical rigor in biomedical research, while also showing how mathematical play can foster engagement and curiosity.
---
### **6. Conclusion**
While creative transformations such as swap, invert, and mirror can generate novel interpretations and visual puns (e.g., "PIE" from π), they cannot replace the precision and empirical grounding of standard mathematical models in cancer research. Nonetheless, such playful approaches can serve as educational tools or inspiration for mathematical outreach, enriching the cultural landscape of science[1][3][4][5][7].
---
**In summary:**
Standard mathematical modeling with π remains essential for cancer research, whereas creative transformations provide an imaginative but non-scientific perspective. Both approaches, when understood in context, contribute to the broader appreciation of mathematics in science.
Citations:
[1] A Review of Mathematical and Computational Methods in Cancer ... https://pmc.ncbi.nlm.nih.gov/articles/PMC9359441/
[2] Optimization Formulations and Algorithms for Cancer Therapy https://amath.washington.edu/research/graduate/optimization-formulations-and-algorithms-cancer-therapy
[3] An Overview of Mathematical Modelling in Cancer Research - MDPI https://www.mdpi.com/2504-3110/7/8/595
[4] A mathematical journey of cancer growth - KEEP https://keep.lib.asu.edu/items/154838
[5] Mathematical modelling of cancer invasion : a review https://research-repository.st-andrews.ac.uk/handle/10023/25867
[6] [PDF] A Comparative Study of Elementary Mathematics Specialists and ... https://scholarworks.waldenu.edu/cgi/viewcontent.cgi?article=4834&context=dissertations
[7] [PDF] MATHEMATICAL MODELING OF CANCER TREATMENTS AND ... https://researchspace.ukzn.ac.za/bitstreams/6baf6127-07e1-4ab1-9e86-f19bb05b76fb/download
[8] Comparing Methods of Teaching Radical Functions in Various ... https://scholarworks.calstate.edu/concern/theses/pr76fb42k