1. Session Overview

The LUMS Math Circle held on December 12, 2025, featured an interdisciplinary session titled “Mathematics of Cancer Growth,” jointly led by Dr. Amir Faisal and Dr. Waqas Ali Azhar. The session was carefully designed to weave together biological insight, guided primarily by Dr. Amir Faisal, with mathematical modelling and quantitative reasoning, led by Dr. Waqas Ali Azhar.

Objectives of the Session

• Present cancer as a biological process grounded in cellular behaviour

• Demonstrate how mathematics can model biological growth

• Encourage active learning through guided discussion and activities

2. Biological Foundations: Cells and Cancer

Dr. Amir Faisal began by anchoring the discussion by establishing the biological foundations necessary to understand cancer growth, ensuring that all students shared a common conceptual framework.

Core Biological Ideas

• Cells as the fundamental units of life
  He explained that all living organisms are composed of cells and that every cell originates from a pre-existing cell, making cell division central to understanding both healthy tissue and disease.

• Cancer originating from a single abnormal cell
  Students learned that most cancers begin with one cell that acquires harmful mutations, a point emphasized by Dr. Amir to counter the misconception that cancer appears fully formed.

• Cancer is a multi-step biological process
  Dr. Amir Faisal highlighted that tumour formation requires several biological stages, including mutation accumulation and changes in growth behaviour, naturally motivating mathematical analysis later in the session.

3. Exponential Growth: The “One Bad Cell” Model

Following the biological setup, Dr. Waqas Ali Azhar introduced the first mathematical growth model based on idealized cell division. Shifting the discussion from biology to numbers with a deceptively simple question: What happens if one bad cell keeps doubling?

• What followed was a striking transformation of perspective. Under Dr. Waqas’s guidance, students translated Dr. Amir’s biological narrative into mathematics, arriving at the exponential growth model:

N(t) = 2^t

• By calculating cell counts after specific numbers of divisions, Dr. Waqas showed how mathematics can estimate when tumours become detectable. Which shows how dangerous “simple doubling” can be.

• Students learned that tumours can remain biologically invisible while still growing exponentially, a conclusion drawn directly from mathematical reasoning.

What had started as a harmless equation now felt urgent.

4. Realistic Growth: Accounting for Death and Constraints

This section intentionally combined both facilitators’ expertise.

Introducing Real-World Complexity

• Biological limits to cell growth
  Dr. Amir stepped back into the discussion, almost deliberately interrupting the mathematical optimism. He reminded student that real biology is messy. Real cells do not divide indefinitely, as many die and others fail to divide due to limited nutrients, space, or immune response.

• Mathematical modelling of partial growth
  Picking up this biological complication, Dr. Waqas re-entered the discussion, this time refining the mathematics to match reality. He introduced growth at 50% per day and helped students derive the recurrence relation
     
  and its explicit formula.

• Comparing idealized and realistic models
  The mood shifted again. Growth was still powerful—but now it felt earned, conditional, and fragile. Dr. Waqas used this moment to highlight a core lesson of modelling: mathematics is only as honest as its assumptions.

5. The Probability Turn: When Rare Becomes Inevitable

With growth models in place, Dr. Amir Faisal changed the direction of the session by asking a deceptively simple question: If cancer requires several mutations, and mutations are rare, why does cancer happen at all?

That question hung in the air—until Dr. Waqas Ali Azhar seized it.

Dice appeared. Probabilities were calculated. Rolling three sixes in a row suddenly became the star of the show. Under Dr. Waqas’s guidance, students discovered that rarity loses its meaning when trials number in the trillions.

What felt impossible on a small scale became unavoidable at the scale of the human body.

6. Numbers Meet Biology: Expected Value and Age

As calculations grew larger, Dr. Waqas Ali Azhar leaned into expected value, showing students how mathematics predicts outcomes not for one cell, but for billions acting over decades.

Then Dr. Amir Faisal returned to ground the numbers in biology, connecting expected value to aging, accumulated cell divisions, and long-term cancer risk. The abstract became personal: time itself emerged as a risk factor.

7. Small Advantages, Huge Consequences

The session took a dramatic turn when Dr. Amir Faisal introduced the idea of selective advantage, describing how a slightly faster-dividing cell can slowly dominate its neighbours.

Moments later, Dr. Waqas Ali Azhar quantified that intuition, writing numbers on the board that showed how tiny growth advantages explode into massive population differences over time.

The room could feel it now: cancer is not about being faster—it’s about being slightly faster for long enough.

8. When the Room Became the Tumor

Perhaps the most memorable moment came when Dr. Amir Faisal transformed the classroom itself into a biological system. Students became cells. Space became tissue. Growth suddenly felt physical.

As crowding increased, Dr. Waqas Ali Azhar stepped in, translating the chaos into the language of carrying capacity and logistic growth. What students experienced with their bodies, he captured with equations.

The lesson landed effortlessly: growth always slows—unless it finds somewhere new to go.

9. Conclusion

The December 12, 2025, LUMS Math Circle was more than a lesson—it was a performance of ideas. Through a dynamic interplay between Dr. Amir Faisal’s biological storytelling and Dr. Waqas Ali Azhar’s mathematical precision, students witnessed how two disciplines converge to explain one of the most complex phenomena in medicine.

By the end, cancer was no longer an abstract fear or a mysterious disease. It was a story—written in cells, shaped by time, and told through mathematics.

The event concluded with the distribution of certificates.

Acknowledgment

Heartfelt thanks are extended to Dr. Amir Faisal and Dr. Waqas Ali Azhar for guiding students from playful exploration of biological ideas to clear and precise mathematical reasoning about cancer growth.

We also express our sincere appreciation to Ms. Noreen Sohail, Ms. Shazia Zafar, Mr. Qamar Abbas, and Mr. Javaid Qayoom for their dedicated support and excellent organization, which contributed greatly to the success of the event.

Here are some highlights from the event: