mathematical-model

Instead of hitting a tumour as hard as possible, adaptive therapy gives just enough drug to keep it in check and stops when it shrinks, so drug-sensitive cells survive to compete with resistant ones. A pilot trial in prostate cancer roughly doubled the time to progression on abiraterone.

Instead of equations for average behaviour, agent-based models simulate every cell as an individual with rules for dividing, moving, dying and signalling, producing virtual tumours in which immune attack, drug delivery and evolution can be watched and tested.

Models of how tumours recruit blood vessels, and of Rakesh Jain's idea that anti-angiogenic drugs at the right dose normalise rather than destroy those vessels, improving drug and oxygen delivery for a window of days.

Chemotherapy doses are usually written per square metre of body surface, a convention from 1958 that scales drug clearance between species and people; it is imprecise, and for many newer drugs flat or weight-based doses have replaced it.

Peter Nowell's 1976 idea that a tumour is an evolving population of competing clones is now measured directly by sequencing several regions or repeated blood samples, and models of branching evolution predict which clones will drive relapse.

Mathematics from population genetics shows that resistant cells almost always exist before treatment in large tumours, and that combining drugs with different resistance mutations from the start can succeed where the same drugs in sequence fail.

Cancer cells are treated as players whose success depends on what neighbouring cells do, which lets researchers predict how a tumour's mix of cell types shifts under treatment and design schedules that steer it.

Resistant cells arise by chance mutation as a tumour grows, so the chance of a cancer already containing resistant cells rises with its size. The model argued for treating early and for alternating non-cross-resistant drugs.

Tumours do not grow exponentially forever: growth slows as they enlarge, following a curve Benjamin Gompertz devised for human mortality in 1825 and Anna Kane Laird fitted to tumours in 1964. It explains why small tumours are the most chemosensitive and why doubling times lengthen.

A dose of chemotherapy kills a constant fraction of cancer cells, not a constant number, so each cycle removes the same proportion, which is why treatment continues after the tumour has disappeared from scans.

Mathematical oncology writes down how tumours grow, evolve, respond to treatment and interact with the immune system as equations or simulations, then uses them to design doses, schedules and trials. This page gathers every model OnCo covers.

From Paget's seed-and-soil idea to models that estimate when metastases were seeded from a primary and how long they lay dormant, these frameworks explain late relapse and argue for treating micrometastases early.

Because tumours regrow fastest when small, the best way to finish them is to give the same chemotherapy doses closer together. The idea, from Larry Norton and Richard Simon, was proved in breast cancer by the CALGB 9741 trial and made two-weekly chemotherapy a standard.

A digital twin is a computer model of one patient's tumour and body, updated with each scan and blood test, used to forecast how the disease will respond to each option before it is tried.

Equations that describe how a drug's concentration rises and falls in the body and how that concentration translates into effect and toxicity; the reason doses are given per square metre, why some drugs are infused over days, and how children's doses are set.

Gliomas grow by both dividing and migrating through the brain, and a two-parameter equation fitted to a patient's MRI scans estimates how far invisible cells have spread, which can guide how much brain to irradiate and how fast the tumour will grow.

Mechanistic computer models that join a drug's pharmacology to the biology of the tumour and the body, used by developers and regulators to pick doses, predict combinations and explain why a trial failed.

The speed at which a molecular marker falls during treatment predicts outcome better than a single level: BCR-ABL halving time in chronic myeloid leukaemia and circulating tumour DNA slopes in solid tumours are now used to judge response within weeks.

Growing tumours compress themselves and their surroundings; models of this solid stress explain collapsed vessels, poor drug delivery and stiff stroma, and point to drugs that soften the tumour so treatment can get in.

Radiotherapy works through repair, redistribution, reoxygenation and repopulation between fractions; the discovery that tumours speed up their regrowth during a course explained why gaps in treatment cost cures and led to accelerated schedules.

Curves that turn a radiation dose into a probability: how likely the tumour is to be eradicated and how likely a nearby organ is to be damaged. They underlie dose constraints, dose escalation trials and comparisons between treatment plans.

How long a tumour takes to double in volume, measured from two scans; it separates cancers from benign nodules in lung screening, sorts aggressive from indolent disease and estimates how long a tumour has been present.

Predator-prey style equations describe how immune cells hunt tumour cells, and they reproduce dormancy, escape and the delayed, sometimes explosive, responses seen with immunotherapy; they now help design combination and scheduling trials.