Applications of mathematics in clinical setting What are some examples of successful mathematical attempts in clinical setting, specifically at the patient-disease-drug level?
To clarify, by patient-disease-drug level, I mean the mathematical work is approved to be used as part of a decision making process to prescribe a specific treatment for a specific patient?
I do not mean general modeling attempts that study and simulate on a more or less theoretical level.
Additionally, are these mathematical work have to be approved by the regulators like the FDA or a national board of physicians?
 A: Amazingly, noone mentionned the use of the inverse Radon transform in the scanner / medical imaging.
A: Machine learning is on its way to provide the type of personalized health care referred to by the OP. In June of this year the FDA (US Food and Drug Administration) has proposed a regulatory framework for machine learning algorithms in medical decision making. 
The document distinguishes "locked algorithms" (basically decision trees) and "adaptive algorithms" (a supervised learning algorithm that will "change its behavior using a defined learning process", as the inputs are given new data). Only the locked algorithms are currently approved by the FDA, and the document tries to set up a pathway for supervised machine learning algorithms to be used in health care. 
A: An example of a simple mathematical/evolutionary game theory model used to determine treatment scheduling in clinical treatment of metastic and castrate resistant prostate cancer can be found at https://www.nature.com/articles/s41467-017-01968-5. While the clinical trial is on-going, initial results show that the model derived treatment schedule offers significant improvement in time to treatment failure when compared to the standard-of-care.
In short, the authors use a 3 population evolutionary game theory model to study the interplay between androgen dependent, androgen independent and androgen producing cancerous cells. They use this model to determine an "adaptive therapy" dosing strategy for abiraterone, a drug that strongly effects androgen dependent cancer cells.Typically, prostate cancer cells develop resistance to abiraterone. Consequently, disease progression and treatment failure is observed in roughly a year (11 months for Prostate Specific Antigen [a biomarker] increase and 16 months for radiographic progression). While small, the study shows that the median time to PSA and radiographic progression is at least 27 months when patients follow the model derived treatment schedule. 
Adaptive therapy is designed to take advantage of the "cost of resistance." When abiraterone concentrations are high, it is hypothesized that the relative cost of resistance is lower than the fitness benefit, so cancerous cells are likely to evolve into abiraterone resistant strains. Adaptive therapy aims to use treatment holidays to protect against the development of these resistant strains. In general, treatment is suspended when PSA reaches half of pre-treatment levels, and only restarts once pre-treatment PSA levels have been reached. 
A: I'm going to conflate mathematics with statistics as Carlo Beenakker did.  Then the earliest application that I know of is that Decision Trees were invented by Breiman et al. to analyze the issue of- among people who have had a heart attack, which patients were most likely to have another heart attack. I also believe that Ed Frenkel, in his autobiography, claimed to have developed a similar methodology to explain to doctors how to triage.  A really beautiful application is something called James-Stein estimation which deals with the inadmissability of the mle for estimating (true) means of many variable.  The short story is that if you have 4 or more series of observations, then the individual estimated mean of each series of observations is not the best estimates of the real means.  This was applied by Efron (mentioned in various books and papers including "Large Scale Inference") to the question of  deciding which genes (via gene expression) were likely to be influential in causing a specific cancer.   
A: From the Automatic Control Laboratory at ETH Zürich, a project on automating anaesthesia:

The first steps to introduce feedback control in anesthesia were undertaken more than ten years ago. The project encompasses all control system aspects, namely

*

*design of experiments and open loop identification


*modeling of biological complex systems


*statistical analysis of large population data


*control design and simulation


*pilot studies and clinical routine validation
Automated Control of Anesthesia was successfully tested during surgery on more than 200 patients and 650 hours.
The successful application of our ideas is made possible by the close cooperation with: the Department of Anaesthesiology at the University Hospital in Berne (Inselspital), where volunteers are enrolled for model identification and controllers are validated in clinical studies.


Papers:

*

*Eleonora Zanderigo, Valentina Sartori, Gorazd Sveticic,
Thomas Bouillon, Peter Schumacher, Michele Curatolo, Manfred Morari, A new model for drug interactions and optimal drug dosing, 27th Annual Conference of the IEEE Engineering in Medicine and Biology, January 2006.


*Eleonora Zanderigo, Antonello Caruso, Thomas Bouillon, Martin Luginbuhl, Manfred Morari, Pharmacodynamic modelling of drug-induced ventilatory depression and automatic drug dosing in conscious sedation, 2006 International Conference of the IEEE Engineering in Medicine and Biology Society.


*Antonello L.G. Caruso, Thomas W. Bouillon, Peter M. Schumacher, Martin Luginbuhl, Manfred Morari, Drug-induced respiratory depression: an integrated model of drug effects on the hypercapnic and hypoxic drive, 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, August 2007.


*Antonello L.G. Caruso, Thomas W. Bouillon, Peter M. Schumacher, Manfred Morari, On the modeling of drug induced respiratory depression in the non-steady-state, 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, August 2008.


*Antonello L. G. Caruso, Thomas W. Bouillon, Peter M. Schumacher, Eleonora Zanderigo, Manfred Morari, Control of drug administration during monitored anesthesia care, IEEE Transactions on Automation Science and Engineering, Volume 6, Issue 2, April 2009.


*Antonello L. G. Caruso, Manfred Morari, Model predictive control for propofol sedation, World Congress on Medical Physics and Biomedical Engineering, September 2009.

Theses:

*

*Valentina Sartori, Optimal modeling and drug administration for anesthesia in clinical practice, ETH Zürich, 2006.


*Eleonora Zanderigo, Optimal administration of analgesics in humans, ETH Zürich, 2006.


*Antonello L. G. Caruso, Personalized drug dosing through modeling and feedback, ETH Zürich, 2009.
A: I know of one mathematical system that was used before the widespread use of CT scans of brain in diagnosing stroke type. That of Scoring methods, which gave at those days a clinical decision that is by way much better than the clinical decision made by highly ranked experienced professional Consultant Neurologists. An example of a score that I myself researched was the Allen Score in differentiating stroke type, i.e. hemorrhagic from ischemic strokes. 
I personally think that along similar lines we can get machines to overcome the experienced human diagnosis in many fields in medicine, albeit not always of course. 
A: The APGAR test. It may not be the most interesting example because it's maths are apparently just a sum but in fact it's a simple decision tree. However, it's approved, useful and widely used.
