Mathematical Methods in Medical Physics, Oncology and Biology

Course Content of a two-week course covering

Radiation Dosimetry, Monte-Carlo simulation of Ionizing Radiation Transport, Cellular Radiobiology, Radiobiological Models of Tumour Control and Normal-Tissue Damage, Optimization of Radiotherapy Treatment Plans, Analytical and Iterative Methods of Image Reconstruction, Correlations Between Imaging and RNA Sequencing Data including Gene Set Enrichment Analysis


I. Radiation Dosimetry for external-beam radiotherapy - theory and practice

Essential background knowledge (approx. two lectures)

The interactions of (ionizing) radiation with matter, with particular emphasis on mass attenuation coefficients, mass-energy transfer and mass-energy absorption coefficients (photons); stopping power, collision and radiation (charged particles).

Quantities, Definitions and Units:

Absorbed Dose (in Gray), Kerma (in Gray), Fluence (in m-2) of ionizing radiations (electrons, photons, protons, neutrons)

Relationships between Fluence, Kerma and Dose

Charged-particle equilibrium (CPE), the Fano theorem; collision kerma; delta-ray equilibrium.

Detector/dosimeter response: the Bragg-Gray cavity concept; the 'large' photon detector; the 'Burlin' or intermediate detector. Expressions for Dmed/Ddet for these 3 categories of dosimeter.

Special emphasis on the mass (collision) stopping-power ratio: the Bragg-Gray and Spencer-Attix formulations.

Practical detectors - corrections for deviations from 'perfect' B-G behaviour, and for charge recombination

International Codes of Practice for the determination of (reference) absorbed dose in radiotherapy beams of megavoltage Photons and Electrons, and of kilovoltage photons (x-rays). 

II. Monte-Carlo simulation of (ionizing) Radiation Transport

Rationale for the Monte-Carlo (MC) approach; basic principles; the selection of a variable from a differential cross section employing pseudo-random numbers.

Photon MC transport - principles, ray-tracing; coding up selection of interaction type, energy loss from Compton scattering, Pair Production, and the photoelectric effect.

Charged-particle MC transport (emphasis on electrons) - the condensed-history approach; class-II codes: ‘continuous’-energy losses and ‘catastrophic’ collisions.

Coupled electron-photon MC transport.

Public-domain MC codes - the EGSnrc code system

Applications of MC in Radiation Dosimetry and Radiotherapy treatment planning.

[3.5 - 4 days including computer lab exercises with the ‘VisualMC’ software (Baker & Nahum)]


The Radiobiology of tumour-cell killing; TCP and NTCP modelling; Radiobiological optimisation of external-beam treatments.

The concept of clonogenic tumour cells; cell-killing by ionising radiation; the linear-quadratic (LQ) model, repair; principles of Fractionation; the alpha/beta ratio of tumours; the role of oxygen and hypoxia; LET effects; repopulation; inter-patient differences in tumour radiosensitivity; the 5 Rs of radiobiology for radiotherapy.

Modelling the probability of (local) tumour control (TCP): Poisson statistics and stochastic models; adaptation to observed outcomes via inter-patient variation in radiosensitivity; incorporating tumour-dose heterogeneity via the (differential) dose-volume histogram (DVH); derivation of TCP-model parameters consistent with observed clinical outcomes; the incorporation of clonogen repopulation; the effect of dose rate.

Modelling the probability of normal-tissue complications (NTCP): the nature of radiation-induced complications (early & late effects); the effect of NT volume; alpha/beta ratios for early and late NT complications; the rationale for fractionation; the Lyman-Kutcher-Burman (LKB) NTCP model; the Relative-Seriality (RS) model. 

Cancer induction by ionising radiation (the work of Uwe Schneider)

The use of NCTP and 'isotoxicity' to individualize the (tumour) prescription dose. Varying the number of (weekday) fractions. Examples of isotoxic clinical protocols. Incorporating isotoxicity into the 'inverse planning' of intensity-modulated radiotherapy (IMRT). The potential of isotoxic dose/fraction-number prescribing for improving clinical outcomes. The 5 levels of radiobiological-model based radiotherapy optimiziation (Nahum-Uzan 2012). IMRT Optimization techniques (mathematical formulations and algorithms)

(2 - 2.5 days including exercises with BioSuite – Uzan & Nahum 2012)


Mathematical methods for Static and Dynamic Radiotherapy Image Reconstruction, including Genomic analysis

Analytical reconstruction theory for CT and PET. (2 lectures)

Iterative reconstruction methods for CT and PET. (1 lecture)

4-D image reconstruction theory for dynamic PET data describing variations of tracer 3D uptake over 1-3 hours post-injection. (1 lecture)

Kinetics analysis of dynamic PET data and of dynamic CT or MRI perfusion imaging. (2 lectures)

Validation of indices obtained from kinetic modelling of tumour PET tracer uptake via studies of correlations with RNA sequencing data derived from tumour biopsies, including gene set enrichment analysis. (1 lecture)

(1.5 days)

Presentation and discussion of some research problems of current interest in module one, two, and three.

A reading list of relevant books, book chapters and key research papers will be provided, covering the subject matter of the three modules.

Nonlinear Operator Theory and Applications

Group 1: Nonlinear Operator Theory

Week 1. Geometric Properties of real Normed Spaces

• Hilbert space identities.

• The normalized duality map and some properties.

• The classical Banach spaces: Lp, lp and the Sobolev spaces W m p (Ω), for 1 < p < ∞.

• Characteristic inequalities in these spaces.

Week 2. Nonlinear Operators and Iterative Algorithms

(a) Nonlinear Operators

• Monotone operators in real Hilbert spaces, motivation and examples.

• Accretive and monotone operators in general real normed spaces: motivation and examples: nonexpansive maps, subgradients of proper convex function, solutions of some partial differential equations (the Heat equation), pseudo-contractive operators.

• Zeros of accretive and monotone operators.

• Generalizations of these operators.

• Hammerstein Integral Equations: Motivation from Physics.

(b) Iterative Algorithms

(i) The Mann Algorithm, the Halpern Algorithm; Proximal Point Algorithm of Martinet and Rockafellar; the Perturbation Algorithm of Chidume; Iterative algorithm of Chidume and Zegeye for Hammerstein Equations.

(ii) Techniques of proof of convergence; weak and strong convergence. Numerical experiments using test examples; comparison of algorithms in terms of number of iteration and CPU time.

(iii) Inertial Algorithm for speeding up convergence. Comparison with noninertial algorithms. Week 3. PhD Theses Problems and Some Current Research Results

• New geometric inequalities in real normed spaces.

• Several appropriate PhD theses research problems: discussion and hints on possible suitable directions.

• Introduction to Forward-Backward Iterative Algorithm for Image Restoration and Signal Processing.

• Presentations of recently published results.

Differential equations, dynamical systems and stochastic analysis

MODULE ONE :     Basic theory of ordinary differential equations.

 1-Exponential formula and linear ordinary differential equations,

2-Nonlinear differential equations : Cauchy-Lipschitz’s theorem,  Continuous dependence with respect to the initial data, local existence and blowing  up phenomena, Peano’s theorem,

3-Resolvent operators and nonautonomous ordinary differential equations,

4-Asymptotic stability : Linear systems, Linearization principle, Lyapunov functions, LaSalle invariance principle.


MODULE TWO : Stochastic Differential Equations

1-Probability theory, random variables, and standard Brownian motion,

2- Ito stochastic integrals,

3-Stochastic differential equations,

4- Applications in finance.


MODULE THREE : Applications of ordinary differential equations

 1-Populations dynamics,

2-Control theory .

Course on Stochastics and Fundamental Probability Theory

School of Stochastics and Fundamental Probability Theory.

Objectives: The trainee will acquire the most solid basis of Probability theory and its applications. The course will help him to teach at Graduate and to have a solid basis for undertaking Research works.

Methodology: Large contents of probability theory are exposed within two weeks and exercises are proposed. But most of the courses are openings only. The students are given full texts to be read between tow years. Most of the contents are also exposed in videos. The trainees can even begin following videos before the beginning of the schools and after. The trainees can reach the objectives only if the follow up with those texts and videos.

Program Year one

The foundation of Probability Theory.

Module 1: Finite dimension probability laws, diffeomorphic change of variables conditional mathematical expectation and conditioning (two days). [Contents: all features of probability laws in general. Complete characterization of finite-dimensional probability laws; main probability inequalities]

Day 1: Exposition of the theory (Gane Samb Lo)

Day 2: Exercises (A. Niang)

Module 2. Limit theorem and weak convergence, comparison. Applications to limit theorem for independent random variables (three days). [Contents: strong and Weak laws of large numbers (WLLN, SLLN), Kolmogorov theory for independent random variables, Kintchine SLLN), Strassen or law of the iterated logarithm (LIL)]

Day 1: Exposition of the theory

Day 2: mastering the proofs

Day 3: Exercises and extension.

Module 3. Lois infinitely decomposable, stable laws and the closure of the CLT question for independent data. (two days).

[Contents: Central limit theorem for square integrable data and non-square integrable data. Characterization of possible limits as infinitely decomposable laws. Representations of Levy-Kintchine. Max stability on R and Fundamental Theorem of Gnedenko (may be treated as exercise). Applications to invariance principles and link to Levy processes]

Module 4. Types of dependences and related results. (Two days)

[Contents: Markov chains, mixing random sequences, associated data, exchangeable sequences, martingale and sisters. SLLN, CLT and invariance principles in the dependent scheme]

Module 5. Foundation of Stochastic processes through Kolmogorov Existence Theorem. Main examples: Poisson and Brownian. Markov processes. Levy Processes. (two days)

Module 6 (special). Regularity of Stochastic processes (cadlag, progressive measurability, Holder paths) and Brownian process.

Module 7. Martingales of continuous times, Levy process and opening to stochastic calculus.

Program Year two

Levy processes, stochastic calculus, application to Finance and Actuarial Sciences.

Module 1. Review of: Martingales, infinitely decomposable laws, stables laws. (two days)

Module 2. Levy process and Stochastics calculus. (five days)

Module 3. Levy process and Stochastics calculus. (Application to Finance and other discipline)

Program Year III

Most advanced Calculus and Some account of the most recent research activities

1. Accounts of the some most advanced “stochastics” are given: Malliavin Calculus, Factionary processes, Sheets (Brownian, Poisson, ect., Self-similar processes, Super Processes, Processes on trees, etc.

2. For each period of three days, one the most recent and important papers published in the highest journals (Annals of Probability, Journal of Poincare, Bernoulli Society, etc..) is entirely studied by the trainees.


The capacity to read the high-level reference books and the papers at the program will be the most valuable indicator of success.

A - The contents of Year One will entirely be ready for students.

1. Lo, G.S.(2016). A Course on Elementary Probability Theory. SPAS Editions. Saint-Louis, Calgary, Abuja. Doi : 10.16929/sbs/2016.0003.


2. Lo, G. S. (2017) Measure Theory and Integration By and For the Learner. SPAS Books Series. Saint-Louis, Senegal - Calgary, Canada. Doi : http://dx.doi.org/10.16929/sbs/2016.0005. ISBN : 978-2-9559183-5-7

3. Lo, G.S.(2018). Mathematical Foundation of Probability Theory. SPAS Books Series. Saint-Louis, Senegal - Calgary, Canada.

Doi : http://dx.doi.org/10.16929/sbs/2016.0008.

Arxiv : arxiv.org/pdf/1808.01713

4. Lo, G.S.(2021). Probability III : Sequences of random variables. SPAS Books Series. Saint-Louis, Senegal - Calgary, Canada. To come in SPAS-EDS.

5. Lo, G.S.(2021). Probability IV : Stochastic processes and introduction to stochastic calculus. Saint-Louis, Senegal - Calgary, Canada. To come in SPAS-EDS.

6. Lo, G.S.(2021). Probability V : Stochastic calculus and application. Saint-Louis, Senegal - Calgary, Canada. To come in SPAS-EDS

B – Electronic versions of highest level in the field will be available.

1. Vladmir I. Bogachev (2007a). Measure Theory I. Springer.

2. Vladmir I. Bogachev (2007b). Measure Theory II. Springer.

3. Michel Loève (1997). Probability Theory I. Springer Verlag. Fourth Edition.

4. Kai Lai Chung (1974). A Course in Probability Theory. Academic Press. New-York.

5. Kai Lai Chung (1980). Lectures from Markov processes to Brownian Motion. Grundlehren der mathematischen Wissenschaften 249 (A Series of Comprehensive Studies in Mathematics). Springer Sciences+Business Media, LLC.

6. Howard M. Taylor and Karlin S. (1998). An Introduction to Stochastic Modeling. 3rd Edition. Academic Press.

7. Kalyanapuram Rangachari Parthasarathy (2005). Introduction to Probability and Measure. Hindustan Book Agency. India.

8. Kuo H. (2000). Introduction to Stochastic Integration. Springer.

9. Allan Gutt (2005). Probability Theory: a graduate course. Springer.

10. Daniel Revuz and Marc Yor (2005). Continuous Martingales and Brownian Motion. Springer, 3thd Edition, 5th printing.

11. David Applebaum (2004). Levy processes and stochastic calculus. Cambridge University Press.

12. Keni-Iti Sato (1999). Levy processes and infinitely divisible distributions. Cambridge University Press.

13. Albert N. Shiryaev (1999). The essentials of Stochastic Finance: Facts, Models, Theory. World Scientific, Singapore.

14. Jan Grandell (1991). Aspects of Risk Theory. Springer

C – Website with video contents: ufrsat.org/imhotep