References Contacts Ettore Lanzarone No textbook. - - PDF document

Lesson 1 1

Introduction to the course. Perspectives: from the physician making decisions by hand; to the support of mathematical models; to the integration of the models in the machines.

Ettore Lanzarone March 11, 2020

MEDICAL SUPPORT SYSTEMS FOR CHRONIC DISEASES

Engineering and Management for Health University of Bergamo

LESSON 1

References Contacts

Ettore Lanzarone ettore.lanzarone@unibg.it ettore.lanzarone@cnr.it www.mi.imati.cnr.it/Ettore/MSS.html Student reception:

• Wednesday from 13:30 to 14:30

Write an email to fix an appointment No textbook. Slides for the first part of the course “common mathematical approaches applied in medicine and possible issues”. Collection of scientific papers for each example examined in the course.

Lesson 1 2

Evaluation

1. written exam dealing with all theoretical background and examples discussed in the course, followed by oral exam to discuss the written exam and including other questions 2. presentation of the group project Presentation of the project consists of providing the report and preparing a presentation

• f 15 minutes. This presentation will be made by the students of the group together.

It can be done both before or after the other part. The mark will be registered after both parts are completed. The mark is a weighted average (weight 75% for exam; 25% for project)

Introduction

Introduction to the course

Lesson 1 3

Introduction

The topic of the course is the development of medical support systems, where these systems are intended to be mathematical tools that support the decision making process. How to use mathematical models as decision support tools to make medical decisions?

• how to develop a mathematical model to support a medical decision?
• how to use patient-specific information to calibrate the model?
• how to validate the model?
• how to insert the model in an automatic decision making process?

Introduction

The course is divided into three parts: 1. Introduction to common mathematical approaches applied in medicine and possible issues:

• stochastic approaches
• differential approaches
• ptimization approaches

2. Set of examples from real-word medical practice 3. Development of students’ projects

Lesson 1 4

Introduction

For each real-word medical: 1. Description of the medical background and problem statement & possible solutions. 2. Practical work:

• students will receive a dataset related to the problem under analysis;
• students will propose and try an approach to meet the problem objectives.

3. Discussion:

• discussion of the solutions proposed by the students;
• analysis of one or more solutions available in the literature;
• discussion about using the solution for medical decision making.

Introduction

Considered medical problems and mathematical models: 1) hemodialysis Hemodialysis (HD) is still associated with non-negligible rate of comorbidities. In particular, due to therapy discontinuity, HD induces considerable changes in osmotic balances and rapid variations in fluid volumes and electrolytic concentrations within patients’ body compartments. Treatment customization/optimization is required to reduce the associated comorbidities, because the individual tolerance to HD may vary from patient to patient also in the presence of the same treatment conditions.

Lesson 1 5

Introduction

Considered medical problems and mathematical models: 1) hemodialysis Customization requires to simulate and predict the patient-specific response to HD treatment in terms of electrolyte and catabolite kinetics. We will refer to a parametric multi-compartment kinetic model of solute, and we will obtain the customization by means of patient-specific model parameters, which modulate the mass and fluid balance across the main membranes involved in HD process. A patient-specific description of the dynamics will allow to optimize the dialyzer parameters and to identify the most suitable therapy for reducing intra-dialysis complications and associated long-term dysfunctions.

Introduction

Considered medical problems and mathematical models: 2) lumped parameter models of the cardiovascular system Fluid-dynamic behavior of blood in vessels can be characterized in terms of a set of lumped parameters in each segment. The resulting mathematical description is equivalent to that

• f an electric circuit.

Lesson 1 6

Introduction

Considered medical problems and mathematical models: 2) lumped parameter models of the cardiovascular system The resulting system of differential equations can be easily studied to evaluate:

• the impact of alterations in

model parameters (e.g., the link between reduced compliance and hypertension);

• the interaction of the system

with external devices (e.g., cardiopulmonary bypass).

Introduction

Considered medical problems and mathematical models: 3) bioimaging analysis Estimation of functional parameters from a set of images acquired on a patient. Example in Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI): how to estimate diffusion and pseudo-diffusion coefficients from a set of images acquired at different b values. Estimated parameters are useful biomarkers for detecting several pathologies.

Lesson 1 7

Introduction

Considered medical problems and mathematical models: 4) aortic compliance evaluation Similar idea of estimating functional parameters from a set of images acquired on a patient.

Introduction

Considered medical problems and mathematical models: 4) aortic compliance evaluation

• Set of aortic Computed Tomography (CT) acquired over the cardiac cycle
• Identification of the radius profile over the cardiac cycle in several sections
• Mechanical model to get the compliance given the radius profile.

Lesson 1 8

Introduction

Considered medical problems and mathematical models: 5) extracorporeal membrane oxygenation (ECMO) The goal is to provide the best set of parameters to the ECMO machine and alarms if the conditions become critical. Mathematical model to describe the ECMO, the patient, and their interactions. Comparison between model outcomes and measurements to evaluate patient’s parameters. Decisions on setting and alarms based on patient’s parameters e predicted evolution from the model.

Introduction

Considered medical problems and mathematical models: 6) artificial pancreas The goal is to perform an automatic blood glucose control. It requires an estimation of the patient’s needs and his/her conditions, together with a decision model to set the released quantity based on the esimated needs.

Lesson 1 9

Introduction

Considered medical problems and mathematical models: 6) machine learning for decision-making models Machine learning and artificial intelligence are spreading in all applicative fields. We will see one of the applications in medicine, e.g., the implantable bioartificial pancrease. How machine learning can support the design and control

• f these artefacts?

Introduction

Calendar 24 lessons

• f 2 hours

Lesson 1 10

Introduction

Development of students’ project:

• Students divided into groups with at maximum 3 students
• Each group chooses one of the examples discussed in the course or propose a new problem

(which needs to be approved)

• Each group proposes a solution and performs a detailed analysis on the dataset
• Outcomes will be presented in a report and a 15 minutes presentation

Introduction

Evaluation: 1. written exam dealing with all theoretical background and examples discussed in the course, followed by oral exam to discuss the written exam and including other questions 2. presentation of the group project Presentation of the project consists of providing the report and preparing a presentation

• f 15 minutes. This presentation will be made by the students of the group together.

It can be done both before or after the other part. The mark will be registered after both parts are completed. The mark is a weighted average (weight 75% for exam; 25% for project)

Lesson 1 11

Medical decision making: history and perspectives

A quick historical view of medical decision making to highlight the future perspectives

Medical decision making: history and perspectives

Timeline

ANCIENT GRECE In ancient Greek medicine illness was initially regarded as a divine punishment and healing as, quite literally, a gift from the gods. No idea of searching for a cause.

Lesson 1 12

Medical decision making: history and perspectives

Timeline

ANCIENT GRECE By the 5th century BCE, attempts to identify the material causes for illnesses rather than spiritual ones. Greek medical practitioners, began to take a greater interest in the body itself and to explore the connection between cause and effect, the relation of symptoms to the illness itself and the success

• r failure of various treatments.

Medical decision making: history and perspectives

Timeline

MIDDLE AGES Medieval doctors did not have an idea of what caused a disease. Magic motivations not related to observations on previous patients.

Lesson 1 13

Medical decision making: history and perspectives

Timeline

EARLY MODERN AGE First scientific discoveries! For example, William Harvey's discovery of the circulation of the blood in 1628, and Anton van Leeuwenhoek's observation of bacteria in 1683. Connection between observations and health conditions.

Medical decision making: history and perspectives

Timeline

EARLY MODERN AGE Idea of observing patients and taking information from previous observations. Patients may share similar behaviors.

Lesson 1 14

Medical decision making: history and perspectives

Timeline

20th CENTURY Development of diagnostic techniques Development of statistics

First x-ray (1895) Ronald Fisher (1890-1962)

Medical decision making: history and perspectives

Timeline

20th CENTURY Empirical expertise of physician based on diagnostic results and literature data processed with statistical techniques. Decision made by the single physician or a small group of physicians.

Lesson 1 15

Medical decision making: history and perspectives

Timeline

NOWADAYS Big data analysis and advanced mathematical methods allow to model complex medical phenomena. Mathematical models for describing large scale phenomena on entire populations or patient-specific conditions.

Medical decision making: history and perspectives

Timeline

NOWADAYS However, in the practice, mathematical tools are only marginally considered in medical decision making. Decision still made by the single physician or a small group of physicians. If you change physician you may receive a different diagnosis.

Lesson 1 16

Medical decision making: history and perspectives

Timeline

FUTURE CHALLENGES Close the gap between methodologies and clinical practice. “opportunity for industrial and systems engineering and operations research methods to assist medical decision making”

Medical decision making: history and perspectives

Timeline

FUTURE CHALLENGES

• To apply mathematical approaches

for quantitative decision making in the clinical practice.

• To integrate the mathematical models

into tools for automatic evaluation of the patients in real time.

Lesson 1 17

Medical decision making: history and perspectives

Timeline

FUTURE CHALLENGES

• Anyway, medical support systems will not

replace physicians.

• Human knowledge is necessary.

Fundamentals

Mathematical background

Lesson 1 18

Mathematical background

Fundamentals of:

• estimation (stochastic approaches)
• dynamic systems (differential approaches)
• decision (optimization approaches)

Stochastic approaches

Estimation

• Stochastic approaches

Lesson 1 19

Stochastic approaches Questions to ask

Let us assume to have a model characterized by a set of parameters

• How to estimate model parameters from data?
• Are there alternative ways of estimating model

parameters from data?

• How to validate and compare different parameter values?
• How to make statements about estimated values and

assumptions (e.g., model structure)?

Approaches

Some largely adoped approaches we will exploit in the course:

• Moment estimation
• Regression
• Maximum likelihood

estimation

• Bayesian approach

Stochastic approaches Moment estimation

The idea is to use the observations to determine the moments of the underlying parameter distribution (which is assumed to be known). Example 1 The underlying model assumes that observations X1, X2, … come from a normal distribution N(m,s2). The parameters of the density are estimated by fitting the empirical moments:

Empirical moments determine the parameters

• f the distribution. In the example, the normal

distribution is characterized by two parameters. This, the first two empirical moments are used (mean and variance) while including corrections to avoid bias.

Lesson 1 20

Stochastic approaches

Example 2 Now, the model assumes that the observations come from a Gamma distribution. For example, we know that only positive values are possible and we want the density to explicitly respect this constraint. The gamma distribution is parameterised by a shape parameter a and a scale parameter b. The mean of the distribution is a/b and the variance is a/b2. Once again, two parameters in the density require to fit the first two moments.

x

e x x f

b a a

a b b a

 

 

1

) ( ) , ; (

Stochastic approaches

b a b ˆ ˆ ) ( ˆ

2 1 1

X X X X

i i n

  





Estimators (with corrections for bias) are: Example of fitting with a = 4.03 and b = 0.14

Lesson 1 21

Stochastic approaches Regression

We assume a function structure that links two variables X and Y for which we have a set of

• bservations (X1,Y1), (X2,Y2), …
• We assume a structure for the function f: Y = f(X)
• We aim at estimating the coefficients of the function.

Note: X and Y can be either single variables or vectors of variables. Let us consider a linear function in the following...

Stochastic approaches

Function: Y = aX + b Observations: (X1,Y1), (X2,Y2), … For each observation: Yi = aXi + b + ei The best coefficient a is the one that minimizes the overall error (the sum of the squared errors): Minimize { Si ei

2 }

ei

Lesson 1 22

Stochastic approaches

Minimize { Si ei

2 }

in the considered simple linear case becomes Minimize { Si (Yi – aXi – b )2 } Result 

OVERLINE MEANS AVERAGE OVER OBSERVATIONS

Stochastic approaches

The same idea can be applied to more general cases:

• Multiple linear regression with several input variables:
• Nonlinear functions…

Lesson 1 23

Stochastic approaches

How to evaluate the quality of a fitting (the goodness of the coefficients)? Parameter R2 R2  [0,1] values close to 1 are associated with fair estimations (typically values >0.9 denote a good matching)

* VALUE COMPUTED WITH THE REGRESSION FUNCTION RATHER THAN TAKEN FROM THE OBSERVATIONS

Stochastic approaches

Why can R2 be smaller than 1?

NOISY OBSERVATIONS OK – KEEP MODEL NOT RIGHT FUNCTION NO – CHANGE MODEL ONE STRANGE POINT ASSOCIATED WITH A HIGH ERROR OK OR NO DEPENDING ON THE CASE

Lesson 1 24

Stochastic approaches

The same information can be obtained looking at the errors/residuals ei ACCEPTABLE NOISE WITHOUT ANY TREND  GAUSSIAN DISTRIBUTION OF NOISES WITH NULL MEAN

• 30
• 20
• 10

10 20 30 40 50 2 4 6 8 10 12 14 16 18

Stochastic approaches

In case of multiple linear regression with several input variables: How to choose the variables to include (model selection)? AIKAKE INFORMATION CRITERION

• it associates a score for a given combination of included input variables;
• it chooses the combination with the lowest score.

Lesson 1 25

Stochastic approaches Likelihood function

We consider independent and identically distributed observations that come from the same density with unknown parameters (density conditioned on the model parameters). Example with one parameter: X1, X2, … come from a Poisson distribution P(l) Example with two parameters: X1, X2, … come from a normal distribution N(m,s2)

Stochastic approaches

The likelihood is defined by the product of the density function over the i.i.d. observations. In each factor of the product, the value Xi is known. Thus, the likelihood function L only depends on the unknown model parameters. Poisson case:

Proportionality is enough for the estimation purpose

Lesson 1 26

Stochastic approaches

Sometimes, instead of the likelihood function L, its logarithm l is considered: Poisson case: Normal case:

To consider equal instead of proportional is accepted, let’s use this notation

Stochastic approaches

The best parameters (the estimates) are those that maximixe the likelihood function L or, equivalently, the log likelihood function l. Poisson case: Normal case: function minimization with two variables

• null gradient
• Hessian matrix automatically gives a maximum in the point with null gradient

This is called MAXIMUM LIKELIHOOD ESTIMATOR (MLE).

First derivative equal to zero; then the second derivative is automatically negative in the point that makes the first derivative null

Lesson 1 27

Stochastic approaches

MOTIVATION AT THE BASIS OF MLE

• Let consider the 4 observations in the figures.
• For each one 3 densities, compute the likelihood.
• The likelihood assumes the highest value for the

red density, i.e., when the observations are close to the peak of the density.

Stochastic approaches

Interestingly, MLE can be applied when the observations depend

• n other observations.

For example, this is the case of dynamic systems (when each observation of a state variable depends on the value at the previous time instant). Let us consider that variable Y follows a conditional probability density function that depends on the value of another variable X. We know the shape of the density and the relationship between X and Y. For example: Y  N(aX + b, s2)

Corresponding to the example considered for the regression

Lesson 1 28

Stochastic approaches

Y  N(aX + b, s2) The likelihood is the product of the conditional densities evaluated in each observation. For example, in the above case, each term of the product is: and the product (the likelihood) is function of the parameters that link X and Y, i.e., a and b. The maximization is performed with respect to these parameters.

Stochastic approaches

In this case, Yi are not i.i.d. but they they are conditionally to Xi. So it is possible to compute the likelihood as their product. Once again, we have to make assumptions on: shape of the density & relationship between X and Y This approach can be used to estimate the coefficients of differential models based on the observations of the state variables. Let us consider a system characterized by:

Lesson 1 29

Stochastic approaches

where f is a given function whose set of coefficient is . Let us consider that we have observations at equally spaced discrete time intervals Dt. The system can be discretized with time step Dt:

Stochastic approaches

Then, we include an error associated to each observation: A typical choice is to consider a Gaussian White Noise for the errors, i.e., each error follows an independent Gaussian density with null mean value and variance s2. When adopting such noise, the resulting stochastic process is named WIENER PROCESS. Thus, the conditional densities are:

Lesson 1 30

Stochastic approaches

The likelihood is the product of the conditional densities at each observation, which is function of the set of coefficients . Once again, the maximization is performed with respect to these coefficients. Extensions (we will see some of them directly in the applications):

• how to do when the discrete time intervals Dt does not guarantee the convergence of the system

discretization?

• how to do when the time interval at which variables are observed is not constant?
• how to do when, in the presence of several state variables, they are observed at different

frequencies?

Stochastic approaches Bayesian estimation

Based on the Bayes’ theorem CONDITIONED DENSITY: PROBABILITY THAT A OCCURS ONCE WE KNOW B HAS OCCURRED

Lesson 1 31

Stochastic approaches

It is applied considering that A and B represent the data and the model parameters to be estimated:

•  denotes the set of parameters to be estimated
• D denotes the data (set of observations)
• The structure of the model is assumed (density to write the likelihood)

Stochastic approaches

Denominator can be removed as the shape of the posterior density is characterized by proportionality (the missed information can be included as the total probability is 1).

•  denotes the set of parameters to be estimated
• D denotes the data (set of observations)
• The structure of the model is assumed (density to write the likelihood)

Lesson 1 32

Stochastic approaches

With respect to the MLE

ADVANTAGES

• It allows to determine the entire probability

density function of model parameters (with MLE an evaluation of the confidence interval can be obtained with the Fisher information under some assumptions).

• It allows to formalize and integrate the

previous knowledge of the parameters via the prior density (useful especially in the presence of few observations).

DISADVANTAGES

• Analytical expression of the posterior

density available only in the cases; numerical determination via Markov Chain Monte Carlo (MCMC) methods in the

• thers, which can be computationally

expensive.

• Sensible to the prior density. Prior

knowledge has to be properly managed.

Stochastic approaches

As for the MLE, it can be applied when the observations depend on

• ther observations and for dynamic systems (when each observation

depends on the value at the previous time instant). MORE IN GENERAL: when the MLE can be applied, the Bayesian approach can be applied for sure and vice versa.