Life Sciences might be seen as the connections between medicine, biology, mathematics, physics, and computer sciences. Further, optimal control might be defined as the extension of static optimization, or even as some comments, the new face of Variational Calculus.
In this talk we present several examples from life sciences analyzed with the support of optimal control. We might apply basically three approaches to optimal control problems, with their own weaknesses and strengthens: the Pontryagin’s Maximum Principle, Dynamic Programming, or Static Optimization. All the problems treated here were analyzed by the Pontryagin’s Maximum Principle.
The problems are solved using numerical schemes implemented in a computer. Topping the bill, we present two cases from a paper on process of publication: phototherapy for infants affected by neonatal jaundice and Feed-Forward Loop Network. We leave as future works comparisons with other approaches such as dynamic programming, or works on constraints on state space. Furthermore, we have concentrated on continuous-deterministic problems.
Malec, T. & Newman, M. (2013). Research methods Building a kn.docxcroysierkathey
Malec, T. & Newman, M. (2013). Research methods: Building a knowledge base. San Diego, CA: Bridgepoint Education, Inc. ISBN-13: 9781621785743, ISBN-10: 1621785742.
Chapter 5: Experimental Designs – Determining Cause-and-Effect Relationships
hapter 5
Experimental Designs—Determining Cause-and-EffectRelationships
Cosmo Condina/Stone/Getty Images
Chapter Contents
· Experiment Terminology
· Key Features of Experiments
· Experimental Validity
· Experimental Designs
· Analyzing Experiments
· Wrap-Up: Avoiding Error
· Critiquing a Quantitative Study
· Mixed Methods Research Designs
One of the oldest debates within psychology concerns the relative contributions that biology and the environment make in shaping ourthoughts, feelings, and behaviors. Do we become who we are because it is hard-wired into our DNA or in response to early experiences? Dopeople take on their parents’ personality quirks because they carry their parents’ genes or because they grew up in their parents’ homes? Thereare, in fact, several ways to address these types of questions. In fact, a consortium of researchers at the University of Minnesota has spent thepast 2 decades comparing pairs of identical and fraternal twins to tease apart the contributions of genes and environment. You can read moreat the research group’s website, Minnesota Center for Twin and Family Research, http://mctfr.psych.umn.edu/.
Creatas Images/Thinkstock
Researchers at the University ofMinnesota work with twins in order tostudy the impact of genetics versusupbringing on personality traits.
An alternative to using twin pairs to separate genetic and environmental influence is through the use of experimental designs, which have the primary goal of explaining the causes of behavior. Recall fromChapter 2 (Section 2.1, Overview of Research Designs) that experiments can speak to cause and effectbecause the experimenter has control over the environment and is able to manipulate variables. Oneparticularly ingenious example comes from the laboratory of Michael Meaney, a professor of psychiatryand neurology at McGill University, using female rats as experimental subjects (Francis, Dioro, Liu, &Meaney, 1999). Meaney’s research revealed that the parenting ability of female rats can be reliablyclassified based on how attentive they are to their rat pups, as well as how much time they spendgrooming the pups. The question tackled in this study was whether these behaviors were learned fromthe rats’ own mothers or transmitted genetically. To answer this question experimentally, Meaney andcolleagues had to think very carefully about the comparisons they wanted to make. It would have beeninsufficient to simply compare the offspring of good and bad mothers—this approach could notdistinguish between genetic and environmental pathways.
Instead, Meaney decided to use a technique called cross-fostering, or switching rat pups from one mother to another as soon as they wereborn. This resulted in four combinations of rats: (1) thos ...
How long should Offspring Lifespan be in order to obtain a proper exploration?Mario Pavone
The time an offspring should live and remain into
the population in order to evolve and mature is a crucial factor
of the performance of population-based algorithms both in the
search for global optima, and in escaping from the local optima.
Offsprings lifespan influences a correct exploration of the search
space, and a fruitful exploiting of the knowledge learned. In
this research work we present an experimental study on an
immunological-inspired heuristic, called OPT-IA, with the aim
to understand how long must the lifespan of each clone be
to properly explore the solution space. Eleven different types
of age assignment have been considered and studied, for an
overall of 924 experiments, with the main goal to determine
the best one, as well as an efficiency ranking among all the
age assignments. This research work represents a first step
towards the verification if the top 4 age assignments in the
obtained ranking are still valid and suitable on other discrete
and continuous domains, i.e. they continue to be the top 4 even
if in different order.
From simulated model by bio pepa to narrative language through sbmlijctcm
Many theoretical works and tools on epidemiological field reflect the emphasis on decision-making tools
by both public health and the scientific community, which continues to increase.
Indeed, in the epidemiological field, modeling tools are proving a very important way in helping to make
decision. However, the variety, the large volume of data and the nature of epidemics lead us to seek
solutions to alleviate the heavy burden imposed on both experts and developers.
Among the important steps of modeling and simulation: model validation. It refers to the process of
determining how well a model corresponds to the system that it intended to represent. So the question is:
what happens if the model is invalid? Do we need to reproduce another one, or just optimize the existing
one?
Pharmacokinetic/Pharmacodynamic Modeling,Evolutionary Algorithms, and Optima...Jorge Pires
This paper is the formal mixture of three different areas of knowledge, two from applied math and engineering, and one from applied medicine and pharmacology.
The aforementioned areas are: optimal control theory, evolutionary algorithms, and finally pharmacokinetic/pharmacodynamic modeling, PK/PD for short.
We design and test a computer program for solving problems in optimal control theory based on general assumption and apply to a problem in PD/PK modeling.
Although the code is not as fast as the traditional counterpart, which cannot be used to solve the problem presented, the code shows interesting results. We make an analysis of variance of the code.
Dear students we many times problems with Advance research theory application so i am just explain by my PPT slides to help the students and application of theories.
An Hybrid Learning Approach using Particle Intelligence Dynamics and Bacteri...IJMER
The foraging behavior of E. Coli is used for optimization problems. This paper is based on a
hybrid method that combines particle swarm optimization and bacterial foraging (BF) algorithm for
solution of optimization results. We applied this proposed algorithm on different closed loop transfer
functions and the performance of the system using time response for the optimum value of PID
parameters is studied with incorporating PSO method on mutation, crossover, step sizes, and chemotactic
of the bacteria during the foraging. The bacterial foraging particle swarm optimization (BFPSO)
algorithm is applied to tune the PID controller of type 2, 3 and 4 systems with consideration of minimum
peak overshoot and steady state error objective function. The performance of the time response is
evaluated for the designed PID controller as the integral of time weighted squared error. The results
illustrate that the proposed approach is more efficient and provides better results as compared to the
conventional PSO algorithm.
Descriptive model: In this type of model, the purpose is to provide a reasonable description of the data in some appropriate way without any attempt at understanding the underlying aspect, that is, the data-producing device.
Mechanistic model: In the mechanistic model, the importance rests in the knowledge of the device of development, it is important to be able to score on a powerful collaboration among scientists, specialists in the field, and statisticians or mathematicians.
Statistical modeling in pharmaceutical research and developmentPV. Viji
Statistical modeling in pharmaceutical research and development , Statistical Modeling , Descriptive Versus Mechanistic Modeling , Statistical Parameters Estimation , Confidence Regions , Non Linearity at the Optimum , Sensitivity Analysis , Optimal Design , Population Modeling
Malec, T. & Newman, M. (2013). Research methods Building a kn.docxcroysierkathey
Malec, T. & Newman, M. (2013). Research methods: Building a knowledge base. San Diego, CA: Bridgepoint Education, Inc. ISBN-13: 9781621785743, ISBN-10: 1621785742.
Chapter 5: Experimental Designs – Determining Cause-and-Effect Relationships
hapter 5
Experimental Designs—Determining Cause-and-EffectRelationships
Cosmo Condina/Stone/Getty Images
Chapter Contents
· Experiment Terminology
· Key Features of Experiments
· Experimental Validity
· Experimental Designs
· Analyzing Experiments
· Wrap-Up: Avoiding Error
· Critiquing a Quantitative Study
· Mixed Methods Research Designs
One of the oldest debates within psychology concerns the relative contributions that biology and the environment make in shaping ourthoughts, feelings, and behaviors. Do we become who we are because it is hard-wired into our DNA or in response to early experiences? Dopeople take on their parents’ personality quirks because they carry their parents’ genes or because they grew up in their parents’ homes? Thereare, in fact, several ways to address these types of questions. In fact, a consortium of researchers at the University of Minnesota has spent thepast 2 decades comparing pairs of identical and fraternal twins to tease apart the contributions of genes and environment. You can read moreat the research group’s website, Minnesota Center for Twin and Family Research, http://mctfr.psych.umn.edu/.
Creatas Images/Thinkstock
Researchers at the University ofMinnesota work with twins in order tostudy the impact of genetics versusupbringing on personality traits.
An alternative to using twin pairs to separate genetic and environmental influence is through the use of experimental designs, which have the primary goal of explaining the causes of behavior. Recall fromChapter 2 (Section 2.1, Overview of Research Designs) that experiments can speak to cause and effectbecause the experimenter has control over the environment and is able to manipulate variables. Oneparticularly ingenious example comes from the laboratory of Michael Meaney, a professor of psychiatryand neurology at McGill University, using female rats as experimental subjects (Francis, Dioro, Liu, &Meaney, 1999). Meaney’s research revealed that the parenting ability of female rats can be reliablyclassified based on how attentive they are to their rat pups, as well as how much time they spendgrooming the pups. The question tackled in this study was whether these behaviors were learned fromthe rats’ own mothers or transmitted genetically. To answer this question experimentally, Meaney andcolleagues had to think very carefully about the comparisons they wanted to make. It would have beeninsufficient to simply compare the offspring of good and bad mothers—this approach could notdistinguish between genetic and environmental pathways.
Instead, Meaney decided to use a technique called cross-fostering, or switching rat pups from one mother to another as soon as they wereborn. This resulted in four combinations of rats: (1) thos ...
How long should Offspring Lifespan be in order to obtain a proper exploration?Mario Pavone
The time an offspring should live and remain into
the population in order to evolve and mature is a crucial factor
of the performance of population-based algorithms both in the
search for global optima, and in escaping from the local optima.
Offsprings lifespan influences a correct exploration of the search
space, and a fruitful exploiting of the knowledge learned. In
this research work we present an experimental study on an
immunological-inspired heuristic, called OPT-IA, with the aim
to understand how long must the lifespan of each clone be
to properly explore the solution space. Eleven different types
of age assignment have been considered and studied, for an
overall of 924 experiments, with the main goal to determine
the best one, as well as an efficiency ranking among all the
age assignments. This research work represents a first step
towards the verification if the top 4 age assignments in the
obtained ranking are still valid and suitable on other discrete
and continuous domains, i.e. they continue to be the top 4 even
if in different order.
From simulated model by bio pepa to narrative language through sbmlijctcm
Many theoretical works and tools on epidemiological field reflect the emphasis on decision-making tools
by both public health and the scientific community, which continues to increase.
Indeed, in the epidemiological field, modeling tools are proving a very important way in helping to make
decision. However, the variety, the large volume of data and the nature of epidemics lead us to seek
solutions to alleviate the heavy burden imposed on both experts and developers.
Among the important steps of modeling and simulation: model validation. It refers to the process of
determining how well a model corresponds to the system that it intended to represent. So the question is:
what happens if the model is invalid? Do we need to reproduce another one, or just optimize the existing
one?
Pharmacokinetic/Pharmacodynamic Modeling,Evolutionary Algorithms, and Optima...Jorge Pires
This paper is the formal mixture of three different areas of knowledge, two from applied math and engineering, and one from applied medicine and pharmacology.
The aforementioned areas are: optimal control theory, evolutionary algorithms, and finally pharmacokinetic/pharmacodynamic modeling, PK/PD for short.
We design and test a computer program for solving problems in optimal control theory based on general assumption and apply to a problem in PD/PK modeling.
Although the code is not as fast as the traditional counterpart, which cannot be used to solve the problem presented, the code shows interesting results. We make an analysis of variance of the code.
Dear students we many times problems with Advance research theory application so i am just explain by my PPT slides to help the students and application of theories.
An Hybrid Learning Approach using Particle Intelligence Dynamics and Bacteri...IJMER
The foraging behavior of E. Coli is used for optimization problems. This paper is based on a
hybrid method that combines particle swarm optimization and bacterial foraging (BF) algorithm for
solution of optimization results. We applied this proposed algorithm on different closed loop transfer
functions and the performance of the system using time response for the optimum value of PID
parameters is studied with incorporating PSO method on mutation, crossover, step sizes, and chemotactic
of the bacteria during the foraging. The bacterial foraging particle swarm optimization (BFPSO)
algorithm is applied to tune the PID controller of type 2, 3 and 4 systems with consideration of minimum
peak overshoot and steady state error objective function. The performance of the time response is
evaluated for the designed PID controller as the integral of time weighted squared error. The results
illustrate that the proposed approach is more efficient and provides better results as compared to the
conventional PSO algorithm.
Descriptive model: In this type of model, the purpose is to provide a reasonable description of the data in some appropriate way without any attempt at understanding the underlying aspect, that is, the data-producing device.
Mechanistic model: In the mechanistic model, the importance rests in the knowledge of the device of development, it is important to be able to score on a powerful collaboration among scientists, specialists in the field, and statisticians or mathematicians.
Statistical modeling in pharmaceutical research and developmentPV. Viji
Statistical modeling in pharmaceutical research and development , Statistical Modeling , Descriptive Versus Mechanistic Modeling , Statistical Parameters Estimation , Confidence Regions , Non Linearity at the Optimum , Sensitivity Analysis , Optimal Design , Population Modeling
2013.11.14 Big Data Workshop Adam Ralph - 1st set of slidesNUI Galway
Adam Ralph from the Irish Centre for High End Computing presented this Introduction to Basic R during the Big Data Workshop hosted by the Social Sciences Computing Hub at the Whitaker Institute on the 14th November 2013
Models for a Multi-Agent System Based on Wasp-Like Behaviour for Distributed ...infopapers
D. Simian, F. Stoica, C. Simian, Models for a Multi-Agent System Based on Wasp-like Behaviour for Distributed Patients Repartition, Proceedings of the 9th WSEAS International Conference on Evolutionary Computing, Sofia, Bulgaria, ISBN 978-960-6766-58-9, ISSN 1790-5109, pp. 82-86, May 2008
Statistical modeling in pharmaceutical research and development.ANJALI
Statistical modeling in pharmaceutical research and development. This modelling is used in pharmaceutical industries to overcome the challenges related to pharmaceutical formulation, to reduce cost and increase quality and speed of pharmaceutical products.
Brain-Inspired Computation based on Spiking Neural Networks ...Jorge Pires
On this live, prof. Kasabov gives us a gentle overview of Spiking Neural Networks, and their current applications
Full live here, with discussion: https://www.youtube.com/watch?v=niAannUB3pc&t=232s
Have fun 😎😂😁😀
Tutorial: entering a live on StreamYard using a link sent by e-mail, Joining...Jorge Pires
On this brief tutorial, for speakers on this channel, I am going to guide you on how to enter a live, using a link sent by email. You either enter the link on the browser or click on it, on your e-mail.
Alguns insights em startups em healthcareJorge Pires
nos últimos anos, tem havido um crescimento, quase exponencial, dos custos com saúde. Como consequência, formas diversas têm sido propostas e implementadas para lidar com as demandas na área de saúde. Nos tempos atuais, estas demandas envolvem tratamentos cada vez mais eficientes, personalizados, acessíveis e de baixo custo. Diante deste cenário, pode-se dizer que as chamadas startups são a grande promessa para lidar com iniciativas, como implementação de novas ideias e produtos. As startups trazem novas formas de pensar e produzir, com um grau de liberdade não presente em grandes empresas. Pretende-se discutir espaços para startups na área de saúde, em diálogo com literaturas disponíveis na internet. O objetivo deste trabalho é evidenciar como as startups podem contribuir para a medicina personalizada alcançar seus objetivos, healthcare, em geral, superando o dilema da ‘personalização vs. custo’. A motivação é que apesar do desenvolvido das universidades, muitos países, sendo o Brasil um desses, não possuem um fluxo contínuo e corriqueiro de transformação de ciência em bens para a sociedade. Conclui-se que as startups se apresentam como a melhor opção para resolver/lidar com vários problemas/dilemas que surgiram nas áreas médicas, nas últimas décadas, como aumento dos custos e demanda cada vez maior por tratamentos especializados/personalidades.
Mathematical modeling in energy homeostasis, appetite control and food intake...Jorge Pires
The elegant ‘interconnected mechanisms’ by which the gastrointestinal (GI) tract regulates food intake are a marvel of biology, but the redundancy (e.g., several hormones seem to have effects in food intake) of both GI (by means of hormones) and central nervous system (CNS, by means of satiety/satiation signals) pathways governing energy homeostasis poses formidable challenges for scientists trying to take a clear glimpse of this machinery, e.g. for designing anti-obesity and alike pharmaceuticals.
The current work is divided into three parts: part I is regarding fundamentals of physiology and mathematical modeling employed all over the work; part II is more generic and concerns several hormones (what we have called a “web of hormones”) and part III (divided into three chapters) is more specific, concerning a single hormone (i.e., ghrelin). The core of the work is part III, and to a certain extent part II, bearing mind we provide a literature review based on papers scattered/dispersed all over the medical science literature.
The main objective of this work is proposing a mathematical model for ghrelin dynamics (Figure 70), a model centered on the gastrointestinal tract (stomach + small intestine, a two-compartment model), with daily-like dynamics, short-term dynamics; and, simultaneously, proposing a prototype for a systems biology like model (Figure 40), a model based on numerous hormones, for understanding mathematically food intake/bodyweight control.
We test several optimization routines for the parameter estimation procedure, hybrid algorithms (global + local search), for parameter estimation, based on data published for humans (three meals a day). For all the routines, the best is a hybrid composed of simulating annealing as global search and pattern search as local search. In the objective function (sum of the squared errors, SSE), we apply artificial neural networks (a two-layer feedforward neural network) for generating new data from the data already published, a strategy adopted to increase the data set. In the last part of the chapter about ghrelin modeling (part III), we propose several prototypes for future works based on the basal models; the model used for parameter estimation is a “minimal/reduced” model; we also provide discussions and future works for the minimal model and parameter estimation.
Key-words. Ghrelin; leptin; mathematical modelling; food intake; appetite; parameter estimation.
Entre as maiores revoluções do século vinte jaz a mutação de como se vê e modela a realidade que nos rodeia. O escopo do trabalho é chamar a atenção de cientistas em geral para a área de modelagem estocástica. A motivação é a aparente falta de interesse pela modelagem estocástica por cientistas de algumas áreas. Umas das maiores questões não-respondidas do século vinte - que continua até os tempos atuais - com raízes no século dezoito e dezenove, encontrar-se na natureza da nossa realidade. Determinística ou estocástica? Modelos estocásticos já é parte da grande área chamada Pesquisa Operacional. Evolução significa em todos os sentidos. "Deus não joga dados". Em biologia Deus joga dados "e nos chama para jogar juntos". A diferença entre mecânica quântica e biologia reside no fato de que em mecânica quântica, teoricamente, mesmo que se aumente a precisão de equipamentos, o princípio da incerteza, de Heisenberg, no diz que ainda assim teríamos incertezas; ao passo que em biologia essa incerteza nasce da nossa "incompetência" como cientistas. Estudos em probabilidade, que culminaram em tanto na teoria estocástica atual como modelou nossa forma de "ver" o mundo, matematicamente, começou com o cientista italiano Gerolamo Cardano no século dezesseis. Durante o século dezoito, probabilidade começa notavelmente a mudar de configuração, o que culminou na nossa visão contemporâneo de aleatório. Existe a necessidade de haver um cálculo estocástico. Para algumas aplicações, previsibilidade se tornou uma relíquia do passado. O sucesso de cada modelagem, ou seja, determinística ou estocástica, depende do quão a componente estocástica contribui para o processo como um tudo. Modelagens estocásticas abrolham tanto da inépcia temporária e espacial de entender o processo posto diante de nós como pendência teórica.
Fisiologia Matemática, Biologia Matemática, e Biomatemática (leptina e a busc...Jorge Pires
Este material foi produzido após o artigo “Fisiologia Matemática, Biologia Matemática, e Biomatemática: leptina e a busca pelo controle de peso” ter sido aceito para publicação, sendo assim o material não passou por uma banca de professores (especialistas).
Posto deste modo, é de total responsabilidade do autor, Jorge Guerra Pires, qualquer problema que possa surgir como consequência desse material adicional.
O artigo, publicado pela Revista Eletrônica Gestão e Saúde (ISSN 1982-4785), deve ser sempre justaposto como material a ser referido formalmente. Este material é meramente para fins educativos.
On the applicability of computational intelligence in transcription network m...Jorge Pires
Gene expression is a quite important tool for mathematical modeling, to understand genetic networks. On the other hand, computational intelligence is a quite important and powerful set of tools, developed mainly on the last decades. On this set of slides we discuss systems biology, computational intelligence and gene expression.
Ghrelin mathematical modeling and beyond (The big glucose model: the quest fo...Jorge Pires
This is a set of slides used on my talk about ghrelin mathematical modeling. Ghrelin is a hormone produced by the stomach and other parts of the body, it has been shown to be correlated with several physiological functions; herein we exploit the orexigenic ones (i.e. appetite stimulant).
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Digital Artifact 2 - Investigating Pavilion Designs
Optimal Control applied to life sciences: a numerical method based presentation
1. Optimal Control
applied to
A numerical method based presentation
Talk under the framework of the project “Stochastic models in medicine and life
science: mathematical analysis, model identification, validation and stability
properties” sponsored by CAPES Foundation, Ministry of Education of
Brazil.
Jorge Guerra Pires
Information Engineering and Science
PhD program
University of L’Aquila/
Institute of Systems Analysis and
Computer Science
jorgeguerrapires@yahoo.com.br
Italy, 2014
2. June/2014 Jorge G Pires 2
Optimal Control applied to
Abstract: Optimal Control Applied to Life Sciences
Life Sciences might be seen as the connections between medicine, biology,
mathematics, physics, and computer sciences. Further, optimal control might be
defined as the extension of static optimization, or even as some comments, the new
face of Variational Calculus.
In this talk we present several examples from life sciences analyzed with the support
of optimal control. We might apply basically three approaches to optimal control
problems, with their own weaknesses and strengthens: the Pontryagin’s Maximum
Principle, Dynamic Programming, or Static Optimization. All the problems treated
here were analyzed by the Pontryagin’s Maximum Principle.
The problems are solved using numerical schemes implemented in a computer.
Topping the bill, we present two cases from a paper on process of publication:
phototherapy for infants affected by neonatal jaundice and Feed-Forward Loop
Network. We leave as future works comparisons with other approaches such as
dynamic programming, or works on constraints on state space. Furthermore, we
have concentrated on continuous-deterministic problems.
Keywords: Life Sciences, applied optimal control, numerical schemes, Runge-Kutta
Method, forward-backward sweep method.
3. June/2014 Jorge G Pires 3
Optimal Control applied to
Cases Considered throughout the endeavor
Toy models;
Mold and Fungicide;
Bacteria ;
Tasmania Devil facial tumour disease;
Optimal production of Protein;
Drug Administration in one-compartment model;
Applied Optimal Control Theory in Phototherapy of Infants;
Cancer;
4. June/2014 Jorge G Pires 4
Optimal Control applied to
Cases Considered throughout the endeavor
Bioreactors;
Predator-Prey Model;
Discrete Time Models;
Cancer Therapy with Gompertzian Growth and one-compartment
model;
Fish Harvesting;
Epidemic Model;
HIV Treatment;
Bear Populations: metapopulation;
Glucose Model;
Timber Harvesting;
5. June/2014 Jorge G Pires 5
Optimal Control applied to
Cases Reported
Cancer Therapy with Gompertzian Growth and one-compartment
model;
Fish Harvesting; (Excluded, but interesting)
Epidemic Model;
HIV Treatment;
Bear Populations: metapopulation; (Excluded, but interesting)
Optimal production of Protein;
Drug Administration in one-compartment model; (Excluded, but
interesting)
Applied Optimal Control Theory in Phototherapy of Infants;
6. June/2014 Jorge G Pires 6
Optimal Control applied to
1. Introduction
A straightforward definition of life sciences is no longer simple. It might
be said that in the past, this scientific domain comprised of a set of
united field such as medicine and biology that rarely interfered with each
other; nonetheless, in the present it is a “unique” branch comprised of
researches from a variety of field such as mathematics, medicine, and
biology.
The inclusion of mathematics and other fields such as computer science
(and information sciences) came as a “rebirth” of the field.
8. June/2014 Jorge G Pires 8
Optimal Control applied to
1. Introduction
Optimal control might be defined as a sub-area of control system in which
we look for the best policy (control) over a period of time. It was born in
the 50s from concerns on the aerospace industry, instead of optimizing a
finite set of variables, today named static optimization, they wanted to
optimization a dynamical system behavior, dynamic optimization.
In the majority of applications we take as granted important properties of
the system such as controllability.
The similarities are such as the fact that we can show necessary and
sufficient conditions based upon the same mathematical framework.
10. June/2014 Jorge G Pires 10
Optimal Control applied to
1. Introduction
Each box is defined as:
The tradeoff function represents the balance between doing something
and something else, such as letting the system evolves on its own
dynamics or control. In general, this is a integral over the whole period of
optimization and a payoff function for the final state;
The dynamical system represents the real system itself, using state
variables and state equations;
Further details comprise further details such as bounds on control or
even maximum total amount of the same all of the period.
In the scheme it was not represented a arrow between the further details’
box and the tradeoff’s once the demands on the tradeoff are in general
mathematically demanded such as limited tradeoff function on the optimal
policy and pathway.
11. June/2014 Jorge G Pires 11
Optimal Control applied to
1. Introduction
12. June/2014 Jorge G Pires 12
Optimal Control applied to
1. Introduction
13. June/2014 Jorge G Pires 13
Optimal Control applied to
1. Introduction
14. June/2014 Jorge G Pires 14
Optimal Control applied to
2. Ordinary Differential Equation and Systems
The term dynamic refers to phenomena that produce time-changing
patterns, the characteristics of the pattern at one time being interrelated
with those at other times. The term is nearly synonymous with time-
evolution or pattern of change. It refers to the unfolding of events in a
continuing evolutionary process. The term system was originated as a
recognition that meaningful investigation of a particular phenomenon can
often only be achieved by explicitly accounting for its environment [7].
Dynamical system might be defined as a single or set of “systems” that
evolve in time, in the case of more than one, they are in general coupled,
for example in social networks. The most common methodology applied to
model those systems are differential equations. It is not clearly defined, but
other areas such as biology work sometimes with their own methodology for
modeling dynamical system. We might say that one of the challenge of
complex networks theory is to unify those field, mainly using dynamical
systems theory.
15. June/2014 Jorge G Pires 15
Optimal Control applied to
2. Ordinary Differential Equation and Systems
Several issues are treated in the theory of dynamical systems, the most
common are bifurcations, stability, and modeling. Bifurcation is the study of
quantitative behavior of a system close to some special values of the
parameters of the model, called bifurcation parameters. Basically several
systems might “change” their dynamical close to several parameterizations.
Stability is a quite important matter, it is related to the fact that a system can
be kept on a particular state, or at least close to it, called equilibrium points.
Modeling is related to the “ability” of a system to represent real plants, this
is extremely important in control theory, once the study is done on the
model, then it must be applied to the real plant, if the model is too far from
the real plant, quite strange behavior will come up.
Predominantly, we have continuous and discrete systems. They differ on
the tools, for continuous systems, we apply differential equations, whereas
for discrete we apply difference equations. Example of discrete systems is
birth and death modeling, and continuous is density modeling.
16. June/2014 Jorge G Pires 16
Optimal Control applied to
2. Ordinary Differential Equation and Systems: gene expression
dynamics
Source: Simulations for BRICS-CCI Brazil 2013
17. June/2014 Jorge G Pires 17
Optimal Control applied to
2. Ordinary Differential Equation and Systems: gene expression
dynamics
Source: Simulations for BRICS-CCI Brazil 2013
18. June/2014 Jorge G Pires 18
Optimal Control applied to
2. Ordinary Differential Equation and Systems: gene expression
dynamics
Source: Simulations for BRICS-CCI Brazil 2013
19. June/2014 Jorge G Pires 19
Optimal Control applied to
2. Ordinary Differential Equation and Systems: gene expression
dynamics
Source: Simulations for BRICS-CCI Brazil 2013
“Type-1 FFL is a sign-sensitive delay
element that can protect against
unwanted responses to fluctuating
inputs. The magnitude of the delay in
the FFL can be tuned over
evolutionary timescales by varying the
biochemical parameters of regulator
protein y [middle gene], such as its
lifetime, maximum expression, and
activation threshold ” Alon (2007,
p.57).
20. June/2014 Jorge G Pires 20
Optimal Control applied to
3. Numerical Schemes
Numerical Scheme can be defined as an human-written sequence of orders
(algorithm) with the purpose to solve problems based upon numbers.
Examples go from a simple root finder scheme to a more complicate scheme for
teaching networks (learning machine), called learning paradigms.
A quite famous application of numerical schemes came with Gauss, the Mean
Squared Error (MSE), when he predicted the trajectory of an unknown planet based
upon observations, this is today the base for numerical schemes, from optimal
control to learning machine and regression.
The second famous application came with Feynman and Collaborators, when they
have been surprised by the result of a numerical scheme, this had confirmed the
corpuscular nature of heat, no chaotic.
It is said that Hodgkin–Huxley had studied their famous model on dynamics of
neuron using a hand-like calculator, the model is based on dynamical systems.
21. June/2014 Jorge G Pires 21
Optimal Control applied to
3. Numerical Schemes
Nowadays we have an almost unanimous consent regarding the importance of
numerical methods; nonetheless some still “resist” such as an old professor of mine
from Russia.
If we compare the simple method of Guass to the ones used today, we get
somehow surprised, such as the studies of an old professor of mine from Russia,
he was modeling the sun’s dynamics by numerical schemes.
Besides we always make use of computer for numerical schemes, the majority of
them, even the new ones are based upon those, were developed before the
computer could have been even dreamt of. The consequence is that we need to
improvise, see the variations of the famous method of Newton. Maybe this time for
us develop our own schemes!
It is rare a case where we have a single numerical scheme for solving a problem,
the variations go around the aim “cost-simplicity-accuracy”. The choice in the
majority of cases is a matter of schools or preferences.
22. June/2014 Jorge G Pires 22
Optimal Control applied to
3. Numerical Schemes: Euler Method or Tangent Line Method
23. June/2014 Jorge G Pires 23
Optimal Control applied to
3. Numerical Schemes: Euler Method or Tangent Line Method
24. June/2014 Jorge G Pires 24
Optimal Control applied to
3. Numerical Schemes: Euler Method or Tangent Line Method
25. June/2014 Jorge G Pires 25
Optimal Control applied to
3. Numerical Schemes: Euler Method or Tangent Line Method
26. June/2014 Jorge G Pires 26
Optimal Control applied to
3. Numerical Schemes: Euler Method or Tangent Line Method
27. June/2014 Jorge G Pires 27
Optimal Control applied to
3. Numerical Schemes: The Runge-Kutta Method
28. June/2014 Jorge G Pires 28
Optimal Control applied to
3. Numerical Schemes: The Runge-Kutta Method
29. June/2014 Jorge G Pires 29
Optimal Control applied to
3. Numerical Schemes: The Runge-Kutta Method
30. June/2014 Jorge G Pires 30
Optimal Control applied to
3. Numerical Schemes: Error
31. June/2014 Jorge G Pires 31
Optimal Control applied to
3. Numerical Schemes: Shooting Method
32. June/2014 Jorge G Pires 32
Optimal Control applied to
3. Numerical Schemes: Shooting Method
33. June/2014 Jorge G Pires 33
Optimal Control applied to
3. Numerical Schemes: The Newton’s Method
34. June/2014 Jorge G Pires 34
Optimal Control applied to
3. Numerical Schemes: The Secant’s Method
35. June/2014 Jorge G Pires 35
Optimal Control applied to
4. Numerical Solutions for optimal control: algorithms
All the numerical schemes presented here are result of the application of
the necessary conditions based on the theory of Hamiltonian, as a result
of applying the Pontryagin’s Maximum Principle. The necessary
conditions used here are those presented in [11].
All the derivations will be presented before any coding is discussed
(omitted). In order to try to simplify the presentations, we will organize the
problems into prototypes.
The prototypes suppose to serve as a reference model. We have chosen
to present those in prototypes once they will demand different schemes
for each of them.
The good news is that they all might be done with an extension of single
code for the prototype 1: Forward-Backward Sweep Method, for short
FBSM.
36. June/2014 Jorge G Pires 36
Optimal Control applied to
4. Numerical Solutions for optimal control: algorithms
All the models treated follow a common pattern given by the following
picture (next slide).
The drawback of this approach is the red arrow: one must first transform
the optimal control problem into a system of differential equation, this is
hand-calculation, and for complicate system it might be cumbersome.
One potential approach is directly integrating the system. One possible
way of doing that is integrating the Hamiltonian directly, but it might
introduce further error on the approximations of the derivatives for the
optimality condition and the adjoint equation.
Furthermore, it might be complicate once we are demanded to optimize
the Hamiltonian function.
37. June/2014 Jorge G Pires 37
Optimal Control applied to
4. Numerical Solutions for optimal control: algorithms
38. June/2014 Jorge G Pires 38
Optimal Control applied to
4. Numerical Solutions for optimal control: algorithms
This is the simplest problem we can formulate in optimal control
that could be applied to real problem. As we will see, this simple approach
sometimes might not be appropriate and further details should be added,
this is the next prototype.
39. June/2014 Jorge G Pires 39
Optimal Control applied to
4. Numerical Solutions for optimal control: algorithms
40. June/2014 Jorge G Pires 40
Optimal Control applied to
4. Numerical Solutions
for optimal control:
The Forward-Backward
Sweep Method
41. June/2014 Jorge G Pires 41
Optimal Control applied to
4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method
42. June/2014 Jorge G Pires 42
Optimal Control applied to
4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method
43. June/2014 Jorge G Pires 43
Optimal Control applied to
4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method for Prototype 4
44. June/2014 Jorge G Pires 44
Optimal Control applied to
4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method
45. June/2014 Jorge G Pires 45
Optimal Control applied to
4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method for prototype 5
46. June/2014 Jorge G Pires 46
Optimal Control applied to
4. Numerical Solutions for optimal control: The Forward-Backward
Sweep Method
47. June/2014 Jorge G Pires 47
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
48. June/2014 Jorge G Pires 48
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
The case study discussed here is the treatment by phototherapy for
newborns affected the syndrome called Neonatal Jaundice. The
neonatal jaundice is caused by the immaturity of the liver, turning
impossible the conjugation of byproduct of the natural process of
hemoglobin breaking down that happens after birth, therefore the
elimination of the undesired wastes, which are toxic, see for example
Pires et al (2009) and references therein.
According to (Pires et al, 2009; Schoof et al 2012), and further
references, jaundice might be seen as the discoloration of skin, mucous,
and sclera due to the accumulation of bilirubin, a waste of the
hemoglobin breaking down, they turn the skin, mucous, and sclera
yellowish.
49. June/2014 Jorge G Pires 49
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
We know from studies with light properties that frequency is what
matters, what differentiate the colors, the properties of x-ray compared to
gamma-rays. We also know that the visible spectrum is relatively small,
something from 200 nm to 700 nm in wavelength.
The sunlight is the most complete illumination system, containing from
ultraviolet to more rare wavelengths. Nonetheless, we must recollect that
what heats the earth is present in the sunlight, namely, infrared, or even
the so-feared ultraviolet. Therefore, side effects might turn it impossible
to use it depending on the phototherapy intensity demanded.
In the case of the neonatal jaundice, the wavelength corresponding to
the “blue” is the demanded one. Thus, a good system of illumination for
this phototherapy must “possess the blue spectrum in abundance.”
50. June/2014 Jorge G Pires 50
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
51. June/2014 Jorge G Pires 51
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
52. June/2014 Jorge G Pires 52
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
Before going on, some remarks might be useful. In Pires et al
(2009), it is not discussed on the real correlation between the sensitivity
of the bilirubin and the polymeric solution to the radiation, just taken as
granted the correlation, based on theory and independent experiments.
In the case of the bilirubin, we have a circulatory system, that is, just the
portion of the blood in the skin will be exposed to the blue light, whereas
in the polymeric solution, the expose is higher. Nonetheless, this type of
issue had not limited the application of optimal control.
In Lenhart e Workman (2007), this is presented a bear-control policy
designed by optimal control. In Lenhart e Workman (2007), it has been
presented a control of bear over regions sharing boundaries, called
metapopulations. The control of a region affects the other by the dynamic
on the boundaries.
53. June/2014 Jorge G Pires 53
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
Simulations for the neonatal phototherapy. The green curve depicts a “bad case” of the syndrome, that is, the body of the
newborn eliminate the bilirubin in a low rate compared to the “production” rate; the red curve represents an arbitrary case, but
better than the green curve; the blue curve represents the case in which the degree of importance to diminish the concentration
of bilirubin is increased by 10 compared to the risks of the phototherapy. We simulate an “intermittent phototherapy;” we apply
the therapy for 1.5 units of time and then stop for another 1.5 units of time. We have used 10 units of concentrations as initial
state. The dashed rectangle in red is the therapeutic window. Source: own elaboration.
54. June/2014 Jorge G Pires 54
Optimal Control applied to
5. Optimal Phototherapy Regimen: the neonatal jaundice case
Simulations for the phototherapy in newborns using a drug X in junction. The blue curve represents the
simulations using just the phototherapy, whereas the red is the simultaneous use of phototherapy and the drug
X. The dashed line represents the goal (superior limit). We apply the treatment for 1.u units of time. Source:
own elaboration.
55. June/2014 Jorge G Pires 55
Optimal Control applied to
6. Optimal Protein Production: the feed-forward loop network motif
56. June/2014 Jorge G Pires 56
Optimal Control applied to
6. Optimal Protein Production: the feed-forward loop network motif
The topic treated here is quite intriguing or even provocative, and indeed
the treatment herein overlooks several important topics.
For instance, from an engineering viewpoint, one might think of a
bioreactor and we aim to optimize the workings the same; but from a
biological viewpoint, we might want to understand how a certain gene or
even its circuit was selected under evolution.
This topic has been exploited in the literature, see for instance Alon
(2007) for several discussion from a different angle. For the mathematical
model used here, see Cacace et al (2012), or even Pires (2012), for more
details. For discussion on potential application of this topic in engineering,
see for example Pires and Palumbo (2012), or as alternative Pires
(2013).
57. June/2014 Jorge G Pires 57
Optimal Control applied to
6. Optimal Protein Production: the feed-forward loop network motif
On this part of the paper, we discuss on the optimal control of protein
production. We suppose that we are lucky enough to reveal the secrets of
god, and we find a function that is always optimized in the protein production
process. See that Alon (2007) comments that one difficulty in optimization
theories, such as the one used here, is that we may not know the fitness
function in the real world. Let’s neglect this issue.
In simple terms, we have three genes. Further, just one produces what we
need, call it Z, but surprisingly, it cannot be controlled directly, a second gene
must be activated for that, call it X. Thus, the more we have activated X, the
more we will have producing, but something starts to go wrong, then we
identify a third gene, call it Y. We find out that Y is activated by X, but it
inhibits Z, our goal. Then we must activate X as much as possible, but keep Y
as silent as possible. It must have an equilibrium point. The communication
between genes is done by a special group of proteins called transcription
factors, something like papers in a company, functionless, but necessary.
What we have described is called a network motif, namely, it is a feed-
forward loop network motif, an incoherent type I.
58. June/2014 Jorge G Pires 58
Optimal Control applied to
6. Optimal Protein Production: the feed-forward loop network motif
59. June/2014 Jorge G Pires 59
Optimal Control applied to
6. Optimal Protein Production: the feed-forward loop network motif
Controls for different situations, parameterizations: protein degradation rate, mRNA degradation rate, protein
production rate, mRNA production rate, degree of importance to produce protein, degree of importance for
minimizing the control, and threshold for the Hill functions “K” on the mathematical model. The lines on
continuous style are different values for the parameters, on the dashed lines we have increased by 10 the
importance of producing proteins compared to the cost, the cost is applied just on gene X, input gene. The
green function depicts what happens if we add the final value of protein to be optimized (a payoff term). The
initial condition for all genes are ‘0’, but gene X which is given a small amount of mRNA in time ‘0’. Source:
own elaboration.
60. June/2014 Jorge G Pires 60
Optimal Control applied to
7. Epidemic: optimal policy
61. June/2014 Jorge G Pires 61
Optimal Control applied to
7. Epidemic: optimal policy
Epidemics can be defined as the breakout of an infection disease.
Related terms are pandemic (world-spread-out) and endemic (steady-
state).
Epidemics is in general a complex network issue.
Source: http://1.bp.blogspot.com/-WSrc1yadP2U/TzFs3dHA-
TI/AAAAAAAABuo/bVbX1lgTnd8/s1600/epidemic_diffusion.jpg
62. June/2014 Jorge G Pires 62
Optimal Control applied to
7. Epidemic: optimal policy
63. June/2014 Jorge G Pires 63
Optimal Control applied to
7. Epidemic: optimal policy
64. June/2014 Jorge G Pires 64
Optimal Control applied to
7. Epidemic: optimal policy
65. June/2014 Jorge G Pires 65
Optimal Control applied to
7. Epidemic: optimal policy
66. June/2014 Jorge G Pires 66
Optimal Control applied to
7. Epidemic: optimal policy
67. June/2014 Jorge G Pires 67
Optimal Control applied to
7. Epidemic: optimal policy
68. June/2014 Jorge G Pires 68
Optimal Control applied to
7. Epidemic: optimal policy
69. June/2014 Jorge G Pires 69
Optimal Control applied to
8. HIV and AIDS
70. June/2014 Jorge G Pires 70
Optimal Control applied to
8. HIV and AIDS
Acquired immunodeficiency syndrome (AIDS) is medically devastating to
its victims, and wreaks financial and emotional havoc on everyone, infected
or not.
Keywords: Viral Replication; Immunology.
“Viruses are very small biological structures whose reproduction requires a
host cell.” They are not considered living things.
Helper T-lymphocytes play a key role in the process of gaining immunity to
specific pathogens.
HIV is an especially versatile virus. It not only inserts its genetic information
into its host’s chromosomes, but it then causes the host to produce new
HIV.
A virus cannot reproduce outside a host cell, which must provide viral
building materials and energy.
71. June/2014 Jorge G Pires 71
Optimal Control applied to
8. HIV and AIDS
The immune system monitors the body for dangerous pathogens.
When it detects pathogens, the immune system computers and
mobilizes the appropriate responses.
The immune system is made of a vast collection of cells that
communicate and interact in myriad ways.
One of the major tools of the immune system is antibodies.
One of the important roles of the immune system is to scan the cells of
the body for antigens – foreign proteins made by pathogens –.
The scanning process is carried out by T-cells.
72. June/2014 Jorge G Pires 72
Optimal Control applied to
8. HIV and AIDS
73. June/2014 Jorge G Pires 73
Optimal Control applied to
8. HIV and AIDS
Viral nucleic acid enters the host cell and redirects the host cell’s
metabolic apparatus to make new viruses.
“Many RNA viruses do not use DNA in any part of their life cycle.”
Living Systems
and some virus
Retrovirus
74. June/2014 Jorge G Pires 74
Optimal Control applied to
8. HIV and AIDS
75. June/2014 Jorge G Pires 75
Optimal Control applied to
8. HIV and AIDS
Key-point 1 : To find an optimal chemotherapy strategy in the treatment
of the human immunodeficiency virus (HIV).
Key-point 2: The model used herein describes the interaction of the
immune system with HIV.
Key-point 3: It is assumed the treatment acts to reduce the infectivity of
the virus for a finite time.
Antiretroviral drugs are medications for the treatment of infection by
retroviruses, primarily HIV. Different classes of antiretroviral drugs act on
different stages of the HIV life cycle. Combination of several (typically
three or four) antiretroviral drugs is known as highly active anti-retroviral
therapy (HAART).
76. June/2014 Jorge G Pires 76
Optimal Control applied to
8. HIV and AIDS
Each T cell attacks a foreign substance which it identifies with
its receptor. T cells have receptors which are generated by randomly
shuffling gene segments. Each T cell attacks a different antigen. T cells
that attack the body's own proteins are eliminated in the thymus. Thymic
epithelial cells express major proteins from elsewhere in the body. First, T
cells undergo "Positive Selection" whereby the cell comes in contact
with self-MHC expressed by thymic epithelial cells; those with no
interaction are destroyed. Second, the T cell undergoes "Negative
Selection" by interacting with thymic dendritic cell whereby T cells with
high affinity interaction are eliminated through apoptosis (to avoid
autoimmunity), and those with intermediate affinity survive.
77. June/2014 Jorge G Pires 77
Optimal Control applied to
8. HIV and AIDS
78. June/2014 Jorge G Pires 78
Optimal Control applied to
8. HIV and AIDS
79. June/2014 Jorge G Pires 79
Optimal Control applied to
8. HIV and AIDS
80. June/2014 Jorge G Pires 80
Optimal Control applied to
8. HIV and AIDS
81. June/2014 Jorge G Pires 81
Optimal Control applied to
9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
Tumor is a state of the body where cells divide (mitosis, multiply) on an
uncoordinated way. This is a type of cancer in some cases. Tumors might
be classified as benign, premalignant, or malignant (cancer). Cancer is so
feared for spreading out, invading neighboring tissues, tumors (premalignant
cancer) does not invade neighboring tissues.
On this section we present the simplest model possible to build applying the
theories presented so far.
The model presented herein is relatively simple. We apply the theory of
optimal control for the treatment of cancer. Using the mentioned theory, we
can optimize the behavior of a set of differential equations, on our case,
ordinary differential equations.
The model discussed here was published by Fister e Panetta (2003) and
studied by (Lenhart e Workman 2007; Swan, 1984); we try to make it better
by adding the dynamic of the drug.
82. June/2014 Jorge G Pires 82
Optimal Control applied to
9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
83. June/2014 Jorge G Pires 83
Optimal Control applied to
9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
See that the first term of the differential equation represents a ‘growth,’ a
natural process, using a dynamics called Gompertz model, and the
second represent our control by means of the treatment.
It should be pointed out that we can use just this equation on the study of
optimal control applied to tumor therapy and this is what is done on
Lenhart e Workman (2007), but we can do better!
What about the dynamics of the drug? This is what we add here. Drugs
might exhibit peculiar behavior and assuming that we can control the
amount exactly of drug that reaches the site might be a mistake.
As Lenhart e Workman (2007) highlights studies of drugs is a rich and
state of the art field, and one of the models lacking are the ones that
studies multiple drugs taken at the same time. That is, models that takes
into account interactions between different drugs; see that the model
presented here is limited in the sense that it is still kinetics, not dynamics,
models for dynamics are much complicate, we skip them.
84. June/2014 Jorge G Pires 84
Optimal Control applied to
9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
Just to provoke these studies on the literature, let’s consider an ideal case of two
drugs taken for eliminating this tumor.
We consider a simple case, the second drug just increase the absorption of the
first, the drug one is the one that really can eliminate the tumor.
This can increase the possibility to maintain the plasma concentration within the
therapeutic window; see that the second drug is increasing the absorption, then it
should eventually increase the concentration above the therapeutic window of the
drug we need to monitor, and this exactly the trick of optimal control, the optimal
control policy will just use what is needed given a goal (therapeutic window).
This is known as bioavailability; this is similar when you are suggested to take a
drug together with milk, milk is not a drug, just increase the bioavailability of the
take drug, or even when you are asked to use a drug with empty stomach. This
type of analysis is concern of noncompartmental models.
Noncompartmental models are considered easier to use due to several reasons
such as it does not change from individual to individual so much as compartment
models. See Rosenbaum (2011) for more details.
85. June/2014 Jorge G Pires 85
Optimal Control applied to
9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
86. June/2014 Jorge G Pires 86
Optimal Control applied to
9. Optimal Tumor Treatment : Cancer Therapy with Gompertzian
Growth and one-compartment model
87. June/2014 Jorge G Pires 87
Optimal Control applied to
References
[1] George W Swan, Applications of optimal control theory in biomedicine, Pure and
Applied Mathematics. Marcel Dekker Inc, 1984.
[2] Wendell H Fleming; Raymond W Rishel. Deterministic and Stochastic Optimal
Control. Applications of Mathematics 1, Springer-Verlag: 1975.
[3] Boyce, William E.; Diprima, Richard C. Elementary differential equations and
boundary value problems seventh edition. John Wiley & Sons, Inc.: 2001.
[4] Betts, J. T. Practical Methods for optimal Control and Estimation Using Nonlinear
Programming. Second Edition. Advances in Design and Control, Society for Industrial
and Applied Mathematics. 2010.
[5] Wikipedia. The Secant Method Online: http://en.wikipedia.org/wiki/Secant_method.
Accessed on May/2014.
[6] Wikipedia. The Newton’s Method Online: http://en.wikipedia.org/wiki/Newton
%27s_method. Accessed on May/2014.
[7] David G. Luenberger, introduction to dynamic systems: theory, models, and
application. John Wiley & Sons, 1979.
[8] Lynch, Stephen. Dynamical systems with applications using Mathematica®.
Birkhäuser Boston, 2007.
[9] Wikipedia. Logistic Map http://en.wikipedia.org/wiki/Logistic_map. Accessed on
May/2014.
88. June/2014 Jorge G Pires 88
Optimal Control applied to
References
[10] Wikipedia. Tent Map http://en.wikipedia.org/wiki/Tent_map. Accessed on May/2014.
[11] Lenhart, S.; Workman, J.T, Optimal Control Applied to biological models, Chapman
& Hall/ CRC, Mathematical and Computational Biology Series, 2007.
[12] Press, William H. ;Teukilsky, Saul A. ;Vetterling, William T. ;Flannerg, Brian P.
Numerical recipes in C: the art of scientific computing. Second edition. Cambridge
University press: 1992.
[13] Ruggiero, Márcia A. Gomes; Lopes, Vera Lúcia da Rocha. Cálculo numérico:
aspectos computacionais. 2° edição. São Paulo: Pearson Markon Books, 1996.
[14] Devries, Paul L. ; Hasbun, Javier E. A first course in computational physics. Second
edition. Jones and Bartlett Publishers: 2011.
89. June/2014 Jorge G Pires 89
Optimal Control applied to
References
ALON, U. An Introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC,
2007.
CACACE, F., GERMANI, A., PALUMBO, P., The state observer as a tool for the estimation of gene
expression, Journal of Mathematical Analysis and Applications, Vol.391, pp.382-396, 2012.
LENHART, S.; WORKMAN, J. T, Optimal Control Applied to biological models, Chapman & Hall/ CRC,
Mathematical and Computational Biology Series, 2007.
PIRES, J. G. Desenvolvimento de programa baseado no problema da mistura. Pesquisa Operacional:
programação matemática. Simpósio de Engenharia de Produção. XVI: 1-12, 2009.
PIRES, J. G.; DUARTE, A. S.; BIANCHI, R. F.; SANTOS, Z. A. DA S.; BIANCHI, A. G. C. Projeto e
desenvolvimento de Produto: proposta e desenvolvimento de dispositivo eletrônico para auxiliar no
tratamento da icterícia. Gestão do Produto: engenharia do produto. Simpósio de Engenharia de Produção.
XVI: 1-12, 2009.
PIRES J. G. On the applicability of Computational Intelligence in Transcription Network Modeling. Thesis of
master of science. Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Poland.
74:1:46. 2012.
PIRES, J. G.; PALUMBO, P. Engenharia de Software: Planejamento e desenvolvimento de programa
baseado em Inteligência Computacional aplicada a Redes de Expressão Genética. Gestão do Produto:
engenharia do produto. Simpósio de Engenharia de Produção. XIX: 1-12, 2012.
PIRES, J. G. Na importância da biologia e em engenharias: biomatemática e bioengenharias. Educação em
Engenharia de Produção: estudo do ensino de engenharia de produção. Simpósio de Engenharia de
Produção. XX: 1-12, 2013.
ROSENBAUM, S. E Basic pharmacokinetics and pharmacodynamics: an integrated textbook and computer
simulations, John Wiley & Sons, 2011.
Visite: http://www.uri.edu/pharmacy/faculty/rosenbaum/basicmodels.
SCHOOF, C. P.: Zschocke, J.: Potocki, L. Human Genetics: from molecules to medicine. Lippincott Williams
& Wilkins: 2012.
WINSTON, W. L. Operations Research: application and algorithms. Third Edition, 1996.
90. June/2014 Jorge G Pires 90
Optimal Control applied to
What is next....?
Dynamics programming;
Optimization based schemes;
Discretization based approaches;
Stochastic counterparts;
Extension of the models presented herein for stochastic framework;
Pharmacogenomics;
Test some in PK/PD;
Pharmacokinetic/pharmacodynamic interactions;
Receptor Interactions;
Report in September;
Java Library for Optimal Control in Life Sciences (JAR Executable)!? !?
91. June/2014 Jorge G Pires 91
Optimal Control applied to
http://ijcai-15.org/index.php/important-dates