Ghrelin Mathematical model Presentation iasi bio mathlab

Ghrelin Mathematical Modeling

BioMathLab Gemelli Ospedale, 2nd December, 2015, Final Talk of 2015 (Second Year of the PhD Pathway)

Jorge Guerra Pires

Research Group. Costanzo Manes, Andrea De Gaetano, Pasquale Palumbo, Alessandro Borri.Some participations of. Simona Panunzi

J.G Pires, A. Borri, C. Manes, P. Palumbo, A. De Gaetano. A Mathematical Model for Ghrelin: Energy homeostasis and appetite control. Computational and Mathematical Methods in Medicine. Submitted

under invitation.

Envisaged final goal

Methodological procedures

Introduction

Ghrelin empirical dynamics

Mathematical modeling(Just model 1,

if we want, it is possible to go on in the blackboard

Map of the main points of the slides



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Ghrelin mathematical model

Insulin mathematical model

Leptin mathematical model

Pre-processing


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Pre-processing

Pre-processing.

• How to transform properly a meal into a mathematical description for the model;

• Possible future endeavors, certainly not for this project: image processing, given a dish, or a sequence, how the model would respond?

• Publications in general make it difficult a proper transformation for model identification. It could be possible to create some laws, then optimize the parameters in function of the inputs and outputs, several parameters would be good for the same model and input-output relations;

• Even the opposite, given a meal pattern, how that would be in real world?


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Ghrelin Mathematical Model

• That is what we are up to discuss. Basically, as we are going to see in a few minutes, it was possible to model the day-like pattern, meal influenced, not the nighttime. We still need to work out the details.

• In order to make clear the «inputs», the model components, I have numbered the models, the model 1 is basically the one «proposed» by Andrea, after my initial proposal.

• The differences between the models is what they want to cast in addition to the lower model, e.g. Model 1 cast the meal-related pattern, whereas model 2 possesses an additional component for the increase of ghrelin all over the day. Some of the additions certainly might reveal themselves as futile. We are going to see it in the upcoming graphs.


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• We already have several models published, just pick one;

• I have not yet tested it, thus I want to start from toy models, than as things run nicely, I shall shift to more complex models, leaving even future works, for post-doc researches;

• The interactions insulin, leptin, and ghrelin is something under study, therefore sometimes we shall need to “fare il furbo”;

• Insulin seems to work in the same time-scale of ghrelin, but not leptin;

• Insulin seems to connect with both hormones leptin and insulin, therefore we need a multiple-time scale approach.


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• We already have one mathematical model for leptin;

• The current model, as we have seen from the previous discussions, is full of flaws, the notorious one being developed to mice, not human;

• Leptin, in order to go on with current mathematical model, assumes a double-dynamics, one for short-term and the order for long-term, a diurnal dynamics against a daily-monthly dynamics;

• Leptin “competes” with ghrelin in order to control appetite, and seems also fat metabolism, the true question, not necessarily answered by us, is when it is concurrency, or when it is lack of knowledge to single out the hormone effects.


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Envisaged final goal What to plot? Those are simple questions, but difficult to answer

properly

• Ghrelin-time concentrations for different meal-patterns, diet adoption or changes;

• Is it possible “indirect” treatments, e.g. some cases of diabetes with leptin rather than insulin, some results show that leptin can help on the glucose processing (metabolism);

• What happens with the appetite, represented by ghrelin, in different insulin or/and leptin treatments?

• What about drugs designed to control appetite?

• What about other means of effecting appetite, e.g. noncaloric content, tastants?

• Can we make a “dynamic programing” approach for a diet? From ghrelin profile to diet proposal?

Methodological procedures


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Methodological procedures Abstra

ction

Theory

Prediction

Observation

Induction

modeling

Deduction

Validation of the model/theory


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Methodological procedures • The equations have been just ordinary differential equations;

• Ghrelin has been the center of everything;

• All the data used comes from papers already published;

• I have used Matlab/Simulink for my simulations;

• The simulation did not require expressive computational efforts;

• I have given an emphasis on physiology rather than mathematics.


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Introduction


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Introduction Ghrelin, produced mainly by the stomach, is a pretty powerful (i.e., minutes) appetite stimulating hormone;

i.e. it triggers our need for food, our appetite, our willingness to start a meal.

It has been discovered in 1999 by Japanese scientists, but largely spread-out by British groups, and so then it has been a quite important piece for taking in the workings of feeding patterns and behaviors.

Ghrelin is an amino acid peptide, related to growth hormone, which is secreted primarily in the stomach but is found throughout the gastrointestinal system and even in the hypothalamus and amygdala, among other sites, such as the heart and pancreas.

Some claims that the name comes from Growth Hormone releasing, by shorting and gathering, we encounter ghrelin. But exactly how ghrelin exerts its effect is not clear, neither how it is produced, e.g. the complete profile for triggering ghrelin activation and inhibition.


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The complex process of eatingWe can say that eating is one of the simplest activities, it is not necessary being an Einstein for eating with style.

Nonetheless, within the body, it is a quite complex process, maybe amongst the most complex ones, given that we are still try to understand it properly for treating medical conditions such as obesity.

Above is a scheme of the hormones involved in the eating control and metabolism; and their respective sites of production and action in the arcuate nucleus.


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The complex process of eating


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Basically we have two areas, the yellowish one, representing the arcuate nucleus, within the brain, and the greyish-yellow one, representing the digestive system.

The diagram depicts how these two areas communicate. In the picture, three clusters of neurons are represented: the neurons that control appetite, metabolism, and communication brain-body.

The important point to mention regarding this picture is that the clusters for the neurons that control hunger and metabolism is a positive feedback loop amongst themselves, any change in one group is going to affect the other. From the picture we can learn:

The arcuate nucleus is an aggregation of neurons in the hypothalamus. The arcuate nucleus includes several important populations of neurons, including: neuroendocrine neurons, centrally projecting neurons, and others.


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The arcuate nucleus is an aggregation of neurons in the hypothalamus. The arcuate nucleus includes several important populations of neurons, including: neuroendocrine neurons, centrally projecting neurons, and others.

Most of the hormones that control eating and metabolism possesses receptors in the brain. Further, some has positive control, represented by the green triangle, and inhibition, represented by the red-x, or even double function, e.g. leptin and insulin;

Leptin and insulin have a double-role, they can either affect the neurons that control appetite or the ones that control metabolism;

Hormones signaling food intake can be produced either the digestive system or other organs such as the pancreas and fat tissues;

PYY decreases appetite, whereas ghrelin increases;

The signals sent from the body regarding feeding states are done in different receptors on the neuronal system;


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What is coming.... On the approaching discussions, we attempt to put together these details and concentrate just on ghrelin, a single piece of the puzzle, mathematically.

This is the model number 1, the starting for all the upcoming models.

First we are going to some important empirical results.


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D.E.Cummings, D.S. Weigle, R. S. Frayor, P. A. Breen, M.K. E. P. Dellinger, J. Q. Purnell. Plasma Ghrelin levels after diet-induced weight loss or gastric bypass. Ghrelin and Regulation of Body Weight. N Engl J Med, Vol. 346, No. 21. May 23, 2002.


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Sleeping related area

meal related area


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Body weight related, energy homeostasis

Meal related

Plasticity Stability Dilemma


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Diurnal rhythm


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Mathematical modeling


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Model 1: the scaffolding Key-points for the

mathematical model1. Ghrelin is produced constantly: this means that ghrelin is produced unceasingly, unless a suppression mechanism is triggered, e.g. glucose, mechanoreceptor. This mechanism is assumed to suppress the production rate, not ghrelin itself, for this last job we assume being macronutrients, e.g. glucose, or tastants, e.g. bitter. The last case we still need to decide whether to account for;

2. Ghrelin production is inhibited by stomach/duodenum stretching: therefore herein we consider mechanoreceptor and chemoreceptors;

3. Ghrelin is eliminated from blood: clearance rate, first order dynamics;

4. Feeding is done on given times, deterministic: it means that we want to replicate the classical meal experiments, e.g. three meals a day (*);

5. Ghrelin does not go to the brain: it means simply that we are considering a single compartment for the brain and bloodstream;


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mathematical model Feeding is done on given times, deterministic: it means that we want to replicate the classical meal experiments, e.g. three meals a day (*);

A potential simulation is to consider stochastic feeding patterns. Each meal time is given by:

Where is a probability density function, e.g. The most promising equation is the hill function:


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mathematical model Feeding is done on given times, deterministic: it means that we want to replicate the classical meal experiments, e.g. three meals a day (*);

This function is demanded bearing in mind that ghrelin was found to “partially” influence our eating wiliness, that

is, it does not seem to have any considerable effect for low concentrations; the hill function causes this effect, it is

almost zero for low concentration and maximum as we go farther and farther from the threshold, . wants to say that

your eating behavior when given unrestrictedly, you eat as a function of your ghrelin concentrations.


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Model 1: the scaffolding Diagram


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Model 1: the scaffolding Equations


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Model 1: the scaffolding Digestive System

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Digestive System


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Model 1: the scaffolding Stomach

• The first equation is for the first compartment the foodstuff passes by through its pathway towards absorption. The stomach herein is seen as classically seen, just a chemical chamber, the same mass amount that comes in, comes out, not more, not less.

• Thus, we do not consider internal chemical transformations, neither it is considered classically in physiology.

• The stomach is a complex three-dimensional chamber, projecting it into an one-dimensional variable is not simple, maybe even impossible, but let’s assume that we can, for instance by taking the surface areas before and after, and using it with a constant to measure stretching.

• Therefore, the state variable S can either be interpreted as stomach stretching or foodstuff within the stomach, it is not important to make the difference, unless we want to consider the plasticity of the stomach.


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Key-topics

• Food Intake: food comes in and food comes out, no mass is created or destroyed;

• Food output (i.e. from stomach to duodenum);

Foodstuff (McDonalds...) Chyme


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Model 1: the scaffolding Stomach The model

• Food comes in, foodstuff, and food comes out, chyme;

• It is applied the simplest function possible for foodstuff, the boxcar function;

• Mass is conserved;

• The input are not trains (comb-like);

• All the math is of first-order;


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SkmdtdS

SD

N

ii

1


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SkmdtdS

SD

N

ii

1

Meal term, each i represents a different meal, for now, as before presented, it is a carbox function

This is the transference rate, first order, see that more complication can be introduced here, e.g. Output control by ghrelin, already shown, or even other hormones, such as amylin.


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Model 1: the scaffolding Duodenum


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The model

• Chyme comes in, and nutrient load, not important now, and waste leaves the systems, both not being distinguished now, just in future versions of the model;

• All the math is of first-order;


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Model 1: the scaffolding Duodenum Key-topics Food comes from the stomach;

Food leaves the duodenum, however it does not matter where it goes;

DkSkdtdD

DXSD


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DkSkdtdD

DXSD

SkmdtdS

SD

N

ii

1

Stomach

Doudenum

Out of the system, compartment XOne directional mass movement, no vomiting!


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Ghrelin


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Ghrelin


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Ghrelin The model

• Ghrelin is produced, ghrelin is eliminated. Simpler is impossible;

• With exception of ghrelin production, all the dynamics is linear, first order;

• On this first model, just the gut plays a role, on future versions, we also have nutrient loads;

• For future works, likely not for this ongoing project, is modelling the central role of tastants, or even other substances that could be found to influence ghrelin dynamics;

• This model just take into account meal-related dynamics, no metabolism, no circadian-related dynamics or even diurnal-nocturnal patterns.


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Ghrelin Key-topics Ghrelin production rate is inhibited by stomach and duodenum stretching simultaneously, in an independent

and cumulative process;

Ghrelin is eliminated like a arbitrary drug, by renal mechanisms: herein we take as granted a first order elimination rate;

Ghrelin in the brain is negligible for our purposes, we have just one compartment, which makes use of the brain and bloodstream as a single compartment.


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Ghrelin


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Ghrelin Production rate

DSkkP

H

H

121

DSP

111

21

Playing with the math

«Andrea-Jorge’s formula»

However, this transformation is not possible if we have the hill function with degree bigger than one; it may be

necessary if we need to model a situation in which ghrelin is not affected by small stretching/chemoreception, or

when we need to model the effect of ghrelin on appetite, it was demonstrated that ghrelin does not have too much

effect on small concentrations.


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DSkkP

H

H

121

The gut’s contribution, convex combination. For α=0, just the duodenum is important, as some papers seems to show,

whereas for α =1, just the stomach is important, as seems present in some «old and informal» literatures.

IMP. For our modeling purposes, the duodenum and jejunum are the same, symbolically called D.


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DSkkP

H

H

121

β is the «basal-maximum» production rate.

Experiments show that it is not homogeneous amongst individuals. Some seems to have lower rate, whereas others seem to have “aggressive” rates.

In the computational simulations, it seems enough to accounting for the variations reported in the literature among individuals.



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DSkkP

H

H

121


KH is assumed for now to be an “individual parameter”.

Mathematically, KH tells us where the curve “shifts down” in order to achieve the mathematical ghrelin production rate. The higher is KH is lesser effective is the suppression, the more we need stretching signals to have an visible ghrelin suppression.

KH marks where we reach half-way of maximum production in terms of local “units”, the higher it is, the more we need to give in order to achieve half-way from maximum suppression.


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DSkkP

H

H

121

γ tells us the central role of stomach contribution to suppression, maybe compared to the duodenum. It differs from α in two points:

1) it does not have to be bounded; 2) 2) it must have some kind of units, the «same» that we use for stomach or

duodenum.

Attention must be paid here due to dimensional problems.



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DSkkP

H

H

121


DSkkP

H

H

1

Gut role on ghrelin suppression

Playing with the math


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Ghrelin Elimination rate

The elimination rate considered herein is extremely simple. It is first order and do not take into account other

facts such as kidneys, GFR Glomular Filtration Rate.

HClE


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Ghrelin Elimination rate


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Ghrelin Actual ghrelin concentration (bloodstream)

EPdt

dH

HClDSk

kdt

dH

H

H

121

By gathering together the pieces

Computer Experiments


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Extra discussions


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Circadian Rhythm or diurnal?

“…In the early 20th century, circadian rhythms were noticed in the rhythmic feeding times of bees...”

https://en.wikipedia.org/wiki/Circadian_rhythm

To be called circadian, a biological rhythm must meet these three general criteria:

The rhythm has an endogenous free-running period that lasts approximately 24 hours. The rhythm persists in constant conditions, (i.e., constant darkness) with a period of about 24 hours. The period of the rhythm in constant conditions is called the free-running period and is denoted by the Greek letter τ (tau). The rationale for this criterion is to distinguish circadian rhythms from simple responses to daily external cues. A rhythm cannot be said to be endogenous unless it has been tested and persists in conditions without external periodic input. In diurnal animals (active during daylight hours), in general τ is slightly greater than 24 hours, whereas, in nocturnal animals (active at night), in general τ is shorter than 24 hours.

The rhythms are entrainable. The rhythm can be reset by exposure to external stimuli (such as light and heat), a process called entrainment. The external stimulus used to entrain a rhythm is called the Zeitgeber, or "time giver". Travel across time zones illustrates the ability of the human biological clock to adjust to the local time; a person will usually experience jet lag before entrainment of their circadian clock has brought it into sync with local time.

The rhythms exhibit temperature compensation. In other words, they maintain circadian periodicity over a range of physiological temperatures. Many organisms live at a broad range of temperatures, and differences in thermal energy will affect the kinetics of all molecular processes in their cell(s). In order to keep track of time, the organism's circadian clock must maintain roughly a 24-hour periodicity despite the changing kinetics, a property known as temperature compensation. The Q10 Temperature Coefficient is a measure of this compensating effect. If the Q10 coefficient remains approximately 1 as temperature increases, the rhythm is considered to be temperature-compensated.

https://en.wikipedia.org/wiki/Endogenous

https://en.wikipedia.org/wiki/Entrainment_(chronobiology)

https://en.wikipedia.org/wiki/Zeitgeber

https://en.wikipedia.org/wiki/Time_zone

https://en.wikipedia.org/wiki/Jet_lag

https://en.wikipedia.org/wiki/Chemical_kinetics

https://en.wikipedia.org/wiki/Q10_(temperature_coefficient)

“A diurnal cycle is any pattern that recurs every 24 hours as a result of one full rotation of the Earth with respect to the Sun...”

https://en.wikipedia.org/wiki/Diurnal_cycle

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Ghrelin Mathematical model Presentation iasi bio mathlab