Ghrelin Mathematical model Presentation iasi bio mathlab

Ghrelin Mathematical Modeling
BioMathLab GemelliOspedale, 2nd December, 2015, FinalTalk of 2015 (SecondYear 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.

Map of the main points of the slides

BioMathLab Gemelli Ospedale, 2nd December, 2015, FinalTalk of 2015 (SecondYear of the PhD Pathway)J G Pires
Envisaged final goal
Ghrelin
mathematical
model
Insulin
mathematical
model
Leptin
mathematical
model
Pre-processing

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?

Ghrelin
mathematical
model
Insulin
mathematical
model
Leptin
mathematical
model
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.

Ghrelin
mathematical
model
Insulin
mathematical
model
Leptin
mathematical
model
Insulin mathematical model
• 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.

Ghrelin
mathematical
model
Insulin
mathematical
model
Leptin
mathematical
model
Leptin mathematical model
• 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.

Ghrelin
mathematical
model
Insulin
mathematical
model
Leptin
mathematical
model

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
Induction
modeling
Deduction
Validation of the
model/theory

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.

Introduction

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.

The complex process of eating
We 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.

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.

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;

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.

Ghrelin empirical dynamics
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.

Sleepingrelatedareamealrelatedarea

Body weight related, energy
homeostasis
Meal related
Plasticity Stability
Dilemma

Diurnal rhythm

Mathematical
modeling

Model 1: the scaffolding
Key-points for the mathematical model
1. 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;

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:
𝑚 = 𝑓(𝐻)
Where 𝑓 is a probability density function, e.g. 𝑓 𝐻 = 𝑈(0, 𝐻) The most promising equation is the hill function:
𝑓 𝐻 = 𝑈(0, 𝑔)
𝑔 𝑥 =
𝑥 𝑛
𝑘 𝑛 + 𝑥 𝑛

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, 𝑘. 𝑓 wants to say
that your eating behavior when given unrestrictedly, you eat as a function of your ghrelin concentrations.

Diagram

Equations

Digestive System

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.

Stomach
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

Stomach

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;

Stomach
Skm
dt
dS
SD
N
i
i  1

Stomach
Skm
dt
dS
SD
N
i
i  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.

Duodenum

Duodenum
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;

Duodenum
Key-topics
 Food comes from the stomach;
 Food leaves the duodenum, however it does not matter where it goes;
DkSk
dt
dD
DXSD 

Duodenum
DkSk
dt
dD
DXSD 
Skm
dt
dS
SD
N
i
i  1
Stomach
Doudenum
Out of the system, compartment X
One directional mass movement,
no vomiting!

Ghrelin

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.

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.

Ghrelin
Production rate
 DSk
k
P
H
H




121
 DS
P




11
1
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.

Ghrelin
Production rate
 DSk
k
P
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.

Ghrelin
Production rate
 DSk
k
P
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.

Ghrelin
Production rate
 DSk
k
P
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.

Ghrelin
Production rate
 DSk
k
P
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.

Ghrelin
Production rate
 DSk
k
P
H
H




121
  DSk
k
P
H
H




1
Gut role on ghrelin suppression
Playing with the math

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 

Ghrelin
Elimination rate

Ghrelin
Actual ghrelin concentration (bloodstream)
EP
dt
dH

 
HCl
DSk
k
dt
dH
H
H





121
By gathering together the pieces

Computer Experiments

Extra discussions

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.

“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

Ghrelin Mathematical model Presentation iasi bio mathlab

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Ghrelin Mathematical model Presentation iasi bio mathlab