Mathematical modeling in energy homeostasis, appetite control and food intake with a special attention to ghrelin

Mathematical modeling in energy homeostasis,
appetite control and food intake with a special
attention to ghrelin
Jorge Guerra Pires
PhD candidate, 2014-2016 (XXIX cycle)
Thesis public discussion
Systems engineering telecommunications and
HW/SW platforms, Modeling and control of cyber-
physical systems
DIPARTIMENTO DI INGEGNERIA E SCIENZE
DELL'INFORMAZIONE E MATEMATICA (DISIM),
Università degli Studi dell'Aquila (UNIVAQ)
Advisor/tutor: prof. Dr. Costanzo Manes (Dipartimento di
Ingegneria e Scienze dell'Informazione e Matematica, DISIM,
UNIVAQ)
Co-advisor/co-tutor: Dr. Pasquale Palumbo (Istituto di Analisi
dei Sistemi ed Informatica “A. Ruberti" (IASI), Consiglio
Nazionale delle Ricerche (CNR))
8th March 2017, L’Aquila, Italy

1. Introduction
1.1. Ghrelin, food intake, bodyweight and appetite control
The “web of hormones”
Pires JG. Some insights into an integrative mathematical model: a prototype-model for bodyweight and energy homeostasis. Revista Gestão & Saúde (ISSN 19824785). v. 7, n. 3 (2016).
4

1. Introduction
Ghrelin: a key player on meal initiation
Ghrelin is an orexigenic (i.e., appetite stimulant) hormone; in fact, the only one of its
kind, a peripheral hormone that can influence centrally one’s propensity to start a meal.
By own work - adapted from
http://www.pdb.org/pdb/files/1p7x.pdb using
PyMOL, Public Domain,
https://commons.wikimedia.org/w/index.php?curi
d=4790168
Its effect seems to be mediated by a group of neurons in the brain (arcuate nucleus), the same
group of hormones aimed by leptin and insulin, to release neuropeptide y (NPY), which is a
quite powerful neuropeptide in food intake positive control, see figure in the next slide.
Additionally, it was found to be also correlated to energy homeostasis, fat utilization control,
signaling the brain when energy efficiency is needed to be increased. Moreover, several other
functions were associated to ghrelin, even if more research is needed to be done to confirm its
pleiotropic attribute.
5

1. Introduction
Ghrelin: a key player on meal initiation
In the arcuate nucleus (hypothalamus), we have
two agglomerates of neurons: orexigenic –
releases neuropeptide y; anorexigenic – releases
αlpha-Melanocyte-stimulating hormone. Those
signals influence the proper neighboring neurons
for the “message” to go on: hunger and
metabolism control. In turn, these neurons after
activated will induce/inhibit food intake and
energy expenditure; this is done mainly by
controling the behavior of muscles, adipose
tissue, and liver. Source: based on(64, p.932).
6

1. Introduction
1.2. On the applicability of mathematical modeling in energy homeostasis, appetite control and food
intake
Most previous mathematical models of metabolic energy regulation have not explicitly modeled the
neuroendocrine feedback system that maintains energy homeostasis.
Tam, Fukumura and Jain (2009)
Mathematical models have been applied to biology and medicine for quite a while. However, in the “physiology
of eating”, it has been rarely applied, when applied, for “macroscopic” views (c.f., Computer and mathematical
modelling), without getting into details, the mechanism behind the physiological behavors we see in most living
system, especially human; the authors provide discussions on some models already published, see also
‘Computer and mathematical modelling’ on ‘Ghrelin: a review’ on the current text.
7
As highlights Jacquier, “Modeling of the effect of hormones on body weight has…been largely ignored.”

1. Introduction
1.3. Motivation
The obesity issue
8
Obesity graphical representations: a) age–standardized prevalence of obesity (boys, 2–19 years); b) age–
standardized prevalence of overweight and obesity (BMI>=25) and obesity (BMI>=30), ages 20+ years, by sex,
1980–2013. Source: Ng et al.

2. Ghrelin Mathematical modeling
9

2.1. Model diagram10
See(41, chapter 18: “The digestive system”). Source: own elaboration .
General view of foodstuff flux throught the gastrointestinal tract.

2.2. Model diagram
11
Box diagram for the ghrelin dynamics presented herein, centered in the gastrointestinal tract. Legend: H – ghrelin; F – foodstuff; S – Stomach; D
– Duodenum; J – Jejunum; I – Ileum; LI – Large Intestine; X- output compartment; ? – control not confirmed/addressed by experimental data. cf.
Figure 72 (focused on the signal for gastric emptying, that will influence ghrelin dynamics, not explored on the current model). Source: adapted
from(75).

2.3. Equations12
The equations for ghrelin dynamics are split up into two groups: digestive system (gastrointestinal tract) and
ghrelin equation (the hormone concentrations in bloodstream).
To a certain extend they are independent, since factors that can affect the gastrointestinal tract does not
necessary effect ghrelin, and vise versa.
Box diagram for the two groups of equations for the ghrelin mathematical model presented on this discussion. The red-x means that we do not
have “reflux”, that is, feedback, the digestive system influences the ghrelin dynamics, but not vice versa. This is a simplification from a physiological
perspective, given that in some cases it was found that ghrelin can influence the stomach emptying rate(78); we shall take that into account in the
model we apply parameter estimation and future works. Source: own elaboration.

2.4. Equations
13
Gastric emptying control. This
model is not explored on this
section, but partially exploited on
the upcoming discussions. Source:
own elaboration.

2.5. Equations14
All the details from the previous figure of the intestines
are “smashed” into a single compartment called SI
(small intestine).
Source: own elaboration.
Gastrointestinal modeling
(simplified)
Simulations using the model presented herein, for
two cases, if we assume that just small intestine
suppresses ghrelin, or both of them; The meals are
given at 8, 12, 20, for one hour (much higher than a
normal meal, it is just for illustrative purposes). cf.
Figure 74.

15
Mathematical description
Stomach
a) Box diagram for foodstuff dynamics at the stomach. By mass conservation
and having the stomach just as a chamber, the mass of foodstuff has to
equalize the mass of the chyme (in=out). b) a more well-elaborated shape for
food intake using a summation of Gaussian functions.
2.5. Equations

16
Stomach dynamics
Dropping out 𝜁
gastrointestinal tract = stomach + small intestine = S + SI
2.5. Equations

17
Stomach dynamics
Gastric feedback
Ghrelin dynamics with ghrelin feedback controlling gastric emptying.
2.5. Equations

18
Stomach dynamics
Gastric feedback
Stomach dynamics with emptying rate
feedback, controlled by a surrogate
measure, small intestine excitation.
Stomach dynamics with emptying rate feedback,
controlled by a surrogate measure, small intestine
excitation.
2.5. Equations

19
Small Intestine
Box diagram for duodenum/small intestine;
remember that the duodenum is not differentiated
from the jejunum and iluem, just one single
compartment. The mass leaves the duodenum and
goes to a sink, herein we are not concerned about
what happens after that. The red line is not
considered herein. Source: own elaboration.
Key-topics
• Food comes from the stomach;
• Food leaves the duodenum, however, it does not
matter where it goes for now; future versions of the model
may be concerned about it;
2.5. Equations

20
Small Intestine
Small intestine dynamics
Stomach dynamics for changing the
duodenum emptying rate.
2.5. Equations

21
Ghrelin Dynamics
Minimal model
2.5. Equations

22 Some simulations for the ghrelin model: analysis of sensitivity
Set of simulations with arbitrary parameters. In
‘a’ we have the stomach dynamics; it was used an
extra compartment for food availability.
‘b’ is the duodenum dynamics, since it receives from
the stomach, it is “smoothed”.
‘c‘ is Ghrelin dynamics, it falls off after meal as the
literatures points out, it reaches a steady state when
no food is provided.
‘d' is the food intake profile, “boxcar” functions; a
more interesting and challenging pattern is a set of
Gaussians. The duodenum represents the small
intestine.
Source: own elaboration using our model for ghrelin
dynamics.
Analysis of sensitivity for food intake.
2.5. Equations

2.6. Parameter Estimation23
Stomach dynamics
(stomach feedback model)
Small intestine dynamics
Ghrelin dynamics
Setting the problem: our dynamical system

24
Meal dynamics
Stomach emptying rate feedback
Ghrelin production rate,
centered in the small intestine
Ghrelin production rate
Day night double dynamics
2.6. Parameter Estimation

25
The minimum/reduced model centered on the gastric emptying feedback.
Source: adapted from group work

26
Food intake function F(t), according to a breakfast, lunch and dinner occurring at 8:00, 12:00 and 17:30,
respectively.
Source: adapted from group work

27
The fitness or loss function is a mathematical relation used to measure how well/badly an algorithm/model
is performing on its task. Thus, for having a fitness function we need to have a way to measure what we want;
i.e. a measure of distance from “bad and good”. Once we have this measure, we just need to
minimize/maximize this relation; in biology, that may become problematic since not always the optimal is
what we can find in real physiological systems.
Residual Sum of Squares
Fitness/loss function

28
Raw data that we exploit herein; see Appendix A for a table; the upper and lower limits are statistical, standard error
SE. Source: own elaboration, with data from Cummings et al.
Raw data

29
Parameter estimation for the model presented on this section

30
Parameter estimation for the model presented on this section, part II

31
Schematic view of a hybrid algorithm (Hybrid SAPSbnd). Source: own elaboration.
Hybrid SAPSbnd: simulating annealing (global search) refined by pattern search (local search)

32
Optimal curve (Hybrid SAPSbnd). Source: own elaboration.
Hybrid SAPSbnd: simulating annealing (global search) refined by pattern search (local search)

Final Remarks and Future works33
Remodeling the gastric feedback: a web of hormones, including ghrelin
The model for gastric feedback we have applied herein is a very big simplification; it can be seen as a
surrogate measure (i.e., a chosen state variable to represent a hidden variable(s)).
It is true that as the small intestine starts to be excited by nutrient load present in foodstuff, the hormones
that control gastric feedback shall be either suppressed or produced, thus we have a positive correlation.
However, the true physiology behind, if we want to have a model closer to reality, must be taken into
account. It means that we must model the hormone individually, or at least some of them. Ghrelin also was
found to control gastric emptying, in an opposite direction, when ghrelin is high, it seems to induce gastric
emptying, rather than suppressing, as most of them do.

34
The “night mode”: how non-physiological is our modeling
The “night model”, as we see it here and may be found in the literature with other names, is a sequence of
hormone changes due to night/sleeping period. One example is the dawn phenomenon, that effects
negatively mainly people with diabetes.
In our model, we have modeled the effect of that on ghrelin production as a “two-state” function (day or
night); in fact, we are assuming that the “night mode” is responsible for the decline in ghrelin concentration
at night, where there is no meal to explain the fall-off.
More interesting modeling would be to take into account “hormones”, being it “fake” (a mathematical trick,
Figure 63 ) or real (based on physiological observations); thus, the concentration of this hormone must grow
throughout the day and become significant high at night.
Some of this fall-off seems to be explained by leptin, that can join independently or as net force on this fall-
off of ghrelin during the night.
Final Remarks and Future works

35
The production rate: a dependence on time?
From the curve fittings, it was possible to observe that the numerical approaches to parameter estimation did
their best; that is, from now on, we need to improve the model if we want to have a better representability of
experimental data.

36
Assessing the model: is there a better or worse model? How much can we explain?
We have proposed fundamentally two models, and left others as future works; however, we just fit one model,
letting the other fitting as future work. How can we select the best model? In fact, there are some techniques for
model selection, e.g. AIC (Akaike Information Criteria).
Therefore, once we have a sequence of models, e.g. the one herein with and without the gastric feedback, we
may compare them numerically.
Notwithstanding, those techniques do not say how “bad” is a model, if two bad models are compared, they may
give you the least worse; the goodness of a model must be judged on physiological grounds, if we want realistic
(mechanism-based) models.

37
Stomach emptying rate unit: Kcal/min or L-1?
Our model gives gastric emptying rate as liter/hour, however, most of the literatures are published as Kcal/min,
since it was found that gastric emptying rate is a function of caloric content; and that may be even seen in ghrelin
dynamics, since it was found to control gastric emptying rate, and ghrelin suppression was demonstrated to
respond to caloric content rather than volume of the foodstuff.
Consequently, in order to use parameters from the literature, or at least being able to compare fitted parameters
with experimental parameters, we need to adapt the model for this physiological detail.
We have used several optimization models. I honestly believe that from an optimization perspective, we did
more than enough, accordingly any future problem is regarding model improvement. We did not test all the
combinations on the “hybrid style”, since we have a considerable amount of local and global search
techniques. Others was not considered, e.g. evolutionary strategies, except for scientific curiosity, I see no
strong reason to elongate further this issue; it seems unlikely to me that other methods may improve the
fitting, if we need it somehow.
Optimization methods: another one?

38
The loss/fitness function: can we make it better?
Residual Sum of Squares
(extended)
Residual Sum of Squares (extended) with weights

3. Ghrelin mathematical modeling:
extended models
39

3. Ghrelin mathematical modeling: extended models
40
On these slides, we shall discuss several versions/extensions/variations of the model for ghrelin dynamics
discussed herein; hereafter called the basal model (see system below), you may want to see the previous
chapter in case of doubt.
Some of the “versions”/ “extensions” are more well-developed than others; it may reflect the degree of
complexity of the proposed variations, or even reflect the state at which it was realized the insight (i.e., close to
the process of writing the thesis), consequently mainly left as future works.
The basic/basal model for ghrelin
dynamics
3.1. Introduction

41 3.1. Introduction LigLab: a possible Leptin- Insulin-Ghrelin Laboratory
A potential simulator for food intake
based on ghrelin, leptin and insulin.
The models are shown in parallel
because they are different endeavors.
The idea is that given a food intake
profile, for instance for a period of
time, what is the effect in important
variable(s)?,
e.g. glucose levels; given a therapy,
what is the desired effects and
undesired side effects (e.g.,
hyperinsulinemia).

42 Weight loss: gastric bypass
It was found on cases of gastric bypass and weight loss that ghrelin levels change. In the case of
weight loss, as one loses weight, the levels of ghrelin go high, also in case of malnutrition.
That would be of great value to extend our models to contemplate those cases, therefore creating a
possible mathematical way to study those cases in a laboratory.
Some incipient simulations on gastric
bypass and weight loss.

43 A simple glucose-ghrelin model
The insulin-glucose model is a classic on the kind of endeavor we have pursued herein. Hence, it is inevitable
to wonder to which extent our model that can be applied to such situations.
A very simple model, a prototype for future works, can be done by using a linear dynamics for glucose, which
is certainly a simplification , but surely a starting point.
Some incipient simulations for a
ghrelin-glucose model.

44 Leptin ghrelin model: a double-time dynamic model
Leptin-ghrelin experimental dynamics; a) the data
alongside ghrelin and leptin, y-axis is concentrations, ng/ml
for leptin (right-hand side) and pg/ml for ghrelin (left-hand
side); b) a Fourier series regression could be used in a
simple model. In all cases, y-axix are concentration, and x-
axis is time in hour.
Schematic view of a multiscale model for ghrelin-leptin.

45 7.2. A model with several repressors/activators
Prototype for a multiple-factors model for ghrelin. Legend: S – stomach; D – small intestine (duodenum +
jejunum + ileum); X – a possible delay compartment; C- carbohydrates; P – protein; G – glucose; I – insulin; H –
ghrelin. Source: own elaboration).
a possible multiple factor dynamics for
ghrelin production rate

46 7.4. A tastant model based on ghrelin dynamics

47 7.5. A stochastic model based on ghrelin dynamics

4. Future works/Discussions/Final remarks
48

4. Future works/Discussions/Final remarks
49 Final remarks
Mathematical biology, the application of mathematical models to life science and medicine, has been
growing considerable in the last decades. Notwithstanding, the “real” work has yet to be done.
One example is the work produced herein, for a hormone discovered in 1999, showing a slow to a certain
extent application of mathematics in biology in medicine.
It is my hope that what we have done herein, with its own limitations that shall likely be overcame in future
works, shall contribute to these efforts, in especial to mathematical models based in mechanism applied to
bodyweight/food intake dynamics, and that all the future works mentioned herein might be pursued also by
others, independent groups; and, certainly, by my future endeavors.

References
They are in the end of the thesis, they are omitted herein.
50

Thank you for your attention,
Muito obrigado pela atenção,
Grazie mille per l'attenzione,
52
jorge.guerrapires@graduate.univaq.it
Borri A, De Gaetano A, Pires JG, Manes C, Palumbo P. A short-term dynamical model for ghrelin. IFAC 2017.
Accepted last week!

Mathematical modeling in energy homeostasis, appetite control and food intake with a special attention to ghrelin

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