In this article, a closed-loop approach to human body weight control is presented. The main purpose of the article is to demonstrate that applying feedback has significant benefits over conventional open-loop techniques suggested in the rich health literature on the subject. In particular, a closed-loop approach is robust to the strongly adaptive mechanisms of the human body and to disturbances of various kinds. Also, in contrast to conventional approaches proposed in the health literature, the presented method based on feedback does not depend on any specific diet. In fact, the approach can be applied to any diet preferred by the subject.

DISCLAIMER: The proposed method has not been approved by medical doctors. If applied incorrectly, it can potentially cause significant health issues. It is strongly recommended not to pursue the experiments described below without consulting a physician.

Background

For a large and increasing proportion of the world population, overweight and obesity cause a wide range of health issues and in particular are leading causes of premature death. The World Health Organization (WHO) defines overweight as a Body Mass Index (BMI) greater than or equal to 25 kg/m2  and obesity as BMIs larger than 30 kg/m2. According to WHO, worldwide obesity has more than doubled since 1980. In 2014, almost two billion adults were overweight. More than 40 million children under the age of five were overweight or obese in 2014.

In part for health reasons, dieting has been recommended by medical doctors and other health experts for centuries, at least dating back to the early 18th century, e.g. by the English doctor George Cheyne, who based on personal experience recommended diets for anyone suffering from obesity or overweight as described in his 1724 report, An Essay of Health and Long Life. There is no shortage of descriptions of diets in contemporary literature, ranging from esteemed medical journals to popular magazines and newspapers. In Western societies, a significant proportion of the population has been following one or several such diets for longer or shorter periods of time. In addition to health challenges, overweight/obesity also have significant psycho-social effects.

In this article, we shall provide a control theory perspective on weight control. A simple feedback algorithm will be described below along with experimental data verifying the algorithm.

Modeling weight gain

The dynamics of human body weight is by far dominated by three factors:

• Food and drink intake, instantaneously causing a (partly temporary) weight gain
• Excretion, instantaneously causing a (partly temporary) weight loss
• Metabolism, slowly but steadily causing (temporary) weight loss

Recent research has shown that exhaust from lungs (part of excretion) is a major factor in weight loss. Burning 10 kg human fat requires inhalation of 29 kg oxygen. This produces 28 kg carbon dioxide and 11 kg water. As food and drinks are temporarily stored in the human stomach and bowels, the body weight is instantaneously increased with the weight of any food or drink consumed. Metabolism is usually divided into catabolism and anabolism, where catabolism is the process of breaking down organic matter and anabolism is the reverse process of constructing proteins and nucleic acids. Metabolism is catalyzed by enzymes and metabolic rates can be strongly time and state dependent, governed by such catalyzing enzymes. In eukaryotes, such as homo sapiens, metabolism is connected to a series of proteins in mitochondria. As a very coarse model, the level of metabolism at any given time is therefore proportional to the number of mitochondria. The number of mitochondria depends strongly on tissue types and therefore on body distribution of these, but in the larger picture the number of mitochondria is positively correlated with the number of cells in the body, which is finally approximately proportional to the body weight. In summary, this rough reasoning leads to the following extremely simple model for body weight dynamics:

$\frac{dw(t)}{dt} = -\alpha(w,t)\cdot w(t) – e(t) + f(t)$

where $$w(\cdot)$$>0 is the body weight, α is a positive parameter, depending on state and time, that governs the metabolism,  $$f(\cdot)$$ is the food/drink intake function, which can only attain non-negative values and $$e(\cdot)$$ is the excretion function, which can only attain non-negative values. Clearly, this model cannot be expected to be accurate in open loop. E.g. it does not capture the difference in dynamics between catabolism and anabolism, which would require a higher order model. Below, however, we shall argue that this very simple model surprisingly suffices to understand and design closed-loop behavior.

Proposed control algorithm

The approach suggested in this article relies on the following assumptions:

• Body weight is a measureable state variable
• Food weight is a controllable input
• Metabolic rates can be time-varying, but are bounded from below, α≥αmin

Based on these assumptions, a simple feedback control law that takes body weight as its measurement and specifies food intake as the control signal can be devised:

$F(t) = r(t+T) – w(t)$

where $F(t) = \int_t^{t+T}f(\tau)\;d\tau$is the weight of food and drinks consumed during a meal starting at time $$t$$ and ending at time $$t+T$$; further $r(t+T)$ is the control reference at time $$t+T$$ (the end of the meal). Since  $$f(\cdot)$$ is a non-negative function, the reference $$r(\cdot)$$ has to be chosen larger than $$w(\cdot)$$ at all times.  In practice, the algorithm can rely on (suitably conservative) estimates for some meals, if an insufficient number of measurements are available, as long as the integral constraint is met during a day – please, see experimental data below.

A consequence of the above is that a reference with a weight loss demand larger than that dictated by the metabolism (catabolism) at any period of time, is infeasible. In practice, however, the reference weight loss should not only be marginally smaller but significantly smaller than the metabolic weight loss between two consecutive meals, as otherwise the body will not get a sufficient amount of nutrients for sustaining normal operation, and potentially health will be challenged. Further, when choosing a reference, it should be taken into account that α tends to be monotonically increasing/decreasing with a monotonically increasing/decreasing w, i.e. metabolism tends to adapt in order to changing weight (this is well-documented in the medical literature).

Experimental verification

The algorithm described above was applied during an experiment with a duration 44 days with the author of this article as the subject. A reference was chosen that had a constant slope for the first 31 days (one month), followed by a constant value. The initial value of the reference was chosen as the initial condition of the body weight. The final value of the reference was chosen as a body weight that would bring the BMI from an initial 26.1 kg/m2 (mild overweight) down to 23.8 kg/m2, i.e. well into the normal (non-overweight) area. In summary, this schedule inferred a weight loss of 7.4 kg during the 31-day weight loss period, i.e. a daily decrement of 239 g, followed by a static weight condition for 13 days.

Figure 1: Results of closed-loop weight control experiment

The experimental results can be seen in Figure 1. In practice, the algorithm was carried out by three daily body weight measurements: a morning measurement, a measurement immediately before the last meal of the day, and a late evening measurement for validation. The breakfast and the lunch meals were chosen to weigh approximately half the margin between the (known) upcoming evening reference and the morning measurement. That left about half the food intake for the evening meal, which was weighed on the plate and calibrated to match the remaining margin up to the scheduled reference. With this approach, the reference was normally reached by each evening measurement. Figure 1 shows a few overshoots. These happened at events where adhering to social code prohibited the meals from being weighed, so estimates had to be applied instead. Also, drinks taken after the last meal were not calibrated, which gave minor deviations. The undershoots in the beginning of the experiment are deliberate.

It is interesting to note that as the actual weight approaches the target, the metabolic cycles decrease significantly in amplitude. This is probably due to body responses that change the metabolic rates. It is likely that such a mechanism has developed evolutionarily to respond to periods of food scarcity. In contrast, metabolism is seen to increase significantly close to the end of the experiment, where the flat part of the reference has allowed a much higher food intake, causing the body to respond by what could perhaps resemble a food surplus scenario in an evolutionary context. Throughout the experiment, the conscious awareness of the subject provided another level of feedback, as the subject gained experience with the impact of his exercise, food composition, etc.

Conclusions

The closed-loop weight control approach proposed in this article has the virtue of offering deterministic results, based on the single assumption that the algorithm is followed strictly. It should be noted that the method is completely independent of any specific composition of the diet. In fact, although the daily weight loss of the experiment was significant, the involved diet throughout the experiment included a proportion of energy intensive food components such as chocolate and red wine (the red wine is discernible in the experimental results, causing the metabolic cycles to reduce significantly in two instances). Also, it should be noted that due to monotonicity properties of metabolic systems, any diet that has the same weight loss as in the described experiment, will have the exact same average food intake, provided the food has the same distribution of proteins/carbs/fats.

A limitation of the proposed approach is that it does not address nutritional adequacy aspects of diets. If one tries to lose weight too fast, or put a goal for a too-low ultimate steady state reference, one’s health will suffer. In practice the reference trajectory should be “reasonable.”

On the other hand, as an important conclusion of this article, feedback can be combined with any given diet, providing a layer of mathematically guaranteed weight loss to a physiology based diet that would typically be composed from a health perspective. The main approach to a healthy body will always be a healthy lifestyle with healthy food and lots of exercise. However, for anyone on a diet, there is simply no reason not to embed the diet in a closed-loop approach and take advantage of the power of feedback!

Article provided by:
Jakob Stoustrup
Department of Electronic Systems
Automation & Control
Aalborg University, Denmark
IFAC Technical Board