INTRODUCTION
For nutrients to have beneficial effects on health, they must be offered in adequate doses, within maximum and minimum limits. The establishment of what is appropriate (maximum and minimum limits) for each patient is determined by the prescriber and constitutes a critical part of the decision-making process. In other words, the decision-making process consists in establishing maximum and minimum ranges not only for the calories but also for each of the macro and micronutrients. Societies of experts compile recommendations based on the best evidence, also expressing them through constraints, with upper and lower limit limits, for example caloric supply between 25 and 30 kcal.kg-1.d-1 (1).
For patients in nutritional therapy industrialized formulas (IF) constitute the means to supply nutrients and to meet nutritional goals is necessarily related to a daily flow given by the volume infused per unit of time.
However, since calories, proteins or any other nutrients that may become clinically relevant are linked to this flow, there is a risk of not meeting the restrictions for a given clinical circumstance. An example is the difficulty of meeting current protein targets with the formulas available in the market, usually with less protein than necessary, thanks to the use of protein supplement (2), complementary modalities such as supplementary parenteral nutrition (3), or acceptance of a nutritional load above the desired level in order to allow the fulfillment of currently important restrictions.
The IF flow prescription is found by trial and error, erratically, until the constraints are met, or by the prescriber’s choice to prioritize one constraint over others. Not infrequently, the final prescription delivers nutrients out of the desired limits.
In order to improve the efficiency of the IF selection process, it is convenient to have prior knowledge of its potential adequacy to the proposed system of restrictions for each patient.
The objective of this study is to show under what conditions it is possible an industrialized formula to meet the clinical restrictions of calories, macro and micronutrients. To do so, we deduce a mathematical relationship that must be met in order to satisfy such constraints.
METHODS
The energy prescriptions place the energy (or caloric) supply between two extremes:
where
𝒱e: volume required by energy ∈ IF
ρe: energy density ∈ IF
e max: superior limit of energy
e min: inferior limit of energy
Rewriting equation I, we obtain the volume effect of the energy restriction:
Similarly, any and all nutrients will be placed between two extremes in the medical prescription:
where:
𝒱j: volume required by nutrient j ∈ IF
ρj: density of nutrient j ∈ IF
njmax: superior limit of nutrient j
njmin: inferior limit of nutrient j
Rewriting equation III, we obtain the volume effect of the nutrient restriction:
The volume administered to the patient, however, is not something hypothetical or a thing of imagination. There is, for example, no negative volume. Thus:
𝒱e: ≥ 0
𝒱j: ≥ 0
In the same way, it is a real and equal quantity for all constraints, be they nutrients or energy. Thus;
To meet all minimum constraints, we have to choose a volume such that:
To satisfy all the maximum constraints, we have to choose a volume such that:
Therefore, we have the following restriction:
Since the chosen volume must be a positive real quantity, the condition of existence of a solution to a formula becomes:
Or stated literally: the condition for an industrialized formula to meet a constraint system is that the maximum volume between the minimum service volumes of each restriction is less than or equal to the minimum volume between maximum volumes to meet each restriction.
Let’s exemplify in clinical vignettes.
CLINICAL CASE 1
Let patient P with the following weights and goals to be met:
Let a proposed enteral formula to meet the goals described above containing
The system of restrictions to be met by the formula will be composed of the following intervals:
Replacing in the equation X the terms, we have in the term of the left or “maximum volume” among minimal volumes:
máx {1.17 l; 1.4 l} = 1.4 liters of IF
Replacing in the equation X the terms, we have in the term of the right or “minimums” among maximal volumes:
mín {1.4 l; 1.75 l } = 1.4 l
Since Max {1.4} ≤ Min {1.4}, then we have that the formula meets the constraint system and can comply.
CLINICAL CASE 2
Let patient P with the following weights and goals to be met:
Let a proposed enteral formula to meet the goals described above containing
The system of restrictions to be met by the formula will be composed of the following intervals:
Replacing in the equation X the terms, we have in the term of the left or “maximum volume” among minimal volumes:
máx {1.73; 2.88} = 2.88
Replacing in the equation X the terms, we have in the term of the right or “minimums” among maximal volumes
mín {1.31; 3.52} = 1.31
Since {2.88} ≤ Mín {1.31} is a false statement, then we have that the formula does not meet the constraint system and cannot comply.
CLINICAL CASE 3
Let patient P with the following weights and goals to be met:
weight = 90 kg
caloric goal e = 28 to 30 kcal.kg-1.d-1
protein goal p = 0.8 to 1.2 g ptn.kg-1.d-1
carbohydrate goal c = 100 to 150 g.d-1 (new restriction added)
Let a proposed enteral formula to meet the goals described above containing
The system of restrictions to be met by the formula will be composed of the following intervals:
Replacing in the equation X the terms, we have in the term of the left or “maximum volume” among minimal volumes:
máx {2.52; 1.8; 1.23} = 2.52
Replacing in the equation X the terms, we have in the term of the right or “minimals” among maximal volumes
mín {2.70; 2.70; 1.85} = 1.85
Since Máx {2.52} ≤ Mín {1.85} is a false statement, then we have that the formula does not meet the constraint system and cannot comply.
DISCUSSION
In the present work we propose an equation to identify the conditions of existence so that a nutritional formula of defined composition meets the clinical restrictions established based on the medical judgment.
In an ideal theoretical model, where we have an ideal nutritional formula, all the proposed restrictions would be met and we would be facing the perfect nutrition. The perfect nutrition is not dependent on via, and encompasses the modalities of parenteral nutrition, oral nutritional therapy and even these strategies combined with each other, as well as use of supplements.
In real world, however, the solution applied by the prescriber is to simplify the number of restrictions considered, usually based only on protein and caloric needs, with the inclusion of new restrictions for other nutrients depending on the clinical conditions. This bedside solution is performed erratically, in which the prescriber can randomly select formulas that do not necessarily meet the constraints and pre-terminate the search by the most appropriate formula. In addition, this method is vulnerable to non-technical elements such as prescription, marketing and industry advertising habits or anchoring in the last prescription made. Furthermore, as stated by Hoffer, “enteral products that have replaced PN are designed for normal people and hence are protein-deficient for critically ill patients. The current situation requires the treating team to choose between 2 adverse consequences: severe protein deficiency during the first 7 to 10 days in the ICU, or toxic calorie overfeeding as the cost of providing a suitable amount of protein” (4).
The presented theorem that denominated “maximum of the minimi and minimum of the maximi” is a more assertive method, and allows to identify immediately among formulas of defined composition in a portfolio, that attend or not to the set of restrictions. The medical decision focuses only on the determination of the intervals, thus being robust against non-technical influences from marketing and propaganda.
The method can also be applied in the field of parenteral nutrition therapy and in a variety of scenarios whenever the clinical condition in the patient demands a change in the system of restrictions, without restricting free choice of the prescriber in the determination of the restrictions.
An additional utility of the method is to allow that when finding the inviability of a formula it is possible to identify which of the restrictions hinders the viability of its use. In this sense, it may still be a useful method for the industry, when studying the ideal types of patients, for example the severe obese patient, to identify in each formula profile in which categories of restrictions there will be no viability.
In the examples studied, calories, proteins and carbohydrates were considered. The method may encompass the incorporation of other categories including micronutrients and fibers.
The theorem was developed assuming the total nutrient load per unit of time, for example, kcal per day. In mathematics, this mode of developing the calculation is based on integral method. However, the infusion of nutritional formulas occurs in the form of flow. Future research results should analyze the properties of the volume infused in a differential perspective and in this sense include other elements such as cost for example.