Developments in the understanding of the stress response to massive burns have led to a significant reduction in mortality. However, burn survivors develop several metabolic abnormalities and arguably the most detrimental is excessive muscle wasting. Burn-induced muscle catabolism leads to reduced strength, delayed rehabilitation, increased morbidity and mortality. Skeletal muscle serves as a protein depot to buffer plasma amino acid concentrations due to heightened demand for acute phase protein synthesis and wound healing.

To meet protein requirements of burn patients, protein intake of $1.5-3.0\ g·kg^{-1}·day^{-1}$ is recommended while higher intake should be avoided since evidence suggest that do not provide additional benefit and may be harmful. However, the current upper limits are based on the findings of only two clinical studies. Those studies had several methodological limitations, and the findings were not definitive. Therefore, the potential beneficial effect of higher than currently recommended protein intake is understudied.

Assessment of muscle wasting requires the accurate evaluation of the protein synthesis and breakdown. Measurement of protein kinetics are optimal for mechanistic investigations on the molecular mechanism of muscle wasting and for evaluating the acute effect on protein turnover of various therapeutic agents. Therefore, high precision in protein kinetics estimation is of primary importance. Stable isotope methods are currently the most accurate and least invasive approach in vivo. Current methodology is based on radioactive isotope methods where the tracer amount used is practically massless; therefore, adaptations have been made to increase the validity of stable isotopes methods. Despite the corrections, experimental measurements often deviate from the theoretically expected values. All stable isotope techniques depend on the accuracy of the stable isotope enrichments in circulatory blood and tissues, especially in very low enrichments such as diluted by high food intake. A minor error in the calculation of enrichment could lead to significant errors in the final results, particularly in the calculation involving decay kinetics.

In Shriners Hospitals for Children in Galveston, hundreds of stable isotope metabolic studies have been conducted over the last two decades, measuring protein kinetics in severely burn pediatric patients. While metabolic study protocols were similar, patients were studied under varying nutritional conditions therefore the effect of protein intake during metabolic study on protein kinetics can be examined. Our specific aims are as follows:

Aim 1: Increase the precision of mass spectrometry isotope enrichment estimation of patient samples by:

1. Establishing a method for the statistical derivation of the isotopic purity of the tracer.
2. Modeling the observed mass spectrum, using the statistical properties of the mass distribution of tracee and tracer.

Aim 2: To determine the effect of dietary intake on muscle protein kinetics in severely burned children by:

1. Retrospectively assess the effect of varying levels of protein intake on muscle protein kinetics in a group of patients studied while receiving protein intake at levels routinely administered in normal clinical conditions.
2. Assessing the impact of applying the method in Aim 1 for the estimation of protein kinetics, on the interpretation of the results.

The retrospective analysis of multitude of protein kinetics data is an excellent opportunity for a cost-effective investigation of the understudied effect of protein intake on severe burns. Given the scarcity of relevant data in the literature and the tremendous debilitating effect of burn induced muscle catabolism it is likely that the findings of this analysis will be of high clinical interest and a significant addition to the existing literature.

Aim 1: Increase the precision of mass spectrometry isotope enrichment estimation of patient samples by:

1. Establishing a method for the statistical derivation of the isotopic purity of the tracer.

• The effect of labeling on the skewness of tracer’s mass distribution is usually underestimated. Obtaining an accurate estimate of the tracer’s isotopic purity is expected to increase accuracy in enrichment estimation.

2. Modeling the observed mass spectrum, using the statistical properties of the mass distribution of tracee and tracer.

• The methods used for calculation of enrichment for mass spectrometry reading do not take into account the statistical properties of the mass distributions of tracee and tracers. Using this additional information is expected increase precision in enrichment estimation.

Aim 2: To determine the effect of dietary intake on muscle protein kinetics in severely burned children by:

1. Retrospectively assess the effect of varying levels of protein intake on muscle protein kinetics in a group of patients studied while receiving protein intake at levels routinely administered in normal clinical conditions.

• Enteral feeding was not under experimental control and patients were studied during the fed state. Thus, we expect protein intake levels among subjects will vary within a range that will allow the examination of our hypothesis. The large sample size allows for inclusion of covariates in the statistical analysis.
• The variability of random differences in protein intake among patients during the study period will partly explain the variability of protein kinetics and higher levels of protein intake will be positively associated with a lower catabolic state.

2. Assessing the impact of applying the method in Aim 1 for the estimation of protein kinetics, on the interpretation of the results.

• Kinetic parameters for Aim 2 will be calculated using both the established and the new method. In metabolic studies with primed constant tracer infusion protocol (such as studies involved in Aim 2), where the steady state enrichment rarely exceeds 7%, the new methodology is expected to provide a mild improvement in accuracy of enrichment estimation.
• Improvement of kinetic parameters estimation in pulse bolus tracer injection metabolic studies (such as those performed in SHC- Galveston from 2008 and on) is expected to be high and critical to the proper interpretation of the derived kinetic parameters.

Detailed assessment of the severity of muscle wasting in burn patients requires the accurate evaluation of the muscle protein synthesis and breakdown. In vivo measurement of protein kinetics is optimal for mechanistic investigations on the molecular mechanism of muscle wasting and for evaluating the acute effect on protein turnover of various therapeutic agents. Therefore, high precision and accuracy in protein kinetics measurements is of primary importance.

Stable isotope methods are currently the most feasible (noninvasive) and accurate in vivo approach to estimate protein kinetics in burn patients(1). The isotopic distribution of atoms in nature (tracee) does not vary much and is well recorded. In metabolic research tracers are molecules where several atoms on their structure have been artificially exchanged with heavier (usually by 1 neutron) isotopes of the same atoms(2). The determination of substrate kinetics is possible through the measurement of enrichment (ratio of tracer-to-tracee) over time.

The attachment of heavier atoms on tracers does not only cause a shift in mass distribution but also changes of skewness of the distribution. The difference in the skewness of mass distributions of tracee and tracer can be amplified by the degree of tracer’s isotopic purity. A tracer with 99% isotopic purity does not mean that 99% of the molecules are labeled and 1% is not. It means that each labeled atom has 99% change of being actually exchanged with the heavier isotope. For example, in a sample of tracer molecules, having 10 labeled atoms with 99% isotopic purity, only 0.99¹⁰ = 90.4% of the molecules will have all 10 atoms labeled.

Currently, methods for the calculation of enrichment from mass spectrometry data(2,3) either underestimate the effect of labeling on tracer’s mass distribution, leading to biased estimates, or/and ignore statistical properties of tracee and tracer mass distribution leading to increased variability. A method that takes into account both factors is expected to improve the accuracy and precision of enrichment estimation.

A major unsolved metabolic problem in burn injury is muscle wasting the loss of protein mass and strength of the skeletal muscle. In critical illness, loss of lean mass is associated with increased morbidity(4) and mortality(5). For patients with respiratory failure, it significantly prolongs the ventilation time and ICU stay(6). Severe burn injury is characterized by increased whole body protein turnover and a pronounced increase in muscle protein breakdown to a degree that cannot be matched by the concurrent increase in muscle protein synthesis, resulting in net loss of muscle mass(7,8). It appears that in severe burns the body sacrifices skeletal muscle to meet the high demand of amino acids necessary for wound healing and for production of acute phase proteins(9,10).

To meet the elevated daily protein requirements of burn patients, the evidence-based guidelines of the American Burn Association (ABA) recommend protein intake of $1.5-3.0 \ g·kg^{-1}·day^{-1}$ (11). The European Society of Parenteral and Enteral Nutrition (ESPEN) recommends intake of $1.5-3\ g·kg^{-1}·day^{-1}$ in burned children and $1.5-2\ g·kg^{-1}·day^{-1}$ in burned adults(12). However, the ABA and ESPEN recommendations were based on evidence from a handful of studies, of which only one was specific to the effect of protein intake on protein kinetics in burn injury. Wolfe et al(13) compared the effects of protein intakes of $1.4
g·kg^{-1}·day^{-1}$ and $2.2\ g·kg^{-1}·day^{-1}$ over a 3-day period on protein metabolism in 6 severely burned adults using a cross-over study design. Protein turnover was significantly greater in the high versus low protein regimen but the difference between protein synthesis and breakdown was non-significant. Urea production rate was elevated only in the high protein regimen, suggesting that additional amount of the administered protein was partly excreted. Patterson et al.1997(14) investigated the effect of protein intake ranging from $1.15\ g·kg^{-1}·day^{-1}$ to $2.92
g·kg^{-1}·day^{-1}$ on protein metabolism of 12 pediatric patients with average burn size of 25% of body area. In agreement with the study by Wolfe et al.(13), higher protein intake was positively correlated with the urea production rate. Endogenous amino acid rate of appearance, an index of whole-body protein breakdown, was not associated with protein intake. However, the study design precluded concurrent measurement of whole body protein synthesis and thus net balance; there does not allow for definitive conclusions.

Even though high protein supplementation is an obvious intervention against muscle catabolism, the upper limits of the current recommendations are based on the findings of only two studies. Apart from small sample sizes, the whole body protein kinetics methods employed in those studies provide information only for substrates that appear in the plasma pool(15), which is an additional limitation of the current evidence. Even if very high protein intake did not provide further improvement in whole body protein kinetics, that might not be the case for muscle mass(15). Therefore, the potential beneficial effect of higher than currently recommended protein intake is understudied.

These preliminary data were included in abstract presentation Experimental Biology 2018, San Diego, California.

Malagaris I., Porter C., Herndon DN., Yu YM. “Method for the Improvement of Enrichment Estimation in Stable Isotope Metabolic Studies”. Session: “Mathematical models of organ systems, tissues or cells”. Experimental Biology 2018, San Diego, California

The isotopic purity of the tracer was estimated to be 99.1% and then a mixture with known enrichment of 20% was prepared. Visual inspection of the figure below shows that the new method seems to improve accuracy (close to the true enrichment) and precision (low spread of values)

Background: The main motivation for this aim is that currently the error in enrichment determination due to the isotopic impurity of the tracers used in kinetics studies is considered negligible and thus completely ignored. As mentioned in the significance section, a 99% isotopic purity means that any single labeled atom in the tracer’s has a probability of 99% to be indeed labeled (17). In Table 2 are displayed the natural isotopic abundances of atoms that are commonly encountered in metabolic research. The fractional abundances of atoms are considered constant; thus, according to the table, 98.9% of carbon atoms encountered in nature have Mass Number 12 and only 1.1% have Mass number 13. The fractional distribution of complex molecules composed of different atoms, each one having a specific number of naturally occurring isotopic species, can be found by polynomial expansion, convolution integral or Fast Fourier Transform algorithm (FFT)(18).

By using computational software the fractional distribution of complex molecules can be estimated. The input for the algorithm is the chemical formula (type and number of atoms), and the natural abundance of atoms, which is given in tables as the one displayed above (16). The chemical formula of the amino acid phenylalanine is $C_9H_{11}NO_2$. GCMS analysis requires chemical processing of the sample, which results in incorporation of additional atoms on the molecule. Moreover, the process of ionization results in the breaking of chemical bonds of the new molecule at specific positions, yielding a characteristic profile of mass spectrum. One abundant fragment of phenylalanine, has the chemical formula $C_{17}H_{30}N_1O_2Si_2$. Using an algorithm written in R programming language, the estimated natural isotopic distribution is the following.

Variance
$$Var(X) = E[(X - \mu_x)^2]$$ $$Var(X) = \sum_{x_k\in\mathbb{R}}(x_k - \mu_X)^2P_X(x_k)$$ $$Var(X) = E[X^2] - [EX]^2$$ $$EX^2 = \sum_{x_k\in\mathbb{R}}x_k^2P_X(x_k)$$

Variance is not linear
$$Var(\alpha X + b) = \alpha^2Var(X)$$
If $X_1, X_2,..., X_n$ are independent random variables, then the variance of the random variable $X = X_1, X_2,..., X_n$ is $$Var(X) = Var(X_1) + Var(X_2) +...+ Var(X_n)$$

Marginal PDFs
$$f_X(x) = \int_{-\infty}^{\infty}f_{XY}(x,y)dy \text{ for all x,}$$ $$f_Y(y) = \int_{-\infty}^{\infty}f_{XY}(x,y)dx \text{ for all y.}$$ $$F_{XY}(x,y) = \int_{-\infty}^y\int_{-\infty}^xf_{XY}(u,v)dudv$$ $$f_{XY}(x,y) = \frac{\partial^2} {\partial x\partial y}F_{XY}(x,y)$$

Law of Total Probability
$$P(A) = \int_{-\infty}^{\infty}P(A|X = x) f_X(x) dx$$

Law of Total Expectation
$$EY = \int_{-\infty}^{\infty}E[Y|X = x]f_X(x)dx$$ $$EY = E [E [Y|X] ]$$ Law of Total Variance $$Var(Y) = E[Var(Y|X)] + Var(E[Y|X])$$ $$Var(Y) \geq E[Var(X|Y)]$$

Convolution integral
If X and Y are independent random variables and Z = X + Y then $$f_Z(z) = \int_{-\infty}^{\infty}f_X(w)f_Y(z-w)dw$$

Theorem
If X and Y are independent and
$X \sim N(\mu_X,\sigma_X^2)$ and
$Y \sim N(\mu_Y,\sigma_Y^2)$ then
$$X + Y \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2)$$