Modified Krzywicki’s Equations to Quantify Myoglobin Forms on Meat Surfaces

Daqing Piao; Ranjith Ramanathan; Morgan L. Denzer; Morgan Pfeiffer; Gretchen Mafi; Daqing Piao; Ranjith Ramanathan; Morgan L. Denzer; Morgan Pfeiffer; Gretchen Mafi

doi:10.22175/mmb.18338

Introduction

Myoglobin, a strong absorber of light over the visible to near-infrared band, is the primary intracellular oxygen-binding protein of mammalian skeletal and cardiac muscle tissue (Schenkman et al., 1997). The color, the most significant sensory indicator of meat quality, dominates consumers’ decisions to purchase meat. In the United States, the discoloration of beef is estimated to cost the industry approximately $3.73 billion annually (Ramanathan et al., 2022). Therefore, it is important to quantitatively assess and predict beef color changes to facilitate timely inventory or restorative procedures to reduce food waste and the collateral environmental impact (Roy et al., 2023).

The perceived color of meat is strongly associated with myoglobin oxygen saturation (Schenkman et al., 1999b), or equivalently, the relative proportions of redox forms of myoglobin including oxymyoglobin (abbreviated as Oxy), deoxymyoglobin (abbreviated as Deoxy), and metmyoglobin (abbreviated as Met). In beef, Oxy imparts bright cherry-red, Deoxy appears purplish-red, and Met appears brownish-red. Meat discoloration starts between the bright red Oxy layer and the interior Deoxy layer, where oxygen partial pressure is inadequate to oxygenate all Deoxy molecules (Denzer et al., 2024), and Met develops due to the postmortem loss of Met reduction mechanism (Ramanathan and Mancini, 2018). Presently, color is assessed using instrumental and objective approaches. For example, most researchers use handheld colorimeters such as HunterLab MiniScan (Holman et al., 2015) or Minolta colorimeter (Henriott et al., 2020) to characterize color changes. These commercial devices of colorimetry operate in optics by illuminating the muscle with a diffuse field of light in the 400–700 nm spectral band and collecting the diffusely reflected light over a narrow aperture oriented at a small angle of approximately 10° with respect to the normal direction of the muscle surfaces. The raw data acquired by a colorimeter, which is called diffuse spectral reflectance, is used to estimate values such as L* (light vs. dark), a* (red vs. green), and b* (yellow vs. blue). The a* and b* are also used to calculate hue and chroma values. These parameters are widely accepted measures of the perceived color, but they indirectly represent the myoglobin forms that inform the underlying biochemical changes. A more robust assessment of the meat color to determine the status of discoloration may be achieved by direct assessment of the percentage number of Met, which requires algorithmic processing capable of resolving the proportions of the 3 forms of myoglobin from the spectral reflectance measured from the surfaces.

The American Meat Science Association (AMSA) guidelines (King et al., 2023) recommend 2 approaches to quantify the percentage numbers of myoglobin forms on meat surfaces. In AMSA method 1, the spectral reflectance at 4 isosbestic wavelengths (474 nm, 525 nm, 572 nm, and 610 nm) is converted to a Kubelka-Munk (KM) function according to the KM model of light-tissue interaction for the geometry (Piao and Sun, 2021) to which the colorimeter complies. The KM functions at the 4 isosbestic wavelengths are then subjected to a set of combination formulae to deduce the percentage numbers of the myoglobin forms. In AMSA method 2, the spectral reflectance at 4 wavelengths (474 nm, 525 nm, 572 nm, and 730 nm but 700 nm in practice due to the limitation of the spectral range of handheld colorimeter) are converted to a spectral reflex attenuance (Krzywicki, 1979) by taking the natural logarithm of the reciprocal of the spectral reflectance (Shibata, 1962). The spectral reflex attenuance is modeled to be proportional to the spectral absorption, based on the assumption that the path length of the light is spectrally independent, i.e., it remains the same over the entire spectrum of the measurement. The resulting spectral reflex attenuance is then subjected to a set of equations to compute the percentage numbers of Met and Deoxy, with which the percentage number of Oxy is obtained by subtracting the 2 former numbers from 100% (Krzywicki, 1979). Krzywicki’s equations are relatively easier to implement than the AMSA method 1 because it does not require creating myoglobin form standards. However, Krzywicki’s equations could be prone to giving unrealistic numbers of myoglobin forms. It is thus imperative to have a more in-depth understanding of Krzywicki’s equations to promote more robust practical uses. In this work, we aim to develop a more comprehensive analysis of Krzywicki’s equations for myoglobin quantification on meat surfaces.

Methods and Materials

The tissue-optics foundation of Krzywicki’s equations

Molar extinction coefficient

The spectral absorption of each myoglobin form can be characterized by molar extinction coefficient (Jacques, 2013) (or molar absorbance) $ɛ M b F o r m (λ)$ , which represents a unit of per length per molar concentration $(m m · M) − 1$ , where M represents the number of solutes in moles per unit volume. An isosbestic point refers to a wavelength at which 2 or 3 pigments have the same molar extinction coefficient. For myoglobin, the molar extinction coefficients of the 3 forms of myoglobin (Mb) are compared in Figure 1 by using the data of 10 nm interval over (480–650 nm) presented in (Piao et al., 2022), which was compiled from (Tang et al., 2004) and grossly extrapolated down to 460 nm and up to 700 nm to cover the isosbestic points used for the estimation of the percentage numbers of myoglobin forms. Over the spectral range of 460 nm to 650 nm, there are 7 wavelengths that are isosbestic for 2 of the 3 forms of myoglobin and one wavelength that is practically isosbestic for all 3 forms of myoglobin. As is marked, the wavelength of 474 nm is isosbestic for Oxy and Met, the wavelength of 572 nm is isosbestic for Oxy and Deoxy, the wavelength of 610 nm is isosbestic for Deoxy and Met, and the wavelength of 525 nm is isosbestic for all the 3 forms of myoglobin. The 3 dual-isosbestic wavelengths at 474 nm, 572 nm, and 610 nm, and the one triple-isosbestic wavelength at 525 nm, are used for the quantification of the percentage numbers of the myoglobin forms by both methods in the AMSA guidelines (King et al., 2023). Table 1 has tabulated the distribution of the isosbestic properties at the 4 wavelengths of 474 nm, 525 nm, 572 nm, and 610 nm, in which the cells marked with the same number of “*” represent sharing the same extinction coefficient (Krzywicki, 1979).

Table 1.

The isosbestic wavelengths over the 460 nm–700 nm for the extinction coefficients of myoglobin forms. Different cells having the same number of asterisks denote that the extinction coefficient at that wavelength for that form of myoglobin is the same. The wavelength of 730 nm is marked as a reference, as it is implemented as a reference point at which the absorption of myoglobin is neglected, according to Krzywicki’s approach. The cells marked by the same number of “*” correspond to having the same extinction coefficient. “NA” indicates “not applicable.”

View Larger Table

Order of the Wavelength	$ɛ Oxy$		$ɛ Deoxy$		$ɛ Met$
$λ 1 = 474$ nm	$ɛ Oxy (λ 1)$	**	$ɛ Deoxy (λ 1)$	NA	$ɛ Met (λ 1)$	**
$λ 2 = 525$ nm	$ɛ Oxy (λ 2)$	**	$ɛ Deoxy (λ 2)$	**	$ɛ Met (λ 2)$	**
$λ 3 = 572$ nm	$ɛ Oxy (λ 3)$	***	$ɛ Deoxy (λ 3)$	***	$ɛ Met (λ 3)$	NA
$λ 4 = 610$ nm	$ɛ Oxy (λ 4)$	NA	$ɛ Deoxy (λ 4)$	*	$ɛ Met (λ 4)$	*
$λ r = 730$ nm (reference)	$ɛ Oxy (λ r)$	NA	$ɛ Deoxy (λ r)$	NA	$ɛ Met (λ r)$	NA

Figure 1.

Spectral absorptions of the myoglobin forms including oxymyoglobin [Oxy], pure myoglobin or deoxymyolgobin [Deoxy], and metmyoglobin [Met]. The data over the spectral range of 480 nm–650 nm were replicated from Piao et al. (2022). The thinner lines at the spectral range over 460 nm to 480 nm and 650 nm to 700 nm are the results of extrapolation.

Constitution of the spectral absorption of meat by myoglobin

Multiplying the molar extinction coefficient $ɛ M b F o r m (λ)$ with the molar concentration (solutes in moles per unit volume) leads to the absorption coefficient $μ a$ in the unit of 1/length (1/mm is used here). Multiplying the absorption coefficient with the path length of light gives the total attenuation (treated as the absorbance in Krzywicki’s method) that appears as the magnitude of the negative exponent in the equation of the Lambert-Beer Law. The absorbance quantifies the reduction of light intensity over the length of propagation in a uniform medium. For a scattering-free medium (thus, no diffuse reflectance is produced other than the specular reflection at the surfaces, and only transmission is possible), the photon propagates without encountering directional change. The total attenuation or the absorbance when comparing the amount of light detected with the amount of light illuminated is proportional to the absorption. For a scattering medium (therefore, diffuse reflectance is available regardless of the presence of specular reflection on the surfaces), the photons take complicated paths of propagation between the locations of illuminating onto and escaping from the surface. The total attenuation or absorbance when comparing the amount of light detected with the amount of light illuminated is the effects of absorption and scattering combined in a coupled non-linear configuration, which requires a customized model for the retrieval of spectral absorption from the spectral reflectance with the consideration of the spectral dependence of scattering (Jacques, 2013).

We hereby use [tMB] to denote the total molar concentration of myoglobin in [ $μ$ M]. [tMB] is the summation of the concentrations of 3 sub-forms: [Oxy] represents the total molar concentration of Oxy in [ $μ$ M], [Deoxy] represents the total molar concentration of Deoxy in [ $μ$ M], and [Met] represents the total molar concentration of Met in [ $μ$ M]. Therefore [tMB] = [Oxy] + [Deoxy] + [Met], and it should remain unchanged for the muscle (Trout, 1990). We then use x, y, and z to represent respectively the percentage numbers of [Oxy], [Deoxy], and [Met] in the total myoglobin content. Specifically, we have the following:

$x = [O x y] [O x y] + [Deoxy] + [M e t] = [O x y] [tMB] = [O x y] %$

(1)

$y = [Deoxy] [O x y] + [Deoxy] + [M e t] = [Deoxy] [tMB] = [Deoxy] %$

(2)

$z = [M e t] [O x y] + [Deoxy] + [M e t] = [M e t] [tMB] = [M e t] %$

(3)

$x + y + z = 1$

(4)

Note that the use of $x$ , $y$ , and $z$ here to represent the percentage numbers [Oxy]%, [Deoxy]%, and [Met]%, respectively, differ from those in Krzywicki’s proposal (Krzywicki, 1979) wherein $x$ represented [Deoxy]% and $y$ represented [Met]%, and $z$ represented [Oxy]%. We chose to represent [Oxy]%, [Deoxy]%, and [Met]% in the order of $x$ , $y$ , and $z$ , following the order of the 3 forms assessed by Piao et al. (2022) and Denzer et al. (2024).

At one of the wavelengths shown in Figure 1 or Table 1, the muscle absorption of light comes from myoglobin and other pigments. If the absorption of pigments other than myoglobin is lumped to be represented by $μ a 0$ , we have the following:

$μ a (λ n) = ɛ O x y (λ n) [O x y] + ɛ D e o x y (λ n) [D e o x y] + ɛ M e t (λ n) [M e t] + μ a 0 (λ n)$

$= [t M B] · {ɛ O x y (λ n) · x + ɛ D e o x y (λ n) · y + ɛ M e t (λ n) · z} + μ a 0 (λ n)$

(5)

Then, the total extinction of exclusively the myoglobin is obtained as follows:

$μ a (λ n) − μ a 0 (λ n) [t M B] = {ɛ O x y (λ n) · x + ɛ D e o x y (λ n) · y + ɛ M e t (λ n) · z}$

(6)

An algorithm to estimate the percentage numbers of the myoglobin forms from the spectral reflectance generally proceeds to recover the myoglobin-exclusive total extinction represented by Equation (6), from which the individual percentage number of each of the myoglobin forms may be calculated by linear transformation involving matrix analytics. Alternatively, derivations based on assessing Equation (6) at multiple isosbestic wavelengths could lead to a convenient set of equations that require only simple 2-stage algebraic operations on the spectral measurements to estimate the percentage numbers of the myoglobin forms. Krzywicki’s equations take this 2-stage approach of algebraic operations that are implementable on an EXCEL spreadsheet. Supplemental Material 1 presents a detailed rederivation leading to the complete set of Krzywicki’s equations, including the one for [Oxy]% that was omitted in the original proposal of Krzywicki (Krzywicki, 1979).

The 2-stage feature of Krzywicki’s original approach to independently calculate the percentage numbers of myoglobin forms

Krzywicki’s original approach to estimating the percentage numbers of myoglobin forms, in reference to Supplemental Material 1, can be summarized as the following 2 stages. Stage 1 operates on the measured spectral reflectance to obtain a value representing the spectral absorptivity, which hereby is denoted by a generic symbol of $U abs$ . Using $U abs$ converted from the spectral reflectance, the values of $U abs$ at 4 isosbestic wavelengths ( $λ 1 = 474 nm, λ 2 = 525 nm, λ 3 = 572 nm, λ 4 = 610 nm$ ) and a reference wavelength ( $λ r = 730 nm$ ) are obtained by weighting configured as follows:

$U abs (λ 1) = 0.6 × U abs (470 nm) + 0.4 × U abs (480 nm)$

(7)

$U abs (λ 2) = 0.5 × U abs (520 nm) + 0.5 × U abs (530 nm)$

(8)

$U abs (λ 3) = 0.8 × U abs (570 nm) + 0.2 × U abs (580 nm)$

(9)

$U abs (λ 4) = U abs (610 nm)$

(10)

$U abs (λ r) = U abs (730 nm) ≈ U abs (700 nm)$

(11)

Stage 2 of Krzywicki’s proposal implements the values of

$U abs$ at the 4 isosbestic wavelengths and a reference wavelength as obtained with Equations (7 )–(11) into the following 3 equations to independently calculate the respective percentage numbers of [Oxy]%, [Deoxy]%, and [Met]%.

$x = 2.159 {1 − 2.375 U abs (λ 4) − U abs (λ r) U abs (λ 2) − U abs (λ r)}$

(12)

$y = 2.375 {1 − U abs (λ 1) − U abs (λ r) U abs (λ 2) − U abs (λ r)}$

(13)

$z = {1.395 − U abs (λ 3) − U abs (λ r) U abs (λ 2) − U abs (λ r)}$

(14)

In realization, Krzywicki recommended deducing [Oxy]% by subtracting [Deoxy]% and [Met]% from 100% over independently calculating [Oxy]% using the originally omitted Equation (12). It remains to be clarified whether this recommendation has been promoted as a convenience or as a necessity.

Two algorithms for calculating the spectral absorptivity properties in stage one of Krzywicki’s approach

We use $R ∞ (λ)$ to denote the spectral reflectance obtained by a colorimeter. In converting the spectral reflectance values of $R ∞ (λ)$ to the spectral absorptivity parameters in stage one of Krzywicki’s approach, we implement 2 algorithms including Krzywicki’s original method concerning reflex attenuance, which is denoted as $A ref (λ)$ . The 2 algorithms for converting the spectral reflectance to spectral absorptivity properties are detailed as follows.

The original implementation of Krzywicki’s approach to calculate the percentage numbers of myoglobin forms by converting the spectral reflectance to the reflex attenuance in stage 1

In reference to Supplemental Material 2.1, stage 1 of Krzywicki’s original approach concerns the following set of equations:

$A ref (λ n) = l n [1 R ∞ (λ n)]$

(15)

$A ref (λ 1) = 0.6 × A ref (470 nm) + 0.4 × A ref (480 nm)$

(16)

$A ref (λ 2) = 0.5 × A ref (520 nm) + 0.5 × A ref (530 nm)$

(17)

$A ref (λ 3) = 0.8 × A ref (570 nm) + 0.2 × A ref (580 nm)$

(18)

$A ref (λ 4) = A ref (610 nm)$

(19)

$A ref (λ r) = A ref (730 nm) ≈ A ref (700 nm)$

(20)

Stage 2 then contains the following set of materialized forms of Equations (12 )–(14):

$x = 2.159 {1 − 2.375 A ref (λ 4) − A ref (λ r) A ref (λ 2) − A ref (λ r)}$

(21)

$y = 2.375 {1 − A ref (λ 1) − A ref (λ r) A ref (λ 2) − A ref (λ r)}$

(22)

$z = {1.395 − A ref (λ 3) − A ref (λ r) A ref (λ 2) − A ref (λ r)}$

(23)

An alternative implementation of Krzywicki’s approach to calculate the percentage numbers of myoglobin forms by converting the spectral reflectance to spectral absorption in stage 1

In reference to Supplemental Material 2.2, we take an alternative set of equations in stage 1, which converts the spectral reflectance to spectral absorption denoted by $μ a (λ)$ . The equations are as follows:

$μ a (λ) = 0.25 [1 (2 π R ∞) 2 + 0.16] 1 2.2 − 0.11875$

(24)

$μ a (λ 1) = 0.6 × μ a (470 nm) + 0.4 × μ a (480 nm)$

(25)

$μ a (λ 2) = 0.5 × μ a (520 nm) + 0.5 × μ a (530 nm)$

(26)

$μ a (λ 3) = 0.8 × μ a (570 nm) + 0.2 × μ a (580 nm)$

(27)

$μ a (λ 4) = μ a (610 nm)$

(28)

$μ a (λ r) = μ a (730 nm) ≈ μ a (700 nm)$

(29)

Stage 2 then contains the following set of materialized forms of Equations (12 )–(14):

$x = 2.159 {1 − 2.375 μ a (λ 4) − μ a (λ r) μ a (λ 2) − μ a (λ r)}$

(30)

$y = 2.375 {1 − μ a (λ 1) − μ a (λ r) μ a (λ 2) − μ a (λ r)}$

(31)

$z = {1.395 − μ a (λ 3) − μ a (λ r) μ a (λ 2) − μ a (λ r)}$

(32)

Summary comparison of the 2 algorithms in stage 1 to implement Krzywicki’s equations

We refer to the configurations shown in Figure 2 to summarize how Krzywicki’s equations are implemented by using the 2 algorithms specified respectively in the sections titled “The original implementation of Krzywicki’s approach to calculate the percentage numbers of myoglobin forms by converting the spectral reflectance to the reflex attenuance in stage 1” and “An alternative implementation of Krzywicki’s approach to calculate the percentage numbers of myoglobin forms by converting the spectral reflectance to spectral absorption in stage 1.” There are 2 paths of the algorithms, as are shown schematically, to be operated on the same spectral reflectance that is represented by $R ∞ (λ)$ marked by the up-directing solid arrow tailing an oblique cylindrical structure representing the geometry of the collecting optics.

The top path converts the spectral reflectance to reflex attenuance according to Equations (15), from which the values at 4 isosbestic points of 474 nm, 525 nm, 572 nm, and 610 nm, and a reference point of 700 nm are deduced using Equations (16 )–(20) for feeding into Equations (21 )–(23) to independently calculate respectively [Oxy%], [Deoxy]%, and [Met]%. To assess the effect of the reference value chosen as the baseline, this path is implemented at 3 levels of the baseline assessed at 730 nm. The first path marked by 1a implements Equations (21 )–(23) by treating $A ref (730 nm) = A ref (700 nm) × 100 %$ , that is, to take the value at 700 nm to approximate the value at 730 nm according to ASMA (King et al., 2023). The second path marked by 1b implements Equations (21 )–(23) by treating $A ref (730 nm) = A ref (700 nm) × 20 %$ , that is, to give a baseline reference value that is 20% of that at 700 nm to make it potentially more realistic in representing the value at the lower-absorbing wavelength of 730 nm according to the original recommendation of Krzywicki (Krzywicki, 1979). The third path marked by 1c implements Equations (21 )–(23) by treating $A ref (730 nm) = A ref (700 nm) × 0 %$ , that is, to assume the baseline reference level at 730 nm to be negligible.

Figure 2.

A general schematic of using a colorimeter produced spectral reflectance to estimate myoglobin forms. Given that a source of the ideally uniform spectral profile of $S in (λ)$ is used to diffusely illuminate the medium, the diffuse reflectance acquired for compliance with the Kubelka-Munk (KM) model is $R ∞ (λ)$ . The 2 shaded blocks represent 2 paths of implementing the same set of x, y, and z of Krzywicki’s formula, but differ in how the absorption variable is inverted from the spectral reflectance. Within each shaded block, 3 paths correspond to 3 different levels of the baseline reference to be implemented in calculating x, y, and z.

The bottom path converts the spectral reflectance to absorption via the new algorithm of Equation (24), from which the values at 4 isosbestic points of 474 nm, 525 nm, 572 nm, and 610 nm, and a reference point of 700 nm, are deduced using Equations (25 )–(29) for feeding into Equations (30 )–(32) to independently calculate respectively [Oxy%], [Deoxy]%, and [Met]%. To assess the effect of the reference value chosen as the baseline, this path is implemented at 3 levels of the baseline assessed at 730 nm. The first path marked by 3a implements Equations (30 )–(32) by treating $μ a (730 nm) = μ a (730 nm) × 100 %$ according to AMSA (King et al., 2023). The second path marked by 3b implements Equations (30 )–(32) by treating $μ a (730 nm) = μ a (700 nm) × 20 %$ , that is, to give a baseline reference absorption that is 20% of that at 700 nm considering that the absorption at 730 nm is likely lower than at 700 nm. Finally, the third path marked by 3c implements Equations (30 )–(32) by treating $μ a (730 nm) = μ a (700 nm) × 0 %$ , that is, to assume the baseline reference level at 730 nm to be negligible.

These combinations of the algorithmic procedures are designed to assess how the same set of equations for computing x, y, and z could be affected by the spectral absorption or absorptivity parameter, and for the same type of absorption or absorptivity parameter, how the choice of the baseline reference level projected at 730 nm affects the quantification of x, y, and z.

Measuring surface color on steaks displayed under retail conditions

Two separate data sets were used to test the revised implementations of Krzywicki’s equations. The sub-aim 1 with the first data set was to compare the performance of the 2 algorithms (Krzywicki’s original and revised equations) on estimating the percentage numbers of the 3 forms of myoglobin at different configurations of the baseline reference (n = 7; longissimus lumborum). This sub-aim helped identify the difference in the effect of the baseline reference between the 2 algorithms of converting the spectral reflectance to spectral absorptivity properties. The sub-aim 2 used 44 loins to specifically assess how the 2 algorithms of sub-aim one at the same setting of neglecting the baseline reference at 730 nm would perform in estimating the increase of [Met]% expected for color-stable and color-labile muscles in retail display. The 44 loins were the combination of the muscles used in the sub-aim one, 3 previous independent studies (Piao et al., 2022; Denzer et al., 2024; Denzer et al., 2025), and one independent procurement. The details of these 5 groups of a total of 44 loins, including the number of steaks, the number of repeated measurements per steak, and the length of the retail display, are included in Supplemental Material Table A1. The choices of loins included psoas major (PM), longissimus lumborum (LL), and semitendinosus (ST). The number of steaks used in each set of retail display was 7 or 8. When combined, there were 14 (7+7) PM muscles, 22 (7+7 + 8) LL muscles, and 8 ST muscles. The number of repeated measurements by colorimeter on each steak was 3, 5, or 6. The number of days of retail display on which the colorimeter measurements were obtained was 6 or 7. The measurements from 7 steaks of LL muscles over a retail display of 6 d (day 0 to day 6) were those used in the sub-aim one.

The sample collections for both data sets were similar. United States Department of Agriculture Choice PM, LL, and ST muscles were collected from a commercial beef purveyor 5 d postmortem. The vacuum-packaged loins were transported on ice to the Oklahoma State University Robert Kerr Food and Agricultural Products Center. Muscle pH was measured, on all lions of the 3 choices, after arrival at 3 random locations across each sub-primal with a pH probe (Handheld HI 99163; probe FC232; Hanna Instruments). All loins were within a normal pH range of 5.5–5.6 (standard error = 0.1 for all 3 muscle types). Approximately 6 d postmortem, muscles were sliced into 2.54-cm thick steak from the anterior end of each loin and were selected for retail display and colorimeter measurements. Steaks were packaged immediately after cutting in white Styrofoam™ trays overwrapped with polystyrene polyvinyl chloride film (15,500–16,275 cm³, O₂/m²/24 h at 23°C, E-Z Wrap Crystal Clear Polyvinyl Chloride Wrapping Film, Koch Supplies, Kansas City, MO, USA). All steaks were placed in a coffin-style retail display case for up to 7 d at 2 ± 1°C, during which the measurements by the colorimeter were conducted daily. There was continuous LED lighting throughout the retail display. The lighting differed between 1,100 LUX (Philips LED lamps, 12 watts, 48 in., color temperature = 3,500 K, 54 Phillips, China) and 1,612 to 2,152 lux (Philips Deluxe Warm White Fluorescent lamps; Andover, MA, USA; color rendering index = 86; color temperature = 3,000°K). All packages were rotated daily to minimize the effects of variation in light intensity or temperature due to location.

Raw color analysis

During the retail display, the daily measurement of the instrumental color of steaks was conducted using a HunterLab 4500L MiniScan EZ Spectrophotometer (2.5-cm aperture, illuminant A, and 10° standard observer angle; HunterLab, Reston, VA, USA). The surface of each steak was read at 3, 5, or 6 randomly selected positions, and the surface color was characterized by the Commission Internationale de l´Eclairage (CIE) L*, a*, and b* values and reflectance from 400 nm to 700 nm. Chroma $(a *) 2 + (b *) 2$ was determined using CIE a* and b* values, representing the red intensity of the color (King et al., 2023).

Offline data-processing implementing the 2 algorithms of calculating the percentage numbers of the forms of myoglobin

The raw spectral profile acquired by HunterLab MiniScan, which covered wavelengths from 400 nm to 700 nm at a 10 nm spectral interval, was transferred to a computer for offline processing of the percentage numbers of myoglobin forms. The offline processing including statistical analyses was programmed in MATLAB R2022a (MathWorks, Natick, MA, USA) or implemented in GraphPad Prism 6 (GraphPad Software, GraphPad Prism V6.0, La Jolla, CA, USA). The spectral reflectance values of $R ∞ (λ)$ were converted to reflex attenuance and spectral absorption, according to Equations (15) and (24), respectively The implementation of Equation (24), as specified in Supplemental Material 2, involved assuming that the reduced scattering of the medium is constant at 1 mm⁻¹. The resulting spectral absorptivity parameters at 470 nm and 480 nm were summed at weights of respectively 60% and 40%, according to Equations (16) or (25), to interpolate the value at $λ 1 = 474 nm$ . The resulted spectral absorptivity parameters at 520 nm and 530 nm were summed at weights of respectively 50% and 50%, according to Equations (17) or (26), to interpolate the value at $λ 2 = 525 nm$ . The resulted spectral absorptivity parameters at 570 nm and 580 nm were summed at weights of respectively 80% and 20%, according to Equations (18) or (27), to interpolate the value at $λ 3 = 572 nm$ . The resulted spectral absorptivity parameter at 610 nm was used as a value at $λ 4 = 610 nm$ . These deduced spectral absorptivity at the 4 wavelengths were then substituted into the respective equations in the section titled “Two algorithms of calculating the spectral absorptivity properties in the stage 1 of Krzywicki’s approach” to calculate [Oxy]%, [Deoxy]%, and [Met]%, at different levels of the baseline reference projected for 730 nm, as is detailed in Figure 2.

Statistical Analysis

The experimental design was a randomized complete block. The loin from each animal served as a block, i.e., each loin was used as a random term. Previous studies reported that LL, PM, and ST muscles have differences in color stability (Seyfert et al., 2006). Hence, data for each muscle type was analyzed separately. The main treatment effect was time. Data were analyzed separately for each variable: [Oxy]%, [Deoxy]%, and [Met]%. HunterLab MiniScan measurements were averaged to calculate the mean and standard deviation (STD). For surface color and pH, the random effect was each loin. For surface color measurements, day was a repeated measure observed in the study. The repeated measures covariance-variance structure was determined by evaluating the Akaike’s information criterion (AICC) output. The GLIMMIX procedure of SAS (SAS 9.4; SAS Inst., Cary, NC, USA) was used to determine the least-squares means, and significance was considered at P < 0.05. The least-squares means were separated using the pairwise differences (PDIFF) option (least-squares difference) and were significant at P < 0.05.

One outcome of this study is assessing how the change of [Oxy]%, [Deoxy]%, or [Met]% over the retail display has been compared with the change of chroma. For such a comparison between the percentage number of one myoglobin form and the chroma, the data groups have the same data structures and sample sizes. These 2 data groups were thus subjected to a 2-tailed unpaired t-test. Another outcome of the study is assessing how the change of specifically [Met]% compared with chroma over the retail display differs among 3 types of muscles. For such comparison, the data groups have the same data structures but different sample sizes. These 3 data groups were then subjected to a 1-way analysis of variance (ANOVA) (Kim, 2014). For both the t-test and ANOVA, the data groups were a priori calculated of the mean value and the standard deviation in MATLAB for loading into GraphPad Prism. The P value of <0.05 was considered statistically significant.

Results

The results of applying each of the 2 paths of algorithms to convert the spectral reflectance of the steaks to spectral absorptivity properties for substituting into Krzywicki’s equations to obtain independently [Oxy]%, [Deoxy]% and [Met]% are presented in the following sections. To make the comparisons between the 2 algorithms more explicit to help assess which algorithm could be more robust than the other and to also inform an optimal practice in setting the baseline reference value projected for 730 nm using the value at 700 nm, the resulting percentage numbers of the myoglobin forms estimated for 7 LL steaks over 6 d of retail display (5 repeated measurements on each steak per day) by each of the 2 algorithms are presented in 2 separate figures, Figures 3 and 4, according to the following general guidelines.

Figure 3.

The percentage forms of Oxy, Deoxy, and Met, calculated by Equations (21), (22), and (23), at 3 different settings of the baseline value of reflex attenuance projected for 730 nm. The top panel corresponds to implementing the algorithm of Aref with the reference reflex attenuance at 730 nm set as 100% of the value obtained at 700 nm. The middle panel corresponds to implementing algorithm of Aref with the reference reflex attenuance at 730 nm set as 20% of the value obtained at 700 nm. The bottom panel corresponds to implementing algorithm of Aref with the reference reflex attenuance at 730 nm set as 0. The second column corresponds to [Oxy]% calculated by subtracting [Deoxy]% and [Met]% from 100%. The shades mark the range of realistic values, 0%–100%.

Figure 4.

The percentage forms of OxyMb, DeoxyMb, and MetMb, calculated by Equations (30), (31), and (32), at 3 different settings of the baseline value of absorption projected for 730 nm. The top panel corresponds to implementing the algorithm of absorption via Piao and Sun’s model with the baseline reference at 730 nm set as 100% of the value obtained at 700 nm. The middle panel corresponds to implementing algorithm absorption via Piao and Sun’s model with the baseline reference at 730 nm set as 20% of the value obtained at 700 nm. The bottom panel corresponds to implementing the algorithm of absorption via Piao and Sun’s model with the baseline reference at 730 nm set as 0. The second column corresponds to [Oxy]% calculated by subtracting [Deoxy]% and [Met]% from 100%. The shades mark the range of realistic values, 0%–100%.

In Figures 3 and 4, the change in the percentage number of myoglobin form is plotted in a bar chart as a function of the day of the retail display. Each bar indicates the mean value and the STD of that day after averaging over 35 measurements (5 positions per sample and 7 samples per d). Each figure contains 4 columns of subfigures. Three columns (ordered as leftmost, second from the rightmost, and right most) contain [Oxy]%, [Deoxy]%, and [Met]% obtained by using the materialized forms of Equations (12), (13) and (14), respectively. A fourth column (ordered as second from the leftmost) represents the [Oxy]% calculated indirectly by subtracting [Deoxy]% and [Met]% from 100%. The abscissa of all subfigures is scaled over the range of −25% to 175% to display the distribution of most but not all values of the estimation as some values have been significantly beyond the realistic range of 0%–100%. The shaded area in each sub-figure represents the realistic range of over 0%–100%. The 3 panels, in descending order from top to bottom, correspond to the implementation of the same algorithm but at different levels of the baseline reference value of the spectral absorptivity properties projected for 730 nm. The top panel corresponds to implementing an algorithm of choice in Krzywicki’s equations with the baseline reference value of spectral absorptivity set as 100% of the value at 700 nm to approximate that at 730 nm according to AMSA (King et al., 2023). The middle panel corresponds to implementing the same algorithm of choice as in the middle panel but with the baseline reference value of spectral absorptivity set as 20% of the value obtained at 700 nm to approximate that at 730 nm, according to the original recommendation of Krzywicki (Krzywicki, 1979) considering the absorbance at 730 nm to be lower than that at 700 nm. The bottom panel corresponds to implementing the same algorithm of choice as the top panel but with the baseline reference value of spectral absorptivity set as 0% of the value obtained at 700 nm to treat it as negligible.

The percentage numbers of myoglobin forms estimated by using algorithm #1 to deduce reflex attenuance in stage 1 for running Krzywicki’s equations

Figure 3 presents the percentages of myoglobin forms estimated by stage 1 implementing the reflex attenuance with Krzywicki’s equations to find the [Oxy]%, [Deoxy]%, and [Met]%. The leftmost column is [Oxy]% by direct computation using Equation (21), and the second column is [Oxy]% by subtracting [Deoxy]% and [Met]% of the right 2 columns from 100%.

The top row corresponds to using 100% of $A ref$ at 700 nm as the baseline reflex attenuance projected for $A ref$ at 730 nm that resulted in [Deoxy]% increasing noticeably from 6.32% ± 1.55% on day 0 to 18.05% ± 15.41% on day 6, and [Met]% increasing from 20.59% ± 1.69% at day 0 to 38.46% ± 10.52% at day 6. Subtracting [Deoxy]% and [Met]% from 100% resulted in [Oxy]% reducing globally from 73.10% on day 0 to 43.49% on day 6. Direct computation of [Oxy]% independent of [Deoxy]% and [Met]% resulted in [Oxy]% decreasing from an above 100% value (108.32% ± 6.43%) on day 0 to a negative value (−35.37% ± 57.04%) on day 6. The values plotted at the top panel of Figure 3 are referred to in Table 2.

Table 2.

The percentage of myoglobin forms estimated by implementing the reflex attenuance within Krzywicki’s equations and using 100% of the reflex attenuance at 700 nm as the baseline reference attenuance at 730 nm.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
x: [Oxy]%	108.32 ± 6.43	92.42 ± 9.28	83.29 ± 13.80	73.77 ± 17.83	53.92 ± 29.40	7.04 ± 60.53	−35.37 ± 57.04
100%−(y+z)	73.10	69.17	66.50	66.56	61.62	54.27	43.49
y: [Deoxy]%	6.32 ± 1.55	8.22 ± 1.69	9.23 ± 1.39	7.68 ± 2.14	10.18 ± 1.73	11.48 ± 6.77	18.05 ± 15.41
z: [Met]%	20.59 ± 1.69	22.61 ± 1.82	24.28 ± 2.30	25.76 ± 3.20	28.21 ± 4.22	34.25 ± 8.00	38.46 ± 10.52

The middle row corresponds to using 20% of $A ref$ at 700 nm as the baseline reflex attenuance for $A ref$ at 730 nm as it may be a better approximation of the lower reflex attenuance at 730 nm than at 700 nm. This resulted in [Deoxy]% increasing noticeably from 4.34% ± 1.02% on day 0 to 11.42% ± 8.94% on day 6, and [Met]% increasing from 26.46% ± 1.34% on day 0 to 38.86% ± 7.00% on day 6. Subtracting [Deoxy]% and [Met]% from 100% resulted in [Oxy]% reducing globally from 69.20% on day 0 to 49.72% on day 6. Direct computation of [Oxy]% independent of [Deoxy]% and [Met]% resulted in [Oxy]% being all negative over the days of display. The values plotted at the top panel of Figure 3 are referred to in Table 3.

Table 3.

The percentage of myoglobin forms estimated by implementing the reflex attenuance within Krzywicki’s equations and using 20% of the reflex attenuance at 700 nm as the baseline reference attenuance at 730 nm.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
x: [Oxy]%	−17.69 ± 12.69	−18.72 ± 10.60	−26.01 ± 13.97	−32.02 ± 18.28	−48.84 ± 25.03	−85.74 ± 49.64	−122.59 ± 45.35
100%−(y+z)	69.20	66.71	64.79	64.86	61.34	56.54	49.72
y: [Deoxy]%	4.34 ± 1.02	5.87 ± 1.21	6.57 ± 1.00	5.48 ± 1.53	7.18 ± 1.17	7.76 ± 4.07	11.42 ± 8.94
z: [Met]%	26.46 ± 1.34	27.42 ± 1.41	28.64 ± 1.77	29.66 ± 2.43	31.48 ± 3.04	35.70 ± 5.58	38.86 ± 7.00

The bottom row corresponds to using 0% of $A ref$ at 700 nm as the baseline reflex attenuance $A ref$ at 730 nm, i.e., treating $A ref$ at 730 nm as 0. That resulted in [Deoxy]% increasing noticeably from 4.02% ± 0.94% on day 0 to 10.47% ± 8.08% on day 6, and [Met]% increasing from 27.40% ± 1.28% on day 0 to 38.91% ± 6.47% on day 6. Subtracting [Deoxy]% and [Met]% from 100% resulted in [Oxy]% reducing globally from 68.58% on day 0 to 50.62% on day 6. Direct computation of [Oxy]% independent of [Deoxy]% and [Met]% resulted in [Oxy]% being all negative values from day 0 to day 6. The values plotted at the top panel of Figure 3 are referred to in Table 4.

Table 4.

The percentage of myoglobin forms estimated by implementing the reflex attenuance within Krzywicki’s equations and not considering a reference reflex attenuance.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
x: [Oxy]%	−37.78 ± 13.06	−37.23 ± 10.73	−44.15 ± 13.88	−49.61 ± 18.14	−65.73 ± 24.08	−±47.27	−135.89 ± 43.02
100%--(y+z)	68.58	66.30	64.50	64.57	61.29	56.87	50.62
y: [Deoxy]%	4.02 ± 0.94	5.47 ± 1.13	6.13 ± 0.93	5.12 ± 1.43	6.68 ± 1.09	7.18 ± 3.70	10.47 ± 8.08
z: [Met]%	27.40 ± 1.28	28.22 ± 1.34	29.36 ± 1.68	30.31 ± 2.30	32.02 ± 2.84	35.94 ± 5.19	38.91 ± 6.47

The percentage numbers of myoglobin forms estimated by using algorithm #2 in stage 1 deducing absorption directly from the spectral reflectance for running Krzywicki’s equation

Figure 4 presents the percentages of the myoglobin forms estimated by implementing the absorption deduced by Piao and Sun’s model (Piao and Sun, 2021) with Krzywicki’s equations to find the [Oxy]%, [Deoxy]%, and [Met]%. The leftmost column is [Oxy]% by direct computation using Equation (30), and the second column is [Oxy]% by subtracting [Deoxy]% and [Met]% of the right 2 columns from 100%.

The top row corresponds to using 100% of the absorption estimated by implementing the absorption at 700 nm using Piao and Sun’s model as the baseline reference absorption at 730 nm. This resulted in [Deoxy]% increasing globally from 15.58% ± 3.27% on day 0 to 33.10% ± 24.29% on day 6, and [Met]% increasing significantly from −5.02% ± 5.06% on day 0 to 35.72% ± 20.53% on day 6. Subtracting [Deoxy]% and [Met]% from 100% resulted in [Oxy]% reducing gradually from 89.43% on day 0 to 31.18% on day 6. Direct computation of [Oxy]% independent of [Deoxy]% and [Met]% resulted in [Oxy]% reducing globally from above 100% on day 0 through day 5 to 63.87% ± 53.85% on day 6. The values plotted at the top panel of Figure 4 are referred to in Table 5.

Table 5.

The percentage of myoglobin forms estimated by implementing the absorption estimated by Piao and Sun’s model (Piao and Sun, 2021) within Krzywicki’s equations and using 100% of the absorption at 700 nm as the baseline reference absorption at 730 nm.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
x: [Oxy]%	172.52 ± 28.52	166.23 ± 5.46	161.14 ± 9.01	156.23 ± 11.23	141.48 ± 21.87	101.80 ± 54.88	63.87 ± 53.85
100%−(y+z)	89.43	81.49	75.52	75.36	64.24	49.84	31.18
y: [Deoxy]%	15.58 ± 3.27	19.23 ± 3.78	21.00 ± 2.68	17.71 ± 4.53	22.07 ± 3.28	23.16 ± 10.87	33.10 ± 24.29
z: [Met]%	−5.02 ± 5.06	−0.72 ± 4.98	3.48 ± 6.34	6.93 ± 8.28	13.69 ± 9.78	27.00 ± 17.04	35.72 ± 20.53

Figure 5.

Comparison of the direct estimation of [Oxy]% with respect to the indirect estimation of [Oxy]% via [Deoxy]% and [Met]% between 2 algorithms. The abscissa represents the [Oxy]% expected by subtracting [Deoxy]% and [Met]% from 100%. The ordinate represents the [Oxy]% calculated directly using the equations of respective pertinence. Both the abscissa and ordinate carry the scale of −100% to 200% to accommodate most of the [Oxy]% valued obtained by direct computation nonassociated with [Deoxy]% and [Met]%. The diagonal line at the middle cell represents perfect agreement between the directly calculated [Oxy]% and the associatively derived [Oxy]% using [Deoxy]% and [Met]% of both being physically feasible. The left panel corresponds to the 4 algorithms implemented by using 100% of the value corresponding to 700 nm as the baseline reference at 730 nm. The middle panel corresponds to the 2 algorithms implemented by using a reduced (20% of the) value at 700 nm as the baseline reference at 730 nm. The right panel corresponds to the 2 algorithms implemented by neglecting baseline reference at 730 nm.

The middle row corresponds to using 20% of the absorption at 700 nm estimated by Piao and Sun’s model as the baseline reference absorption at 730 nm as it may be a better approximation of the lower absorption at 730 nm than at 700 nm. This resulted in [Deoxy]% increasing globally from 14.60% ± 3.10% on day 0 to 30.22% ± 21.54% on day 6, and [Met]% increasing significantly from −2.21% ± 5.05% on day 0 to 36.01% ± 18.97% on day 6. Subtracting [Deoxy]% and [Met]% from 100% resulted in [Oxy]% reducing gradually from 87.61% on day 0 to 33.77% on day 6. Direct computation of [Oxy]% independent of [Deoxy]% and [Met]% resulted in [Oxy]% reducing globally from above 100% on day 0 through day 4 to 37.04% ± 55.52% on day 6. The values plotted at the top panel of Figure 4 are referred to in Table 6.

Table 6.

The percentage of myoglobin forms estimated by implementing the absorption estimated by Piao and Sun’s model (Piao and Sun, 2021) within Krzywicki’s equations and using 20% of the absorption at 700 nm as the baseline reference absorption at 730 nm.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
x: [Oxy]%	142.62 ±10.70	140.70 ± 7.82	135.82 ± 11.33	131.70 ± 13.66	115.82 ± 22.98	76.19 ± 55.56	37.04 ± 55.52
100%−(y+z)	87.61	80.33	74.71	74.58	64.06	50.82	33.77
y: [Deoxy]%	14. 60 ± 3.10	18.18 ± 3.64	19.83 ± 2.56	16.75 ± 4.31	20.77 ± 3.09	21.52 ± 9.66	30.22 ± 21.54
z: [Met]%	−2.21 ± 5.05	1.48 ± 4.85	5.45 ± 6.12	8.67 ± 7.94	15.17 ± 9.24	27.66 ± 15.99	36.01 ± 18.97

Figure 6.

The changes of L*, a*, and b* over the day of display are presented in the left 3 figures. The rightmost figure displays the [Met]% resolved by 2 algorithms by neglecting the baseline value at 730 nm, versus the chroma averaged for 5 measurements per steak and 7 steaks per d, on each day over the duration of the retail display.

The bottom row corresponds to using 0% of the absorption at 700 nm estimated by Piao and Sun’s model as the baseline reference absorption at 730 nm, i.e., setting the baseline absorption at 730 nm as 0. This resulted in [Deoxy]% increasing globally from 14.37% ± 3.07% on day 0 to 29.58% ± 20.94% on day 6, and [Met]% increasing significantly from a near-zero value of −1.57% ± 5.05% on day 0 to 35.07% ± 18.61% on day 6. Subtracting [Deoxy]% and [Met]% from 100% resulted in [Oxy]% reducing gradually from 87.19% on day 0 to 34.35% on day 6. Direct computation of [Oxy]% independent of [Deoxy]% and [Met]% resulted in [Oxy]% reducing globally from above 100% on day 0 through day 4 to 30.99% ± 54.64% on day 6. The values plotted at the top panel of Figure 4 are referred to in Table 7.

Table 7.

The percentage of myoglobin forms estimated by implementing the absorption estimated by Piao and Sun’s model (Piao and Sun, 2021) within Krzywicki’s equations and not considering the baseline reference absorption at 730 nm.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
x: [Oxy]%	135.75 ± 11.79	134.77 ± 8.42	129.93 ± 11.82	125.99 ± 14.16	109.87 ± 23.24	70.32 ± 55.75	30.99 ± 54.64
100%−(y+z)	87.19	80.01	74.52	74.39	64.02	51.03	34.35
y: [Deoxy]%	14.37 ± 3.07	17.94 ± 3.61	19.56 ± 2.53	16.53 ± 4.26	20.47 ± 3.05	21.15 ± 9.40	29.58 ± 20.94
z: [Met]%	−1.57 ± 5.05	2.00 ± 4.82	5.91 ± 6.08	9.08 ± 7.86	15.51 ± 9.12	27.81 ± 15.75	35.07 ± 18.61

Figure 7.

Comparison of the [Met]% estimated by 2 algorithms of implementing Krzywicki’s equations against a* (left panel) and chroma (right panel) over the duration of the display. The value of a* or chroma as the abscissa is displayed in descending order to be consistent with the change of it over the retail display. The [Met]% as the ordinate is displayed in an ascending order to be consistent with the expected change of it over the retail display. Within each of the left or right panels, the top figure corresponds to the PM muscle, the middle figure LL muscle, and the bottom figure ST muscle. Within each figure of a specific muscle type, the trace marked by black empty diamonds corresponds to the algorithm of the bottom panel of Figure 3, and the trace marked by purple solid circles corresponds to the algorithm of the bottom panel of Figure 4, The middle panel illustrates the key algorithm for estimating [Met]% by each of the 2 algorithms compared in each of the figures on the left and right panels.

Assessing the reality of the percentage numbers of myoglobin forms estimated by the 2 algorithms of inversion for implementing Krzywicki’s equations

The derivations detailed in Supplemental Material 1 demonstrate 3 independent equations for calculating respectively [Oxy]%, [Deoxy]%, and [Met]%. These 3 equations are grossly parallel with respect to each other, and it would be reasonable to expect that they perform evenly in estimating the respective percentages of the myoglobin forms. In an ideal scenario of accurate analytics and accurate quantification of the spectral absorptivity parameters, the [Oxy]% obtained directly by Equation (12) will be as accurate as the [Deoxy]% obtained directly by Equation (13) and the [Met]% obtained directly by Equation (14). Should this be the case, the estimation of the [Oxy]% by subtracting [Deoxy]% and [Met]% from the total 100% myoglobin should be as good as the [Oxy]% calculated parallelly with [Deoxy]% and [Met]%.

In the 2 algorithms investigated, the calculations of [Deoxy]% and [Met]% seemed to be resilient to the level of reference baseline projected at 730 nm in producing realistic values. However, the direct calculation of [Oxy]% is very sensitive to the level of reference baseline projected at 730 nm in producing realistic values. The [Oxy]% estimated indirectly by subtracting the realistic values of [Deoxy]% and [Met]% from 100% by the 2 algorithms are all in a realistic range, regardless of the level of reference baseline projected at 730 nm. As suggested by these 2 cases, it becomes imperative to examine how the [Oxy]% estimated indirectly by subtracting the values of [Deoxy]% and [Met]% from 100% may compare with the [Oxy]% calculated directly just like how [Deoxy]% and [Met]% are obtained. These comparisons are shown in Figure 5, wherein the [Oxy]% calculated directly is plotted against [Oxy]% calculated indirectly for the same set of samples.

In Figure 5, both abscissa and ordinate are scaled over a range of −100% to 200% to accommodate most of the data points corresponding to 1a versus 2a, 1b versus 2b, and 1c versus 2c of Figure 2. The abscissa represents what remains in 100% after subtracting [Deoxy]% and [Met]%, that is the indirect estimation of [Oxy]% or the associative estimation of it based on the physical constraint that summing it with [Deoxy]% and [Met]% must make 100%. The ordinate represents the direct estimation of [Oxy]% or the independent estimation of it as estimating [Deoxy]% and [Met]% with no physical constraint on the reality of it. The shaded area represents the realistic range of [0% 100%] of that parameter. The diagonal line in the middle cell represents an ideal matching between the directly calculated [Oxy]% and the indirectly deduced [Oxy]% corresponding to 100% subtracting [Deoxy]% and [Met]% combined.

There are 2 patterns of distribution of the data to be examined in Figure 5. One is how close the data is distributed within the central cell framed by the box of green indicating a realistic range. The other is the diagonality of data distribution. Out of the algorithms that take 100% of the value at 700 nm as the baseline reference projected for 730 nm or the set a, the results via algorithm 1a have some distribution in the central cell and extend far into the below-0% area of the abscissa representing indirectly calculated [Oxy]%. In comparison, the results via algorithm 2a have some distribution in the central cell and extend far into the above-100% area of the abscissa representing indirectly calculated [Oxy]%. Out of the algorithms that take 20% of the value at 700 nm as the baseline reference projected for 730 nm or set b, the results via algorithm 1b have barely any distribution within the central cell. In comparison, the distributions of results via algorithm 2b extend into both the above-100% and below-0% areas of the abscissa representing indirectly calculated [Oxy]%. Out of the algorithms that take 0% of the value at 700 nm as the baseline reference projected for 730 nm, i.e., neglecting the baseline reference at 730 nm or the set c, the results via algorithm 1c have no distribution within the central cell. In comparison, the results via algorithm 2c have the distribution slightly closer to the central cell than the set 2b. In terms of the diagonality of the distributions, the results via algorithms 2a in set a, 2b in set b, and 2c in set c are not too different in terms of the grossly steeper angle with respect to the abscissa when compared to the reference diagonal line. Since the diagonality is not significantly different among the 3 sets, the closeness of the distribution to the central cell and the symmetry of the distributions with respect to the central cell may favor algorithm 2c.

The change of metmyoglobin of 7 LL steaks over retail display in comparison to the color-change resolved by colorimetry

Further assessment of the performance between algorithms 1c and 2c at the same configuration of neglecting the reference baseline for 730 nm is given in Figure 6, by referring to the L*, a*, and b* presented in Figure 6A, B, and C, respectively, that were recovered by the colorimetry. In (D), the combination of a* and b* in chroma is set as the abscissa, and the increase of [Met]% calculated via the algorithms 1c and 2c are plotted with respect to the chroma that reduced from day 0 to day 6. Both traces revealed linearly increasing [Met]% with respect to the decreasing chroma. The [Met]% via the algorithm 1c started at a high value near 30% on day 0, which is incongruent with the measurement on samples prepared approximately 6d postmortem. The [Met]% via algorithm 2c started at ∼0% on day 0, which is congruent with the measurement on samples prepared approximately 6d postmortem. The values of L*, a*, b* in mean ± STD and chroma in mean only, as plotted in Figure 5A–C, are also tabulated in Table 8.

Table 8.

L*, a*, b*, and chroma of the 7 steaks measured over 6 d of retail display.

View Larger Table

Mean $±$ STD	Day of Display
Mean $±$ STD	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6
L*	41.64 ± 2.74	42.31 ± 2.85	41.54 ± 1.82	40.71 ± 2.81	41.00 ± 3.80	39.58 ± 4.29	37.18 ± 4.30
a*	32.26 ± 2.21	31.20 ± 1.63	30.14 ± 2.19	29.13 ± 2.53	26.20 ± 3.49	21.67 ± 6.11	18.09 ± 5.25
b*	24.04 ± 2.28	23.22 ± 2.10	22.06 ± 2.52	21.39 ± 2.83	19.17 ± 2.63	16.69 ± 3.93	14.13 ± 3.36
Chroma	40.23	38.89	37.35	36.14	32.46	27.36	22.95

Additionally, Supplemental Materials have also compared the change of the directly estimated [Oxy]%, [Deoxy]%, and [Met]% of the 7 LL steaks over the retail display versus the change of the chroma. Figure A1 compared the results obtained by algorithm set 1 at 3 levels of the reference baseline projected for 730 nm, and Figure A2 compared the results obtained by algorithm set 2 at 3 levels of the reference baseline projected for 730 nm. In both figures, the P value <0.05 corresponds to the statistically significant difference between the means of the 2 data groups. In Figure A1, only the data groups of increasing [Deoxy]% resulted in P < 0.05 when compared to the decrease of the chroma. This may be interpreted as that, in comparison with the chroma that decreased over the duration of the display, the increase of [Deoxy]% was significant but that of [Met]% was not, even though it also increased over the duration of the display. Comparatively, in Figure A2, both the data groups of increasing [Deoxy]% and increasing [Met]% resulted in P < 0.05 when compared with the decrease of the chroma. This may be interpreted as that, in comparison with the chroma that decreased over the duration of display, both the increase of [Deoxy]% and increase of [Met]% were significant. Comparison between Figure A1 and A2 may also favor algorithm set 2c, particularly for estimating [Met]% that would likely start at nearly 0% at the beginning of the retail display which was approximately 6d postmortem.

The changes of metmyoglobin of 44 steaks of 3 types over retail display in comparison with the colorimetry results

Previous practices have suggested that algorithm 2c for stage 1 results in more realistic estimations of [Met]% over the retail display, than algorithm 1c for stage 1, at the same configuration of neglecting the baseline reference value projected for 730 nm. Encouraged by this finding, we have tested the 2 algorithms of 1c and 2c for stage 1 in estimating [Met]% of the total 44 muscles of 3 types. The results are presented in Figure 7.

Figure 7 contains 3 panels of left, middle, and right. The left panel compares the [Met]% estimated by 2 algorithms (1c and 2c of Figure 2) of implementing Krzywicki’s equations against CIE a* over the duration of the display. The right panel compares the [Met]% estimated by 2 algorithms (1c and 2c of Figure 2) of implementing Krzywicki’s equations against chroma over the duration of the display. The middle panel summarizes the most relevant equations of each of the 2 algorithms (1c and 2c of Figure 2) for implementing Krzywicki’s equations.

In both left and right panels, the value of a* or chroma as the abscissa is displayed in a descending order to be consistent with the change of it over the retail display. The left and right panels use the same sets of estimated [Met]% as the ordinates and at an ascending order to be consistent with the expected change of it over the retail display. Within each of the left or right panels, the top figure corresponds to 14 PM muscles combining 2 groups, the middle figure corresponds to 22 LL muscles combining 3 groups, and the bottom figure corresponds to 8 ST muscles of one group. Within each figure of a specific muscle type, the trace marked by black empty diamonds corresponds to applying the algorithm (1c of Figure 2) resulting in the bottom panel of Figure 3, and the trace marked by purple solid circles corresponds to applying the algorithm (2c of Figure 2) resulting in the bottom panel of Figure 4, The middle panel illustrates the key steps for estimating [Met]% by each of the 2 algorithms (1c and 2c of Figure 2) compared in each of the sub-figure of the left and right panels.

In both the left and right panels of Figure 7, a shaded horizontal strip is added to indicate a negative value of [Met]% as it is nonrealistic. In contrast to this zone of nonrealistic values, some observations can be made for the association of [Met]%, specifically the chroma change in the right panel. For the 8 ST muscles of one group, both the chroma and [Met]% remain relatively stable over the duration (7 d) of retail display. For the 22 LT muscles of 3 groups combined, the chroma reduced steadily and [Met]% increased steadily over the durations (5 or 6 d) of retail display. For the 14 PM muscles of 2 groups combined, the chroma reduced steadily over the durations (5 or 6 d) of retail display, and [Met]% increased grossly over the duration of display but it had a pronounced change appearing as a sudden increase (jump) of [Met]% as the chroma reduced to slightly below 30.

The 2 algorithms showed similar patterns in terms of the gross change of [Met]%. However, some differences are easily noted. For both the PM and LL muscles, the terminal values of [Met]% estimated by the 2 algorithms are comparable. However, for the ST muscles, the terminal value of [Met]% of ∼30% estimated by algorithm 1c is substantially greater than the values of <10% estimated by algorithm 2c. The overall range of the change of [Met]% for PM and LL muscles combined is significantly less by estimation using algorithm 1c than by 2c.

Figure 8 has also presented an inter-type comparison of the [Met]% among the PM, LL, and ST muscles, as was estimated using algorithm 1c and algorithm 2c, where the comparison of the means was analyzed by 1-way ANOVA and the comparison of the standard variations was given by Brown-Forsythe test. Figure 8A compared [Met]% estimate of the 3 types of muscles by algorithm 1c, which converted the reflex attenuance from spectral reflectance for implementation in Krzywicki’s equation. The differences among the 3 sets of means are statistically significant (P = 0.0012), and the differences among the 3 sets of standard deviations are also statistically significant (P = 0.0397). Figure 8B compared [Met]% estimate of the 3 types by algorithm 2c, which converted the reduced scattering scaled absorption from spectral reflectance for implementation in Krzywicki’s equation. The differences among the 3 sets of means are statistically significant (P = 0.0033) but the differences among the 3 sets of standard deviations are statistically insignificant (P = 0.0879). The statistically significant difference in the mean values among the 3 data sets confirms that the pattern of the increase of [Met]% over retail display differed significantly among the PM, LL, and ST muscles.

Figure 8.

Comparison of the [Met]% of the 3 types of muscles. (A) The [Met]% was estimated by the algorithm of converting the reflex attenuance from spectral reflectance for implementation in Krzywicki’s equation. The differences among the 3 sets of means are statistically significant (P = 0.0012) by one-way ANOVA, and the differences among the 3 sets of standard deviations are also statistically significant (P = 0.0397) by the Brown-Forsythe test. (B) The [Met]% was estimated by the algorithm of converting the reduced scattering scaled absorption from spectral reflectance for implementing Krzywicki’s equation. The differences among the 3 sets of means are statistically significant (P = 0.0033) by one-way ANOVA, but the differences among the 3 sets of standard deviations are statistically insignificant (P = 0.0879) by the Brown-Forsythe test. The statistically significant difference in the mean values among the 3 data sets confirms that the pace of the increase of [Met]% over retail display differed significantly among the 3 types of muscles that were known to differ in color liability.

Discussion

The raw measurement of the colorimeter that it acquires from the meat surfaces, the spectral reflectance, characterizes the change of the spectrum of light diffusely reflected from the meat’s superficial layer in comparison with the spectrum of light irradiated onto the surfaces. Therefore, the colorimeter is inherently viable for assessing the spectrally significant pigments of meat that contribute to the measured spectral modulation when comparing to the spectrum of the diffuse irradiation, providing that an algorithm associating the spectral components of meat with the spectral reflectance in a forward projection or an inverse process (Piao et al., 2022) is available.

Both methods recommended by the AMSA (King et al., 2023) can be conveniently implemented, using mathematical operators such as EXCEL, directly on the spreadsheet that can be generated from the colorimeter; however, these approaches need to consider physical constraints such as that the percentages of the myoglobin forms must not be below 0% or greater than 100%. In practice, it is beneficial to know which method may be simpler while being more robust in terms of being less prone to error or producing less nonrealistic reading. Hernández et al. (2015) have offered a much-needed detailed comparison of the 2 methods. The AMSA method 2 or the one proposed by Krzywicki is appreciably much simpler than the AMSA method 1, not only in the algebraic formula but also in not needing reference measurements from myoglobin form standards that require meticulous and laborious processing. However, as has been demonstrated by Hernández et al. (2015), the absolute values of the percentage numbers of myoglobin forms given by ASMA method 1 and AMSA method 2 can differ greatly. The mismatching between the 2 methods of the AMSA guidelines entails understanding the 2 methods in greater detail or depth than has been available. Regarding AMSA method 2, the equation for [Met]% originally proposed by Krzywicki has been shown to often produce negative values (Viriyarattanasak et al., 2011). Since AMSA method 2 recommends deducing [Oxy]% indirectly by subtracting [Deoxy]% and [Met]% from 100%, the so-calculated [Oxy]% may then become overestimated if [Met]% has been below 0%. A question then arises concerning how the [Oxy]% obtained by direct deduction using an equation parallel to that of [Deoxy]% or [Met]% compares against the [Oxy]% calculated indirectly via [Deoxy]% and [Met]%. Additionally, it is unknown if there can be better practice than using the recommended baseline reference at 700 nm as the value projected at 730 nm for calculating the percentage numbers of myoglobin forms. These unknowns have encouraged this work.

One aim of this work has been to examine if there is a better strategy to apply the exceptionally convenient equations of Krzywicki’s to mitigate the under or overestimations of the percentages of myoglobin forms to improve the certainties in assessing the biochemical changes of discoloration of meat. It is worth noting that the convenience of Krzywicki’s equations may need to be taken advantage of with a better understanding of pitfalls that may affect the soundness of the recovered proportions of myoglobin forms. To facilitate such an examination, Equations (12), (13), and (14) in materialized modular forms are represented in the following by referring to Supplemental Material 1:

$[Oxy] % = α O x y {β O x y − γ O x y μ a (λ 4) − μ a 0 (λ 4) μ a (λ 2) − μ a 0 (λ 2)}$

(33)

$[Deoxy] % = α Deoxy {β Deoxy − γ Deoxy [μ a (λ 1) − μ a 0 (λ 1)] [μ a (λ 2) − μ a 0 (λ 2)] − {μ a 0 (λ 1) − μ a 0 (λ 2)} [μ a (λ 2) − μ a 0 (λ 2)]}$

(34)

$[M e t] % = α M e t {β M e t − γ M e t μ a (λ 3) − μ a 0 (λ 3) μ a (λ 2) − μ a 0 (λ 2)}$

(35)

Where

$α O x y = 2.159$ ,

$α D e o x y = 2.375$ ,

$α M e t = 1.0$ ,

$β O x y = β D e o x y = 1.0$ ,

$β M e t = 1.395$ ,

$γ O x y = 2.375$ , and

$γ D e o x y = γ M e t = 1.0$ .

There are a couple of assumptions that underlie Krzywicki’s algorithms. One is that myoglobin does not contribute to the absorption of the steak at 730 nm. The other is that the ratio of reflex attenuance measured from the sample at 2 wavelengths accurately represents the ratio of absorption of the myoglobin present in the sample at the 2 wavelengths. The first assumption leads to the practice of using an absorptivity measured at 730 nm as a baseline reference value to be subtracted from the absorptivity measured at any isosbestic wavelengths of myoglobin. Krzywicki’s equations can accurately recover the percentage numbers of myoglobin forms, should the true baseline absorptivity of the steaks relating to non-myoglobin pigments be available to be placed in the equations. Although the myoglobin absorption is low at 730 nm, there is still a non-negligible absorption of myoglobin at 730 nm. Therefore, subtracting any reference value, regardless of, if it is measured at 700 nm or 730 nm, from an absorptivity at an isosbestic wavelength, will cause an underestimation of the absorptivity exclusive to the myoglobin at that isosbestic wavelength. Because Krzywicki’s equations are based on the ratio of the absorptivity of myoglobin at 2 isosbestic wavelengths regardless of the forms, the underestimation of both the numerator and denominator could lead to complex patterns of over or underestimation of the percentage of the individual form of myoglobin. This would make the estimation sensitive to the level of reference baseline projected for 730 nm, as was observed with the direct estimation of [Oxy]% by using Krzywicki’s original approach that gave rise to negative values of [Oxy]% over the duration of display, with the baseline reference level projected at 730 nm to be much smaller than that at 700 nm.

The practices shown in this work have demonstrated direct calculations using Equations (21 )–(23) with Krzywicki’s original approach led to the estimation of [Oxy]% that was sensitive to the reference baseline level projected to 730 nm, somewhat stable estimation of [Deoxy]%, and stable but seemingly underestimation of the change of [Met]% in comparison to the change of chroma. A few observations can be promptly made with Equations (33 )–(35) to appreciate the conditions that may affect the accuracy and physical soundness of the calculated percentage numbers of the myoglobin forms. Equation (34) indicates that the correct estimation of [Deoxy]% is needed to consider the second fractional term in the bracket. This second fractional term vanishes if the absorption contributed by non-myoglobin pigments is spectrally flat between 474 nm and 525 nm. This is unlikely to be true for muscle tissue since non-myoglobin pigments including water, cytochrome C, etc. have different absorptions over 474 nm and 525 nm. This second fractional term has a denominator that is the difference of the absorption at the same wavelength of 525 nm between the myoglobin-containing sample and myoglobin-excluding sample. This value will depend upon the actual molar concentration of myoglobin, and the smaller the myoglobin content is, the greater the effect it has on this second fractional term. Therefore, using Krzywicki’s Equation (14) (Krzywicki, 1979) or Equation (34) in this manuscript by neglecting the second fractional term in the bracket will result in underestimation or overestimation of [Deoxy]% if the non-myoglobin tissue absorption is lower or higher in 474 nm than in 525 nm and the underestimation or overestimation would become greater for muscle containing a lesser amount of myoglobin.

The condition that each of x, y, and z must remain bounded within 0% and 100% constraints what ranges the fractional terms within each of Equations (33 )–(35) must reside. For example, Equation (33) dictates that $μ a (λ 4) − μ a 0 (λ 4) μ a (λ 2) − μ a 0 (λ 2)$ must be bounded within [0.226, 0.421] to arrive at [Oxy]% that is realistic and Equation (34) dictates that the 2 fractional terms combined must be bounded within [0.579, 1] to keep [Deoxy]% to be realistic. Similarly, Equation (35) dictates that $μ a (λ 3) − μ a 0 (λ 3) μ a (λ 2) − μ a 0 (λ 2)$ must be bounded within [0.395, 1.395] to keep [Met]% to be realistic. These constraints infer that, if the error introduced to the conversion of the spectral absorption parameters from the spectral reflectance measured by a calorimeter drives a pertinent fractional term beyond the permitted range, the associated proportional form of myoglobin will become either negative or greater than 100% to become nonrealistic. Therefore, the robustness of the mathematical model that associates or maps the tissue’s properties determines the robustness of the implementation of Krzywicki’s equations. However, one should expect that, even with the implementation of an accurate model to invert the spectral reflectance to the true spectral absorption, there is a limit to the accuracy of the percentages of myoglobin forms estimated by using Krzywicki’s equations due to another confounding factor that is difficult if not impossible to mitigate. Accurate measurement of myoglobin forms in muscle by optical spectroscopy is challenging (Arakaki et al., 2007), because of the co-presence of hemoglobin and myoglobin (Arakaki et al., 1996; Schenkman et al., 1999; Arakaki et al., 2010). Because the spectral absorption properties of hemoglobin are very close to those of myoglobin, one can anticipate that the recovered total absorption will always contain the contribution of hemoglobin, and therefore, the percentage numbers of myoglobin forms will be an accurate representation of the true myoglobin oxygenation status if and only if there is no phase difference between the myoglobin oxygenation change and the hemoglobin oxygenation change during the display. This condition is unlikely to be true since hemoglobin confined in the blood vessels of stopped blood circulation may be less affected by the change of the diffusion of oxygen as myoglobin does.

It is worth noting that the pathlength of a photon’s propagation in muscle is wavelength dependent because of the weak yet consistent wavelength dependence of scattering in the visible wavelength (Geladi et al., 1985; Gussakovsky et al., 2008) over which Krzywicki’s equations operate. Therefore, given that the other assumption regarding the baseline reference value of the spectral absorptivity at 730 nm is valid, the accuracy or robustness of Krzywicki’s equation is still affected by the potential spectral dependence of scattering. Neglecting the wavelength dependence of scattering has also become a limitation of the practices of the newly introduced algorithms shown in this work. Accurate estimation of the percentage numbers of myoglobin forms thus requires using another modality of tissue measurement to assess the spectral scattering properties of the meat over the same spectrum of diffuse reflectance obtained by the colorimeter. It is reasonable to expect that incorporating the spectral scattering properties to algorithms 2c engaging Piao and Sun’s model (Piao and Sun, 2021) may improve the soundness and usefulness of the percentage numbers of myoglobin forms calculated by Krzywicki’s equations than has been shown.

Between the 2 approaches of algorithm 1c and algorithm 2c, we are inclined to use algorithm 2c. As algorithm 1c does, algorithm 2c takes only one step of conversion from the spectral reflectance to the spectral absorption, with an assumption of spectrally invariant scattering, to implement Krzywicki’s equations. The algebraic form of algorithm 2c can be readily implemented in a simple mathematical processer like EXCEL, reflex to Krzywicki’s original approach of calculating the reflex attenuance. Even with the presence of spectrally variant scattering, the implementation of algorithm 2c with a simple mathematical processor like EXCEL is uncomplicated if the spectral variance of the scattering can be represented in simple algebraic form. However, the validity of the proposed algorithm 2c as stage 1 of Krzywicki’s equations needs to be examined for muscles of known conditions, such as containing pure forms of myoglobin, muscles of other types, such as dark-cutter and meat from other species, and colorimeters of other models or configurations in future works.

Conclusions

The American Meat Science Association has provided 2 algorithms to quantify the percentage numbers of myoglobin forms with the use of spectral reflectance compatible with measurements by handheld colorimeter. Out of the 2 algorithms, the algorithm originally proposed by Krzywicki is simpler to implement, but questions remain about the robustness and limitations of the algorithm. By detailing the derivations based on tissue optics, we have identified the approximations, and thereby potential sources of errors, of the individual equations for [Oxy]%, [Deoxy]%, and [Met]%. The rederivation of Krzywicki’s equations has promoted us to also introduce one additional algorithm for estimating the spectral absorptivity properties needed in the process of estimating [Oxy]%, [Deoxy]%, and [Met]%. The newly introduced algorithm is operable with the spreadsheet of the colorimeter spectral data using simple mathematical processors like EXCEL. The colorimeter measurements from 7 LL steaks over 6 d of retail display have been used to examine if the new algorithms may improve the quantification of the percentage numbers of myoglobin forms by using Krzywicki’s equations. Krzywicki’s original algorithm is resilient for [Deoxy]% and [Met]% but quite sensitive for [Oxy]% in terms of the dependence on the baseline reference of reflex attenuance projected for 730 nm by using the value obtained at 700 nm. The newly introduced algorithm of deducing spectral absorption from the spectral reflectance for feeding into Krzywicki’s equations outperforms Krzywicki’s original proposal of using the reflex attenuance, without the need of a baseline reference at 730 nm. However, neither of the 2 algorithms for directly estimating [Oxy]% gives physically feasible values throughout the entire duration of the display. This reassures that [Oxy]% based on any variations of Krzywicki’s approach shall be deduced by subtracting [Deoxy]% and [Met]% from 100%. The 2 algorithms have also been applied to a total of 44 steaks of 3 different muscle types including PM, LL, and ST in retail display. The new algorithm revealed a pronounced sudden increase of [Met]% in 14 PM steaks as the chroma reduced to approximately 30, which is unobserved from the 22 LL steaks and 8 ST steaks at similar values of the chroma.

Acknowledgements

The authors thank Anuj Sharma for assisting with the acquisition of colorimetry data.

Funding

This research was supported by the U.S. Department of Agriculture National Institute of Food and Agriculture (100005825) (2022-67017-36538) grant program.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Supplemental Materials

1. Re-derivation of Krzywicki’s Equations

If the same muscle has been processed to contain 100% of only one form of myoglobin as would be required in the AMSA method 1, the absorption baseline of $μ a 0$ that is caused by tissue components other than myoglobin must remain, which will make the absorption measured from muscle of pure forms of the myoglobin to become the following:

$μ a O x y (λ n) = [t M B] · ɛ O x y (λ n) + μ a 0 (λ n)$

(A1)

$μ a D e o x y (λ n) = [t M B] · ɛ D e o x y (λ n) + μ a 0 (λ n)$

(A2)

$μ a M e t (λ n) = [t M B] · ɛ M e t (λ n) + μ a 0 (λ n)$

(A3)

For convenience in the subsequent analytics, the following substitutions of the symbols will be used when necessary:

$ɛ O x y (λ n) = A (λ n) = A n$

(A4)

$ɛ D e o x y (λ n) = B (λ n) = B n$

(A5)

$ɛ M e t (λ n) = C (λ n) = C n$

(A6)

For further analytical convenience and in referring to Figure 1 of the main part of this work, we note that the A and C at 474 nm are nearly identical to the A and B and C at 525 nm (Krzywicki, 1979). Using Q to represent the triple-isosbestic value at 525 nm allows representing other isosbestic values in the following scaled forms:

$A 1 = C 1 = [A ‖ C] 1 ≍ Q$

(A7)

$A 2 = B 2 = C 2 = [A ‖ B ‖ C] 2 = Q$

(A8)

$A 3 = B 3 = [A ‖ B] 3 = K (3 ∶ 2) · Q$

(A9)

$B 4 = C 4 = [B ‖ C] 4 = K (4 ∶ 2) · Q$

(A10)

Where the two unitless coefficients in Eqs. (A9) and (A10) are respectively

$k (3 ∶ 2) = A 3 A 2 = B 3 B 2 = 1.395$

(A11)

$k (4 ∶ 2) = C 4 C 2 = B 4 B 2 = 0.421$

(A12)

And at the four isosbestic points of 474 nm, 525 nm, 572 nm, and 610 nm, we have

$A 1 · x + B 1 · y + C 1 · z = [μ a (λ 1) − μ a 0 (λ 1)] / [t M B]$

(A13)

$A 2 · x + B 2 · y + C 2 · z = [μ a (λ 2) − μ a 0 (λ 2)] / [t M B]$

(A14)

$A 3 · x + B 3 · y + C 3 · z = [μ a (λ 3) − μ a 0 (λ 3)] / [t M B]$

(A15)

$A 4 · x + B 4 · y + C 4 · z = [μ a (λ 4) − μ a 0 (λ 4)] / [t M B]$

(A16)

By using Eqs. (A7 )-(A10), Eqs. (A13 )-(A16) can appear as

$Q · x + B 1 · y + Q · z = [μ a (λ 1) − μ a 0 (λ 1)] / [t M B]$

(A17)

$Q · x + Q · y + Q · z = [μ a (λ 2) − μ a 0 (λ 2)] / [t M B]$

(A18)

$k (3 ∶ 2) · Q · x + k (3 ∶ 2) · Q · y + C 3 · z = [μ a (λ 3) − μ a 0 (λ 3)] / [t M B]$

(A19)

$A 4 · x + k (4 ∶ 2) · Q · y + k (4 ∶ 2) · Q · z = [μ a (λ 4) − μ a 0 (λ 4)] / [t M B]$

(A20)

By referring to Eq. (4) of the main part of this work, Eq. (A18) may be replaced by a simpler form

$Q = μ a (λ 2) − μ a 0 (λ 2) [t M B]$

(A21)

We introduce the following symbols to facilitate discussions for subsequent needs:

$β n = μ a (λ n) − μ a 0 (λ n) μ a (λ n) = 1 − μ a 0 (λ n) μ a (λ n)$

(A22)

$(β n β 2) Sample = μ a (λ n) − μ a 0 (λ n) μ a (λ 2) − μ a 0 (λ 2) · μ a (λ 2) μ a (λ n)$

(A23)

Equation (A22) represents the extent of the baseline absorption contributed by the non-myoglobin pigments to the absorption measured from the muscle. What Eq. (A23) represents is how accurate the relative absorption at two wavelengths is, when not considering the baseline absorption of non-myoglobin pigments, in comparison to that with consideration of the baseline absorption of non-myoglobin pigments.

1.1 Re-derivation of Krzywicki’s equation of x or [Oxy]%

Regrouping the terms of Eq. (A17) and (A19) makes the following sets

$Q · x + Q · (y + z) = [μ a (λ 2) − μ a 0 (λ 2)] / [t M B]$

(A17*)

$A 4 · x + k (4 ∶ 2) · Q · (y + z) = [μ a (λ 4) − μ a 0 (λ 4)] / [t M B]$

(A19*)

Subtracting (A17*) from

$k (4 ∶ 2) ·$ (A19*) leads to the following evolution of the algebraic while referring to Eq. (A21):

$x = {k (4 ∶ 2) · μ a (λ 2) − μ a 0 (λ 2) [t M B] − μ a (λ 4) − μ a 0 (λ 4) [t M B]} (k (4 ∶ 2) · Q − A 4) = {k (4 ∶ 2) · [μ a (λ 2) − μ a 0 (λ 2)] − [μ a (λ 4) − μ a 0 (λ 4)]} (k (4 ∶ 2) · Q − A 4) 1 [t M B] = [{k (4 ∶ 2) · [μ a (λ 2) − μ a 0 (λ 2)]} (k (4 ∶ 2) · Q − A 4) − {μ a (λ 4) − μ a 0 (λ 4)} (k (4 ∶ 2) · Q − A 4)] 1 [t M B] = [k (4 ∶ 2) (k (4 ∶ 2) · Q − A 4) − 1 (k (4 ∶ 2) · Q − A 4) μ a (λ 4) − μ a 0 (λ 4) μ a (λ 2) − μ a 0 (λ 2)] [μ a (λ 2) − μ a 0 (λ 2) [t M B] = [k (4 ∶ 2) (k (4 ∶ 2) · Q − A 4) − 1 (k (4 ∶ 2) · Q − A 4) μ a (λ 4) − μ a 0 (λ 4) μ a (λ 2) − μ a 0 (λ 2)] Q [M b T] [t M B] = [k (4 ∶ 2) Q (k (4 ∶ 2) · Q − A 4) − Q (k (4 ∶ 2) · Q − A 4) μ a (λ 4) − μ a 0 (λ 4) μ a (λ 2) − μ a 0 (λ 2)] = k (4 ∶ 2) Q (k (4 ∶ 2) · Q − A 4) {1 − 1 k (4 ∶ 2) · μ a (λ 4) − μ a 0 (λ 4) μ a (λ 2) − μ a 0 (λ 2)}$

(A24)

Utilizing Eq. (A23), Eq. (A24) becomes

$x = k (4 ∶ 2) Q (k (4 ∶ 2) · Q − A 4) {1 − 1 k (4 ∶ 2) (β 4 β 2) Sample · μ a (λ 4) μ a (λ 2)}$

(A25)

and noting that

$K (4 ∶ 2) = 0.421$ and

$A 4 = 0.226 · Q$ , the following terminal form of

$x$ , which was absent from Krzywicki’s original work, is obtained:

$x = 2.159 {1 − 2.375 (β 4 β 2) Sample μ a (λ 4) μ a (λ 2)}$

(A26)

1.2 Re-derivation of Krzywicki’s equation of y or [Deoxy]%

Subtracting Eq. (A17*) from Eq. (A18) makes the following evolution of the algebraic while referring to Eq. (A22):

$y = μ a (λ 2) − μ a (λ 1) Q − B 1 1 [t M B] ≈ {μ a (λ 2) − μ a 0 (λ 2) − μ a (λ 1) + μ a 0 (λ 1)} Q − B 1 1 [t M B] − {μ a 0 (λ 1) − μ a 0 (λ 2)} Q − B 1 1 [t M B] = {μ a (λ 2) − μ a 0 (λ 2) (Q − B 1)} 1 [t M B] − {μ a (λ 1) − μ a 0 (λ 1) (Q − B 1)} 1 [t M B] − {μ a 0 (λ 1) − μ a 0 (λ 2)} Q − B 1 1 [t M B] = {Q (Q − B 1)} − {μ a (λ 2) − μ a 0 (λ 2) (Q − B 1)} μ a (λ 1) − μ a 0 (λ 1) μ a (λ 2) − μ a 0 (λ 2) 1 [t M B] − {μ a 0 (λ 1) − μ a 0 (λ 2)} Q − B 1 1 [t M B] = {Q (Q − B 1)} − {Q (Q − B 1)} μ a (λ 1) − μ a 0 (λ 1) μ a (λ 2) − μ a 0 (λ 2) − {μ a 0 (λ 1) − μ a 0 (λ 2)} Q − B 1 1 [t M B] = Q (Q − B 1) [1 − μ a (λ 1) − μ a 0 (λ 1) μ a (λ 2) − μ a 0 (λ 2) − {μ a 0 (λ 1) − μ a 0 (λ 2)} Q 1 [t M B]] = Q (Q − B 1) [1 − μ a (λ 1) − μ a 0 (λ 1) μ a (λ 2) − μ a 0 (λ 2) − {μ a 0 (λ 1) − μ a 0 (λ 2)} [μ a (λ 2) − μ a 0 (λ 2)]]$

(A27)

Utilizing Eq. (A23), Eq. (A27) becomes

$y = Q (Q − B 1) [1 − (β 1 β 2) Sample μ a (λ 1) μ a (λ 2) − {μ a 0 (λ 1) − μ a 0 (λ 2)} [μ a (λ 2) − μ a 0 (λ 2)]]$

(A28)

Substituting the values of Q/

$B 1$ = 7.6/4 in Eq. (A28) leads to the following:

$y = 2.375 {1 − (β 1 β 2) Sample μ a (λ 1) μ a (λ 2) − {μ a 0 (λ 1) − μ a 0 (λ 2)} [μ a (λ 2) − μ a 0 (λ 2)]}$

(A29)

If

$μ a 0 (λ n) = μ a 0 (λ r)$ , or equivalently

$μ a 0 (λ 1) = μ a 0 (λ 2)$ , Eq. (A29) relaxes to Eq. (14) of Krzywicki’s (Krzywicki, 1979).

1.3 Re-derivation of Krzywicki’s equation of z or [Met]%

Multiplying Eq. (A18) with $k (3 ∶ 2)$ and then subtracting Eq. (A19*) makes the following evolution of the algebraic while referring to Eq. (A23):

$z = {k (3 ∶ 2) · [μ a (λ 2) − μ a 0 (λ 2)] − [μ a (λ 3) − μ a 0 (λ 3)]} {(k (3 ∶ 2) · Q − C 3)} 1 [t M B] = {k (3 ∶ 2) · [μ a (λ 2) − μ a 0 (λ 2)] (k (3 ∶ 2) · Q − C 3)} 1 [t M B] − {μ a (λ 3) − μ a 0 (λ 3) (k (3 ∶ 2) · Q − C 3)} 1 [t M B] = {k (3 ∶ 2) (k (3 ∶ 2) · Q − C 3)} Q − {1 (k (3 ∶ 2) · Q − C 3)} μ a (λ 3) − μ a 0 (λ 3) [M b T] = {k (3 ∶ 2) · Q (k (3 ∶ 2) · Q − C 3)} − {Q (k (3 ∶ 2) · Q − C 3)} · μ a (λ 3) − μ a 0 (λ 3) μ a (λ 2) − μ a 0 (λ 2) = {k (3 ∶ 2) · Q (k (3 ∶ 2) · Q − C 3)} − {Q (k (3 ∶ 2) · Q − C 3)} · (β 3 β 2) Sample · μ a (λ 3) μ a (λ 2)$

(A30)

Noticing that

$k (3 ∶ 2) · Q (k (3 ∶ 2) · Q − C 3) = 10.6 (10.6 − 3.0) = 1.395$

(A31)

$Q (k (3 ∶ 2) · Q − C 3) = 7.6 (10.6 − 3.0) = 1$

(A32)

Eq. (A30) becomes

$z = {1.395 − (β 3 β 2) S a m p l e · μ a (λ 3) μ a (λ 2)}$

(A33)

if

$μ a 0 (λ n) = μ a 0 (λ r)$ , Eq. (A33) relaxes to Eq. (10) of Krzywicki’s (Krzywicki, 1979).

2. Methods for stage one of Krzywicki’s equations to convert spectral reflectance to spectral absorptivity properties

The derivations given in Supplemental Material 1 in revisiting the derivations of Krzywicki’s equations dictate that the proportional forms of myoglobin are exclusively determined by the absorption of the medium containing myoglobin at the four isosbestic wavelengths. Once the absorption properties of the medium at the four isosbestic wavelengths are available, the myoglobin percentage forms are straightforwardly and accurately determined by the sets of three equations of (A26), (A29), and (A33), which in practice relax to the set of Eq. (12 –14) in the main part of this work. A central task or challenge of estimating the percentage forms of myoglobin by using the spectral reflectance acquired with colorimeter or any other device configurations is to then accurately deduce the absorption properties of the medium at the four isosbestic wavelengths using the spectral reflectance. In the following, we list the algorithm of to deduce reflex attenuance, and a new algorithm applicable to the spectral reflectance of colorimeter. As these algorithms, including Krzywicki’s one, are to use the spectral reflectance to reversely obtain the absorptivity properties of the medium, the models for these conversions will be referred to as “inversion model”. In the following, the spectral reflectance obtained by colorimeter is denoted as $R ∞$ , according to the notation conventional to the Kubelka–Munk (KM) model of light–tissue interaction that is the basis of utilizing the reflectance in colorimeter. The KM model ideally applies to a diffuse or blanketed illumination over a medium of infinite thickness, and a narrow-aperture acquisition of the diffusely reflected light over an optical axis oblique to the normal direction of the medium surfaces, for the rejection of specular reflection of the irradiation normal to the medium surfaces.

2.1 Inversion model #1 that converts the spectral reflectance to reflex attenuance to be implemented with Kryziwicki’s equations

In Kryzwicki’s original proposal, the $U abs (λ n)$ of Eqs. (12 –14) in the main part of this work is replaced with a reflex attenuance that is obtained from the spectral reflectance according to the equation of the following:

$A ref (λ n) = ln [1 R ∞ (λ n)]$

(A34)

Subsequently, replacing

$μ a (λ n)$ in Eqs. (A26), (A29), and (A33) with

$A ref (λ n)$ leads to direct calculations of, respectively, [Oxy]%, [Deoxy]%, and [Met]%.

2.2 A new model of converting the spectral reflectance to reduced scattering scaled absorption

The diffuse optical acquisition of colorimeter to which the AMSA methods apply is based on the KM model of light–tissue interaction. A light–tissue interaction model projects how the medium’s absorption and scattering properties affect the photon escaping the medium after propagating in the medium. A light–tissue interaction model is needed even for measuring on a shallow surface. Due to lack of a general analytical approach to light–tissue interaction from the perspective of electromagnetic wave interacting with tissue medium, which is further complicated by the need to account for boundary conditions, the light–tissue interaction model is often developed by customized approximations applying to the individual geometry of the applicator probe. Since the 1940s (Kubelka, 1948), applications of diffuse reflectance that deals with significantly varying relative strength of absorption over scattering of the medium, as is applicable to assessing myoglobin forms, have favored the KM theory or model for its convenience. In the simplest geometry modeled with an infinitely thick opaque medium, the percentage of photon flux re-emitting from the medium over the photon flux entering the medium, which is denoted as $R ∞$ , is connected to a lumped medium property called KM function, as follows:

$[K M] (λ) = {R K M − 1} 2 2 R K M = {R ∞ − 1} 2 2 R ∞$

(A35)

Where K is the medium’s absorption property and S is the medium’s scattering property. The KM model has been very popular to implement, due to the simplicity of Eq. (A35).

In the past decade, more developments toward better understanding of the KM approach, based on the more accurate and rigorous radiative transfer model of light–tissue interaction than the flux theory leading to the KM approach, have suggested that the KM function shall contain in its scattering proportion a coupling with the absorption as follows (Sandoval & Kim, 2014):

$[K M] (λ) = 8 μ a (λ) 3 μ s ′ (λ) − μ a (λ) (1 – 3 g)$

(A36)

Where

$μ a$ is the absorption coefficient [unit: 1/mm] as in Eq. (5) of the main part of this work,

$μ s ′ (λ)$ is a reduced scattering coefficient [unit: 1/mm], and g is the scattering anisotropy factor [unit-less].

$μ s ′ (λ)$ may also be expressed as

$μ s (λ) (1 − g)$ , of which

$μ s (λ)$ is the scattering coefficient [unit: 1/mm].

The scattering coefficient $μ s (λ)$ is the reciprocal of the average pathlength of light propagation between two consecutive events of scattering in the tissue that is anisotropic and forward biased. The anisotropy factor $g$ of Eq. (A36) represents the first-order directionality of the scattering. The reduced scattering coefficient $μ s ′ (λ)$ of Eq. (A36) is the reciprocal of the average pathlength of light propagation between two events of scattering counted, separated by many forward-biased scatterings in the middle, to have the direction of the second scattering event counted become completely random in comparison to the first scattering event counted. Because the forward-biased scattering needs to accumulate more scattering events to become randomized in the direction of scattering in comparing to the initial direction, the pathlength taken between two consecutive forward-biased scattering events can be much smaller than the pathlength taken to randomize the scattering direction. Therefore, the reciprocal of the pathlength taken to randomize the scattering direction, which becomes $μ s ′ (λ)$ in the unit of 1/length, is usually much smaller than the reciprocal of the pathlength taken between two consecutive forward-biased scattering events, which becomes $μ s (λ)$ also in the unit of 1/length. And for biological tissue, it is known that $μ s ′ (λ)$ varies relatively slowly and reasonably smoothly over the visible and NIR spectral band, and is very close in value to 1.0 mm⁻¹ (Jacques, 2013) at the neighborhood of 600 nm. For the benefit of simplicity, we assume $μ s ′ (λ) = 1 m m − 1$ in this work.

Note that the KM model that uses the KM function of Eq. (A36) is accurate for collections of diffused reflection under diffuse irradiation, such as irradiating using an integrating sphere or a blanketed irradiation like that of a colorimeter. However, the applicability of the KM approach using the KM function of Eq. (A36) degrades for a relatively high absorption (Lindberg & Laude, 1974). This can be appreciated with Eq. (A36) by noticing that, since $g$ for the forward-biased scattering of biological tissue is usually greater than 0.8, as $μ a$ increases to become much greater than $μ s ′ (λ)$ , the KM function will saturate at $8 / (3 g − 1)$ . The saturation will make the KM function stop responding to further increase of absorption, which will subsequently “level” the model prediction of KM to make it unsuitable for predicting diffuse photon reflectance from a highly absorbing medium (Piao & Sun, 2021). Yet high absorption is what is exactly needed for producing greater contrast of measurements of the spectrally significant pigments like myoglobin over the background tissue. Therefore, the KM model has a limit in robustly mapping high absorption to diffuse reflectance. To improve the validity of the KM function in representing the tissue’s spectral properties using the spectral reflectance, a modified KM function has been demonstrated (Piao & Sun, 2021) as follows:

$[K M] (λ) = 32 μ a (λ) 3 μ s ′ (λ) − μ a (λ) (1 – 1.75 g)$

(A37)

Where

$[K M] (λ)$ is obtained from the spectral reflectance the same way as that of Eq. (A35). Forward model operation based on this modified KM function has shown to noticeably outperform that based on KM function in predicting the diffuse reflectance involving strong absorption, when governed by the same Eq. (A35) in the forward process. In assessing diffuse photon remission from a thick opaque medium with respect to the absorption coefficient to which the KM model of light–tissue interaction may be applicable, Piao & Sun (Piao & Sun, 2021) also introduced an empirical model independent from the KM function, as follow:

$R ∞ (λ) = 1 2 π (4 μ a (λ) μ s ′ (λ) + 0.475) 2.2 − 0.16$

(A38)

Forward model operation based on Eq. (A38) has shown to significantly outperform that based on implementing the modified KM function of Eq. (A37) with Eq. (A35) in predicting the diffuse reflectance involving strong absorption. By treating

$μ s ′ (λ) = 1 m m − 1$ , we have from Eq. (A38) the spectral absorption via the Piao & Sun model (Piao & Sun, 2021) as:

$[μ a (λ)] O S U = 0.25 [1 (2 π R ∞) 2 + 0.16] 1 2.2 − 0.11875$

(A39)

Equation (A39) is the basis of the new algorithm introduced in this work, which takes the same number of a single step of converting the spectral reflectance to a spectral absorptivity term as the spectral reflex attenuance is converted, for implementing with the set of equations of A27, A29, and A33.

Table A1.

Choice of loins, number of steaks, number of repeated measurements per steak, and number of days of retail display of the total 44 muscles that have been used towards this work. “X” indicates that colorimeter measurement has been taken on that day.

View Larger Table

Table A1a. LL.
No. of steaks	No. of repetition	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6	Day 7
7^{^*}	5	X	X	X	X	X	X	X
7	6	X	X	X	X		X	X	X
8	3	X	X	X	X	X	X

Table A1b. PM.
No. of steaks	No. of repetitions	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6	Day 7
7	3	X	X	X	X	X	X
7	6	X	X	X	X	X	X	X

Table A1c. ST.
No. of steaks	No. of repetitions	Day 0	Day 1	Day 2	Day 3	Day 4	Day 5	Day 6	Day 7
8	6	X	X	X	X	X	X	X

Represents the set used leading to Figures 3–6.

Figure A1.

The percentages of the myoglobin forms [Oxy]% (column A), [Deoxy]% (column B), and [Met]% (column C) of the 7 LL muscles over 6 d of retail display, calculated by the equations sets (21), (22), and (23) at three different settings (column D) of the baseline value of the spectral absorptivity, represented by reflex attenuance of Eq. (A34), projected for 730 nm, in comparison to the chroma. The top panel corresponds to implementing the algorithm of reflex attenuance with the baseline reference at 730 nm set as 100% of the value obtained at 700 nm. The middle panel corresponds to implementing the algorithm of reflex attenuance with the baseline reference at 730 nm set as 20% of the value obtained at 700 nm. The bottom panel corresponds to implementing the algorithm of reflex attenuance with the baseline reference at 730 nm set as 0% of the value obtained at 700 nm, i.e., neglecting the base reference at 730 nm. The P value of each sub-figure is the power of two-tailed unpaired t-test between the two groups of data. (P < 0.05) indicates that the two groups are significantly different. The checkmark is placed if the two groups of data are statistically significantly different (P < 0.05) and the values of the percentages of the myoglobin form are within the physically realistic range of 0% to 100%. The check mark essentially indicates that the change (e.g., increase) of the myoglobin form over the duration of display is significantly opposite to the change (e.g., decrease) of the chroma over the duration of display. The increase of [Met]% over the duration of display estimated by the algorithm of reflex attenuance with Krzywicki’s equation is statistically insignificant, when compared to the decrease in chroma.

Figure A2.

The percentages of the myoglobin forms [Oxy]% (column A), [Deoxy]% (column B), and [Met]% (column C) of the 7 LL muscles over 6 d of retail display, calculated by the equations sets (30), (31), and (32) at three different settings (column D) of the baseline value of the spectral absorptivity, represented by reduced scattering scaled absorption of Eq. (A39), projected for 730 nm, in comparison to the chroma. The top panel corresponds to implementing the algorithm of reduced scattering scaled absorption with the baseline reference at 730 nm set as 100% of the value obtained at 700 nm. The middle panel corresponds to implementing the algorithm of reduced scattering scaled absorption with the baseline reference at 730 nm set as 20% of the value obtained at 700 nm. The bottom panel corresponds to implementing the algorithm of reduced scattering scaled absorption with the baseline reference at 730 nm set as 0% of the value obtained at 700 nm, i.e., neglecting the base reference at 730 nm. The P value of each sub-figure is the power of two-tailed unpaired t-test between the two groups of data. (P < 0.05) indicates that the two groups are significantly different. The checkmark is placed if the two groups of data are statistically significantly different (P < 0.05) and the values of the percentages of the myoglobin form are within the physically realistic range of 0% to 100%. The check mark essentially indicates that the change (e.g., increase) of the myoglobin form over the duration of display is significantly opposite to the change (e.g., decrease) of the chroma over the duration of display. The increase of [Met]% over the duration of display estimated by the algorithm of reduced scattering scaled absorption with Krzywicki’s equation is statistically significant, when compared to the decrease in chroma.

Literature Cited

Jacques, S. L. (2013). Optical properties of biological tissues: A review. Phys. Med. Biol. 58:R37–R61. https://doi.org/10.1088/0031-9155/58/11/R37

Krzywicki, K. (1979). Assessment of relative content of myoglobin, oxymyoglobin and metmyoglobin at the surface of beef. Meat Sci. 3:1–10. https://doi.org/10.1016/0309-1740(79)90019-6

Kubelka, P. (1948). New contributions to the optics of intensely light-scattering materials. Part I. J. Opt. Soc. Am. 38:448. https://doi.org/10.1364/JOSA.38.000448

Lindberg, J. D., & Laude, L. S. (1974). Measurement of the absorption coefficient of atmospheric dust. Appl. Optics 13:1923. https://doi.org/10.1364/AO.13.001923

Piao, D., & Sun, T. (2021). Diffuse photon remission from thick opaque media of the high absorption/scattering ratio beyond what is accountable by the Kubelka–Munk function. Opt. Lett. 46:1225. https://doi.org/10.1364/OL.415650

Sandoval, C., & Kim, A. D. (2014). Deriving Kubelka–Munk theory from radiative transport. J. Opt. Soc. Am. A 31:628. https://doi.org/10.1364/JOSAA.31.000628

Modified Krzywicki’s Equations to Quantify Myoglobin Forms on Meat Surfaces

Abstract

Funding

Introduction

Methods and Materials

The tissue-optics foundation of Krzywicki’s equations

Molar extinction coefficient

Constitution of the spectral absorption of meat by myoglobin

The 2-stage feature of Krzywicki’s original approach to independently calculate the percentage numbers of myoglobin forms

Two algorithms for calculating the spectral absorptivity properties in stage one of Krzywicki’s approach

The original implementation of Krzywicki’s approach to calculate the percentage numbers of myoglobin forms by converting the spectral reflectance to the reflex attenuance in stage 1

An alternative implementation of Krzywicki’s approach to calculate the percentage numbers of myoglobin forms by converting the spectral reflectance to spectral absorption in stage 1

Summary comparison of the 2 algorithms in stage 1 to implement Krzywicki’s equations

Measuring surface color on steaks displayed under retail conditions

Raw color analysis

Offline data-processing implementing the 2 algorithms of calculating the percentage numbers of the forms of myoglobin

Statistical Analysis

Results

The percentage numbers of myoglobin forms estimated by using algorithm #1 to deduce reflex attenuance in stage 1 for running Krzywicki’s equations

The percentage numbers of myoglobin forms estimated by using algorithm #2 in stage 1 deducing absorption directly from the spectral reflectance for running Krzywicki’s equation

Assessing the reality of the percentage numbers of myoglobin forms estimated by the 2 algorithms of inversion for implementing Krzywicki’s equations

The change of metmyoglobin of 7 LL steaks over retail display in comparison to the color-change resolved by colorimetry

The changes of metmyoglobin of 44 steaks of 3 types over retail display in comparison with the colorimetry results

Discussion

Conclusions

Acknowledgements

Funding

Disclosures

Data Availability

Literature Cited

Supplemental Materials

1. Re-derivation of Krzywicki’s Equations

1.1 Re-derivation of Krzywicki’s equation of x or [Oxy]%

1.2 Re-derivation of Krzywicki’s equation of y or [Deoxy]%

1.3 Re-derivation of Krzywicki’s equation of z or [Met]%

2. Methods for stage one of Krzywicki’s equations to convert spectral reflectance to spectral absorptivity properties

2.1 Inversion model #1 that converts the spectral reflectance to reflex attenuance to be implemented with Kryziwicki’s equations

2.2 A new model of converting the spectral reflectance to reduced scattering scaled absorption

Literature Cited