Package {MFO}

Title:	Maximal Fat Oxidation and Kinetics Calculation
Version:	0.2.1
Date:	2026-11-11
Description:	Computes the maximal fat oxidation (MFO), the exercise intensity that elicits it (Fatmax), the Fatmax zone and the crossover point from indirect calorimetry data of a graded exercise test. Substrate oxidation is estimated with the stoichiometric equations of Frayn (1983) <doi:10.1152/jappl.1983.55.2.628>, Peronnet and Massicotte (1991, Can J Sport Sci 16(1):23-29) or Jeukendrup and Wallis (2005) <doi:10.1055/s-2004-830512>, and Fatmax is located with a third- or second-degree polynomial, the measured values or the MRER method. Fat oxidation kinetics are described with the SIN model of Cheneviere et al. (2009) <doi:10.1249/MSS.0b013e31819e2f91>, which returns dilatation, symmetry and translation, together with the intensity at which fat oxidation becomes negligible (Fatmin). Applications of these methods are given in Achten et al. (2002) <doi:10.1097/00005768-200201000-00015> and Montes de Oca et al. (2021) <doi:10.1080/17461391.2020.1788650>.
License:	MIT + file LICENSE
Encoding:	UTF-8
Language:	en-GB
LazyData:	true
RoxygenNote:	7.3.3
URL:	https://github.com/JorgeDelro/MFO, https://jorgedelro.github.io/MFO/
BugReports:	https://github.com/JorgeDelro/MFO/issues
Imports:	stats, dplyr, ggplot2, tibble, minpack.lm, openxlsx, readxl, tidyr
Depends:	R (≥ 4.1.0)
Suggests:	covr, knitr, rmarkdown, testthat (≥ 3.0.0), withr
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-11-12 09:10:10 UTC; jorge
Author:	Jorge R Fernandez-Santos [aut, cre], Jesus G Ponce-Gonzalez [ctb], Cristina Casals [ctb], Jose L Gonzalez-Montesinos [ctb], Juan Corral-Perez [ctb], Alejandro Perez-Bey [ctb]
Maintainer:	Jorge R Fernandez-Santos <jorgedelrosario.fernandez@uca.es>
Repository:	CRAN
Date/Publication:	2026-11-12 14:40:08 UTC

Maximal Fat Oxidation & FatMax Function

Description

Calculate Maximal Fat Oxidation (MFO), the exercise intensity that elicits MFO (FatMax), the FatMax zone, and the crossover point. Supports multiple methods to locate MFO (P3, P2, MV, MRER) and stoichiometric equations (Frayn, Jeukendrup, Peronnet).

Usage

MFO(
  step_time,
  db_MFO,
  db_basal,
  db_graded = NULL,
  cv_var,
  author,
  VO2max = NULL,
  method = "P3",
  step_duration = 180,
  steady_state_duration = 60,
  RER_cutoff = NULL,
  negative_FAT = "truncate",
  body_mass = NULL,
  fat_free_mass = NULL,
  HRmax = NULL,
  vo2max_window = 30,
  include_basal = FALSE,
  energy_equivalents = c("classic", "source"),
  use_incomplete_step = FALSE,
  min_steps = 6
)

Arguments

Value

Returns a list which contains:

• MFO_db: database with Load, VO2, VCO2, HR, RER, CHO, FAT, Kcal, porc_VO2 (one row per exercise step, plus the resting row first if 'include_basal = TRUE').

• MFO_plot: ggplot object with the MFO plot including FAT and CHO curves.

• MFO: Maximal fat oxidation (g/min).

• MFO_relative: MFO relative to body mass (mg/kg/min), if body_mass provided.

• MFO_FFM: MFO relative to fat-free mass (mg/kgFFM/min), if fat_free_mass provided.

• FAT_MAX: Intensity that elicits MFO (% VO2max).

• FAT_MAX_HR: Heart rate at FatMax (bpm), interpolated from data.

• FAT_MAX_porcHRmax: FatMax as % HRmax, if HRmax provided.

• FatMax_zone_lower: Lower bound of FatMax zone (% VO2max).

• FatMax_zone_upper: Upper bound of FatMax zone (% VO2max).

• FatMax_zone_method: "fitted curve", or "linear interpolation" for 'method = "MV"'.

• crossover_point: Intensity at which CHO > FAT (% VO2max).

• R_squared: R-squared of the fitted model ('NA' for "MV").

• RMSE: Root mean squared error of the fitted model ('NA' for "MV").

• method: Method used for curve fitting.

• energy_equivalents: energy equivalents used for 'Kcal' ("classic" or "source").

• VO2max: VO2max used to express intensity (ml/min).

• VO2max_window: effective averaging window for VO2max (s); NA if VO2max was supplied.

• VO2max_window_requested: 'vo2max_window' as requested (s); NA if VO2max was supplied.

• x_CHO: carbohydrates in basal metabolism (g/min), with the same 'author' equations as the test.

• x_FAT: fat in basal metabolism (g/min).

• x_Kcal: Kcal in basal metabolism (kcal/min).

Quality warnings

Besides the warnings on inputs, 'MFO()' warns when the result needs checking (SPEC 9.5, 14):

• fewer than 'min_steps' exercise steps enter the fit;

• Fatmax is at the edge: the exercise step nearest to Fatmax is the first or the last one (resting row excluded), so the maximum may lie outside the measured range (the test started too high or stopped too early);

• the last step is incomplete (see 'use_incomplete_step');

• HR is missing: 'FAT_MAX_HR' and 'FAT_MAX_porcHRmax' are 'NA', or interpolated from fewer steps. HR never removes steps from the fit.

Methods

• '"P3"', '"P2"': least-squares polynomial of FAT on % VO2max; Fatmax is the maximum of the fitted curve within the measured range (SPEC 6.2, 6.3).

• '"MV"' (measured values): Fatmax is the step with the highest observed FAT, computed with the 'author' equations; MFO is that FAT. No model is fitted, so 'R_squared' and 'RMSE' are 'NA'; the curve drawn and the Fatmax zone use linear interpolation between steps ('FatMax_zone_method = "linear interpolation"'). SPEC 6.1.

• '"MRER"': procedure of Cheneviere et al. (2009, p. 1617, doi:10.1249/MSS.0b013e31819e2f91), after Perez-Martin et al. (2001). (1 - RER) and VO2 are fitted as straight lines of % VO2max, 1 - RER = a1 + b1 x and VO2 = a2 + b2 x, so fat oxidation 1.7 (1 - RER) VO2 is a parabola whose vertex, x = -(a1 b2 + a2 b1) / (2 b1 b2), gives Fatmax, and its value MFO (g/min). This uses the 1.7 (1 - RER) VO2 simplification of Peronnet & Massicotte whatever 'author' is. If the parabola has no maximum or the vertex lies outside the measured range, '"MV"' is used with a warning. SPEC 3, 6.1.

Up to v0.2.0, '"MRER"' returned the step with the highest observed 1.7 (1 - RER) VO2, which is a measured-values method: use '"MV"' for that behaviour (same step; MFO from the 'author' equation). SPEC 19, finding 4.

Examples

## Not run: 
# Read dfs
data(list = c("basal_df", "MFO_df", "VO2max_df"), package = "MFO")

# Convert to data.frame
basal_df <- data.frame(basal_df)
MFO_df <- data.frame(MFO_df)
VO2max_df <- data.frame(VO2max_df)

# Calculate MFO and Fatmax (P3 method)
result_MFO <- MFO(step_time = 20,
                 db_MFO = MFO_df,
                 db_basal = basal_df,
                 db_graded = VO2max_df,
                 cv_var = "RER",
                 author = "Frayn",
                 method = "P3")

# With all options
result_MFO <- MFO(step_time = 20,
                 db_MFO = MFO_df,
                 db_basal = basal_df,
                 db_graded = VO2max_df,
                 cv_var = "RER",
                 author = "Frayn",
                 method = "P3",
                 RER_cutoff = 1.0,
                 body_mass = 75,
                 fat_free_mass = 60,
                 HRmax = 190)

## End(Not run)

MFO test dataframe

Description

Synthetic breath-by-breath data from an incremental exercise test with 8 stages (25-200 W, 3 min per stage). Simulates a moderately trained male cyclist covering ~20-75% VO2max. The fat oxidation pattern shows a clear inverted U-shape with a peak around 35-40% VO2max, suitable for demonstrating P3, P2, MRER, and SIN model fitting.

Usage

data(MFO_df)

Format

A data frame with 72 rows and 8 variables:

Time

time test, in seconds (20s intervals)

HR

heart rate, in beats/min

VO2

volume of oxygen consumption, in mL/min

VCO2

volume of exhaled carbon dioxide, in mL/min

RER

respiratory exchange ratio

BF

breathing frequency, in breaths/min

VE

ventilation, in L/min

PETCO2

end-tidal carbon dioxide pressure, in mmHg

Maximal Fat Oxidation Kinetics (SIN Model)

Description

Fit the SIN model (Cheneviere et al., 2009) to describe fat oxidation kinetics and extract three parameters: dilatation (d), symmetry (s), and translation (t). Also calculates Fatmin (the intensity where fat oxidation reaches zero), which is a unique capability of the SIN model over polynomial methods.

Usage

MFO_kinetics(MFO_data, fatmin_threshold = NULL)

Arguments

Details

The model is equation \[14\] of Cheneviere et al. (2009, Med Sci Sports Exerc 41: 1615-1625, p. 1618, doi:10.1249/MSS.0b013e31819e2f91):

\%MFO = \sin\{[\pi^{1/s} / (\pi + 2d) \cdot (Kx + d + t)]^s\}, \quad K = \pi/100,

x in % VO2max. Up to v0.2.0 the package computed (pi^(1/s)/pi) + 2d, so its 'd' (and 't') are not comparable with published values (SPEC 6.5, 19).

Value

Returns a list which contains:

• MFO_kinetics_data: 'MFO_data' plus normalized %MFO ('porc_MFO', 100 = highest observed FAT).

• MFO_kinetics_plot: ggplot object with the observed %MFO, the basic SIN and the fitted SIN.

• d: dilatation parameter (width of the curve).

• t: translation parameter (horizontal shift along intensity axis).

• s: symmetry parameter (controls curve shape asymmetry; <1 left-skew, >1 right-skew).

• Fatmin: exercise intensity (% VO2max) at the second zero of the curve, or at 'fatmin_threshold' of MFO if given.

• FatMax_SIN: exercise intensity (% VO2max) at peak fat oxidation from SIN model.

• R_squared: R-squared of the SIN model fit (observed proportions of MFO).

• RMSE: Root mean squared error of the SIN model fit (proportion of MFO).

• d_se: standard error of dilatation parameter.

• s_se: standard error of symmetry parameter.

• t_se: standard error of translation parameter.

• convergence: logical, whether the SIN model converged.

Fitting procedure

As in Cheneviere et al. (2009), the model is fitted by non-linear least squares (Levenberg-Marquardt, [minpack.lm::nlsLM()], start d = 0, s = 1, t = 0) directly to the observed rows of 'MFO_data', with fat oxidation expressed as a proportion of the highest observed value (%MFO / 100). There is no smoothing and no synthetic grid: v0.1.0/v0.2.0 fitted the SIN to a fine grid predicted by a third-degree polynomial, which inflated n and shrank the standard errors of d, s and t (SPEC 19, finding 7). R-squared and RMSE are computed on the observed proportions. At least 4 rows are needed for 3 parameters: with fewer the function warns and returns 'convergence = FALSE'; with 4 or 5 it fits but warns that the fit is unreliable and that at least 6 steps are recommended (SPEC 6.5).

'FatMax_SIN' and 'Fatmin' come from closed forms derived from eq. \[14\]: Fatmax = \[(pi + 2d) 2^(-1/s) - d - t\] / K and, by default, Fatmin = second zero of the curve = (pi + d - t) / K, Cheneviere's definition (SPEC 19, finding 3). Fatmin can lie beyond the last observed step (extrapolation).

Which rows are fitted

'MFO_kinetics()' fits every row it receives. Cheneviere et al. (2009) fitted exercise steps only, and averaged only steps with RER <= 1.0:

• 'MFO(..., RER_cutoff = 1)' reproduces their step selection;

• 'MFO(..., include_basal = TRUE)' adds the resting row to 'MFO_db', so it is also fitted here. This departs from Cheneviere's procedure in the SIN as well as in the P3.

The SIN is sensitive to the last steps because the curve must reach zero: in the example data ('author = "Frayn"') the last step (74.7 % VO2max, RER 1.03) has negative fat oxidation.

• 'negative_FAT = "truncate"' (default) sets it to 0 and that point anchors the second zero: d = -0.385, s = 0.910, t = 0.355, FatMax_SIN = 36.2, Fatmin = 76.4 % VO2max.

• 'negative_FAT = "exclude"' or 'RER_cutoff = 1' drops it: d = -0.440, s = 0.539, t = 0.023, FatMax_SIN = 33.2, Fatmin = 85.3 % VO2max.

Report which option was used when publishing SIN parameters.

Examples

## Not run: 
# Read dfs
data(list = c("basal_df", "MFO_df", "VO2max_df"), package = "MFO")

# Convert to data.frame
basal_df <- data.frame(basal_df)
MFO_df <- data.frame(MFO_df)
VO2max_df <- data.frame(VO2max_df)

# Calculate MFO and Fatmax
result_MFO <- MFO(step_time = 20,
                 db_MFO = MFO_df,
                 db_basal = basal_df,
                 db_graded = VO2max_df,
                 cv_var = "RER",
                 author = "Frayn")

# Calculate MFO Kinetics
result_kinetics <- MFO_kinetics(result_MFO$MFO_db)

# Fatmin at 5 % of MFO instead of the second zero
MFO_kinetics(result_MFO$MFO_db, fatmin_threshold = 0.05)$Fatmin

## End(Not run)

Maximal Fat Oxidation calculation of multiple databases

Description

Process multiple participants' MFO data in batch, generating plots and summary results. Supports all parameters from the MFO() and MFO_kinetics() functions.

Usage

MFOs(
  from = c("folder", "files"),
  path,
  db_basal_name,
  db_MFO_name,
  db_graded_name,
  step_time,
  cv_var,
  author,
  method = "P3",
  VO2max = NULL,
  step_duration = 180,
  steady_state_duration = 60,
  RER_cutoff = NULL,
  negative_FAT = "truncate",
  body_mass = NULL,
  fat_free_mass = NULL,
  HRmax = NULL,
  remove_rows = NULL,
  col_name_VO2 = "VO2",
  col_name_VCO2 = "VCO2",
  col_name_RER = "RER",
  col_name_HR = "HR",
  save_plot = TRUE,
  save_result = TRUE,
  output_format = "xlsx",
  vo2max_window = 30,
  include_basal = FALSE,
  fatmin_threshold = NULL,
  energy_equivalents = c("classic", "source"),
  use_incomplete_step = FALSE,
  min_steps = 6
)

Arguments

Details

Every warning raised for a participant is re-issued with the participant ID first (e.g. '"P02: No HR data in db_MFO: ..."'), so the quality warnings of [MFO()] (missing HR, few steps, Fatmax at the edge, incomplete last step) can be traced in a batch. A participant whose data cannot be processed gives a row of 'NA' and a warning with the error.

Value

Data frame with one row per participant containing:

• ID: Participant identifier.

• MFO_g_min: Maximal fat oxidation (g/min).

• MFO_relative: MFO relative to body mass (mg/kg/min), if body_mass provided.

• MFO_FFM: MFO relative to fat-free mass (mg/kgFFM/min), if fat_free_mass provided.

• VO2max: VO2max used (ml/min).

• VO2max_window: effective averaging window for VO2max (s); NA if VO2max was supplied.

• FAT_MAX_perVO2max: FatMax as % VO2max.

• FAT_MAX_HR: Heart rate at FatMax (bpm).

• FAT_MAX_porcHRmax: FatMax as % HRmax, if HRmax provided.

• FatMax_zone_lower: Lower bound of FatMax zone (% VO2max).

• FatMax_zone_upper: Upper bound of FatMax zone (% VO2max).

• FatMax_zone_method: "fitted curve", or "linear interpolation" for "MV".

• crossover_point: Crossover intensity (% VO2max).

• R_squared: R-squared of the curve fit.

• method: method requested.

• energy_equivalents: "classic" or "source".

• d: SIN dilatation parameter.

• t: SIN translation parameter.

• s: SIN symmetry parameter.

• Fatmin: Fatmin from SIN model (% VO2max).

• FatMax_SIN: FatMax from SIN model (% VO2max).

• SIN_R_squared: R-squared of SIN model.

• SIN_convergence: whether the SIN model was fitted.

• CHO_g_basal: Basal CHO oxidation (g/min).

• FAT_g_basal: Basal FAT oxidation (g/min).

• Kcal_g_basal: Basal energy expenditure (kcal/min).

Examples

## Not run: 
# MFOs function
results <- MFOs(from = "folder",
                path = "/path/to/participant_data",
                db_basal_name = "basal_df",
                db_MFO_name = "MFO_df",
                db_graded_name = "graded_df",
                step_time = 20,
                cv_var = "RER",
                author = "Frayn",
                method = "P3")

## End(Not run)

Graded exercise test dataframe

Description

Synthetic breath-by-breath data from a graded exercise test to exhaustion (10 stages, 75-300 W). Simulates a moderately trained male cyclist reaching VO2max ~3200 mL/min and HRmax ~185 bpm. Used to determine VO2max when this value is not known a priori.

Usage

data(VO2max_df)

Format

A data frame with 30 rows and 9 variables:

Time

time test, in seconds (20s intervals)

HR

heart rate, in beats/min

Load

workload, in watts

VO2

volume of oxygen consumption, in mL/min

VCO2

volume of exhaled carbon dioxide, in mL/min

RER

respiratory exchange ratio

BF

breathing frequency, in breaths/min

VE

ventilation, in L/min

PETCO2

end-tidal carbon dioxide pressure, in mmHg

Basal test dataframe

Description

Synthetic breath-by-breath data from a 15-minute resting metabolic assessment. Simulates a moderately trained male cyclist (VO2max ~3200 mL/min) with an initial stabilization period followed by steady-state resting metabolism.

Usage

data(basal_df)

Format

A data frame with 45 rows and 8 variables:

Time

time test, in seconds (20s intervals)

HR

heart rate, in beats/min

VO2

volume of oxygen consumption, in mL/min

VCO2

volume of exhaled carbon dioxide, in mL/min

RER

respiratory exchange ratio

BF

breathing frequency, in breaths/min

VE

ventilation, in L/min

PETCO2

end-tidal carbon dioxide pressure, in mmHg

Calculate step means from raw data

Description

Generalized function that works with any step_time and step_duration. Extracts the last minute of each step (steady-state) for averaging.

Usage

calculate_vars(
  step_time,
  db_MFO,
  VO2max,
  author,
  step_duration = 180,
  steady_state_duration = 60,
  RER_cutoff = NULL,
  energy_equivalents = "classic",
  use_incomplete_step = FALSE
)

Arguments

Details

With 'author = "Jeukendrup"', a single warning lists the steps (after 'RER_cutoff') whose intensity is outside 40-75 % VO2max, the range of the Jeukendrup & Wallis (2005, Table 5) equations (SPEC 3, 9.5).

Value

data frame with columns: VO2, VCO2, HR, RER, CHO, FAT, Kcal (one row per step)

Fat oxidation with the RER simplification (MRER)

Description

FAT (mg/min) = 1.7 (1 - RER) VO2 (ml/min), the simplification of the Peronnet & Massicotte (1991) equation used by Perez-Martin et al. (2001) and Cheneviere et al. (2009, p. 1617). Used by 'method = "MRER"' in [MFO()].

Usage

fat_oxidation_MRER(VO2, RER)

Arguments

Value

fat oxidation rate in g/min

Basal metabolic rate

Description

Calculate basal metabolism from rest test data using a rolling window to identify the most metabolically stable 5-minute period.

Usage

met_basal(
  step_time,
  db,
  cv_var,
  window_duration = 300,
  author = "Frayn",
  energy_equivalents = "classic"
)

Arguments

Preprocess breath-by-breath data into fixed intervals

Description

Averages breath-by-breath data into fixed time intervals.

Usage

preprocess_bxb(db, target_interval = 20)

Arguments

Value

data frame with data averaged into fixed intervals.

Read databases for MFO package

Description

Read databases for MFO package

Usage

read_MFO_databases(
  from = c("folder", "files"),
  path,
  db_basal_name,
  db_MFO_name,
  db_graded_name,
  col_name_VO2 = "VO2",
  col_name_VCO2 = "VCO2",
  col_name_RER = "RER",
  col_name_HR = "HR",
  remove_rows = NULL
)

Arguments

Value

Returns 3 databases:

• participant_db_basal: database with basal metabolism.

• participant_db_MFO: database with MFO test.

• participant_db_graded: graded exercise test.

Substrate oxidation equations

Description

Calculate fat and carbohydrate oxidation from VO2 and VCO2 using different stoichiometric equations.

Usage

substrate_oxidation(
  VO2,
  VCO2,
  author,
  VO2max = NULL,
  phase = c("exercise", "rest"),
  energy_equivalents = c("classic", "source")
)

Arguments

Details

Rule: a coefficient printed in the source is used as printed; a derived coefficient is used only where the source prints no equation (SPEC 19).

**Frayn (1983)** (J Appl Physiol 55: 628-634, eqs. 5 and 6, n = 0), as printed: CHO = 4.55 VCO2 - 3.21 VO2, FAT = 1.67 VO2 - 1.67 VCO2. Deriving them from his Table 1 constants (glucose 0.746/0.746, fat 2.03/1.43 L/g) gives 4.535/3.195 and 1.667/1.667; this is only a cross-check in the tests.

**Peronnet & Massicotte (1991)** (Can J Sport Sci 16: 23-29) print no equations. The package derives them from the O2/CO2 constants of their Table 1 (glucose 0.7455/0.7426, fatty acid 2.0092/1.4136 L/g) by solving VO2 = 0.7455 CHO + 2.0092 FAT and VCO2 = 0.7426 CHO + 1.4136 FAT for CHO and FAT: CHO = 4.58519 VCO2 - 3.22597 VO2, FAT = 1.69469 VO2 - 1.70130 VCO2. Up to v0.2.0 the rounded 4.585 / 3.2255 / 1.6946 / 1.7012 were used (SPEC 19, finding 6).

**Jeukendrup & Wallis (2005)** (Int J Sports Med 26 Suppl 1: S28-S37), with the coefficients printed in their Table 5:

• below 50 % VO2max (40-50 % set): CHO = 4.344 VCO2 - 3.061 VO2;

• at or above 50 % VO2max (50-75 % set): CHO = 4.210 VCO2 - 2.962 VO2;

• FAT = 1.695 VO2 - 1.701 VCO2 in both.

With 'phase = "rest"' there is no printed equation: Jeukendrup & Wallis propose none. As a package decision (pure glucose, no glycogen use at rest), the pair is derived from the O2/CO2 constants of their Table 4 (glucose 0.7455/0.7426, fatty acid 2.0092/1.4136 L/g) by solving the 2x2 system: CHO = 4.58519 VCO2 - 3.22597 VO2, FAT = 1.69469 VO2 - 1.70130 VCO2. The same derivation with 50/50 and 20/80 glucose/glycogen reproduces the Table 5 coefficients within 0.005 and is kept as a cross-check in the tests.

The Table 5 sets are defined for 40-75 % VO2max only. Below 40 % the 40-50 % set and above 75 % the 50-75 % set are extrapolated; [calculate_vars()] warns when an exercise step falls outside the range (SPEC 3, 9.5). Resting values ('phase = "rest"') never trigger that warning.

**Energy equivalents** ('Kcal = CHO * e_cho + FAT * e_fat', SPEC 19, finding 1):

• "classic": 4 kcal/g CHO and 9 kcal/g FAT for every equation (v0.2.0). Default in 0.2.x; the default will change to "source" in 0.3.0.

• "source", 'author = "Peronnet"': glucose 3.8683 and fat 9.7460 kcal/g (Peronnet & Massicotte 1991, Table 1).

• "source", 'author = "Frayn"': Frayn (1983, p. 632) gives no energy equivalents; as a package decision the Peronnet & Massicotte values are used.

• "source", 'author = "Jeukendrup"': Jeukendrup & Wallis (2005, Table 5), CHO 3.95 kcal/g below 50 % VO2max (40-50 % set) and 4.07 kcal/g at or above 50 % (50-75 % set), fat 9.75 kcal/g. At rest ('phase = "rest"', pure glucose pair, package decision) the Peronnet & Massicotte glucose and fat values are used, because they are the constants of that pair. 'Kcal' jumps at 50 % VO2max with the CHO value.

Value

Named list with CHO (g/min), FAT (g/min), Kcal (kcal/min)