Bluck-Coward self-corrected half-emptying time
Source:R/breathtest_parameters.R
t50_bluck_coward.RdUses Newton's method to solve the self-corrected Bluck-Coward equation for 1/2 to compute the half-emptying time t_50.
See also equation G(n,t) in
Bluck LJC, Jackson S, Vlasakakis G, Mander A (2011) Bayesian hierarchical methods to interpret the 13C-octanoic acid breath test for gastric emptying. Digestion 83_96-107, page 98.
Arguments
- cf
Named vector of coefficients; only
kandbetaare required. In this package,kis measured in units of 1/min (e.g. 0.01/min), in publications it is often quoted as 1/h (e.g. 0.6/h).
Value
Time where value is 1/2 of the maximum, i.e. t_50 or t_1/2 in minutes; in the publication by Bluck et al, the parameter is called t_1/2(in).
Examples
# From table 3 and 4 in Bluck et al.; values for \code{k} and \code{beta}
# (nls, bayesian) are entered and checked against the tabulated values of
# t_{1/2(in)}.
# Most errors are small, but there are some outliers; errors in paper table?
# Parameters and Bluck et al. results:
# table 3 of Bluck et al.
cf3 = data.frame(
method = rep(c("nls", "bayesian")),
group = rep(c("lean", "obese"),each=2),
k = c(0.576,0.606,0.529,0.608),
beta = c(5.24, 5.79, 5.95, 7.54),
t12 = c(3.67, 3.63, 4.23, 3.99),
t12in = c(2.076, 2.110, 2.422, 2.466),
tlag = c(2.88, 2.88, 3.34, 3.26),
tlagin = c(1.632, 1.724, 1.92, 2.101)
)
cf3 = dplyr::mutate(cf3,
t50_maes_ghoos = t50_maes_ghoos(cf3),
t50_bluck_coward = t50_bluck_coward(cf3),
tlag_maes_ghoos = tlag_maes_ghoos(cf3),
tlag_bluck_coward = tlag_bluck_coward(cf3),
err_t50_maes_ghoos = round(100*(t50_maes_ghoos-t12)/t12, 2),
err_t50_bluck_coward =
round(100*(t50_bluck_coward-t12in)/t12in, 2),
err_lag_maes = round(100*(tlag_maes_ghoos-tlag)/tlag,2),
err_lag_bluck_coward =
round(100*(tlag_bluck_coward-tlagin)/tlagin,2)
)
cf3
#> method group k beta t12 t12in tlag tlagin t50_maes_ghoos
#> 1 nls lean 0.58 5.2 3.7 2.1 2.9 1.6 3.6
#> 2 bayesian lean 0.61 5.8 3.6 2.1 2.9 1.7 3.6
#> 3 nls obese 0.53 6.0 4.2 2.4 3.3 1.9 4.2
#> 4 bayesian obese 0.61 7.5 4.0 2.5 3.3 2.1 4.0
#> t50_bluck_coward tlag_maes_ghoos tlag_bluck_coward err_t50_maes_ghoos
#> 1 2.1 2.9 1.7 -1.21
#> 2 2.1 2.9 1.8 -0.81
#> 3 2.5 3.4 2.1 -1.34
#> 4 2.5 3.3 2.2 0.26
#> err_t50_bluck_coward err_lag_maes err_lag_bluck_coward
#> 1 0.58 -0.15 2.5
#> 2 1.37 0.62 1.8
#> 3 3.18 0.94 7.3
#> 4 3.20 1.92 3.9
# table 4
# there are large differences for mj3, both using the bayesian (26%)
# and the nls method (16%). The other data are within the expected limits
cf4 = data.frame(
method = rep(c("nls", "bayesian"),each=3),
group = rep(c("mj1", "mj2", "mj3")),
k = c(0.585, 0.437, 0.380, 0.588, 0.418, 0.361),
beta=c(4.35, 4.08, 4.44, 4.49, 4.30, 4.29),
t12 = c(3.39, 4.25, 4.82, 3.40, 4.61, 5.09),
t12in = c(1.77, 2.16, 2.19, 1.81, 2.34, 2.43),
tlag = c(2.56, 3.17, 3.39, 2.58, 3.40, 3.62),
tlagin = c(1.30, 1.53, 1.33, 1.35, 1.65, 1.57)
)
cf4 = dplyr::mutate(cf4,
t50_maes_ghoos = t50_maes_ghoos(cf4),
t50_bluck_coward = t50_bluck_coward(cf4),
tlag_maes_ghoos = tlag_maes_ghoos(cf4),
tlag_bluck_coward = tlag_bluck_coward(cf4),
err_t50_maes_ghoos = unlist(round(100*(t50_maes_ghoos-t12)/t12)),
err_t50_bluck_coward =
round(100*(t50_bluck_coward-t12in)/t12in,2),
err_lag_maes = round(100*(tlag_maes_ghoos-tlag)/tlag,2),
err_lag_bluck_coward =
round(100*(tlag_bluck_coward-tlagin)/tlagin,2)
)
cf4
#> method group k beta t12 t12in tlag tlagin t50_maes_ghoos
#> 1 nls mj1 0.58 4.3 3.4 1.8 2.6 1.3 3.3
#> 2 nls mj2 0.44 4.1 4.2 2.2 3.2 1.5 4.2
#> 3 nls mj3 0.38 4.4 4.8 2.2 3.4 1.3 5.1
#> 4 bayesian mj1 0.59 4.5 3.4 1.8 2.6 1.4 3.3
#> 5 bayesian mj2 0.42 4.3 4.6 2.3 3.4 1.6 4.6
#> 6 bayesian mj3 0.36 4.3 5.1 2.4 3.6 1.6 5.3
#> t50_bluck_coward tlag_maes_ghoos tlag_bluck_coward err_t50_maes_ghoos
#> 1 1.8 2.5 1.3 -3
#> 2 2.2 3.2 1.6 0
#> 3 2.8 3.9 2.1 6
#> 4 1.8 2.6 1.4 -3
#> 5 2.4 3.5 1.8 -1
#> 6 2.8 4.0 2.1 4
#> err_t50_bluck_coward err_lag_maes err_lag_bluck_coward
#> 1 -0.56 -1.8 2.2
#> 2 2.82 1.5 6.6
#> 3 26.03 15.7 57.8
#> 4 -0.50 -1.0 1.9
#> 5 4.19 2.6 11.0
#> 6 15.93 11.4 34.6