Microbial growth models: A general mathematical approach to obtain μ max and λ parameters from sigmoidal empirical primary models

Longhi, Daniel Angelo; Dalcanton, Francieli; Aragão, Gláucia Maria Falcão de; Carciofi, Bruno Augusto Mattar; Laurindo, João Borges

doi:10.1590/0104-6632.20170342s20150533

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Brazilian Journal of Chemical Engineering

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Bioprocess Engineering • Braz. J. Chem. Eng. 34 (2) • Apr 2017 • https://doi.org/10.1590/0104-6632.20170342s20150533 copy

Microbial growth models: A general mathematical approach to obtain μ _max and λ parameters from sigmoidal empirical primary models

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Empirical sigmoidal models have been widely applied as primary models to describe microbial growth in foods. In predictive microbiology, the maximum specific growth rate (µ _max ) and the lag phase (λ) are the parameters of some models and have been considered as biological parameters. The objective of the current study was to propose mathematical equations to obtain the parameters μ _max and λ for any sigmoidal empirical growth model. In a case study, the performance was compared of two models based on empirical parameters and two models based on biological parameters. These models were fitted to experimental data for Lactobacillus plantarum in six isothermal conditions. Some advantages of the proposed approach were the practical and biological interpretation of these parameters, and the useful information of the secondary modeling describing the dependence of µ _max and λ with the temperature.

Keywords
predictive microbiology; mathematical modelling; secondary models; food safety

Power Type Growth	Shifted Logistic Function
$t_{i f x} = \sqrt[n]{\frac{b (n - 1)}{n + 1}}$	$t_{i f x} = t_{c}$
$y_{ifx} = \frac{A(n-1)}{2n}$	$y_{ifx} = \frac{A}{2} - \frac{A}{1+ \exp {(kt}_{c})}$
$μ_{max} = \frac{A {(n-1)}^{(\frac{n-1}{n})} {(n+1)}^{(\frac{n+1}{n})}}{4n \sqrt[n]{b}}$	$μ_{max} = \frac{Ak}{4}$
$λ = \sqrt[n]{b} {(\frac{n - 1}{n + 1})}^{(\frac{n + 1}{n})}$	${λ=t}_{c} - \frac{1}{k} [2- \frac{4}{1+ \exp ({kt}_{c})}]$

Model	Parameter	Temperature
Model	Parameter			4 °C	8 °C	12 °C	16 °C	20 °C	30 °C
Shifted Logistic Function	A ^a	1.39 (± 0.21)	2.01 (± 0.15)	2.20 (± 0.09)	2.26 (± 0.06)	2.61 (± 0.18)	2.63 (± 0.10)
	k^a [h^-1]	0.010 (± 0.003)	0.040 (± 0.008)	0.103 (± 0.014)	0.173 (± 0.014)	0.252 (± 0.047)	0.592 (± 0.066)
	t_c^a [h]	161.2 (± 18.8)	70.1 (± 4.1)	30.0 (± 1.2)	16.9 (± 0.4)	9.5 (± 0.7)	4.8 (± 0.2)
	µ_max [h^-1]	0.004	0.020	0.057	0.098	0.164	0.389
	λ [h]	26.3	25.4	12.3	6.5	2.9	1.8
Power Type Growth	A	1.62 (± 0.43)	2.09 (± 0.19)	2.28 (± 0.05)	2.31 (± 0.11)	2.55 (± 0.06)	2.73 (± 0.21)
	b [hⁿ]	7,180 (± 10,246)	9,550 (± 106,230)	22,320 (± 10,880)	4,069 (± 4,084)	394.5 (± 162.7)	72.94 (± 48.33)
	n	1.57 (± 0.34)	2.65 (± 0.51)	2.91 (± 0.15)	2.89 (± 0.37)	2.57 (± 0.19)	2.63 (± 0.48)
	µ_max [h^-1]	0.004	0.021	0.060	0.106	0.187	0.406
	λ [h]	24.3	25.5	12.0	6.7	3.3	1.7
Logistic Modified	µ_max^a [h^-1]	0.004 (± 0.001)	0.021 (± 0.003)	0.062 (± 0.008)	0.103 (± 0.007)	0.181 (± 0.031)	0.417 (± 0.048)
Logistic Modified	λ^a [h]	48.4 (± 15.5)	29.3 (± 6.7)	14.0 (± 2.2)	7.3 (± 0.8)	3.7 (± 1.2)	2.1 (± 0.4)
Gompertz Modified	µ_max^a [h^-1]	0.004 (± 0.001)	0.020 (± 0.002)	0.058 (± 0.003)	0.099 (± 0.007)	0.177 (± 0.014)	0.390 (± 0.038)
Gompertz Modified	λ^a [h]	31.3 (± 12.7)	25.0 (± 5.3)	11.8 (± 0.8)	6.2 (± 0.8)	3.2 (± 0.6)	1.6 (± 0.3)

Model	Parameter	m (°C⁻¹ h^−0.5 )	T_min (°C)	R ² _adj	RMSE (h^-0.5 )
Shifted Logistic Function	k^a [h^-1]	0.0256 (± 0.0013)	0.01 (± 0.88)	0.998	0.010
	(1/t_c)^a [h^-1]	0.0151 (± 0.0015)	−0.55 (± 1.92)	0.993	0.011
	µ_max [h^-1]	0.0218 (± 0.0007)	1.37 (± 0.47)	0.999	0.005
	(1/λ) [h^-1]	0.0238 (± 0.0075)	-1.89 (± 5.95)	0.939	0.056
Power Type Growth	µ_max [h^-1]	0.0224 (± 0.0010)	1.31 (± 0.78)	0.999	0.008
Power Type Growth	(1/λ) [h^-1]	0.0237 (± 0.0066)	-1.85 (± 5.24)	0.952	0.049
Logistic Modified	µ_max^a [h^-1]	0.0226 (± 0.0009)	1.33 (± 0.68)	0.999	0.007
Logistic Modified	(1/λ) [h^-1]	0.0227 (± 0.0043)	-1.03 (± 3.44)	0.977	0.032
Gompertz Modified	µ_max^a [h^-1]	0.0219 (± 0.0010)	1.28 (± 0.76)	0.999	0.008
Gompertz Modified	(1/λ) [h^-1]	0.0249 (± 0.0055)	-1.19 (± 4.05)	0.969	0.041

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[1] *To whom correspondence should be addressed. E mail: bruno@enq.ufsc.br.