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Primary and secondary models

Primary and secondary models. ComBase Workshop, 13 October 2009. Growth of bacterial populations. Growth rate (µ max ). lag. Growth of bacterial populations. Model of Baranyi and Roberts (1994). µ max =0.045 h -1. Lag=77 h. Growth of bacterial populations.

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Primary and secondary models

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  1. Primary and secondary models ComBase Workshop, 13 October 2009

  2. Growth of bacterial populations Growth rate (µmax) lag

  3. Growth of bacterial populations Model of Baranyi and Roberts (1994) µmax=0.045 h-1 Lag=77 h

  4. Growth of bacterial populations

  5. Modelling the bacterial growth rate Growth rates of L.monocytogenes as a function of temperature

  6. Modelling the lag timeHistory-dependence of lag Growth of Listeria monocytogenes in broth, at 15 oC, after different subculturing procedures. The max.spec. growth rates are the same, but the lag times are different.

  7. Lag: Quantify the delay with the number of missing generations m lag h0=mmax*lag work to be done during the lag. lnx0 h0

  8. Interpretation of h0 , the “work to be done”, when adjusting to the new environment (i.e. during the lag period) More favourable environment: shorter lag, higher rate, but h0 lag/doubl.time is constant (Listeria monocytogenes at 5 and 10 oC).

  9. Secondary Models • 2 types of secondary models • Response surface models Ex: sqrt(μmax)= a0+a1T+a2pH+a3T*pH+a4T2+a5pH2 • Non linear models Ex: sqrt(μmax)= b0(T-Tmin)(pH-pHmin) +ε +ε

  10. Predictive model x (cell.conc.) (log scale) Temperature pH % salt, … Models x0 time lag

  11. Response surface models • Identify a0, a1, a2, a3, a4, a5,… • Linear regression • Minimizing the Sum Square of Residuals

  12. Response surface models • Outputs from software tools • Evaluation of fit • Parameter estimates • Significance of parameters • Confidence intervals,… Calculations based on assumptions

  13. Checking model assumptions • The residuals follow a normal distribution • A normal probability plot of the residuals can be used to test normality (the points should lie along a straight line). • The residuals are independent, their variance is constant • Plot of residuals against the response variable and the explanatory variables in the model.

  14. Checking model assumptions Transformation: Sqrt (spec.growth rate) Or Ln (spec. growth rate) residuals residuals 0 0 specific growth rate (h-1) Sqrt (specific growth rate (h-1))

  15. Software outputsTest of hypothesis • a0=a1=a2=a3=a4=a5,…=0? • F test. Rejected (at 0.05%) if p<0.05 • ai=0? (0≤i≤p) • Rejected (at 0.05%) if p<0.05

  16. Examples… Ln μ (1/h) NaCl (%) pH

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