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Baranyi Model

Three-Phase linear Model

Richards’ Model

Weibull Model

Logistic Model

Gompertz Model

Von Bertalanffy Model

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- 1. Modeling and parameter estimation of bacterial growth
- 2. Bacterial Growth Four stages: Lag, Exponential, Stationary, Death Actual Physiological adaptation during lag phase is too complex Most Mathematical models focus on log and lag phases
- 3. Mathematical Models Baranyi Model Three-Phase linear Model Richards’ Model Weibull Model Logistic Model Gompertz Model Von Bertalanffy Model
- 4. Baranyi Model Explained by Jószef Baranyi Mathematician and statistician Worked for the institute of food research, UK for two decades Further advances in food microbiology would be impossible without a dedicated mathematician. Focuses on lag phase calculations
- 5. Modeling Population Growth in Baranyi Model A non- autonomous model where an α(t) factor, a so-called adjustment function describes the transition from the lag to the exponential phase. dx/dt = α(t)μ(x) (0≤t<∞; 0<x) x(0) = x0 where 0<x0<xmax 0 ≤α(t)≤1 (0≤t<∞) α(t) → 1 monotone increasingly as t → ∞ α(t) is a sufficiently smooth function. It can depend on the pre-inoculation environment
- 6. Functions for α(t): α(t) can be expressed using many functions to suit different conditions. Using Hill function to model Adaptation period: α(t)= tn/(λn+tn) where n is +ve & λ is a parameter which is a suitable definition of lag time. Further, this function has an inflection point: Λn= λn√(n-1)/(n+1)
- 7. Functions for α(t): (cont.) Using Michaelis-Menten kinetics: α(t)= P(t)/Kp+P(t) KP is the Michaelis-Menten constant and after inoculation, the accumulation of P(t) follows a first order kinetic process. dP/dt = vb P where vb is the characteristic of the environment.
- 8. Estimating the distribution parameters in Baranyi Model: To quantify the physiological state of the initial population α is introduced. α= e -μ λ The physiological state of the inoculum, however, is equal to the arithmetical mean of the physiological states of the individual cells, the αi=e -μτi quantities. This is physiological state theorem ANOVA is developed using this theorem. Initial number of cells (N0) in a well follows a Poisson Distribution.
- 9. Three-phase linear model
- 10. THREE PHASE LINEAR Imposes a horizontal line fits on the lag and stationary periods and uses least square regression to allocate observations to three phases as well as to the co-ordinates of the three lines themselves. This extension of early graphical methods of microbiologists is mechanistic and very simple and is a fully adequate for use as a primary model for support multi-parameter environmental modeling of bacterial growth.
- 11. MODELING BACTERIAL GROWTH IN THREE-PHASE LINEAR MODEL:
- 12. MODELING BACTERIAL GROWTH IN THREE-PHASE LINEAR MODEL: (CONT.) Most food micro- biology applications are not overly interested in the stationary phase. In reality, if the stationary phase is reached, the food is either spoiled if the microorganism is non-pathogenic or a threat to public health if it is a pathogenic species. Liquid cultures, particularly when they are agitated, have rather rapid transition between exponential and stationary growth.
- 13. Richards’ Model
- 14. Richards’ Model: The generalised logistic curve or function, also known as Richards' curve is a widely used and flexible sigmoid function for growth modelling, extending the well-known logistic function. It has six parameters: A: the lower asymptote; K: the upper asymptote. If A=0 then K is called the carrying capacity; B: the growth rate; ν>0 : affects near which asymptote maximum growth occurs. Q: depends on the value Y(0) M: the time of maximum growth if Q=ν
- 15. A particular case of Richard's function is: which is the solution of the so-called Richard's differential equation (RDE): with initial condition Where provided that ν > 0 and α > 0.
- 16. Parameter Estimation: When estimating parameters from data, it is often necessary to compute the partial derivatives of the parameters at a given data point t
- 17. Weibull Model:
- 18. The Weibull growth model is described by the equation: where l = length, (or weight, height, size) and t = time. The four parameters are: β, is the lower asymptote; L∞, is the upper asymptote; k, is the growth rate and δ, is a parameter that controls the x-ordinate for the point of inflection.
- 19. The point of inflection on the x axis lies at: If δ = 1 the Weibull is a simple exponential growth curve.
- 20. ω(t) = (β0-β1exp(-β2tβ3)) + ε Parameters can be estimated by partial derivatives of ω with respect to β0, β1, β2, β3 β0 is the asymptote or the potential maximum of the response variable; β1 is the biological constant; β2 is the parameter governing the rate at which the response variable approaches its potential maximum;
- 21. Logistic Model:
- 22. Logistic Model: The logistic growth curve (sometimes called the Verhulst model as it was first proposed as a model of population growth Pierre Verhulst 1845, 1847) is one of the simplest of the S-shaped growth curves. A biological population with plenty of food, space to grow, and no threat from predators, tends to grow at a rate that is proportional to the population -- that is, in each unit of time, a certain percentage of the individuals produce new individuals. If reproduction takes place more or less continuously, then this growth rate is represented by this distribution.
- 23. Modeling Growth: where t is time, l is length (size), K is the growth rate and delta a term which expresses the rate at which growth declines with size. After integration and some rearrangement we arrive at the 3 parameter logistic growth curve: I is the age at the inflection point and L∞ is the upper asymptote (maximum size reach after infinite growing time).
- 24. The 3 parameter logistic has a lower asymptote of 0. The point of inflection on the y-axis occurs at This last formula states that the point of inflection is always at at 50 % of the asymptotic size (L∞). This does not hold true for all growth processes. You should consider using the Logistic growth curve to model sigmoid growth processes in which the point of inflection is approximately 1/2 of the maximum possible size. If a non-zero asymptote is required then the 4 parameter version of the equation is required this is expressed by the equation: where a is the lower asymptote and d is a shape parameter that determines the steepness of rising curve.
- 25. Gompertz Model:
- 26. Modeling Growth: The Gompertz curve was originally derived to estimate human mortality by Benjamin Gompertz (Gompertz, B. "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies.") Charles Winsor (1932) presented an early description of the use of this equation to describe growth processes.
- 27. where K is the growth rate and L∞, termed 'L infinity', is the asymptotic length at which growth is zero.. Integrating this becomes: where t is age and I is the age at the inflection point. The equation above is the 3 parameter version of the Gompertz growth curve (see below for an example plot). Growth II can also fit the 4 parameter version: in which A is the lower asymptote (see below for an example plot) and B is the upper asymptote minus A.
- 28. The point of inflection on the y-axis occurs at This last formula states that the point of inflection is always at about 36.8 % of the asymptotic size (L∞). This does not hold true for all growth processes. You should consider using the Gompertz growth curve to model sigmoid growth processes in which the point of inflection is approximately 1/3 of the maximum possible size.
- 29. von Bertalanffy Model
- 30. von Bertalanffy derived this equation in 1938 from simple physiological arguments. It is the most widely used growth curve and is especially important in fisheries studies. In its simplest version the so-called von Bertalanffy growth equation is expressed as a differential equation of length (L) over time (t): K is the growth rate and L∞, termed 'L infinity' in fisheries science, is the asymptotic length at which growth is zero.
- 31. Integrating this becomes: The parameter t0 is included to adjust the equation for the initial size of the organism and is defined as age at which the organisms would have had zero size. Thus to fit this equation you need to fit 3 parameters (L∞, K and t0 ) by nonlinear regression.
- 32. To fit this curve we must therefore estimate 3 parameters, L∞, K and t0. While this was once done graphically, it is now accomplished using the Levenberg-Marquardt Method for non-linear regression.
- 33. And Finally!!!

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