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\begin{center}
\Large{\bf MODELLING GROWTH  }
\end{center}
The expression  growth function generally denotes an analitical funtion that relates the dry weight W (or similar properties like fresh weight,live weight,
leaf area,etc.) of an organizm to time t. In other words,
\begin{equation}
W = f(t)
\end{equation}
The traditional approach to dealing with growth functions is to address the rate
of growth, e.g $dW/dt$ as in the case of dry weight W.Differentiation of 
Eqn.(1) with respect to time becomes\\
\begin{equation}
\frac{dW}{dt} = g(t),where \hspace{.2in} g(t)=\frac{df(t)}{dt}=h(W) 
\end{equation}
If we eliminate the time variable t between Eqn.(1) and (2),we obtain the following relationship in the form of "rate $=$ a function of state":
\begin{equation}
\frac{dW}{dt} = h(W) 
\end{equation}
whre h is a function operator.Occasionally it may be preferable to show rate
of growth as\\
\begin{equation}\frac{dW}{dt} = u(W,t) \end{equation}
where u denotes a function of W and t.Units of growth rate may take many forms,
but let us make the following choices:day as the  unit of time and kg as the unit of weight, therefore (kg/day) is the unit of growth rate.\\
\mbox{}\\
More commonly encountered growth equations are divided by integration of a differential equation of the form of either Eqn(3) or (4).It has become a common
practice to plot growth data on semilog scales.Caution must be exercised, however, because it is possible, for instance, sigmoid curve to appear non-sigmoid on semilog scales.Therefore,as we examine different forms of growth equations,we will show both linear and semilog scales.\\
\mbox{}\\
Normally growth is viewed as a closed two compartment model such that shown in Figure 1.the system has no inputs or outputs. The growth process transfers meterial from the substrate compartment to the second compartment without loss.
Different growth equations evolve based on the various assumptios used concerning how growth rate depens on W and S.\\
\mbox{}\\
\fbox{\rule[0.39in]{0in}{0.12in}\Large{\bf Substrate,S}}
$\Rightarrow$\fbox{\rule[0.39in]{0in}{0.12in}\Large{\bf Growth}}
$\Rightarrow$\fbox{\rule[0.39in]{0in}{0.12in}\Large{\bf Dry Weight,W}}\\
{\bf Figure 1.}Closed 2-compartment growth model.\\
\mbox{}\\
The assamption that there is no gain or loss of meterial from the system implies that
\[\frac{dW}{dt} = -\frac{dS}{dt}\]
\begin{equation}
\frac{dW}{dt}+\frac{dS}{dt}=\frac{d}{dt}(W + S) = 0
\end{equation}
such that\\
\begin{equation}
W + S = a \hspace{.1in} constant = W_o + S_o = W_f + S_f = C
\end{equation}
where $W_o$ and  $S_o$ are the initial values of W and S at time t$ = 0$;$W_f$ and $S_f$ are the final values as t $\longrightarrow \infty$ (assuming that a steady state is aventually reached ); C is constant. Next, we write the rate of growth as a function $\nu$ of W and S to get \\
\begin{equation}
\frac{dW}{dt} = \nu (W,S)
\end{equation}
From Eqn.(6), we can substitute $S =C - W $ into Eqn.(7)
\begin{equation}
\frac{dW}{dt} = \nu (W,C-W) = h(W) 
\end{equation}
where h is a function of the single variable W as in Eqn.(3).The determinig factor now is the assumtion made about the form of the function $\nu$ in Eqn.(7)
because it establishes the form of Eqn.(8) and after integration the form of the growth equation(Eqn.(1)) itself.
\newpage
\begin{enumerate}
\item {\bf Simple Exponential Growth with Abrupt Cut-off}\\
\mbox{}\\
 In this initial model the assumptions are;
\begin{enumerate}
\item The quantity of the growth is proportional to dry weight, 
\item Growth occurs at maximal rate as long as substrate is available,
\item Growth is irreversible,
\item Growth stops when the substrate is exhausted.
\end{enumerate}
\mbox{}\\
Eqn.(7) takes the form
\begin{equation}
\frac{dW}{dt} = \mu W
\end{equation}
where $\mu$ is the {\em specific}, or {\em relative, growth rate}.
The parameter $\mu$ depends on the proportion of W which constitutes growth and the efficiency or speed with which the growth process can function. Intagration of Eqn.(9) and application of rules of logarithms results in 
\begin{equation}
W = W_o e^{\mu t} \hspace{.2in} for \hspace{.2in} 0 \leq t \leq t_f
\end{equation}
\begin{equation}
W = W_f \hspace{.2in} for \hspace{.2in}  t \geq t_f
\end{equation} 
When $ W = W_f, S_f = 0 $. From Eqn.(6),
\begin{equation}
W_f = W_o +  S_o
\end{equation}
Growth stops aptruptly,and from Eqn.(10),
\begin{equation}
t_f = {ln[(W_o + S_o)/W_o]}/ \mu
\end{equation}
Figure 2 includes both linear and semilog plots of simple exponential growth limited by the amount of substrate available. The growth curve is linear in the semilog plot since from Eqn.(10),
\begin{equation}
ln W = ln W_o + \mu t
\end{equation}
\newpage
\item {\bf Monomolecular Growth Model }\\
The monomolecular model describes the process of a simple irreversible first order chemical reaction.The assumptions are
\begin{enumerate}
\item The quantity of the growth portion is constant ans indipendent of the dry weight W.
\item The growth rate is proportional to the substrate level S.
\item Growth is irreversible.
\end{enumerate}
Instead of Eqn.(9),
\begin{equation}
\frac{dW}{dt} = kS
\end{equation}
where k is constant.Using Eqn.(6) with $S_f = 0,S = W_f -W$ can be subtituted
into Eqn.(15) resulting in
\begin{equation}
\frac{dW}{dt} = k(W_f-W)
\end{equation}
Separating variables in this latter equation and integrating
\begin{equation}
\int_{W_o}^{W_f} \frac{dW}{W_f-W} = \int_{0}^{t} kdt 
\end{equation}
Thus
\begin{equation}
ln \left( \frac{W_f-W_o}{W_f-W} \right) = kt
\end{equation}
By the properties of logarithms we can also write Eqn.(18) in the form
\begin{equation}
W_f-W-(W_f-W_o) e^{-kt} = 0
\end{equation}
Now if the initial dry weight $W_o =0$,Eqn.(19) will take the form
\begin{equation}
W = W_f(1-e^{-kt})
\end{equation}
Figure 3 illustrates the linear and semilog plots of this model. The growth
rate decreases continuously with no point of inflection as can be seen through 
derivative which is only zero for t$\rightarrow$ $\infty$ or W $\rightarrow$ $W_f$.
\begin{equation}
\frac{d^2 W}{dt^2} = -k^2(W_f-W)=-(W_f-W_o)k^2e^{-kt}
\end{equation}
\newpage
\item {\bf Logistic Growth Model}\\
\mbox{}\\
To this point we have considered two extremes of conditions. In Eqn.(9) growth rate depends on dry weight W and not on the availability of substrate S, and in Eqn.(15) growth rate depends on only the substrate level and not on the dry weight.Derivation of the growth equation, however, uses a composite assumption:
\begin{enumerate}
\item The quantity of the growth is proportional to the dry weight.
\item The growth rate is proportional to the amount of the subtrate S.
\item The growth is irreversible.
\end{enumerate}
Consequently,
\begin{equation}
\frac{dW}{dt} = k^{\prime} WS
\end{equation}
where k $\prime$ is a constant. Again, with $S = W_f-W$ and $S_f = 0$
\begin{equation}
\frac{dW}{dt} = k^{\prime} W(W_f-W)
\end{equation}
 For convenience, we can define a constant $\mu$ in terms of $k^{\prime}$
\begin{equation}
k^{\prime} = \frac{\mu}{W_f} 
\end{equation}
and substitute for $k^{\prime}$ in Eqn.(23) so that
\begin{equation}
\frac{dW}{dt} = \mu W \left(1-\frac{W}{W_f} \right)
\end{equation}
Using partial fractions and integrating
\begin{equation}
\int_{W_o}^{W} \left( \frac{1}{W_f-W}+\frac{1}{W} \right) dW = \int_{0}^{t} \mu dt
\end{equation}
After some manupilation, we can find the following equation;
\begin{equation}
W= \frac{W_o W_f e^{\mu t}}{W_f - W_o + W_o e^{\mu t}} 
\end{equation}
A more common way of writing the logistic equation is
\begin{equation}
W= \frac{W_o W_f}{W_o + (W_f - W_o) e^{- \mu t}}
\end{equation}
\newpage
In either of these last two equation,for $W_o << W_f$ and for low values of t
where we can put $W_o = 0$ in the denominator, dry weight approximates
\begin{equation}
W \approx W_o e^{\mu t}
\end{equation}
Initialy, there is exponential growth with a relative growth rate of $\mu$. As
$t \rightarrow \infty,W \rightarrow W_f$ such that there is growth limited by
an assimptote.\\
Differentiating Eqn.(25) yields a second derivative as
\begin{equation}
\frac{1}{\mu} \frac{d^2 W}{dt^2} = \frac{dW}{dt}\left( 1-\frac{2W}{W_f} \right)
\end{equation}
which,when equated to zero, establishes an inflexion point at
\begin{equation}
W = \frac{1}{2} W_f
\end{equation}
Substituting Eqn.(31) into Eqn.(28) and solving for $t = t^{*}$ at which the inflexion occurs,we find that
\begin{equation}
t^{*} = \frac{1}{\mu} ln \left( \frac{W_f - W_o}{W_o} \right)
\end{equation}
The logistic growth curves are shown in linear and semilog plots in Figure 4.
Note that this is the first model to yield a smooth sigmoid shape.
\newpage
\item {\bf Gompertz Growth Model}\\
Unlike the logistic growth model, the Gompertz equation is derived under the following assumptions:
\begin{enumerate}
\item The substrate is nonsaturating,e.g. the process is always saturated with
the substrate.
\item The growth is proportional to the dry weight W as the constant $\mu$.
\item The effectiveness of the growth decays exponentially with the time according to the first$-$order kinetics. 
\end{enumerate}
Possible causes of the exponential decay may be degradation of enzymes,senescence or development and differentiations.The underlying assumptions for the Gompertz equation are incorporated in an expression similiar
to Eqn.(9),but with the specific growth rate $\mu$ no longer a constant.It is now determined as
\begin{equation}
\frac{d \mu }{dt} = -D \mu
\end{equation}
where D is a new parameter referring to decay in the specific growth rate.Integration of Eqn.(33) results in
\begin{equation}
\mu = \mu_{o} e^{-Dt}
\end{equation}
where $\mu_{o}$ is the value of $\mu$ at $t=0.$Substituting Eqn.(34) into Eqn.(9)
results in 
\begin{equation}
\frac{dW}{dt} = \mu_{o} W e^{-Dt}
\end{equation}
which is a function of both the dry weight W and the time t as mentioned in Eqn.(4).Integrating this last equation,
\begin{equation}
\int_{W_o}^{W} \frac{dW}{W} = \mu_{o} \int_{0}^{t} e^{-Dt} dt
\end{equation}
results in,
\begin{equation}
ln \left( \frac{W}{W_o} \right) = \frac{\mu_o}{D} \left(1-e^{-Dt} \right)
\end{equation}
which by the rules of the logarithms can be written as
\begin{equation}
W = W_o e^{[\mu_o (1-e^{-Dt})/D]}
\end{equation}
For small t,$e{-Dt} \approx 1-Dt.$ Thus $1-e^{-Dt} \approx Dt$ and the growth
with
\begin{equation}
W \approx W_o e^{\mu_o t}
\end{equation}
The asymptote $W = W_f$ is approached as $t \rightarrow \infty$ where
\begin{equation}
W_f = W_o e^{\mu_o /D}
\end{equation}
Differentiating Eqn.(35) yields,
\begin{equation}
\frac{1}{\mu_o} \frac{d^2 W}{dt^2}=\frac{dW}{dt} e^{-Dt} -DW e^{-Dt} 
\end{equation}
which when equated to zero with substitution from Eqn.(35) for $dW/dt$ indicates
that there is a point of inflexion at time $t = t^{*}$ where
\begin{equation}
t^{*} = \frac{1}{D} ln \left( \frac{\mu_o}{D} \right)\hspace{.2in} and \hspace{.2in}W(t=t^{*}) = \frac{W_f}{e} =\frac{W_f}{2.72}
\end{equation}
Eqn.(35) can be presented in the form of a "rate as a function of state"(Eqn.(3)) by substituting for $e^{-Dt}$ from Eqn.(37) to render
\begin{equation}
\frac{dW}{dt} = \mu_o W \left[ 1- \frac{D}{\mu_o} ln \left( \frac{W}{W_o} \right) \right]
\end{equation}
With the aid of Eqn.(40),D or $\mu_o$ can be replaced by $W_f$,resulting in two alternative, frequently encountered forms of the Gompertz equation:
\begin{equation}
\frac{dW}{dt} = \mu_o W \left[ \frac{ln(W_f /W)}{ln (W_f /W_o)} \right] = DW ln \left( \frac{W_f}{W} \right)
\end{equation}
Figure 5. illustrates the Gompertz model.Note that it has the same number of parameters,3, as the logistic growth model,but the inflection point occurs at
$1/e$ times the final weight rather that at $ W_f/2$.Comparing Figure 4 and 5,we can see that for the same initial and final weights and a similiar time of inflexion, the Gompertz model predicts faster early growth, yet a slower approach to the asymptote and a longer linear peiod around the inflextion point.
\newpage
\item {\bf Richard's Growth Model }\\
Richards (1959,1969) adapted to plant growth modelling an equation that had been developed initially by von Bertalanffy (1957) to describe growth animals.Although empirical in approach,this equation has a certain level of generality that may be advantageous. Utilizing an additional parameter n,
the Richard's model incorporates the three previously considered growth models;
the monomolecular model for $n=-1$,the logistic model for $n = 1$ and the Gompertz model for $n = 0$.It may be written as the following differential equation:
\begin{equation}
\frac{dW}{dt} = \frac{kW(W_{f}^{n}-W^n)}{nW_{f}^{n}}
\end{equation}
where k,n and $W_f$ are constant and k and $W_f$ are positive.Since n$<-1$ results in infinite growth rate as W$\rightarrow 0$,which is certainly $non-physiological$,we limit n$\geq -1$.\\
When $n = -1$,Eqn.(45) becomes,
\begin{equation}
\frac{dW}{dt} = k(W_f - W)
\end{equation}
which is identical with Eqn.(16),the monomolecular model.With $n=-1$,it takes the form
\begin{equation}
\frac{dW}{dt}= kW \left( 1-\frac{W}{W_f} \right)
\end{equation}
which is identical to Eqn.(25) where k is replaced by $\mu$.This is,therefore,
the logistic model.With n$=0$,we need to exercise care in taking the limit as
n$\rightarrow 0$ and use for $W_{f}^{n}$ and $W^{n}$ the expantion,
\begin{equation}
x^n = e^{nlnx} = 1+nlx+\frac{1}{2!} (nlnx)^2+...
\end{equation}
Under these latter precautions it can be shown that,
\begin{equation}
\frac{dW}{dt} = kWln \left( \frac{W_f}{W} \right)
\end{equation}
which is identical with the Gompertz model in Eqn.(44),where k is replaced by D.\\
Integrating Eqn.(45) as shown below,
\begin{equation}
n \int_{W_o}^{W} \left( \frac{1}{W} + \frac{W^{n-1}}{W_f^n-W^n} \right) dW = \int_{0}^{t} kdt
\end{equation}
results in,
\begin{equation}
W_o^n W_f^n = W^n[W_o^n + (W_f^n-W_o^n)e^{-kt}]
\end{equation}
which can also be written as,
\begin{equation}
W = \frac{W_o W_f}{[W_o^n + (W_f^n - W_o^n)e^{-kt}]^{1/n}}
\end{equation}
Note that at $t=0$,$W = W_o$ and as $t \rightarrow \infty$,$W \rightarrow W_f$.
If relevant,inflexion points can be found by further differentiating Eqn.(45) to obtain
\begin{equation}
\frac{d^2W}{dt^2} = \frac{k}{nW_f^n} \frac{dW}{dt} [W_f^n - (n+1)W^n]
\end{equation}
The time of the inflexion can befound by equating Eqn.(53) to zero,which implies that $dW/dt = 0$ or
\begin{equation}
W(t=t^*) = W_f \left( \frac{1}{n+1} \right)^{1/n}
\end{equation}
Substituting Eqn.(54) into Eqn.(53) leads to the time of inflexion $t^{*}$ as
\begin{equation}
t^{*} = \frac{1}{k} ln \left( \frac{W_f^n - W_o^n}{nW_o^n} \right)
\end{equation}
Varying the value of n leads to the family of curves illustrated in Figure 6.
Note from Eqn.(54) that in addition to the previous forms of the logistic and Gompertz equations,now the point of inflexion can occur at any fraction of the final dry weight $W_f$ as n varies over the range $-1< n < \infty$.
\newpage
\item {\bf Chanter Growth Model }\\
A hybrid of the logistic and Gompertz functions was devised by Chanter (1976). Its parameters are similiarly interpreted.Analygous to Eqns.(25) and (35), the Chanter equation looks thusly:
\begin{equation}
\frac{dW}{dt} = \mu W \left( 1- \frac{W}{B} \right) e^{-Dt}
\end{equation} 
where $\mu$,B and D are constants.Note that the specific growth rate $\mu$ from Eqn.(56) is modified from the traditional value $(1/W)(dW/dt)$ by two factors.First, the factor $(1-W/B)$ depends linearly on the substrate level as in the logistic model (Eqn.(22) and, secondly,the factor $e^{-Dt}$ depends on the passage of time.The exponential factor may represents physiological events 
such as differentiation,development or senescence, as in the Gompertz model (Eqn.(35)).\\
Integration of Eqn.(56) leads to
\begin{equation}
W= \frac{W_o B}{W_o + (B-W_o) e^{ \{-[ \mu (1-e^{-Dt})/D] \} } }
\end{equation}
Although not identical, this last equation is similiar to the Richard's model
 in Eqn.(52). Biological meaning can be assigned to each of the parameters $W_o$
 ,$\mu$,D,B.\\
At time $t = 0$,the initial weight $W=W_o$.The final weight $W_f$ as $t \rightarrow \infty $ is more complicated as,
\begin{equation}
W_f = \frac{W_o B}{W_o +(B-W_o) e^{-\mu /D}}
\end{equation}
Either substrate availability B or differentiation D, or a combination of both,
may influence the final weight $W_f$.For example,with
\begin{equation}
W_L = B\hspace{.2in} and \hspace{.2in} W_G = W_o e^{\mu/D}
\end{equation}
as the limiting final weights obtained with the logistic or Gompertz models acting alone,Eqn.(58) can be written as
\begin{equation}
W_f = \frac{W_L W_G}{W_L + W_G -W_o}
\end{equation}
such that the final weight always lies below both $W_L$ and $W_G$,yet may approach
the lower of $W_L$ and $W_G$ very closely.\\
The following expression can be derived for B from Eqn.(58)
\begin{equation}
B = \frac{W_f W_o ( e^{\mu D} -1)}{W_o e^{ \mu /D} - W_f}
\end{equation}
and substituted for B in Eqn.(57) to obtain a growth equation in terms of $\mu$,D,$W_o$ and $W_f$.\\
Eqn.(56),which is now a function of both W and t in the manner of Eqn.(4),can be converted to the form of Eqn.(3),which is in the standard " rate as a function of state" form,by a series of selected manipulations.First,Eqn.(57) can be used to solve for $e^{-Dt}$.Then this value can be substituted directly in to Eqn.(56) to yield,
\begin{equation}
\frac{dW}{dt} = \mu W \left( \frac{B-W}{B} \right) \left[ 1- \frac{D}{\mu } ln  \left( \frac{W}{B-W} \right) \left( \frac{B-W_o }{W_o }\right)  \right]
\end{equation} 
For the case of no senescence,i.e. the limit as $D \rightarrow 0$,Eqns.(57) and (62) become identical to the logistic model's  Eqns.(28) and (25), respectively. 
On the other hand, with no substrate limitation (i.e. in the limit as B $\rightarrow \infty$ ), this pair of equations become identical to Eqns.(38) and (43), respectively, for the Gompertz model.\\
Computer numerical differentiation is needed to find the time $t^*$ at which the inflexion point occurs and to determine the dry weight at the inflexion point,i.e. W ($t=t^*$).\\
Figure 7 illustrates the family of curves formed by the Chanter model.Note that in these plots of Eqn.(57) the values of initial and final weights and the specific growth rate are constant ($W_0 = 1,W_f = 100$ and $\mu = 0.5$). The logistic model curve L is established as D $\rightarrow 0$ ,the intermediate case at D$=0.05$ and the Gompertz model G as D $\rightarrow \mu /[ln(W_f/W_o)] =
0.1086$. The value of B established from Eqn.(61) for each assumed value of D and substituted into Eqn.(57) to solve for the W(t) response.For a set of values of $W_o$, $W_f$ and $\mu$, the value of D can only exist in the range of 
\begin{equation}
0 \leq D \leq \mu / [ln( W_f /W_o)]  
\end{equation} 
The upper limit set by the vanishing denominator of Eqn.(61) which makes B$= \infty$. As D is varied in this range, a family of curves is created which lie between the corresponding logistic and Gompertz models.Each curve in the family
has the same initial and final weights the same initial specific growth rate.
As D increases, the points of inflexion  shift to later times. 
\end{enumerate}
\end{document}

