Simultaneous identification of growth law and estimation of its rate parameter for biological growth data: a new approach

Abstract

Scientific formalizations of the notion of growth and measurement of the rate of growth in living organisms are age-old problems. The most frequently used metric, “Average Relative Growth Rate” is invariant under the choice of the underlying growth model. Theoretically, the estimated rate parameter and relative growth rate remain constant for all mutually exclusive and exhaustive time intervals if the underlying law is exponential but not for other common growth laws (e.g., logistic, Gompertz, power, general logistic). We propose a new growth metric specific to a particular growth law and show that it is capable of identifying the underlying growth model. The metric remains constant over different time intervals if the underlying law is true, while the extent of its variation reflects the departure of the assumed model from the true one. We propose a new estimator of the relative growth rate, which is more sensitive to the true underlying model than the existing one. The advantage of using this is that it can detect crucial intervals where the growth process is erratic and unusual. It may help experimental scientists to study more closely the effect of the parameters responsible for the growth of the organism/population under study.

Appendices

Appendix A

Modified estimate of RGR For Gompertz law let us consider,

$$\begin{array}{@{}rcl@{}} X_{t_{1}} = X_{0}\exp{\left[\frac{b(1 - e^{-ct_{1}})}{c}\right]}\\ X_{t_{2}} = X_{0}\exp{\left[\frac{b(1 - e^{-ct_{2}})}{c}\right]}\\ X_{t_{3}} = X_{0}\exp{\left[\frac{b(1 - e^{-ct_{3}})}{c}\right]}\\ \end{array} $$

where $t_{1}, t_{2}$ and $t_{3}$ are consecutive time points. Then taking the ratio of the logarithm of $\frac {X_{t_{2}}}{X_{t_{1}}}$ and $\frac {X_{t_{3}}}{X_{t_{2}}}$ we obtain, $\ln \left (\frac {X_{t_{2}}}{X_{t_{1}}}\right )/\ln \left (\frac {X_{t_{3}}}{X_{t_{2}}}\right )$ and this implie

$$c = \frac{1}{\Delta t}\ln\left[\frac{\ln(d_1)}{\ln(d_2)}\right],$$

where $d_{1}$ and $d_{2}$ are defined as in Section 2.4. Now putting this estimate into the equation of $\frac {X_{t_{2}}}{X_{t_{1}}}$, we can get the estimate of b. Putting these estimates in the RGR equation for Gompertz law, which is $be^{-ct_{1}}$, we can get the ISRP. Similarly for the logistic equation, the interval estimate for RGR can be calculated.

Appendix B

Proof of Theorem 1 We have,

$$\begin{array}{@{}rcl@{}} \frac{\Delta \mu_{l}}{ \Delta \mu_{e}} = \left[\frac{2(\alpha + \beta)}{\ln\left[\frac{(1+\alpha)(1+\beta)}{(1-\alpha)(1-\beta)}\right] }\right] \left[\frac{(X_{2}+X_{1})^{2}-(X_{2}-X_{1})^{2}}{(X_{2}+X_{1})^{2} - (X_{2}\beta - X_{1}\alpha)^{2}} \right] = m.n \end{array} $$

Lemma 2

m < 1, always true.

Proof

Let, $\psi (\beta ) = \ln \left (\frac {1+\beta }{1-\beta }\right ) - 2\beta \Rightarrow \psi ' (\beta ) = \frac {2\beta ^{2}}{1-\beta ^{2}} > 0$ as $0<\beta <1,$ which implies ψ(β) is an increasing function in β. This implies, ψ(β) > ψ(0) and so,

$$ \ln\left[\frac{1+\beta}{1-\beta}\right] - 2\beta > 0 $$

(21)

Similarly,

$$ \ln\left[\frac{1+\alpha}{1-\alpha}\right] - 2\alpha > 0 $$

(22)

Condition (21) and (22) implies $\frac {2(\alpha +\beta )}{\ln \left [\frac {(1+\alpha )(1+\beta )}{(1-\alpha )(1-\beta )}\right ]} < 1 \Rightarrow m < 1$ □

Lemma 3

X ₂(1 − β) > X ₁(1 − α) ⇒ n < 1

Proof

n < 1 ⇒ | X ₂ − X ₁ | > | X ₂ β − X ₁ α | ⇒ X ₂(1 − β) > X ₁(1 − α)

Lemma 2 and Lemma 3 imply that Δ μ _e < Δ μ _l if X ₂(1 − β) > X ₁(1 − α), i.e., if the lowest value of X ₂ due to experimental errors is greater than that of X ₁. Let us consider the following particular cases:

• 1. (α = β = 1)

  Then n = 1 ⇒ Δ μ _e = Δ μ _l

• 2. $\frac {X_{2}}{X_{1}} > 1 > \frac {\alpha }{\beta }$

  $\Rightarrow X_{2}(1-\beta ) > X_{1}(1-\alpha ) \Rightarrow \Delta \mu _{e} < \Delta \mu _{l}$

• 3. $\frac {\alpha }{\beta } > \frac {X_{2}}{X_{1}} > 1$

  $\Rightarrow X_{1}\alpha - X_{2} > X_{2}\beta - X_{1} \Rightarrow X_{2}(1+\alpha ) > X_{1}(1+\beta ) \Rightarrow \Delta \mu _{e} < \Delta \mu _{l}$

• 4. $\frac {X_{2}}{X_{1}} > \frac {\alpha }{\beta } > 1$

  $\Rightarrow X_2-X_{2}\beta > X_{1} - X_{1}\alpha \Rightarrow X_{2}(1+\alpha ) > X_{1}(1+\beta ) \Rightarrow \Delta \mu _{e} < \Delta \mu _{l}$ but $\frac {1}{2}\leq m \leq 1$ and $n \geq 2$, then $\Delta \mu _{e} > \Delta \mu _{l}$

□

Lemma 4

$\frac {\Delta \mu _{e}}{\Delta \mu _{p}} = \frac {\ln \left ( \frac {1+a(1+\Delta t)}{1+at}\right )}{\Delta t} < 1$

Proof

We have, $\frac {\Delta \mu _{e}}{\Delta \mu _{p}} = \frac {1}{\Delta t} \ln \left (\frac {1+a (t_{1} + \Delta t)}{1 + a t_{1}}\right ) = \frac {1}{\Delta t} \ln \left (1 + \frac {a t_{1}}{1+a t_{1}} + \frac {a \Delta t}{1 + a t_{1}}\right )$ $\approx \ln \left (1 + \frac {at_{1}}{1+at_{1}}\right ) (\text {when }\Delta \text {t small}) < \ln (2) < \ln (e) =1$ □

Lemma 5

$\frac {\Delta \mu _{e}}{\Delta \mu _{g}} < 1$

Proof

$\frac {\Delta \mu _{e}}{\Delta \mu _{g}} = \frac {\exp {(c\Delta t)}-1}{\Delta t c \exp {(ct_{2})}}$ Let, $\lambda (c) = \exp {(c\Delta t)} - \Delta t c \exp {(ct_{2})} \Rightarrow \lambda '(c) = \Delta t [\exp {(c\Delta t)} - (1+ct_{2})\exp {(ct_{2})} ]$.

Now, $t_{2} > \Delta t$ (always true) $\Rightarrow (1+ct_{2})\exp {(ct_{2})} > \exp {(c\Delta t)}$. As $t_{2} >0$ is always true and $c>0$ for Gompertz law, this implies, $\lambda '(c) < 0 \Rightarrow \lambda (c)\downarrow c \Rightarrow \lambda (c) < \lambda (0) \Rightarrow \left (\frac {\exp {(c\Delta t)}-1}{\Delta t c \exp {(ct_{2})}}\right ) < 1 \Rightarrow \frac {\Delta \mu _{e}}{\Delta \mu _{g}} < 1$ □

Lemma 6

Δμ _glc − Δμ _e > 0

Proof

$\Delta \mu _{glc} - \Delta \mu _{e} = \frac {1}{2\Delta t}\ln P$. To prove the lemma, we have to prove that, P > 1. We have, $(1-\alpha ) < (1+\alpha ) \Rightarrow \left ( a^{\frac {1}{d}} - [X_{1}(1-\alpha ) ]^{\frac {1}{d}}\right ) > \left ( a^{\frac {1}{d}} - [X_{1}(1+\alpha ) ]^{\frac {1}{d}}\right ) \Rightarrow \left (\frac {\left ( a^{\frac {1}{d}} - [X_{1}(1-\alpha ) ]^{\frac {1}{d}}\right )}{\left ( a^{\frac {1}{d}} - [X_{1}(1-\alpha ) ]x^{\frac {1}{d}}\right )}\right ) >1$. Similarly, $\left (\frac {\left ( a^{\frac {1}{d}} - [X_{1}(1-\beta ) ]^{\frac {1}{d}}\right )}{\left ( a^{\frac {1}{d}} - [X_{1}(1+\beta ) ]^{\frac {1}{d}}\right )}\right ) >1$. This too implies that, P > 1. □

Now the proof of the theorem follows easily from Lemmas 2, 3, 4, 5 and 6.