Stochastic Modeling of In Vitro Bactericidal Potency

Abstract

We provide a Galton–Watson model for the growth of a bacterial population in the presence of antibiotics. We assume that bacterial cells either die or duplicate, and the corresponding probabilities depend on the concentration of the antibiotic. Assuming that the mean offspring number is given by $m(c) = 2 / (1 + \alpha c^\beta )$ for some $\alpha , \beta$, where c stands for the antibiotic concentration we obtain weakly consistent, asymptotically normal estimator both for $(\alpha , \beta )$ and for the minimal inhibitory concentration, a relevant parameter in pharmacology. We apply our method to real data, where Chlamydia trachomatis bacterium was treated by azithromycin and ciprofloxacin. For the measurements of Chlamydia growth quantitative polymerase chain reaction technique was used. The 2-parameter model fits remarkably well to the biological data.

Appendix

Proof of Lemma 1

Conditioning on $\mathbf {X}_n$

$$\begin{aligned} \mathbf {E}\left[ \mathbf {X}_{n+1} | \mathbf {X}_n \right] = \begin{pmatrix} m X_n \\ p_0 X_n + Y_n \end{pmatrix} = \mathbf {X}_n \mathbf {M}, \end{aligned}$$

thus

$$\begin{aligned} \mathbf {E}\mathbf {X}_n = \mathbf {X}_0 \mathbf {M}^n. \end{aligned}$$

We have, by induction on n that

$$\begin{aligned} \mathbf {M}^n = \begin{pmatrix} m^n &{} p_0 (1 + \cdots + m^{n-1}) \\ 0 &{} 1 \end{pmatrix}, \end{aligned}$$

thus

$$\begin{aligned} \begin{aligned} \mathbf {E}Z_n&= m^n + p_0 ( 1 + m + \cdots + m^{n-1} ) \\&= {\left\{ \begin{array}{ll} m^n \left( 1 + \frac{p_0 }{m-1} \right) - \frac{p_0 }{m-1}, &{} \text {if } m \ne 1, \\ 1 + n p_0 , &{} \text {if } m = 1, \end{array}\right. } \end{aligned} \end{aligned}$$

as claimed. $\square$

Proof of Proposition 3

In what follows all the iterated limits are meant as first $x_0 \rightarrow \infty$ and then $N \rightarrow \infty$. By Proposition 2 and the delta method

$$\begin{aligned} \begin{aligned}&\frac{\sqrt{N} m(c_i) (2-m(c_i)) \mu _n'(m(c_i))}{2 \sigma _\varepsilon \mu _n(m(c_i)) \log 2} \left( h_i - \log \left( \frac{2}{m(c_i)} - 1 \right) \right) \\&= \frac{\sqrt{N}}{k_i} \left( h_i - \log \left( \frac{2}{m(c_i)} - 1 \right) \right) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} N(0,1), \end{aligned} \end{aligned}$$

for $i=1,2,\ldots , K$. Recall the notation in (11). Then using the independence of $h_i$’s

$$\begin{aligned} \sqrt{N} \sum _{i=1}^K \left( h_i - \log \left( \frac{2}{m(c_i)} - 1 \right) \right) \left( K \ell _i - L_1 \right) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} N(0, s_n^2), \end{aligned}$$

with $s_n^2 = \sum _{i=1}^K k_i^2 (K \ell _i - L_1)^2$. Substituting back into (10)

$$\begin{aligned} \sqrt{N} ({\widehat{\beta }} - \beta ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} N( 0, \sigma _\beta ^2). \end{aligned}$$

(15)

Similarly

$$\begin{aligned} \sqrt{N} \left( \log {\widehat{\alpha }} - \log \alpha \right) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} N(0, z_n^2), \end{aligned}$$

(16)

with $z_n^2 = \sum _{i=1}^K k_i^2 (L_2 - L_1 \ell _i)^2 / (KL_2 - L_1^2)^2$, which implies

$$\begin{aligned} \sqrt{N} ( {\widehat{\alpha }} - \alpha ) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} N( 0, \sigma _\alpha ^2). \end{aligned}$$

The statement for the covariance follows the same way. From (15) and (16) we obtain

$$\begin{aligned} \sqrt{N} \left( {\widehat{\vartheta }} - \vartheta \right) {\mathop {\longrightarrow }\limits ^{\mathcal {D}}} N( 0, \sigma _\vartheta ^2), \end{aligned}$$

as claimed. $\square$