Novel Rate-Area-Shape Modeling Approach To Quantify Bacterial Killing and Regrowth for In Vitro Static Time-Kill Studies

Abstract

In vitro static concentration time-kill (SCTK) studies are a cornerstone for antibiotic development and designing dosage regimens. However, mathematical approaches to efficiently model SCTK curves are scarce. The currently used model-free, descriptive metrics include the log₁₀ change in CFU from 0 h to a defined time and the area under the viable count versus time curve. These metrics have significant limitations, as they do not characterize the rates of bacterial killing and regrowth and lack sensitivity. Our aims were to develop a novel rate-area-shape modeling approach and to compare, against model-free metrics, its relative ability to characterize the rate, extent, and timing of bacterial killing and regrowth from SCTK studies. The rate-area-shape model and the model-free metrics were applied to data for colistin and doripenem against six Acinetobacter baumannii strains. Both approaches identified exposure-response relationships from 0.5- to 64-fold the MIC. The model-based approach estimated an at least 10-fold faster killing by colistin than by doripenem at all multiples of the MIC. However, bacterial regrowth was more extensive (by 2 log₁₀) and occurred approximately 3 h earlier for colistin than for doripenem. The model-free metrics could not consistently differentiate the rate and extent of killing between colistin and doripenem. The time to 2 log₁₀ killing was substantially faster for colistin. The rate-area-shape model was successfully implemented in Excel. This new model provides an improved framework to distinguish between antibiotics with different rates of bacterial killing and regrowth and will enable researchers to better characterize SCTK experiments and design subsequent dynamic studies.

INTRODUCTION

The increasing prevalence of multidrug-resistant (MDR) bacteria presents a significant threat to human health globally and is compounded by a severe lack of effective antibiotics (1, 2). This has forced researchers to develop novel strategies on how to use available antibiotics optimally against MDR bacteria, such as Acinetobacter baumannii, which is one of the most dangerous Gram-negative pathogens (3, 4). Understanding the relationship between antibiotic exposure and antibacterial activity underpins the rational design and optimization of dosage regimens (4–6).

In vitro static concentration time-kill (SCTK) studies are one of the most commonly used experimental models to assess antibiotic activity (7–11). Data from SCTK often are employed to determine the relative activity of different antibiotics (12–14). Despite the ubiquity of SCTK studies, modeling approaches that efficiently characterize the rate, extent, and timing of bacterial killing and regrowth are scarce (15, 16). Antibiotic activity from SCTK studies is routinely quantified by model-free, descriptive metrics (Fig. 1), such as the log₁₀ change in CFU over 24 h (ΔCFU₂₄) (7, 8, 12) and the area under the bacterial curve (AUBC_0–24) from 0 to 24 h (14, 17). The ΔCFU₂₄ and AUBC_0–24 are easy to compute and widely used. However, these metrics usually only quantify one time interval and do not assess the rates of bacterial killing and regrowth (Fig. 2); moreover, they do not distinguish between susceptible and resistant bacteria. Initial slopes sometimes are calculated from SCTK data to characterize bacterial killing (15, 16, 18); however, this analysis ignores bacterial regrowth. Therefore, these approaches have limitations.

FIG 1.

Commonly applied model-free metrics (ΔCFU₂₄ and AUBC_0–48) used to quantify antibiotic activity from in vitro static time-kill studies.

FIG 2.

Time-kill curves for an antibiotic with rapid killing and rapid regrowth (curve A; continuous line) and an antibiotic with slow killing and slow regrowth (curve B; dashed line).

To extend the ΔCFU and AUBC data, these model-free metrics can be calculated over multiple time intervals (e.g., from 0 to 5 h and 0 to 24 h). However, the ranking of an antibiotic with rapid killing and regrowth compared to an antibiotic with slow killing and regrowth can change considerably depending on the time interval (Fig. 2). Therefore, it is difficult to translate ΔCFU and AUBC results from SCTK studies to clinical practice.

Mechanism-based pharmacodynamic (PD) modeling is a growing field that expands the analysis of in vitro data, providing in-depth insights into the dynamics and mechanisms underpinning bacterial growth, killing, and emergence of resistance (18–21). However, the widespread adoption of these methods is limited by the need for specialized software, advanced training in pharmacostatistical methods and numerical calculus, and detailed knowledge of mechanisms of antibiotic action and bacterial resistance.

Our first objective was to develop a novel, rate-area-shape modeling approach for efficient analysis of SCTK data. As a second objective, we compared the ability of the new model and of the model-free ΔCFU and AUBC methods to characterize the rate, extent, and timing of bacterial killing and regrowth. We assessed antibiotics with very rapid (colistin) and slower killing (doripenem) as probe drugs against six A. baumannii strains. The calculations for the rate-area-shape model were implemented in a widely available software program (Microsoft Excel).

(Parts of this work were presented at the 5th International Pharmaceutical Federation [FIP] Pharmaceutical Sciences World Congress, April 2014, Melbourne, Australia.)

MATERIALS AND METHODS

Rate-area-shape model development.

The proposed rate-area-shape model (equation 1) was developed to characterize the rate, extent, and timing of bacterial killing and regrowth. This model consisted of an exponential function describing bacterial killing (first term on the right side) and a logistic component representing bacterial regrowth (second term).

$${\log \hspace{0.17em}}_{10}\hspace{0.17em}\text{CFU/ml}\hspace{0.17em}=\hspace{0.17em}\text{A}\cdot {\text{e}}^{-K_d\cdot t}+\frac{B}{1+e^{-K_r\cdot (t-C)}}$$ (1)

where the parameters A, B, K_d, and K_r all are greater than 0. The variable t is time, A is the extent of bacterial killing, and B is the extent of bacterial regrowth. Both A and B are on a log₁₀ scale. K_d characterizes the rates of bacterial killing and K_r the rate of regrowth. The parameter C describes the time delay of bacterial regrowth. The model for growth controls lacked the term for bacterial killing:

$${\log \hspace{0.17em}}_{10}\hspace{0.17em}\text{CFU/ml}=\frac{B}{1+e^{-K_r\cdot (t-C)}}$$ (2)

Estimation.

The log₁₀ viable count data from SCTK experiments were used to estimate the model parameters by ordinary least-squares regression via the built-in SOLVER function in Microsoft Excel (22). We fitted the data for each strain and antibiotic concentration separately (i.e., without pooling). To illustrate the exposure-response relationship, the individually fitted parameter estimates were plotted against antibiotic concentration (expressed as a multiple of the MIC). For antibiotic-containing time-kill curves, the bacterial regrowth rate (K_r) for each strain was fixed to the estimate for the growth control of the respective experiment. This improved the robustness of the estimation and was important due to the limited number of observed viable counts during bacterial regrowth (i.e., between approximately 6 and 24 h in our SCTK studies).

The bacterial killing rate (K_d) was constrained to ensure that the estimate did not exceed the maximum theoretically observable killing rate (K_{d max}) for the chosen experimental design (i.e., no colonies at the first observation time). The K_{d max} was calculated based on the initial inoculum and the first observation time by solving the first component of equation 1 for K_d.

$$K_{d\hspace{0.17em}\max \hspace{0.17em}}=\frac{1}{t_1}\cdot \ln \hspace{0.17em}\frac{N_0}{N_{\text{LD}}}$$ (3)

To calculate K_{d max}, we set A to the initial inoculum (N₀) of 6 log₁₀ CFU/ml and assumed that a high antibiotic concentration can achieve a maximum of 6 log₁₀ killing. The first observation time was set to 0.5 h (t₁) and the limit of quantification for viable counting (N_LD) to 1.3 log₁₀ CFU/ml (equivalent to one colony per agar plate). This yielded a K_{d max} of 3.06 h⁻¹, which was used as the upper limit for K_d during estimation.

Derivation of model-based metrics.

The rate-area-shape model enables the calculation of four summary metrics based on the estimated model parameters. These metrics were derived by disaggregating the rate-area-shape model (equation 1) into its two components, bacterial killing and regrowth. The calculation of these model-based metrics assumed that bacterial killing and regrowth can be considered independently. We calculated the time to x_{log10 killing} (t_{x log10 kill}) based on the first component of equation 1:

$$A-x_{\log \hspace{0.17em}10\hspace{0.17em}\text{killing}}=A\cdot e^{-K_d\cdot t}$$ (4)

Under the assumption that the maximum magnitude of killing (i.e., A) is greater than x_{log10 killing}, rearranging equation 4 yields

$$t_{x\hspace{0.17em}\log \hspace{0.17em}10\hspace{0.17em}\text{kill}}=-\frac{1}{K_d}\cdot \ln \hspace{0.17em}\left(1-\frac{x_{\log \hspace{0.17em}10\hspace{0.17em}\text{killing}}}{A}\right)$$ (5)

We then calculated the time to x_{log10 regrowth} based on the second component of equation 1 as

$$x_{\log \hspace{0.17em}10\hspace{0.17em}\text{regrowth}}=\frac{B}{1+e^{-K_r\cdot (t-C)}}$$ (6)

If the maximum magnitude of regrowth (i.e., B) is greater than x_{log10 regrowth}, then t_{x log10 regrowth} can be calculated as

$$t_{x\hspace{0.17em}\log \hspace{0.17em}10\hspace{0.17em}\text{regrowth}}=C+\frac{1}{K_r}\cdot \ln \hspace{0.17em}\left(\frac{x_{\text{Log}10\hspace{0.17em}\text{regrowth}}}{B-x_{\text{Log}10\hspace{0.17em}\text{regrowth}}}\right)$$ (7)

In the analysis of our SCTK data for colistin and doripenem, we described bacterial killing by the time to 2 log₁₀ CFU/ml killing and bacterial regrowth by the time to 5 log₁₀ CFU/ml regrowth. As the last observation time in our time-kill studies was 48 h, time to 5 log₁₀ CFU/ml regrowth was reported as 60 h in cases where the calculated value exceeded 60 h.

Additionally, we derived the analytical solution for the area under the bacterial killing curve (AUBKC_t) and the area under the bacterial regrowth curve (AUBRC_t) from time zero to t. The integration of the first component of equation 1 from time zero to t yields

$${\text{AUBKC}}_t={\displaystyle {\int }_0^{t}A\cdot e^{-K_d\cdot t}dt=\frac{A}{K_d}\cdot (1-e^{-K_d\cdot t})}$$ (8)

Therefore, as time approaches infinity, the AUBKC becomes A/K_d. Integration of the second component of equation 1 from time zero to t yields

$${\text{AUBRC}}_t={\displaystyle {\int }_0^t\frac{B}{1+e^{-K_r\cdot (t-C)}}\hspace{0.17em}dt=B\cdot \left(t+\frac{1}{K_r}\cdot \ln \hspace{0.17em}\frac{1+e^{-K_r\cdot (t-C)}}{1+e^{K_r\cdot C}}\right)}$$ (9)

The sum of AUBKC and AUBRC equals the AUBC for the model-generated curve fit. For the SCTK study with colistin and doripenem described below, the AUBKC₂₄ and AUBRC₂₄ were evaluated from 0 to 24 h.

Bacterial strains and in vitro SCTK studies.

A reference strain and 5 clinical A. baumannii isolates were studied. These were A. baumannii ATCC 19606, FADDI-AB009, FADDI-AB016, FADDI-AB030, FADDI-AB051, and FADDI-AB156. The colistin MIC was 0.5 mg/liter for all strains except FADDI-AB009 (MIC, 2 mg/liter; previously identified as A. baumannii isolate 9 by Li et al. [23]) and FADDI-AB156 (MIC, 16 mg/liter). Strains ATCC 19606 and FADDI-AB009 were colistin heteroresistant (23). The doripenem MIC was 0.25 mg/liter for FADDI-AB009, 1 mg/liter for ATCC 19606, 8 mg/liter for FADDI-AB030 and FADDI-AB016, 16 mg/liter for FADDI-AB051, and 32 mg/liter for FADDI-AB156. The experimental protocol for the in vitro SCTK studies was reported previously (10, 11). Viable counts were observed at 0 (i.e., within 10 min before dosing), 0.5, 1, 2, 4, 6, 24, and 48 h. Experimental arms included a growth control and colistin sulfate (Sigma-Aldrich, MO, USA) or doripenem (Janssen Pharmaceuticals, NJ, USA) at concentrations ranging from 0.5 to 64 mg/liter.

Model-free analysis of experimental data.

The model-free metrics ΔCFU and AUBC were calculated based on log₁₀-transformed viable counts. The log₁₀ transformation improved the sensitivity of these model-free metrics, especially if rapid and extensive bacterial regrowth yielded large AUBC values that masked the effect of the initial bacterial killing. The AUBC method was implemented in Microsoft Excel using the linear trapezoidal rule to integrate the log₁₀ CFU/ml versus time data via a point-to-point approach. The ΔCFU was calculated as the difference between the observed log₁₀ CFU/ml at a given time point and that at 0 h (initial inoculum).

Statistical analysis.

The relationships between the colistin and doripenem concentration and the two model-free metrics (ΔCFU and AUBC), as well as the four model-based summary metrics (time to 2 log₁₀ killing [T2LK], time to 5 log₁₀ regrowth [T5LR], AUBKC₂₄, and AUBRC₂₄) were assessed using Spearman's rank correlation coefficient (ρ). At each multiple of the MIC, the activity of colistin and doripenem against the six strains was compared using the Wilcoxon rank-sum test. To distinguish between activity and regrowth for colistin and doripenem, a pooled analysis over all multiples of the MIC was performed via analysis of covariance (ANCOVA). We used an α of 0.05 for all significance testing. Statistical calculations were performed using the R computing package (version 3.0.1).

RESULTS

Signature profiles for the rate-area-shape model (equation 1) illustrate the impact of model parameters A, B, C, K_d, and K_r on the shape of the viable count profiles (Fig. 3). Inserting a time of zero into equation 1 yields a modeled initial inoculum equal to A + B/[1 + exp(K_r · C)]. For profiles with significant killing (Fig. 3), the inoculum is determined largely by A. The rate-area-shape model could describe a large variety of shapes for viable count profiles. Curve fits of the 78 individual colistin and doripenem SCTK profiles by the rate-area-shape model were precise and unbiased, with coefficients of determination (r²) above 0.923 for all but two profiles (median, 0.977).

FIG 3.

Signature profiles of the rate-area-shape model for different values of the magnitude of bacterial killing (A), magnitude of bacterial regrowth (B), delay in bacterial regrowth (C), bacterial killing rate (D), and bacterial regrowth rate (E). The units for model parameters A, B, C, K_r, and K_d are the same as those shown in Table 1.

Estimated model parameters.

Against the six tested A. baumannii strains, colistin achieved much more rapid bacterial killing than doripenem (Fig. 4). Consistent across all multiples of the MIC against strain FADDI-AB009, the killing rate (K_d) was 13.8- ± 4.44 times greater (average ± standard deviation from all antibiotic-containing arms for strain FADDI-AB009) for colistin than for doripenem (Table 1). Regrowth was more extensive for colistin [B = (8.10 ± 1.73) log₁₀ CFU/ml] than for doripenem [B = (6.10 ± 1.52) log₁₀ CFU/ml] and occurred earlier for colistin (C = 13.9 ± 4.62 h) than for doripenem (C = 16.1 ± 5.99 h). The extent and delay of regrowth were noticeably different between both antibiotics at 2 and 8 times the MIC (Table 1).

FIG 4.

Summary (means ± standard errors [SE]) of in vitro time-kill study data for colistin (•) and doripenem (Δ) for the growth controls (A) and concentrations of 0.5× MIC (B), 1× MIC (C), 2× MIC (D), 8× MIC (E), and 32× MIC (F) for A. baumannii strains. Lines (continuous line, colistin; dashed line, doripenem) show curve fits of the rate-area-shape model through the means from the experimental data. Static time-kill studies were conducted at concentrations between 0 and 64 mg/liter; when grouped by MIC, the number of strains in each group varied due to differences in baseline MIC. Points have been shifted slightly to improve the clarity of error bars.

TABLE 1.

Parameter estimates of the rate-area-shape model obtained from static time-kill studies with colistin and doripenem against A. baumannii strain FADDI-AB009^b

Dose (×MIC)	Estimate for model:						Kda (h−1)
	A (log10 CFU/ml)		B (log10 CFU/ml)		C (h)
	COL	DOR	COL	DOR	COL	DOR	COL	DOR
0			8.82	8.18	−2.98	−1.85
0.5	5.10	6.18	9.13	7.39	9.64	7.86	1.27	0.114
1	5.63	6.30	8.35	8.00	18.4	22.0	0.950	0.0990
2	5.14	6.28	9.31	5.81	10.8	13.0	2.98	0.143
8	4.72	6.27	8.43	4.81	11.2	16.0	3.06	0.202
32	5.39	6.11	5.05	4.62	19.4	21.7	3.06	0.253

^aThe model (from equation 1) fits the viable count data on a log₁₀ scale. Therefore, the numerical values of K_d and K_r cannot be compared to the values of first-order bacterial killing and growth rate constants from a differential equation model.

^bThe growth rate was 0.228 h⁻¹ for all colistin profiles and 0.410 h⁻¹ for all doripenem profiles.

Exposure-response relationships.

As expected, colistin and doripenem displayed exposure-response relationships over the concentration range from 0.5 to 64 times the MIC. Correlations between the PD metrics (Fig. 5) and the antibiotic concentration (expressed as multiples of the MIC) could be identified by both the model-free (ΔCFU and AUBC) and model-based metrics, as expected. All correlations were statistically significant (P < 0.05).

FIG 5.

Summary of model-free (A to D) and model-based summary metric values (E to H) obtained from time-kill studies at increasing concentrations of colistin (black bars) and doripenem (gray bars) against six strains of A. baumannii. Data are presented as means ± SE (n = 2 to 6). (A and B) ΔCFU₂₄ and ΔCFU₄₈; (C and D) AUBC_0–24 and AUBC_0–48; (E) time to 2 log₁₀ killing; (F) time to 5 log₁₀ regrowth; (G) AUBKC_0–24; (H) AUBRC_0–24. Asterisks highlight statistically significant (P < 0.05) differences between colistin and doripenem.

Differences between both antibiotics assessed via model-free metrics.

The ΔCFU₂₄ and ΔCFU₄₈ results indicated that doripenem achieved more killing than colistin at 2 times the MIC (P < 0.05) (Fig. 5A and B), but the differences at the other multiples of the MIC were not significant. The ANCOVA of ΔCFU₂₄ and ΔCFU₄₈ across all concentrations showed better overall activity (P < 0.01) for doripenem (n = 29) than for colistin (n = 49). Importantly, model-free AUBC_0–24 and AUBC_0–48 lacked sensitivity, as neither metric showed a statistically significant difference (P > 0.05) between both antibiotics at any multiple of the MIC or when assessed simultaneously via ANCOVA across all studied antibiotic concentrations.

Comparison of colistin and doripenem via model-based metrics.

The rate of killing as represented by T2LK was substantially faster for colistin than for doripenem consistently across all six strains (Fig. 5E). Colistin reached a near-maximal killing rate at approximately 8× MIC, whereas the rate of killing by doripenem tended to increase over the entire concentration range. The T2LK was significantly (P < 0.05) shorter for colistin (n = 6) than for doripenem (n = 4) at 1× MIC (colistin, 53.7 ± 18.6 min; doripenem, 157 ± 36.9 min) and 4× MIC (colistin, 16.6 ± 4.77 min [n = 5]; doripenem, 129 ± 28.6 min [n = 4]), as well as for an overall comparison across all multiples of the MIC (P < 0.01).

The T5LR showed a trend toward better regrowth suppression for doripenem than for colistin, especially for antibiotic concentrations of 2× MIC and higher. At 2× MIC, the T5LR (colistin, 10.3 ± 1.90 h [n = 6]; doripenem, 21.7 ± 2.27 h [n = 3]) was significantly (P < 0.05) shorter for colistin. The overall analysis of T5LR across all multiples of the MIC yielded a P value of <0.01 in favor of doripenem.

The model-based AUBKC could clearly distinguish between colistin and doripenem, whereas the model-free AUBC could not (Fig. 5C and D). The AUBKC values significantly favored colistin over doripenem (P < 0.05) at 1, 2, and 4× MIC (Fig. 5G) and for an overall comparison across all multiples of the MIC (P < 0.01).

The AUBRC showed a trend toward better regrowth suppression for doripenem compared to colistin. At 2× MIC, the AUBRC values (colistin, 121 ± 14.0 log₁₀ [CFU/ml] · h; doripenem, 43.2 ± 12.9 log₁₀ [CFU/ml] · h) significantly (P < 0.05) favored doripenem over colistin. An overall ANCOVA analysis did not show a significant difference between AUBRC of colistin and doripenem (P = 0.083), likely because doripenem yielded better regrowth suppression only for concentrations of 2× MIC and higher (Fig. 5H).

DISCUSSION

The objectives of this work were to develop and evaluate a novel rate-area-shape model for the analysis of SCTK profiles. The commonly applied model-free metrics ΔCFU and AUBC did not account for the rate and extent of bacterial killing or the timing of regrowth. While colistin yielded much faster (Table 1) and more extensive killing of six A. baumannii strains (Fig. 4), substantial and early bacterial regrowth yielded inferior ΔCFU₂₄ and ΔCFU₄₈ compared to those for doripenem (Fig. 5). Similarly, the model-free AUBC could not differentiate between colistin and doripenem (Fig. 4 and 5), since the enhanced killing by colistin was offset by more extensive and earlier regrowth compared to doripenem. Overall, the model-free AUBC had relatively poor sensitivity (Fig. 5).

To generate insights not provided by the model-free metrics ΔCFU and AUBC (17, 24–31) and the less commonly used analyses of the initial slope (15, 16, 18) of time-kill profiles, we developed the rate-area-shape model to quantify the rates and extents of bacterial killing and regrowth and the timing of regrowth. The rate-area-shape model utilizes all experimental data to estimate its parameters. This new model could identify clear differences in bacterial killing and regrowth between colistin and doripenem (Fig. 4). Bacterial killing was at least 10-fold faster for colistin than for doripenem at all multiples of the MIC (Table 1). However, regrowth was considerably less extensive and occurred approximately 3 h later for doripenem than for colistin at both 2× and 8× MIC (Table 1). The model-based summary metrics T2LK, T5LR, and AUBKC could clearly differentiate between both antibiotics (Fig. 5). The derivation of these summary metrics assumes that the killing of susceptible bacteria is (largely) independent of the regrowth of resistant bacteria. This assumption seems appropriate, in particular for infection models at normal (i.e., low) bacterial densities that have no or minimal cell-to-cell communication. While this study examined colistin and doripenem against A. baumannii as a clinically important pathogen, the rate-area-shape method is applicable to data for other pathogens and antibiotics.

Ease of implementation, accessibility, and efficiency were important considerations during the development of the rate-area-shape model. Parameter estimation of this model and calculation of the model-based summary metrics (Fig. 5) can be performed readily in Excel (and other packages) and does not require advanced numerical techniques typically needed for mechanism-based modeling. Closed-form solutions for the model-based metrics enhance the proposed approach and characterize relevant aspects of viable count profiles (e.g., time to 2 log₁₀ kill and time to 5 log₁₀ regrowth) that are not captured by the model-free ΔCFU and AUBC.

Estimation of the rate-area-shape model requires at least 5 observations per viable count profile (including the 0-h time point). To capture relevant aspects of the time-kill profiles (i.e., bacterial killing and regrowth), the associated observation times should be carefully chosen. Profiles with fewer observations (e.g., only 2 or 3 samples per profile) could be modeled simultaneously via a population modeling approach to address this potential limitation. Such an estimation can be implemented in any population modeling package and requires minimal estimation time, as no differential equations are needed. Population estimation methodology also allows one to simultaneously estimate exposure-response relationships for the empirical model parameters for one or multiple strains (e.g., by expressing drug concentrations as multiples of the MIC).

Analysis of data sets with nonmatching observation times is problematic for the model-free ΔCFU and AUBC methods, since differences in observation times can considerably affect the interpolated areas calculated via the trapezoidal method. This limitation may force researchers to repeat experiments to match experimental designs. The rate-area-shape model is fully suitable for the analysis of data sets with nonmatching time points, as this model fits all viable counts simultaneously. This facilitates the analysis of published data sets from different studies.

While the accessibility and simplicity of the rate-area-shape model are advantages, one needs to recognize potential limitations of this empirical modeling approach. The current form of the rate-area-shape model does not account for more complex microbiological phenomena, such as specific mechanisms of bacterial killing or of resistance. The rate-area-shape model also does not formally distinguish between susceptible and resistant bacteria; however, it distinguishes between bacterial killing and regrowth (Fig. 5E to H), whereas the model-free ΔCFU and AUBC approaches do not.

The current version of the rate-area-shape model was not developed to describe viable count profiles for in vitro studies with repeated dosing that can be generated in one-compartment or hollow fiber in vitro infection models (9, 32). However, the empirical model can be extended to describe viable count data for dynamic infection models if each dosing interval is fitted separately. In this case, regrowth can be caused either by declining antibiotic concentrations or by bacterial resistance. Finally, when applied to combination therapy, our new approach can empirically describe bacterial killing and regrowth for each studied combination of antibiotic concentrations.

In conclusion, we developed and evaluated a novel rate-area-shape model for the efficient, robust, and sensitive analysis of in vitro SCTK data. This new model can quantify the rate, extent, and timing of bacterial killing and regrowth and could clearly differentiate between colistin and doripenem. In contrast, the widely used model-free metrics (ΔCFU and AUBC) could not identify the rapid killing and more extensive and earlier regrowth for colistin compared to that for doripenem. The proposed model and its associated summary metrics provided good sensitivity and were implemented in a widely accessible spreadsheet program (Excel). This novel approach will provide the scientific community with improved insights into the rate, extent, and timing of bacterial killing and regrowth in in vitro SCTK experiments.