Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, USA

Department of Electrical and Computer Engineering, Prairie View A&M University, Prairie View, TX 77446, USA

Computational Biology Division, Translational Genomics Research Institution, Phoenix, AZ 85004, USA

Department of Bioinformatics and Computational Biology, University of Texas M.D. Anderson Cancer Center, Houston, TX 77030, USA

Abstract

Background

Molecularly targeted agents (MTAs) are increasingly used for cancer treatment, the goal being to improve the efficacy and selectivity of cancer treatment by developing agents that block the growth of cancer cells by interfering with specific targeted molecules needed for carcinogenesis and tumor growth. This approach differs from traditional cytotoxic anticancer drugs. The lack of specificity of cytotoxic drugs allows a relatively straightforward approach in preclinical and clinical studies, where the optimal dose has usually been defined as the "maximum tolerated dose" (MTD). This toxicity-based dosing approach is founded on the assumption that the therapeutic anticancer effect and toxic effects of the drug increase in parallel as the dose is escalated. On the contrary, most MTAs are expected to be more selective and less toxic than cytotoxic drugs. Consequently, the maximum therapeutic effect may be achieved at a "biologically effective dose" (BED) well below the MTD. Hence, dosing study for MTAs should be different from cytotoxic drugs. Enhanced efforts to molecularly characterize the drug efficacy for MTAs in preclinical models will be valuable for successfully designing dosing regimens for clinical trials.

Results

A novel preclinical model combining experimental methods and theoretical analysis is proposed to investigate the mechanism of action and identify pharmacodynamic characteristics of the drug. Instead of fixed time point analysis of the drug exposure to drug effect, the time course of drug effect for different doses is quantitatively studied on cell line-based platforms using system identification, where tumor cells' responses to drugs through the use of fluorescent reporters are sampled over a time course. Results show that drug effect is time-varying and higher dosages induce faster and stronger responses as expected. However, the drug efficacy change along different dosages is not linear; on the contrary, there exist certain thresholds. This kind of preclinical study can provide valuable suggestions about dosing regimens for the

Introduction

Drug development is currently an expensive and prolonged process with high attrition rate. The rate of new drug approvals in the U. S. has remained essentially constant since 1950, while the costs of drug development have soared

The focus of anticancer drug development in recent years has shifted from cytotoxic drugs to targeted therapy

While the lack of specificity of the traditional cytotoxic anticancer agents allows a relatively straightforward, well-established approach, developing a paradigm to better analyze the efficacy of molecularly targeted agents (MTAs) is substantially more complex

Firstly, the optimal dose has usually been defined as the "maximum tolerated dose" (MTD) for conventional cytotoxic anticancer drugs rather than the dose that produces a quantifiable therapeutic effect. This toxicity-based dosing approach is founded on the assumption that the therapeutic anticancer effect and toxic effects of the drug increase in parallel as the dose is escalated

A hypothetical dose-effect curve for targeted therapy

**A hypothetical dose-effect curve for targeted therapy**.

Secondly, the pharmacodynamics (PD) of drugs have been extensively investigated _{max }

Thirdly, traditional design of the dosing regimen to achieve some desired target goal such as relatively constant serum concentration may not be optimal because MTA targets mostly sit in interacting complex dynamical regulatory networks and such complex target contexts pose significant challenges for assessing mechanisms of action for MTAs

In sum, it is difficult and expensive to optimize dosing regimens using strictly empirical methods for MTAs. A novel preclinical model combining experimental methods and theoretical analysis is proposed in this study to investigate the mechanisms of action and identify pharmacodynamic characteristic of MTAs. As a first step, the time courses of drug effect for different doses are quantitatively studied on cell line-based platforms using system identification, where a tumor cell's response to investigational drugs through the use of fluorescent reporters is sampled frequently over a time course. A dynamic model is proposed to study the time course of drug efficacy for MTAs and then the experimental data are analyzed by our proposed model using a Kalman filter. Through such preclinical study, valuable suggestions about dosing regimens may be furnished for the

Methods

The proposed approach is an integration of experiment and theory to investigate regulatory process dynamics by combining multiple complementary disciplines, including: (i) using fluorescent reporters in molecular technology to study cells' transcriptional activities under drug perturbation; (ii) these being captured by an automatic epifluorescent microscope over a time course; and (iii) such data being processed by large-scale image processing for dynamic analysis. A truly multi-dimensional dynamics of tumor cell response to drugs can be characterized through systematic perturbations to test different combinations of cell types, reporters, and drugs/dosages, augmented by iterative systematic theoretical analysis. This methodology differs from high-throughput technique like RNA expression profiling with microarrays, which provide a snapshot of an aspect of the system at one time point.

Experimental methodology

Understanding cell response to a drug requires experimental designs that ask very specific questions about what is happening in a cell in the absence of a drug and how the cell activities change when the drug is present. The objective of the experimental protocol is to efficiently capture cell process dynamics in response to drugs and thereby obtain a deeper understanding of the genetic regulatory mechanisms, the point being to make preclinical research more predictive. Fluorescent reporters have long been used in molecular technology to study cells' transcriptional activities or the cellular localization of components, either in a population of cells or a single cell

In this experimental set-up, using different wells to test different combinations of cell type, GFP reporter and experimental condition allows this approach to provide a multi-dimensional examination of the cells' responses to a variety of stimuli. Not only can it follow multiple genes simultaneously, but it can also compare cellular activities under various conditions. Furthermore, it captures the dynamics of transcriptional regulation. This produces data on ~200-400 individual cells per well that can be analyzed both individually, as a distribution, or in aggregate, as an average. Fluorescent intensity data can be extracted from these images using specialized image analysis tools developed for this application

Image processing

Typical fluorescent images are shown in Figure

Time course response to lapatinib by HCT116 with reporter for MKI67: Left panels show 2 typical fluorescent images (nuclei: blue, GFP: green) sampled for the same site in a 48-hour lapatinib treatment

**Time course response to lapatinib by HCT116 with reporter for MKI****67**: Left panels show 2 typical fluorescent images (nuclei: blue, GFP: green) sampled for the same site in a 48-hour lapatinib treatment. a) The upper panels show the case before any drug is applied. b) The lower panels show the case 48 hours after lapatinib was added. The right panels show the log2(GFP) intensity histogram for each time point. The fraction of the total population having a particular intensity is shown on the y-axis and the log2 intensity of the eGFP fluorescence measured for the cell is shown on the x-axis. The distribution before drug is shown as a thick yellow line at the upper right panel. The lower right panel shows the profile that is color-coded with time, starting with red, changing to yellow and then green, and ending with blue. The profile at the ending time points is shown with bold yellow line.

To facilitate automatic processing of the experiment results, the transcriptional levels of the fluorescent images need be properly extracted, quantized, and saved and the image processing algorithm should be fast with good balance between performance and robustness _{2 }transform before being exported.

Segmentation Results: a) left panel: nuclear channel, where red lines are the identified nuclei boundaries; b) right panel: reporter channel, where green lines are identified cell boundaries, while the red objects are the nuclei used as markers

**Segmentation Results**: a) left panel: nuclear channel, where red lines are the identified nuclei boundaries; b) right panel: reporter channel, where green lines are identified cell boundaries, while the red objects are the nuclei used as markers.

Experimental set-up for the dosing study

The dosing study is carried out on the colon cancer cell-line HCT116 with a reporter for the MKI67 gene, a nuclear antigen tightly correlated with proliferation ^{14}. One of the time courses from experiment (dosage = 8_{2}(GFP) intensity histogram for each time point.

Since MKI67 is turned on during proliferation and off when the cells are not cycling, it is expected to show a binary, switch-like histogram of cell intensities, rather than a graded transition. This behavior is observed in Figure

Mathematical model formulation

The experimental results provide information on the percentage of cells shifted as a consequence of the drug activity. The measurements facilitate asking important questions in drug development. For instance, does dosing alter the extent of response, the timing of response, or both? In addition to qualitative questions, we are interested in modeling the drug effect

Because there are different numbers of cells in different wells (the range is about ~200-400 cells per well), we perform normalization to calculate the percentage of cells shifted. Since there are many factors including drug effect that contribute to the cell shifting, calibration is performed by comparing to the control group to exclude other contributing factors. The notations used in this work are listed below

•

• _{1}(

• _{1}(_{1}(

• _{c}

• _{1c}(

• _{1c}(_{1c}(_{c}

• _{1}(_{1c}(

• _{av}

• _{i}

We justify _{1}(

N_{1}(t) is a Gaussian process when the number of cells per well is large enough

In general, _{c}_{j }_{i}_{j}_{i}_{j}_{i}_{j}

where 0 ≤ _{j }_{1 }has the binomial PMF given by

When the number of cells per well is large, say _{1 }at any given time instant can be accurately approximated by the Gaussian distribution due to the central limit theorem. Next we show that _{1}(

**Proposition 1**. _{1}(

_{0}, _{1}(_{0}) is a Gaussian random variable. For any sampling point, at time _{j}_{1}(_{j}) can be expressed as

where _{1}(_{j}_{−1}) is the total number of shifted cells at time _{j}_{−1}, and the additional number of shifted cells in the time interval [_{j}_{−1}, _{j}

If _{1}(_{j}_{−1}) is sufficiently large, _{1}(_{j}_{−1}) > 32, then Δ_{1}(_{j}_{1}(_{0}) is Gaussian, _{1}(_{j}

Modeling the cell shifting process

From our previous experimental observation, the cell shifting process on colon cancer cell-line HCT116 with a reporter for the MKI67 gene under lapatinib treatment shows a binary shifting characteristic. It is assumed that the number of shifting cells is related to: (i) the drug effect corresponding to different dosages; and (ii) the number of proliferating cells (non-shifted cells, _{1}). Since _{1}(_{1}(_{1}(_{1c}(_{1c}(_{c}_{1}(_{1c}(_{av}

where _{av}_{av}

In this model, we assume that both _{1}, decreases exponentially with the factor

The noise terms account for the various uncertainties introduced by the experiment. For instance, the cells may not be at the same cell cycle during the experiments, and thus may not be affected by the drug if some of the cells are actually dormant. This kind of uncertainties are modeled by process noise

To observe the relationship between the drug effect coefficient

System identification from time-series data using Kalman filter

Kalman filtering

where the 2-dimensional state vector (containing the parameters to be estimated) is

The implementation of the Kalman filter is given by the following equations

where ^{- }and ^{+ }indicate the

In general, a Kalman filter may be interpreted as a one-step predictor with an appropriate gain calculator

Convergence of the Kalman filter is an important issue

In practice, noise statistics (such as the covariance matrices) may not be known and need to be estimated. The Kalman filter is sensitive to the estimation error of noise statistics. Poor estimates of the noise covariance can result in filter divergence. An alternative would be using an _{∞ }

Results

Two-step analysis is performed to evaluate the drug effect study for different dosages. Firstly, we performed a proof-of-concept experiment using Monte Carlo simulation to demonstrate that the proposed model can mimic experimental observation. Secondly, we analyzed the time-varying drug effect for different dosages based on real experimental data from Dr. Bittner's lab at Translational Genomics Research Institution (TGen).

Proof-of-concept experiment using Monte Carlo simulation

It is assumed that a group of 200 cells has mean GFP intensity at 2^{18}. When the drug is applied, each cell determines whether to shift to a lower intensity or not individually by flipping a coin (Bernoulli trial) at each time point, as we assumed in the theoretical model. The histograms of percentage of cells at intensity in the range of [2^{14}, 2^{19}] along time are shown in Figure

The change of histogram of percentage of cells at intensity [2^{14}, 2^{19}] under drug intake along time using Monte Carlo simulation

**The change of histogram of percentage of cells at intensity [2 ^{14}, 2^{19}] under drug intake along time using Monte Carlo simulation**.

Drug effect analysis for the dosing study performed at TGen

For the experiments performed on the cell-line at TGen, there are 6 different dosages tested for the drug laptinib, from 1

The estimate of the drug effect coefficient along time for 6 different dosages

**The estimate of the drug effect coefficient along time for 6 different dosages**.

The estimate of the balancing factor along time for 6 different dosages

**The estimate of the balancing factor along time for 6 different dosages**.

It is observed from Figure

Unlike

Figure

The Convergence result of the the proposed algorithm using Kalman filter

**The Convergence result of the the proposed algorithm using Kalman filter**.

Post data processing for the dosing study performed at TGen

From Figure

The smoothed drug effect coefficient along time for 6 individual dosage

**The smoothed drug effect coefficient along time for 6 individual dosage**.

The smoothed drug effect coefficient along time for 6 different dosages

**The smoothed drug effect coefficient along time for 6 different dosages**.

The smoothed balancing factor coefficient along time for 6 individual dosage

**The smoothed balancing factor coefficient along time for 6 individual dosage**.

The smoothed balancing factor coefficient along time for 6 different dosages

**The smoothed balancing factor coefficient along time for 6 different dosages**.

Conclusions and future work

The ultimate goal of target-based cancer drug development is to improve the efficacy and selectivity of cancer treatment by exploiting the differences between cancer cells and normal cells. The current cancer drug development process is confronting huge challenges, such as how to better understand the target in context and develop predictive preclinical models to better understand the molecular mechanisms of the biological systems they target and hence reduce the attrition rate. An integrated experimental and theoretical approach is proposed to assess the efficacy of molecularly targeted agents based on cell-line platforms. As a first step, drug efficacies for different dosages are characterized along time. Specifically, tumor cell's responses are analyzed through the use of fluorescent reporters sampled frequently over a time course; quantification is done by microscopic scanning of cells in culture in multi-well plates using the automated epifluorescent imager; fluorescent intensity data are extracted from these images using specialized large-scale image analysis tools developed for this application; the dynamics of drug efficacy for different dosages are studied using dynamic modeling; and time-varying parameters are estimated using system identification techniques. It is observed that the drug efficacy is time and dosage dependent. The objectives are two-fold: (i) The dosing study for MTAs should be based on both efficacy and toxicity consideration to find the biologically effective dose (BED) instead of the maximum tolerated dose (MTD) for cytotoxic agents. The time course of drug effect for different dosages can provide information on the gradient of drug effect vs. dosage, and thus on the BED. (ii) Instead of a fixed time point pharmacodynamics study of MTA, characterization of the entire time course of drug effect provides insight into designing an optimal schedule for drug administration.

Based on a similar experimental set-up and measurements to follow the cell/drug (dosages) dynamics, a truly multi-dimensional dynamics of tumor cell responses to drugs can be characterized through systematic perturbations to test different combinations of cell types, reporters, and drugs/dosages, augmented by iterative systematic theoretical analysis. Such an approach would facilitate the study of optimal dose and schedule, such as whether short pulses of higher dose, persistent dosing with lower dose, or some other regimen would have the most favorable outcomes. Moreover, the complex target context can be inferred with multi-dimensional cell response dynamics with the help of advanced system identification methods. In sum, better intervention strategies can be designed. Such topics are either currently being pursued or will be in future projects.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

XL and LQ developed and implemented the algorithm, conducted all simulations and data processing and wrote the initial draft of the paper. JH performed the image analysis. MB performed the experiments. ED advised XL on algorithm development and revised the paper. All authors read and approved the final manuscript.

Acknowledgements

Based on “Assessing the efficacy of molecularly targeted agents by using Kalman filter”, by Xiangfang Li, Lijun Qian, Michael L Bittner and Edward R Dougherty which appeared in

Xiangfang Li has been supported by the National Cancer Institute (2 R25CA090301-06). The experimental and image analysis work was supported in part by the W. M. Keck Foundation and Predictive Biomarker Sciences.

This article has been published as part of