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In this paper, we established a multiscale mechanistic model for studying drug delivery, biodistribution, and therapeutic effects of cancer drug therapy in order to identify optimal treatment strategies. Due to the specific characteristics of cancer, our proposed model focuses on drug effects on malignant solid tumor and specific internal organs as well as the intratumoral and regional extracellular microenvironments. At the organ level, we quantified drug delivery based on a multicompartmental model. This model will facilitate the analysis and prediction of organ toxicity and provide important pharmacokinetic information with regard to drug clearance rates. For the analysis of intratumoral microenvironment which is directly related to blood drug concentrations and tumor properties, we constructed a drug distribution model using diffusion-convection solute transport to study temporal/spatial variations of drug concentration. With this information, our model incorporates signaling pathways for the analysis of antitumor response with drug combinations at the extracellular level. Moreover, changes in tumor size, cellular proliferation, and apoptosis induced by different drug treatment conditions are studied. Therefore, the proposed multi-scale model could be used to understand drug clinical actions, study drug therapy-antitumor effects, and potentially identify optimal combination drug therapy. Numerical simulations demonstrate the proposed system's effectiveness.

Cancer disease remains a leading cause of high morbidity and mortality in both adults and children. Despite extensive efforts to discover and develop effective drugs, very few promising candidates currently exist in the development pipeline. Traditional methods for developing and testing new drugs usually involve a series of controlled experiments on groups of selected healthy volunteers to establish the relationship between dose and therapeutic effects for a given drug candidate. Traditional drug development methods are time and resource intensive and carry substantial risk for adverse effects. Hence a more efficient approach is urgently needed.

Fortunately, cancer research has undergone dramatic changes recently. One of the most important changes is the application of computational modeling for pharmacokinetics/pharmacodynamics (PK/PD) analysis [

Integrated multi-scale mechanistic model. (a) Drug effects on tumor microenvironment and different type of tumor cells via signaling pathways like EGF-EGFR, TNF

Due to the urgency of cancer therapy and great demand to improve cancer drug development and therapeutic effects, we propose a multi-scale mechanistic model that integrates both PK and PD to study drug dose-effect relationships and to identify the optimal drug treatment strategies. The model aims at three distinct levels of detail that include the target cell, the tumor microenvironment, and whole organs where drug metabolism and/or excretion may occur. To address how drugs distribute through the whole body, we adopt a multicompartmental model to describe dynamic drug concentrations as a function of time. For unknown parameters, we refer to experimental data to make reasonable estimation which in turn may predict organ drug concentrations under different treatment conditions. With predicted blood drug concentrations, we perform a diffusion-convection solute transport model within the intratumoral space to calculate the temporal and spatial variation of drug concentration within tumor. We then use this drug distribution information as input data for the next component of the model that assesses the signaling pathway through drug binding effects. At this extracellular level, we selected the EGF-EGFR (endothelial growth factor) signaling pathway to study drug binding effects on cell proliferation and apoptosis to determine the drug treatment-therapeutic effect relationship. Our model’s detailed description of the complete mechanistic system can potentially predict the therapeutic effect and possible organ toxicity of a given drug treatment strategy, thereby enabling optimization of drug selection and dosing.

The remainder of this paper is organized as follows: Section

In this section, we globally introduce the structure of an integrated system with emphasis on the function of our proposed multi-scale mechanistic model. This model aims to search for the optimal cancer therapy drug dose and treatment strategies.

The system begins at the level of macroscale drug delivery model. We use reported experimental data from molecular imaging and physiological experiments to estimate optimal parameters for the drug delivery and followed by using an optimized drug delivery system to predict concentration-time curves in different organs. Therefore, with this model we can easily change input drug dose and treatment intervals to study different therapeutic effects. With blood concentrations of drug we can further calculate spatial distribution inside a solid tumor by solving diffusion-convection partial differential equations (PDEs). Basically, drug inside tumor will accumulate and decay in accordance with drug administration and clearance. The module at the intratumoral level actually offers a way to study drug therapeutic effects by understanding how regional variations in drug distribution may affect overall antitumor effects.

The final component of the modeling approach targets intracellular mechanisms in which therapeutic response is directly linked to signaling pathway blockade or alteration. This module can provide a high-resolution overview of drug action in the targeted tumor cell. Finally, the system is designed to yield a prediction of tumor progression or positive treatment response by inputting known drug efficacy data, which includes drug dose and schedule. Hence, the first two parts in our multi-scale model which targets the organ level and intratumoral microenvironment are directly coupled to pharmacokinetics analysis which is linked to the last component of the model that analyzes the pharmacodynamic effects at the tumor cell level.

The flowchart in Figure

Multi-scale mechanistic model structure.

The aforementioned introduce the overall structure of our proposed multi-scale mechanistic model and its primary objective for cancer therapy. Now in this part we will introduce the three modules separately to show how each modules works at different level.

The first part of our model is mainly concerned with the dynamic drug concentration changes over time inside various major organs such as blood, liver, spleen, heart, and kidneys, which is essentially what constitutes PK analysis. Pharmacokinetics includes four distinct factors: drug absorption, distribution, metabolism, and excretion, which are often referred to as ADME analysis in pharmacology [

Pharmacokinetics system figure. (a) Different drug administration methods including IV, oral, topical, inhalation, IM, SC, and IP. (b) Drug distribution process between blood and other organs.

For the drug distribution process, we have used a multicompartment model (Figure

In the ODE system, we need to estimate the unknown parameter, where we refer to the time course of drug biodistribution data reported in literatures. The experimental data should have a series of observations on different time points for each compartment like

This is a typical least square estimation objective function. But due to underlying strong noise in experimental data, some traditional least square methods (LSMs) like Gradient Descent, Gauss-Newton method, and Levenberg-Marquardt algorithm are not able to converge, yet they are prone to fall into many local optimums. We therefore use Genetic Algorithm (GA) [

When implementing this model, we firstly choose a way for drug administration, by either IV or oral. IV administration is easy to study by using the proposed ODE dynamic system since drug comes into plasma directly, either by bolus injection or by intravenous infusion; when it comes to oral administration, drug will undergo a smoother absorption process which makes it take longer time to reach peak concentration in blood and also prolongs its clearance time. We proposed the following model to fit typical oral administration blood drug concentration-time curves as shown in Figure

Blood drug concentration-time curve (Oral administration).

By setting optimized parameters, the drug delivery model is able to predict drug concentration versus time curves in blood and different organs, such as

In the previous section, we discussed drug delivery at the organ level which is related to the drug concentration in blood. When drug reaches the tumor site, it will form a temporal and spatial drug distribution inside the tumor as determined by concentration changes in blood, microenvironment, and tumor properties. As solid tumors grow, development of heterogeneous neovasculature results in variable areas of rapid growth where the blood supply is adequate or necrosis at sites that are oxygen and nutrient-poor. Due to this heterogeneity, solid tumor shows heterogeneous growth patterns and has prompted particular attention on tumor angiogenesis modeling [

As shown in Figure

(a) Cross-section of solid tumor embed in normal tissue. (b) Diffusion-convection drug transport model in solid tumor. (c) Drug transport mechanism in intratumoral space (to be continued).

Also according to mass conservation law [

There are two boundary conditions for solving this equation. For the outer boundary, we assume that the tumor tissue pressure equals the normal tissue pressure. Since the tumor has a nonflux center due to its spatial sphere symmetry, we set

After we have studied the intratumoral fluid, we are ready to address drug transport based on the information acquired previously. Since the heterogeneous distribution of drug is mainly induced by drug diffusion and intratumoral fluid convection, we can use convection-diffusion equation to study the drug transport process. As shown in Figure

We will assess drug effects on a targeted tumor cell in this section. In the previous literature [

Normally, binding of EGF to EGFR receptor triggers complex signaling cascade with interactions between activated receptors, recruited proteins, and plasma membrane molecules, ultimately resulting in multiple down-stream effects that control cell proliferation and survival. However, in the presence of a targeted drug, this EGF-EGFR signaling pathway will be blocked and ultimately resulting in cell death in a drug dose-dependent manner. This process could be depicted as follows:

We use the steady state of (

Based on (

Due to the binding loss of EGF:EGFR, the effective

The effective EGF-EGFR activates down-stream effectors. High

EGF-EGFR signaling pathway and drug effects on cell proliferation and apoptosis rates.

In order to show the effectiveness and advantages of our proposed model, we performed a series of experiments to illustrate the properties of our model. We analyzed both IV method and oral drug treatment since they are the most widely used administration routes. The proposed multi-scale model gives us a detailed description at different levels of drug action on the whole body and particularly in solid tumor.

To predict drug concentration changes in different organs, we built up the drug delivery system. For IV administration study, the system performance was to a high extent determined by system parameters introduced in (

In this work, we selected four compartmental data to test the model, namely, the liver, lung, tumor, and blood. After the drug delivery ODE system was set up according to (

Experimental data fitting by Drug delivery ODE system.

Figure

IV administration drug concentration curves. (a) Bolus injection. (b) Intravenous infusion.

However, if we choose drugs that are orally administration, concentration in blood will rise smoothly and fall more gradually. It takes hours to reach peak value and even longer to be cleared. This process could be described by a biexponential function model introduced in (

(Oral treatment) Experimental measured drug concentration curve and the fitting curve predicted by proposed model.

Moreover, when treated with an oral drug, patients usually need to take a dose every several hours to maintain an adequate drug concentration. The proposed oral treatment model could be used to predict drug concentration curves in plasma when treated with various timing intervals. For example, with a drug treatment interval of 12 hours, drug concentration in blood gradually rises and oscillates at a predictable level (Figure

Blood drug concentration curve of a fixed time interval dosing.

With the blood concentrations, drug distribution inside tumors can be readily calculated. This can be achieved by integrating the intratumoral fluid pressure and velocity in (

Baseline parameter values used for intratumoral PDE system.

Parameters | Baseline values | Unit | References |
---|---|---|---|

cm^{2}/mm Hg-sec | [ | ||

cm/mm Hg-sec | [ | ||

200 | cm^{−1} | [ | |

15.6 | mm Hg | [ | |

0.90 | [ | ||

0.82 | mm Hg | [ | |

15 | mm Hg | [ | |

cm^{2}/sec | [ | ||

cm/sec | [ | ||

0.9 | [ | ||

Sec^{−1} | [ |

Intratumoral fluid pressure and velocity distribution.

After obtaining pressure and velocity values, we can calculate the drug concentration based on drug transport equation (

Intermittent intravenous infusion treatment. The treatment time is 2 hrs and treatment interval is 3 hrs.

Drug distribution results provide detailed information for predicting and studying drug effects on different region inside a tumor. In this part, we use the EGFR signaling pathway model to evaluate relationship between drug concentration and treatment effects. The first part of signaling pathway model is focused on drug binding effects through the EGF receptor. Literatures clearly demonstrated that the EGF-EGFR signaling pathway activates tumor cell proliferation and specific blockade of this signaling results in the tumor inhibition [

Baseline parameter values used for signaling pathway model.

Parameters | Baseline values | Unit | References |
---|---|---|---|

EGF | mol/l | Estimated | |

EGFR | 800,000 | [ | |

mol/l | [ | ||

mol/l | [ | ||

0.0143 | Hour^{−1} | [ | |

0.010 | Hour^{−1} | Estimated | |

4800 | [ | ||

Estimated |

Based on (

Drug concentration-cell growth rate relationship.

This process is described by the Hill function, and based on this model, we can calculate net cell growth rate

When studying different administration methods for cancer treatment, we first generated blood drug concentration curves such as that shown in Figures

Different oral drug treatment intervals and related minimum peak concentration for inhibiting tumor growth.

When it comes to IV administration, we also focus on outer boundary drug concentration curve such as that shown in Figure

Minimum drug concentration for different treatment strategy*.

I.V.Time (hr) | Minimum concentration (nM) | |||||
---|---|---|---|---|---|---|

Interval (hr) | ||||||

4 | 8 | 12 | 16 | 18 | 24 | |

1 | 150 | 492 | 1453 | 2000 | 2000 | 2000 |

2 | 86 | 242 | 693 | 2000 | 2000 | 2000 |

3 | 64 | 158 | 460 | 1264 | 2000 | 2000 |

4 | 54 | 116 | 317 | 848 | 2000 | 2000 |

5 | 48 | 90 | 230 | 652 | 1756 | 2000 |

*The minimum drug concentration search space is from 1 nM to 2000 nM.

The results indicate that a longer treatment period can be used with relatively longer dosing intervals. As an example, for an IV administered drug, the infusion time of 5 hours and dosing interval of less than 16 hours results in a positive cell net death rate which is predicted to inhibit tumor growth. However if the dosing interval is longer than 16 hours, the calculated cell net death rate is negative, resulting in tumor progression.

In addition, Figure

Mean cell death evaluation-IV strategy relationship. (

In this part, a series of experiments have been done to demonstrate the effectiveness of the proposed multi-scale model. Experimental results indicate that the proposed model is able to serve as a powerful analysis tool to find out proper strategy for drug testing or clinical treatment.

In this work we propose a multi-scale mechanistic model for cancer drug therapy study, in which we study drug action at mechanistically distinct levels that span the macroscopic effects in major organs to the subcellular alterations in signaling cascades within tumor cells. This multiscale model provides a detailed and comprehensive view of drug action and provides important predictive relationships that may be used to cost-effectively guide both preclinic and clinical trials. However, our model is still at its early stage and several aspects must be further refined. For example, drug mechanism models at the organ level should also incorporate specific properties of drug such as composition, specific targets, and mode of action. In our initial work the ideal tumor was treated as a homogeneous tiny sphere; however tumors in practice are highly heterogeneous in size, shape, vascularity, and mode/site of invasion; these factors pose a significant challenge for accurate modeling. As already alluded, tumor growth, regional variability, and response to varying drug concentrations within tumors remain a challenge to model and to accurately determine experimentally. However such information perhaps obtainable by molecular imaging techniques may facilitate further refinements to the model. We have used a well-studied signaling pathway for the PD module in our model; however further improvement awaits greater detail in our understanding of multiple biochemical and signaling pathways involved in tumorigenesis. Nonetheless there is sufficient pharmacological and biochemical information at this time to provide a reasonably detailed framework for a comprehensive model as we have presented. Our approach provides an important foundation for analyzing and predicting drug action and antitumor effects as well as systemic toxicity to facilitate the drug discovery and development process.

The author thank Dr. Brian E. O’Neill for his kind help and suggestion and Drs. Zhengzheng Shi, Zheng Li, and Tao Peng for their discussion. This work was partially supported by the NIH Grants 5R01EB009009-03 (Li), 1R01LM010185-01 (Zhou), and NSFC Grants 61133010, 31071168, 61005010, 60905023 & 60975005 (Huang).