Effect of tumor size on drug delivery to lung tumors M. Soltani, M. Sefidgar, H. Bazmara C. Marcus, R. M. Subramaniam, A. Rahmim

Abstract– Drug delivery to solid tumors can be expressed physically using biomechanical phenomena such as convection, diffusion of drug in extracellular matrices, and drug extravasation from microvessels. Applying computational methods to solve governing conservation equations clarifies the mechanisms of drug delivery from the injection site to a solid tumor. In this study, multiple tumor geometries were obtained from PET/CT images. An advanced numerical method was used to solve fluid flow and solute transport equations simultaneously to investigate the effect of tumor size on drug delivery to lung tumors. Data from 20 patients with lung tumors were analyzed and the tumor geometrical information including size, shape, and aspect ratios were classified. In order to investigate effect of tumor size, tumors with similar shapes but different sizes ranging from 1 to 28.6 cm3 were selected and analyzed. A hypothetical tumor, similar to one of the analyzed tumors but scaled to reduce its size to 0.2 cm3, was also analyzed. An ideal bolus injection was considered for the model. The effects of two transport mechanisms, namely convection and diffusion, were considered in this study. The results show because of size of considered lung tumor, the diffusion transport rate is higher than convection transport rate. Based on governing equations, the diffusion transport is only depended on concentration gradient and independent of size of tumor, therefore the predicted concentration profile for considered tumors are similar. When size of tumor is decreased significantly, the drug concentration is also significantly increased.

I. INTRODUCTION

Lung cancer is the most common cause of cancer-related death in men and women, and was responsible for 1.56 million deaths annually, as of 2012 [1]. Chemotherapy is one of the ways widely used for cancer treatment. Based on the findings from clinical applications, most cancer treatments with drugs fail to eliminate solid tumors completely [2]. The computational method can investigate why systemic administration cannot distribute drug uniformly in tumors[3]. The drug exchange between microvessels and extracellular

Manuscript received December 05, 2015. M. Soltani is with Division of Nuclear Medicine, Department of Radiology

and Radiological Science, School of Medicine, Johns Hopkins University, MD 21287, USA (e-mail: ma_soltani@yahoo.com), and Mechanical Engineering, KNT University, Tehran, Iran.

M. Sefidgar is with Department of Mechanical Engineering, Pardis Branch Islamic Azad University, Pardis new city, Iran (e-mail: sefidgar@gmail.com).

H. Bazmara is with Department of Mechanical Engineering, Pardis Branch Islamic Azad University, Pardis new city, Iran (e-mail: bazmara@gmail.com).

C. Marcus is with the Department of Radiology, Johns Hopkins University, Baltimore, MD, USA 21287 (e-mail: cmarcus7@jhmi.edu).

M. Subramaniam is with the Departments of Radiology, Oncology, Otolaryngology, Head and Neck Surgery, and Health Policy and Management, Johns Hopkins University, Baltimore, MD, USA 21287 (e-mail: rsubram4@jhmi.edu).

A. Rahmim is with the Departments of Radiology, and Electrical & Computer Engineering, Johns Hopkins University, Baltimore, MD, USA 21287 (telephone: 410-502-8579, e-mail: arahmim1@jhmi.edu, webpage: www.jhu.edu/rahmim).

matrices, drug removal by lymphatic system, drug diffusion and convective transport in extracellular matrices should be included by mathematical simulation. Computational fluid dynamics (CFD) can model the whole drug delivery process and clarify the mechanisms of drug delivery from the injection site to absorption by a solid tumor [4].

Baxter and Jain, based on the theoretical framework in their 1D mathematical method, found the effective factors on drug delivery such as microvessel permeability, interstitial fluid pressure (IFP), and interstitial fluid velocity (IFV) [5]–[8].

Wang et al. [9]–[11] developed a simulation framework of drug delivery to tumors by considering the complex 3D geometry. Wang and Li [9] used modified MRI images for tumor geometry. They considered interstitial fluid flow with blood and lymphatic drainage in their model. Wang et al. [10] studied the effect of elevated interstitial pressure, convective flux, and blood drainage on the delivery of specified solute to brain tumors.

The study of tissue transport property effect on drug delivery has been considered in a number of recent studies. Zhao et al. [12] used a 3D computational model to predict the distribution of IFV, IFP, and solute transport through a tumor. Arifin et al. [13], [14] studied the sensitivity of drug distribution to physiochemical properties in realistic models of brain tumors. A specific tumor captured by MRI was used by Pishko et al. [15]–[17] for modeling drug distribution in tissue with spatially-varying porosity and vascular permeability. The sensitivity of solute distribution to tumor shape and size was not considered in the above-mentioned works.

Developed models of fluid flow in tumors have also focused on incorporating spatial and temporal variations in blood flow using microvascular network models [18]–[22].

Soltani et al. [23] investigated the effect of tumor shape and size on drug delivery by modeling interstitial fluid flow and assuming that drug particles flow within the interstitial fluid. Sefidgar et al. [24] further developed the mathematical model by adding the solute transport equation to previously developed models [25]–[29]. New governing equations were subsequently investigated to find drug concentration in interstitial fluid (DCIF). Spherical and non-spherical tumors and their surrounding normal tissue were modeled assuming rigid porous media. We applied [30], [31] this numerical model to investigate drug distributions in clinically imaged pancreatic tumors. We also implemented [32]–[34] this mathematical method to multi scale models of tumor microvasculature for investigation of drug and tracer delivery distributions. Other parameters related to cancer fluid flow and solute transport were also investigated by our group [35], [36].

In this study, multiple tumor geometries as obtained from clinical PET/CT images of the lung were considered. An

advanced numerical method was used to simultaneously solve fluid flow and solute transport equations in order to investigate the effect of tumor shape and size on drug delivery to solid tumors. Data from n=20 lung cancer patients with non-resectable locoregional disease were analyzed, and geometrical information from the tumors including size were classified. To investigate effect of tumor size, tumors with similar shapes but different sizes, ranging from 1 to 28.6 cm3, were selected and analyzed. A hypothetical tumor similar to one of the analyzed tumors, but scaled to reduce its size to 0.2 cm3, was also analyzed. An ideal bolus injection was considered for the model. Solving fluid flow and solute transport equations simultaneously, the effects of lung tumor size on drug delivery to solid tumor are also investigated.

II. METHODS

A. Image Data Extraction PET/CT images from 20 patients with lung tumor were

obtained and processed to extract the geometrical configuration of tumors. Tumors volume range was from 2.8 to 28.6 cm3. The tumors used in this study are shown in Fig. 1.

Tumor #2 Volume=2.8cm3 Tumor #3 Volume=28.6cm3

Tumor #8 Volume=14.7cm3 Tumor #10 Volume=23.7cm3

Tumor #11 Volume=7.7cm3 Tumor #18 Volume=12.2cm3

Tumor #20 Volume=6.4cm3

Fig. 1 A systematic flow chart of computational technique

B. Mathematical Method: A systematic flow chart is illustrated in Fig. 2 to clarify the

computational techniques involved in this work. Next, the governing equation is introduced.

Fig. 2 A systematic flow chart of computational technique

1) Mathematical Model of Interstitial Flow Since tumor tissue has characteristics the same as porous

media, fluid flow behavior is defined by coupling the fluid flow governing equations. Combination of Darcy’s law and the mass balance’s law results in [37]–[39]:

2i B LPκ φ φ− ∇ = − (1)

where iP : Interstitial fluid pressure,

κ : hydraulic conductivity of the interstitium,

Bφ : The source term, extravasation from microvessels, and

Lφ : The drainage term, elimination by lymphatic system. In biological tissues, the fluid source is evaluated through Starling's law as follows[40]–[42]:

( )( )B BP

i s b iL S

P PV

φ σ π π= − − − (2)

where BP : Blood pressure in microvessel,

SV

: The surface area per unit volume of tissue for transport in

the interstitium, Bπ : Microvessel oncotic pressure,

iπ : Interstitial oncotic pressure,

pL : The hydraulic conductivity of the microvessel wall, and

sσ : Osmotic reflection coefficient. and the lymphatic system is related to the pressure

difference between the interstitium and the lymphatic vessels and is considered only for normal tissues [4], [43]:

( ) ( )0

PL Li L

L

L SP P Normal tissue

r VTumor tissue

φ⎧ −⎪= ⎨⎪⎩

(3)

where

Lφ : The volumetric flow rate into the lymphatic,

PL LL SV

: The lymphatic filtration coefficient, and

LP : The hydrostatic pressure of the lymphatic.

2) Solute Transport: The interstitial transport of drug is governed by the

convection diffusion equation; therefore, the general equation for the mass balance of solutes can be written as[44]–[46]:

( ) ( ) ( )eff i B LC D C v Ct

∂ = ∇⋅ ⋅∇ − ∇⋅ + Φ − Φ∂

(4)

where C : The solute concentration based on tissue volume, ΦB : The rate of solute transport per unit volume from microvessel into the interstitial space, ΦL : The rate of solute transport per unit volume from the interstitial space into lymphatic vessels, and Deff : The effective diffusion tensor. For an isotropic and uniform diffusion in tissues, equation (4) can be written as:

( ) ( )2eff f B L

C D C v Ct

∂ = ∇ −∇⋅ + Φ − Φ∂

(5)

The solute transport rate across the lymphatic vessels can be considered as [38], [47], [48]:

0L

L

C NormalTissueTumorTissue

φ⎧Φ = ⎨⎩

(6)

The solute transport rate across the microvessel can be represented by Patlak equation[18], [49]:

( ) ( )11B B f P P Pe

PS PeC C CV e

φ σΦ = − + −−

(7)

where ( )1B f V

PePS

φ σ−= (8)

Bφ : The fluid flow rate per unit volume of tissue across the microvessel wall, σf : The filtration reflection coefficient, P : The microvessel permeability coefficient, S/V : The microvessel surface area per unit volume of tissue, and Cp : Solute concentration in the plasma.

The detail of solution and value of parameter used in above equation are mentioned in our previous work[24], [50].

C. Model parameterization The interstitial transport properties for normal and tumor tissue are listed in Table 1. These values are used as baseline and some of them are investigated and changed in specified ranges for sensitivity analysis.

The parameters of solute transport model taken from Baxter and Jain [43] are listed in Table 2. Although, the numerical

model is applicable for any type of drug, in present study the properties of Fragment antigen-binding (F(ab’)2) as a sample is used.

TABLE 1. PARAMETER USED IN THIS SIMULATION [24]

Parameter Description value

LP[m/Pa s] Hydraulic conductivity of vessel 2.10×10-11

κ [m2/Pa s] Hydraulic conductivity of the interstitium 3.10×10-14

S/V[m-1] Surface area per unit volume of

tissue for transport in the interstitium

20000

PB[Pa] Intravascular blood pressure 2075 πB[Pa] Capillary oncotic pressure 2666 πi[Pa] Interstitial oncotic pressure 2000 σ Osmotic reflection coefficient 0.82

PL[Pa] Hydrostatic pressure of the lymphatic 0

LPL SL/V [1/Pa s]

Lymphatic filtration coefficient 1×10-7

σf Filtration reflection coefficient 0.9 Deff [m2/s] Effective diffusion tensor 2.0× 10-12

P[m/s] Microvessel permeability coefficient 17.3× 10-11

III. RESULTS DCIF is simulated in case of the bolus injection in which

the plasma concentration decreases with time exponentially ( 0 t

p pC C e τ= ), in which τ is the drug half-life in plasma

(6.1hr). DCIF are non-dimensionalized by 0pC .

The Non-dimensionalized DCIF distribution is shown in Fig. 3 at 7 hr post injection, for different sizes of tumor. The red color in Figure 2 shows the highest drug concentrations, which encompasses mostly the tumor boundary. Fig. 4 shows DCIF for tumor of a patient in which the size of the tumor was intentionally reduced to 0.2 cm3. Fig. 5 shows the temporal variation of average drug concentration in each tumor. There is no significant difference in drug update between tumors with different sizes.

One important metric of disease development and response to cancer via drug delivery is volume of tumors. Based on our previous research [2, 5] when the tumor volume is in the order of >1 cm3 (for all considered tumor in this study), the sensitivity of drug concentration to tumor size decreases. Only for a hypothetical tumor which has size less than 1 cm3, DCIF is high and convection rate has effect on drug delivery. In the analyzed lung tumors, since the interstitial pressure is high in the tumor region, the convection rate vanishes and the diffusion rate reaches a constant value, and consequently the sensitivity of drug concentration to tumor size reduces.

Patient 2,

Patient 3,

Patient 8,

Patient 10,

Patient 11,

Patient 18,

Patient 20,

Fig. 3 DCIF for different types of tumor at 7 hr post injection for a bolus well-perfused injection

Fig. 4 DCIF for the hypothetical tumor at 7 hr post injection for a bolus

well-perfused injection

Fig. 5 Average drug concentration for different sizes of tumors shown in

Fig. 1.

IV. DISCUSSION Drug transport in tissue is depended on two mechanisms,

diffusion and convection. These mechanisms affect drug transport from vessels to the extracellular matrix, and transport within the extracellular matrix. The convection mechanism is depended on interstitial velocity for transport in tissue and on differences between intravascular and interstitial pressures for transport from vessels. Our previous studies [24], [35] indicated these parameters (velocity and pressure) as depended on shape and size of tumor. The diffusion mechanism is only

dependent on the concentration gradient and therefore is independent of shape and size.

In large tumor (order of 10 cm3), the interstitial velocity in tumors goes to zero and pressure differences are minimal [25], [38]. Therefore, the effect of convection transport is reduced and sometimes negligible in large tumors.

In the considered lung tumors, because of the large sizes involved, drug delivery is only depended on diffusion mechanisms, and size of tumors does not affect drug delivery.

For the hypothetical tumor, because of small size, the effect of convection transport is higher than in lager tumors, and as such, drug delivery in increasingly small tumors is greater than in large tumors.

V. CONCLUSION In this study, we investigated drug delivery to different lung

tumor size. Since, in large tumors, the convection mechanism is negligible and diffusion mechanism independent of tumor size is dominant, and the investigated tumors in size were of the order of 10 cm3, drug concentrations are relatively similar. Only for smaller tumors with size lower than 1 cm3, the convection mechanism impacts drug delivery, and drug concentration is subsequently increased more than 20%.

ACKNOWLEDGMENT We gratefully acknowledge support by Siemens Medical

Solutions.

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