Effect of tumor size on drug delivery to lung tumors

Effect of tumor size on drug delivery to lung tumors M. Soltani, M. Sefidgar, H. azmara C. Marcus, R. M. Subramaniam, A. Rahmim Abstract Drug delivery to solid tumors can be exressed hysically using biomechanical henomena such as convection, diffusion of drug in extracellular matrices, and drug extravasation from microvessels. Alying comutational methods to solve governing conservation equations clarifies the mechanisms of drug delivery from the injection site to a solid tumor. In this study, multile tumor geometries were obtained from PET/CT images. An advanced numerical method was used to solve fluid flow and solute transort equations simultaneously to investigate the effect of tumor size on drug delivery to lung tumors. Data from 20 atients with lung tumors were analyzed and the tumor geometrical information including size, shae, and asect ratios were classified. In order to investigate effect of tumor size, tumors with similar shaes but different sizes ranging from 1 to 28.6 cm 3 were selected and analyzed. A hyothetical tumor, similar to one of the analyzed tumors but scaled to reduce its size to 0.2 cm 3, was also analyzed. An ideal bolus injection was considered for the model. The effects of two transort mechanisms, namely convection and diffusion, were considered in this study. The results show because of size of considered lung tumor, the diffusion transort rate is higher than convection transort rate. ased on governing equations, the diffusion transort is only deended on concentration gradient and indeendent of size of tumor, therefore the redicted concentration rofile for considered tumors are similar. When size of tumor is decreased significantly, the drug concentration is also significantly increased. I. INTRODUCTION ung cancer is the most common cause of cancer-related death in men and women, and was resonsible for 1.56 million deaths annually, as of 2012 [1]. Chemotheray is one of the ways widely used for cancer treatment. ased on the findings from clinical alications, most cancer treatments with drugs fail to eliminate solid tumors comletely [2]. The comutational method can investigate why systemic administration cannot distribute drug uniformly in tumors[3]. The drug exchange between microvessels and extracellular Manuscrit received December 05, 2015. M. Soltani is with Division of Nuclear Medicine, Deartment of Radiology and Radiological Science, School of Medicine, Johns Hokins University, MD 21287, USA (e-mail: ma_soltani@yahoo.com), and Mechanical Engineering, KNT University, Tehran, Iran. M. Sefidgar is with Deartment of Mechanical Engineering, Pardis ranch Islamic Azad University, Pardis new city, Iran (e-mail: sefidgar@gmail.com). H. azmara is with Deartment of Mechanical Engineering, Pardis ranch Islamic Azad University, Pardis new city, Iran (e-mail: bazmara@gmail.com). C. Marcus is with the Deartment of Radiology, Johns Hokins University, altimore, MD, USA 21287 (e-mail: cmarcus7@jhmi.edu). M. Subramaniam is with the Deartments of Radiology, Oncology, Otolaryngology, Head and Neck Surgery, and Health Policy and Management, Johns Hokins University, altimore, MD, USA 21287 (e-mail: rsubram4@jhmi.edu). A. Rahmim is with the Deartments of Radiology, and Electrical & Comuter Engineering, Johns Hokins University, altimore, MD, USA 21287 (telehone: 410-502-8579, e-mail: arahmim1@jhmi.edu, webage: www.jhu.edu/rahmim). matrices, drug removal by lymhatic system, drug diffusion and convective transort in extracellular matrices should be included by mathematical simulation. Comutational fluid dynamics (CFD) can model the whole drug delivery rocess and clarify the mechanisms of drug delivery from the injection site to absortion by a solid tumor [4]. axter and Jain, based on the theoretical framework in their 1D mathematical method, found the effective factors on drug delivery such as microvessel ermeability, interstitial fluid ressure (IFP), and interstitial fluid velocity (IF) [5] [8]. Wang et al. [9] [11] develoed a simulation framework of drug delivery to tumors by considering the comlex 3D geometry. Wang and i [9] used modified MRI images for tumor geometry. They considered interstitial fluid flow with blood and lymhatic drainage in their model. Wang et al. [10] studied the effect of elevated interstitial ressure, convective flux, and blood drainage on the delivery of secified solute to brain tumors. The study of tissue transort roerty effect on drug delivery has been considered in a number of recent studies. Zhao et al. [12] used a 3D comutational model to redict the distribution of IF, IFP, and solute transort through a tumor. Arifin et al. [13], [14] studied the sensitivity of drug distribution to hysiochemical roerties in realistic models of brain tumors. A secific tumor catured by MRI was used by Pishko et al. [15] [17] for modeling drug distribution in tissue with satially-varying orosity and vascular ermeability. The sensitivity of solute distribution to tumor shae and size was not considered in the above-mentioned works. Develoed models of fluid flow in tumors have also focused on incororating satial and temoral variations in blood flow using microvascular network models [18] [22]. Soltani et al. [23] investigated the effect of tumor shae and size on drug delivery by modeling interstitial fluid flow and assuming that drug articles flow within the interstitial fluid. Sefidgar et al. [24] further develoed the mathematical model by adding the solute transort equation to reviously develoed models [25] [29]. New governing equations were subsequently investigated to find drug concentration in interstitial fluid (DCIF). Sherical and non-sherical tumors and their surrounding normal tissue were modeled assuming rigid orous media. We alied [30], [31] this numerical model to investigate drug distributions in clinically imaged ancreatic tumors. We also imlemented [32] [34] this mathematical method to multi scale models of tumor microvasculature for investigation of drug and tracer delivery distributions. Other arameters related to cancer fluid flow and solute transort were also investigated by our grou [35], [36]. In this study, multile 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 transort equations in order to investigate the effect of tumor shae and size on drug delivery to solid tumors. Data from n=20 lung cancer atients with nonresectable locoregional disease were analyzed, and geometrical information from the tumors including size were classified. To investigate effect of tumor size, tumors with similar shaes but different sizes, ranging from 1 to 28.6 cm 3, were selected and analyzed. A hyothetical tumor similar to one of the analyzed tumors, but scaled to reduce its size to 0.2 cm 3, was also analyzed. An ideal bolus injection was considered for the model. Solving fluid flow and solute transort 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 atients with lung tumor were obtained and rocessed to extract the geometrical configuration of tumors. Tumors volume range was from 2.8 to 28.6 cm 3. The tumors used in this study are shown in Fig. 1. Fig. 1 A systematic flow chart of comutational technique. Mathematical Method: A systematic flow chart is illustrated in Fig. 2 to clarify the comutational techniques involved in this work. Next, the governing equation is introduced. Fig. 2 A systematic flow chart of comutational technique Tumor #2 olume=2.8cm 3 Tumor #3 olume=28.6cm 3 Tumor #8 olume=14.7cm 3 Tumor #10 olume=23.7cm 3 Tumor #11 olume=7.7cm 3 Tumor #18 olume=12.2cm 3 Tumor #20 olume=6.4cm 3 1) Mathematical Model of Interstitial Flow Since tumor tissue has characteristics the same as orous media, fluid flow behavior is defined by couling the fluid flow governing equations. Combination of Darcy s law and the mass balance s law results in [37] [39]: 2 κ P i = φ φ (1) P i : Interstitial fluid ressure, κ : hydraulic conductivity of the interstitium, φ : The source term, extravasation from microvessels, and φ : The drainage term, elimination by lymhatic system. In biological tissues, the fluid source is evaluated through Starling's law as follows[40] [42]: S P φ = ( P Pi σ s ( π b π i )) (2) P : lood ressure in microvessel, S : The surface area er unit volume of tissue for transort in the interstitium, π : Microvessel oncotic ressure, π : Interstitial oncotic ressure, i : The hydraulic conductivity of the microvessel wall, and s σ : Osmotic reflection coefficient. and the lymhatic system is related to the ressure difference between the interstitium and the lymhatic vessels and is considered only for normal tissues [4], [43]:

PS ( Pi P) Normaltissue φ ( r ) = (3) 0 Tumor tissue φ : The volumetric flow rate into the lymhatic, PS : The lymhatic filtration coefficient, and P : The hydrostatic ressure of the lymhatic. 2) Solute Transort: The interstitial transort 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]: C = D C vc + Φ Φ (4) ( eff ) ( i ) ( ) t C : The solute concentration based on tissue volume, Φ : The rate of solute transort er unit volume from microvessel into the interstitial sace, Φ : The rate of solute transort er unit volume from the interstitial sace into lymhatic vessels, and D eff : The effective diffusion tensor. For an isotroic and uniform diffusion in tissues, equation (4) can be written as: C 2 = D eff C ( v f C ) + ( Φ Φ ) (5) t The solute transort rate across the lymhatic vessels can be considered as [38], [47], [48]: φ C Normal Tissue Φ = (6) 0 Tumor Tissue The solute transort rate across the microvessel can be reresented by Patlak equation[18], [49]: PS Pe Φ = φ( 1 σ f ) CP + ( CP C) (7) Pe e 1 ( 1 σ ) φ f Pe = (8) PS φ : The fluid flow rate er unit volume of tissue across the microvessel wall, σ f : The filtration reflection coefficient, P : The microvessel ermeability coefficient, S/ : The microvessel surface area er unit volume of tissue, and C : Solute concentration in the lasma. The detail of solution and value of arameter used in above equation are mentioned in our revious work[24], [50]. C. Model arameterization The interstitial transort roerties 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 secified ranges for sensitivity analysis. The arameters of solute transort model taken from axter and Jain [43] are listed in Table 2. Although, the numerical model is alicable for any tye of drug, in resent study the roerties of Fragment antigen-binding (F(ab ) 2 ) as a samle is used. TAE 1. PARAMETER USED IN THIS SIMUATION [24] Parameter Descrition value P [m/pa s] Hydraulic conductivity of vessel 2.10 10-11 κ [m 2 Hydraulic conductivity of the /Pa s] interstitium 3.10 10-14 S/[m -1 ] Surface area er unit volume of tissue for transort in the 20000 interstitium P [Pa] Intravascular blood ressure 2075 π [Pa] Caillary oncotic ressure 2666 π i [Pa] Interstitial oncotic ressure 2000 σ Osmotic reflection coefficient 0.82 P [Pa] Hydrostatic ressure of the lymhatic 0 P S / ymhatic filtration coefficient [1/Pa s] 1 10-7 σ f Filtration reflection coefficient 0.9 D eff [m 2 /s] Effective diffusion tensor 2.0 10-12 P[m/s] Microvessel ermeability coefficient 17.3 10-11 III. RESUTS DCIF is simulated in case of the bolus injection in which the lasma concentration decreases with time exonentially 0 t ( C = C e τ ), in which τ is the drug half-life in lasma 0 (6.1hr). DCIF are non-dimensionalized by C. The Non-dimensionalized DCIF distribution is shown in Fig. 3 at 7 hr ost injection, for different sizes of tumor. The red color in Figure 2 shows the highest drug concentrations, which encomasses mostly the tumor boundary. Fig. 4 shows DCIF for tumor of a atient in which the size of the tumor was intentionally reduced to 0.2 cm 3. Fig. 5 shows the temoral variation of average drug concentration in each tumor. There is no significant difference in drug udate between tumors with different sizes. One imortant metric of disease develoment and resonse to cancer via drug delivery is volume of tumors. ased on our revious research [2, 5] when the tumor volume is in the order of >1 cm 3 (for all considered tumor in this study), the sensitivity of drug concentration to tumor size decreases. Only for a hyothetical tumor which has size less than 1 cm 3, DCIF is high and convection rate has effect on drug delivery. In the analyzed lung tumors, since the interstitial ressure 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 20, Patient 3, Fig. 3 DCIF for different tyes of tumor at 7 hr ost injection for a bolus well-erfused injection Patient 8, Fig. 4 DCIF for the hyothetical tumor at 7 hr ost injection for a bolus well-erfused injection Patient 10, Patient 11, Fig. 5 Average drug concentration for different sizes of tumors shown in Fig. 1. Patient 18, I. DISCUSSION Drug transort in tissue is deended on two mechanisms, diffusion and convection. These mechanisms affect drug transort from vessels to the extracellular matrix, and transort within the extracellular matrix. The convection mechanism is deended on interstitial velocity for transort in tissue and on differences between intravascular and interstitial ressures for transort from vessels. Our revious studies [24], [35] indicated these arameters (velocity and ressure) as deended on shae and size of tumor. The diffusion mechanism is only

deendent on the concentration gradient and therefore is indeendent of shae and size. In large tumor (order of 10 cm 3 ), the interstitial velocity in tumors goes to zero and ressure differences are minimal [25], [38]. Therefore, the effect of convection transort is reduced and sometimes negligible in large tumors. In the considered lung tumors, because of the large sizes involved, drug delivery is only deended on diffusion mechanisms, and size of tumors does not affect drug delivery. For the hyothetical tumor, because of small size, the effect of convection transort is higher than in lager tumors, and as such, drug delivery in increasingly small tumors is greater than in large tumors.. CONCUSION In this study, we investigated drug delivery to different lung tumor size. 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