Spatiotemporal distribution modeling of PET tracer uptake in solid tumors

Transcription

1 Ann Nucl Med (2017) 31: DOI /s ORIGINAL ARTICLE Spatiotemporal distribution modeling of PET tracer uptake in solid tumors Madjid Soltani 1,2 Mostafa Sefidgar 3 Hossein Bazmara 3 Michael E. Casey 4 Rathan M. Subramaniam 5 Richard L. Wahl 6 Arman Rahmim 1,7 Received: 6 January 2016 / Accepted: 18 October 2016 / Published online: 5 December 2016 The Japanese Society of Nuclear Medicine 2016 Abstract Objective Distribution of PET tracer uptake is elaborately modeled via a general equation used for solute transport modeling. This model can be used to incorporate various transport parameters of a solid tumor such as hydraulic conductivity of the microvessel wall, transvascular permeability as well as interstitial space parameters. This is especially significant because tracer delivery and drug delivery to solid tumors are determined by similar underlying tumor transport phenomena, and quantifying the former can enable enhanced prediction of the latter. Methods We focused on the commonly utilized FDG PET tracer. First, based on a mathematical model of angiogenesis, the capillary network of a solid tumor and normal tissues around it were generated. The coupling mathematical method, which simultaneously solves for blood flow in the capillary network as well as fluid flow in the interstitium, is used to calculate pressure and velocity distributions. Subsequently, a comprehensive spatiotemporal distribution model (SDM) is applied to accurately model distribution of PET tracer uptake, specifically FDG in this work, within solid tumors. Results The different transport mechanisms, namely convention and diffusion from vessel to tissue and in tissue, are elaborately calculated across the domain of interest and effect of each parameter on tracer distribution is investigated. The results show the convection terms to have negligible effect on tracer transport and the SDM can be solved after eliminating these terms. Conclusion The proposed framework of spatiotemporal modeling for PET tracers can be utilized to comprehensively assess the impact of various parameters on the spatiotemporal distribution of PET tracers. & Madjid Soltani msoltani@jhu.edu & Arman Rahmim arahmim1@jhmi.edu Mostafa Sefidgar sefidgar@gmail.com Hossein Bazmara bazmara@gmail.com Michael E. Casey michael.e.casey@siemens.com Rathan M. Subramaniam Rathan.Subramaniam@UTSouthwestern.edu Richard L. Wahl wahlr@mir.wustl.edu Department of Mechanical Engineering, KNT University, Tehran, Iran Department of Mechanical Engineering, Pardis Branch, Islamic Azad University, Pardis, Iran Siemens Medical Solutions, Knoxville, TN 37932, USA Department of Radiology, University of Texas Southwestern Medical Center, Dallas, TX, USA Department of Radiology, Washington University, St. Louis, MO 63110, USA Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA 1 Division of Nuclear Medicine, Department of Radiology, School of Medicine, Johns Hopkins University, Baltimore, MD 21287, USA

2 110 Ann Nucl Med (2017) 31: Keywords Positron emission tomography Solid tumors Kinetic modeling Spatiotemporal modeling Convection diffusion reaction equations Introduction Molecular imaging methods especially utilizing positron emission tomography (PET) have found increasing applications in different cancers and for different tasks, namely diagnosis, initial staging, restaging, prediction and monitoring of treatment response, surveillance and prognostication [1]. Routine clinical applications include qualitative interpretation of PET studies. At the same time, PET imaging also enables quantitative measurements of radioactivity concentrations over time across the body and in the tumor(s) of interest. The most commonly utilized techniques include (1) static (single time-frame) imaging invoking the standardized uptake value (SUV), and (2) dynamic imaging, invoking simplified tracer kinetic modeling using Patlak modeling [2 7]. Critical knowledge of the tumor environment can be obtained by linking the tissue time activity curve (TAC) to the underlying tumor physiology, which in turn can help determine the most sensible course of action. Conventional compartment models [8, 9] have the limitation that they do not model transport of tracer between compartments at different physical locations, e.g., through diffusion or convection. Furthermore, it is difficult to use these models to investigate the consequences of perturbing the underlying physiology, since actual physical quantities, such as permeability and the local vascular supply, can be hidden within compound parameters. The spatiotemporal distribution model (SDM) which includes convection, diffusion and reaction equation is widely used to simulate drug delivery [10]. Baxter and Jain, based on the theoretical framework in their 1D mathematical method, found the effective factors on drug delivery [11, 12]. Magdoom et al. [13] used a simplified model of SDM for predicting albumin tracer distribution in the lower limb of a mouse. They used a dynamic contrast enhanced (DCE)-MRI based computational model [14, 15] to estimate the spatial variation of transport properties in intestinal space. Stylianopoulos and Jain [16] used the SDM to investigate effects of vascular normalization for improving perfusion and drug delivery in solid tumors. They extended their work [17] by considering the effect of particle size, drug release rate and binding affinity on the distribution and efficacy of nanoparticles to derive optimal design rules. The ability to predict sensitivity of a given tumor to specific therapeutic agents is the holy grail in personalized cancer medicine. We note that tracer delivery and drug delivery to solid tumors, the former used in diagnostic imaging and the latter in therapy, are determined by similar underlying tumor transport phenomena. To this end, we have utilized the spatiotemporal distribution model that we have previously used for assessment of drug delivery [10, 18 20], to comprehensively model the spatiotemporal distribution of PET tracers in solid tumors, focusing in the present work on the tracer FDG. This approach is based on the use of partial differential equations (PDEs), in contrast to ordinary differential equations (ODEs) as commonly utilized for tracer kinetic modeling (compartment model), and enables the assessment of tracer distribution in both time and space. This in turn allows quantification of the impact of various parameters, including tumor angiogenic factors, microvessel and interstitial pressures, and hydraulic conductivity and permeability, amongst others, on the distributions. Such comprehensive modeling may then be utilized in inverse methods, including the use of dynamic FDG imaging, potentially aided by other modalities such as CT or MRI, to enable enhanced prediction of drug delivery to solid tumors. There are a few studies in the area of modeling PET tracer spatiotemporal distribution in biological tissues. Kelly et al. [21] used a simplified form of the spatiotemporal distribution model for investigation of FMISO. Monnich et al. [22, 23] modified Kelly s approach to investigate the influence of acute hypoxia on FMISO retention and the potential to distinguish between retention from chronic and acute hypoxia in serial (or single dynamic) clinical PET scans. In these studies, the spatial scale is in the order of a few millimeters. Furthermore, the applications do not include convection transport from vessel to tissue or within the tissue. Alessio et al. [24] used an axially distributed model for 13 N-ammonia PET. This approach does not include within-tissue convection, and included 1-dimensional diffusion for myocardial perfusion imaging, which is not appropriate for solid tumor imaging. We also note that in these studies, the dynamic (spatially variable) structure of the microvascular networks is not incorporated to calculate the pressure/flow distributions across the networks. The present study uses a general, comprehensive framework, wherein, compared to compartmental modeling, distributions in space are not modeled as independent from one another, and partial differential equations, involving both time and space, are invoked, to collectively model solute distribution over time and space. It involves spatially correlated modeling incorporation both diffusion and convection phenomena models, while considering intravascular flow in a dynamic capillary network coupled with interstitial flow. The proposed framework is applicable to different tracers, and tumors of varying sizes, and for

3 Ann Nucl Med (2017) 31: images spanning short, intermediate or long durations of time post-injection. We elaborate upon the methodology in the next section, followed by presentation of our results and various analyses to assess the performance of our framework. Materials and methods Compartmental modeling of FDG Compartmental models are widely used to describe PET tracer data, as they enable quantification of biochemical and physiological phenomena [25]. While further simplifications to a 3 K (two-tissue) model are commonplace, spectral analysis reveals that a 5 K (three-tissue) model is resolvable from the [ 18 F]FDG data and is a more accurate model to describe [ 18 F]FDG kinetics [26]. The 5 K model is shown in Fig. 1 and can be viewed as an extension of the classic model by Sokoloff et al. [8] (Fig. 2), originally proposed. The value of the 5 K model lies in its explicit accounting of an extracellular compartment; i.e., it assumes that it is possible to distinguish the kinetics steps of delivery of [ 18 F]FDG to the extracellular space, its transport from the extracellular to the intracellular space, and its intracellular phosphorylation. The 5 K model is described by the following ordinary differential equations: dc i dt ¼ K 1 C p ðk 2 þ k 3 ÞC i þ k 4 C e ; dc e dt dc m dt ¼ k 3 C i ðk 4 þ k 5 ÞC e ; ¼ k 5 C e ; ð1þ where C p is [ 18 F]FDG plasma arterial concentration, C i is extracellular concentration of [ 18 F]FDG normalized to extracellular volume, C e is [ 18 F]FDG intracellular concentration, C m is [ 18 F]FDG 6-phosphate ([ 18 F]FDG-6-P) intracellular concentration, K 1 and k 2 are exchange rate parameters between plasma and extracellular space, k 3 and k 4 are transport rate parameters into and out of the cell, and k 5 is the phosphorylation rate. Spatiotemporal distribution model In the spatiotemporal distribution modeling compared to compartmental modeling, distributions in space are not modeled as independent from one another, and partial differential equations, involving both time and space, are invoked, to collectively model solute distribution over time and space. In SDM, it is assumed that the tracer concentration in the second compartment is spatially correlated via diffusion and convection phenomena. Figure 3 describes how spatiotemporal distribution is added to the compartmental model. The general model for solute transport in tissue includes transport in interstitium by diffusion and convection in tissue, transport rate across vessels related to diffusion and convection process and other mechanisms such as reaction, binding to cells. The detail and derivations of equations in the spatiotemporal distribution model (SDM) are provided in our previous work [10, 11, 16, 18, 20]. This model is also referred to as the convection diffusion reaction (CDR) equation. In the present work, we only modified the CDR equation based on the process of [ 18 F]FDG transport. The general form of the SDM in tissue is as follows: oc i ¼ D eff r 2 C i v i rðc i ÞþU V U L L 3 C i þ L 4 C e ot ð2aþ oc e ¼ L 3 C i L 4 C e L 5 C e ð2bþ ot oc m ¼ L 5 C e ; ð2cþ ot where D eff is the effective spatially invariant diffusion coefficient, v i is the interstitial fluid velocity (IFV) vector, U V and U L are the rates of solute transport per unit volume from blood vessels into the interstitial space, and from the interstitial space into lymph vessels, respectively. The first term in the right hand side of Eq. (2a) determines diffusion transport of tracer in tissue. The second term models convection transport of tracer in tissue. Solute transport across the vessel walls and through the interstitium takes place due to both diffusion and convection. Concentration gradients result in diffusive transport. Fig. 1 Four-compartment 5-rate-constant (5 K) model for measuring the metabolic rate of glucose with [ 18 F]fluorodeoxyglucose ([ 18 F]FDG)

4 112 Ann Nucl Med (2017) 31: Fig. 2 Three-compartment model for measuring the metabolic rate of glucose with [ 18 F] FDG Fig. 3 Relationship between the compartmental model and SDM. The SDM is same as three-tissue compartmental model with the difference that tracer distribution is additionally spatially coupled Movement of fluid molecules caused by pressure gradients results in convective transport of solute molecules. A complete description of solute transport based on the pore model of membrane is given by the so-called Kedem Katchalsky equation [27, 28]: U V ¼ J S V S ¼ P m V C Pe ð P C i Þ e Pe 1 S þ L P ½ð V P b P i Þ r s ðp b p i Þ ð1 r f ÞC P ; ð3þ where the various parameters are defined in Table 1. The solute transport rate across the lymphatic vessels can be considered as [15]: U L ¼ / L C; ð4þ where / L, the drainage term, i.e. elimination via the lymphatic system, is calculated by: / L ¼ F LðP i P L Þ Normal Tissue ; ð5þ 0 Tumor Tissue where F L is the lymphatic filtration coefficient, and P L is the hydrostatic pressure of the lymphatic system. These also appear in Table 1. In fluid flow simulations, three fronts are considerable: blood flow in vessels, fluid flow in lymphatic systems and fluid flow in tissue. In the CDR equation, fluid flow in tissue is considered. The simulation of fluid flow in tissue is same as simulation of fluid flow in porous media. One of the earliest formulations for flow transport in porous media is Darcy s law which is appropriate for flow in tissue. In this paper, the IFV is calculated by Darcy s equation, as follows: v i ¼ jrp i ; ð6þ while the interstitial fluid pressure is calculated by a procedure defined in our previous work [35 37]. In an effort to arrive at a combined meaning for these various terms, re-arranging various terms in Eqs. (2a 2c), (3) and (4), the expressions for rate constants L 1 and L 2 as shown in Fig. 3 are given by: L 1 ¼ L PS V ð P b P i r s ðp b p i ÞÞð1 r f L 2 ¼ P ms Pe V e Pe þ / 1 L ; Þþ P ms V Pe e Pe ; 1 ð7þ ð8þ K 1 k 5 and L 1 L 5 rate constants in Figs. 1 and 3 have the same meaning and same values, but the SDM model additionally incorporates diffusion and convection phenomena, and explains contributions of various transport parameters to the rate constants, e.g. as seen in Eqs. (7) and (8). The numerical values of the various terms implemented for evaluating in our simulations are shown in Tables 1 and 2. Computational domain To simplify the calculations, a 2D domain (shown in Fig. 4) is considered for computational simulation as a 6 9 3cm 2 rectangle, that is appropriate to represent one tumor and the surrounding normal tissues. 3D modeling

5 Ann Nucl Med (2017) 31: Table 1 Summary of parameters related to the SDM model Parameter Description Value References r f Filtration reflection coefficient 0.9 [10] S/V (m -1 ) Surface area per unit volume of tissue for transport in the 7000 (normal) [10] interstitium (tumor) P i Interstitial fluid pressure Distribution computed according to [18, 37] v i Interstitial fluid velocity vector Distribution computed according to [18, 37] P b Intravascular blood pressure Distribution computed according to [18, 37] p b (Pa) Capillary oncotic pressure 2670 (normal) 2670 (tumor) [10] p i (Pa) Interstitial oncotic pressure 1330 (normal) 2000 (tumor) L p (m/pa s) Hydraulic conductivity (normal) (tumor) r s Osmotic reflection coefficient 0.91 (normal) 0.82 (tumor) Pe ¼ LPðPb Pi rsðpb piþþð1 rf Þ P m Peclet number Distribution computed F L [1/Pa s] Lymphatic filtration coefficient (normal) [10] D eff (m 2 /s) Effective diffusion coefficient (n) [66, 67] (t) P m (m/s) Microvessel permeability coefficient (n) [21] (t) P L (Pa) Hydrostatic pressure of lymphatic vessels 0 (Normal) [10] [10] [10] [10] Table 2 Summary of parameters related to the kinetic model Parameter Value References U L (1/s) 4.3e-4 [26] k 2 (1/s) 54e-4 [26] L 3 or k 3 (1/s) 8.2e-4 [26] L 4 or k 4 (1/s) 6.7e-4 [26] L 5 or k 5 (1/s) 5.3e-4 [26] complicates the numerical simulation, though changing from 2D model to 3D model can be performed in a relatively straightforward manner. A 2D region is also used in other works such as Anderson and co-workers [29, 30], Jain and co-workers [16, 17, 31]. The tumor is located at the center of the domain with the radius of 1.2 cm. The tumor size is chosen around the size by which tumor initiates the angiogenesis process. This size is implemented in other works as well [32 34]. The two parent vessels from which the vascular network grows are located at the left and right edges of the domain. Numerical simulation procedure An advanced mathematical model is used to calculate interstitial velocity and pressure and intravascular pressure, Fig. 4 A schematic of the solution domain some of the parameters in Eq. (3) are related to interstitial fluid flow and blood flow in capillaries. The general algorithm of the method is shown in Fig. 5. For capillary network, an elaborative model is incorporated based on our previous work [35 37] to generate a microvascular network induced by tumor angiogenesis. Here we elaborate this framework, though the reader may skip this part, without loss of flow in the rest of the presentation. This mathematical model captures the capillary formation by tracking the motion of endothelial cells in capillary sprout tips. On both sides of the domain, parent vessels are considered to be sources of new capillaries. Capillaries start to migrate within the domain and reach the tumor. The details of rules for sprouting angiogenesis and algorithms for this method are outlined in our previous

6 114 Ann Nucl Med (2017) 31: Checking the maximum relative error of IFP, IBP and D with a criteria solution which is considered to be If the maximum relative error is lower than the threshold (10-6 ), the solution procedure of IBP and IFV is stopped and these values are used to calculate CDR. Otherwise, the procedure is repeated until finding new values of IBP and IFP. Fig. 5 The general algorithm used in this study work [38], and by Anderson and co-workers [39, 40], and some results are shown in section III. Calculation of fluid flow (blood flow) in capillary vessels includes a set of non-linear equations. For this reason, an iterative method is applied to solve the fluid flow and remodeling equations in a capillary network. The procedure is as follows: 1. Considering initial guesses for interstitial fluid pressure (IFP), intravascular blood pressure (IBP), blood viscosity (l), hematocrit (H), and capillary diameter (D). 2. Calculating IFP and IBP iteratively based on fluid flow equations mentioned in [18, 37]. 3. Finding H and updating l at each vessel with approach introduced by Secomb and Pries [41]. 4. Calculating hemodynamic and metabolic stimuli related to capillary network remodeling. Detail is described in [18, 42]. 5. Updating vessel diameter D N based on new value calculated in step Calculating the relative error for IFP, IBP and D by maxðx N X 0 Þ=X 0, where X can be each of IFP, IBP and D. The CDR equation (Eq. 2a 2c) is solved by an element based finite volume method (EB-FVM). The EB-FVM has the capability of the finite element method (FEM) in handling complex geometries and also the sound physicalbased properties of the finite volume method (FVM) [43]. The discretized form of the governing equations, in their general form, is then linearized and solved implicitly. Finally, the converged form of the solution is calculated using an iterative method. The criterion for the convergence is to reduce the residual by 4 orders of magnitudes. The Darcy equation and CDR equation are modeled by triangular elements. To check the grid-independency of the code, the results for three different grids were compared, indicating the conservative property of the numerical method. Final choice of the grid included triangular elements (Fig. 6), with an average area of 0.13 mm 2. The CPU configuration was Intel(R) Core(TM) i7-2670qm 2.20 GHz and Memory is 8 GB. The details and mathematics of the framework are explained in our previous studies [10, 35 37, 44, 45]. Boundary condition In numerical simulation, three sets of boundary condition are used. These sets are shown in Fig. 7. To simulate blood flow through capillary network, the inlet and outlet intravascular pressures of parent vessels shown in Fig. 7 (left top) are defined as boundary conditions. Inlet and outlet pressures are chosen in the hope of guaranteeing that the average intravascular pressure is around mmhg, based on the physiological values reported in the literature [46]. These values are also used in other works such as Wu et al. [47 49], Stephanou et al. [40] and Stylianopoulos and Jain [16, 33]. Parent vessel 1! P b;inlet ¼ 25 mmhg P b;outlet ¼ 5 mmhg! P b;inlet ¼ 35 mmhg : P b;outlet ¼ 10 mmhg Parent vessel 2 ð9þ For interstitial flow, Fig. 7 (top right), the continuity of pressure and velocity is applied for the boundary between tumor and normal tissues. For 2D domain boundaries two

7 Ann Nucl Med (2017) 31: Fig Triangular elements used in simulation of interstitial flow and tracer distribution (SDM) Fig. 7 Boundary condition used in numerical simulation, top left figure is boundary condition used in intravascular flow modeling, top right figure is boundary condition used for interstitial flow modeling, and bottom figure is boundary implemented to SDM types of boundary conditions are implemented. For top and bottom boundaries, the periodic boundary condition is applied and the Dirichlet type of boundary condition (the interstitial pressure is zero) is used for lateral boundaries [45]. In the modeling of tracer distribution in tissue with capillary network (Fig. 7, bottom), Eq. (3) is implemented wherever the capillary exists and for other places these terms are set to zero. For the boundary between tumor and normal tissues, the continuity of concentration and its flux are considered as boundary conditions: Dteff rci þ vi Ci Xt ¼ Dteff rci þ vi Ci Xn ; Ci jxt ¼ Ci jxn ; ð10þ where Xt and Xn indicate the tumor and normal tissues at their interface. The open boundary condition is used in 2D domain boundaries for the CDR Equation [50]. The open boundary condition is used to set up mass transport across boundaries where both convective inflow and outflow can occur and is defined by Eq. (11):

8 116 Ann Nucl Med (2017) 31: Fig. 8 Intravascular pressure distribution in domain for the two considered networks Fig. 9 Interstitial pressure distribution in domain for the two considered networks Fig. 10 Interstitial fluid velocity distribution in both considered networks n rc ¼ 0; ð11þ where n is the normal vector. Since the SDM deals both the intracellular and the extracellular components in the same pixel, the boundary condition is not used. Results Two different networks generated by the discrete angiogenesis method are used in the calculations. One of the networks is produced by 3 sprouts as initial condition for capillary network generation with low density vessel. The other network is initialized by 10 sprouts in two parent vessels for producing a particular capillary network in tumor domain similar to experimental observations [51]. These types of networks are only generated to investigate the effects of transport parameters related to vessel distribution. Then fluid flow in capillary and tissue are simulated to compute convection terms in Eq. (2a). The intravascular pressure distributions in the network and interstitial pressure in tissues are shown in Figs. 8 and 9, respectively. The interstitial pressure has its greatest values in the tumor region, since in this region there is no lymphatic system, and blood vessels are highly leaky. The maximum interstitial pressure for both networks is around 2000 Pa as we see in the work of Boucher et al. [52] and Hopland et al. [53]. The interstitial fluid velocity is shown in Fig. 10. As shown in the result, the velocity has very low values in most spaces. The effect of these values on the SDM is investigated in the following section.

9 Ann Nucl Med (2017) 31: Fig. 11 Comparison of SDM results with experimental results. The average concentration of FDG in tumor region for both networks is compared with experimental observations of Backes et al. [54]. C p is the plasma activity curve used in experimental data Model validation Figure 11 shows the comparison of average concentration of FDG in tumor region with experimental observation of Backes et al. [54]. Total uptake including both extracellular and intracellular contributions is computed, as measured in radionuclide imaging. Since the domain and condition of experimental and modeling are different, the results do not exactly match. However, the total uptake of FDG in tumor for numerical simulation for both networks display roughly similar trend to experimental data. We emphasize that this comparison with experimental data is for demonstration purposes, and that future work (see Discussion ) consists of focusing on the inverse problem, namely to estimate parameters that produce fits to experimental data. For example, we implemented our model in FMISO and estimated diffusion coefficient for FMISO in tissue [55]. At the same time, in the next part, we perform further analysis including comparisons with conventional ODE-based analytical methods, with similar modeled parameters. Tracer distribution In this section, the tracer distribution of FDG is investigated. A plasma activity curve obtained from Backes et al. [54] is used in this part of the analysis. The spatiotemporal concentration distributions of FDG at different post-injection times for networks 1 and 2 are shown in Figs. 12 and 13, respectively. The second network shows more uniform distribution. This is due to a higher number of vessels (high vessel density) in the network, which acts as source terms for the tracer. To better determine the effects of SDM terms, the FDG transport is simulated by elimination of each terms and keeping the other terms. The diffusion from vessel wall, diffusion in tissue, convection in tissue and convection from vessel wall are key parts which are considered in this study. The convection terms are related to interstitial flow and diffusion terms are related to concentration gradient. The results related to the first and second networks are shown in Figs. 14 and 15, respectively. As shown in Figs. 14 and 15, the convection terms related to interstitial flow are negligible. In our previous work, we showed that these terms only affect on the special shape tumors and small tumors. This effect is also studied for real tumors (pancreatic and lung tumors) [56, 57]. In the compartment model, the diffusion terms from vessels are considered but the diffusion term in tissue is assumed instantaneous. The contribution of diffusion terms has significant effect when the density of capillary is low (network 1). With increasing capillary density, the effect of terms related to spatiotemporal transport decreases. We also considered the extreme case where the capillary network is uniformly distributed in the whole domain. This model is referred as the without network model. The effects of different terms are shown in Fig. 16. Results show that in this case the FDG transport in only dependent on diffusion from vessels. The base of this approach is close to kinetic model because in both models, the solutes transport from vessel to tissue has main effect on solute distribution. Figure 17 depicts the average FDG uptake across the tumor tissue over time. The results include those from the proposed comprehensive SDM-based approach. Furthermore, the results are compared with the compartmental method (standard kinetic modeling) using ODEs (Eq. (1)), though the latter lacks sophistication of simultaneous spatiotemporal modeling. For a trend comparison with the ODE-based analytical method used in conventional kinetic modeling wherein the K s of Eq. (1) are average values as reported in [26], the SDM results were also averaged in space to eliminate dependency of tracer values on position thus obviously lacking the details shown in Figs. 14 and 15. All concentrations (C e, C i, C m, C total ) are compared for the 2 considered networks, the compartment model and the evenly distributed network. As expected, results of the compartment model are very close to the results of the evenly distributed network. This is because the effect of transport in tissue is strongly reduced due to the uniform distribution of the capillary network. Generally, compartment model predicts higher value of concentration than spatiotemporal distribution

10 118 Ann Nucl Med (2017) 31: C i C e C m C total Time Fig. 12 Contour of FDG concentration parameters (C i is extracellular concentration, C e is intracellular concentration, C m is intracellular concentration and C total is total concentration) defined in Eq. (2a 2c) at 60, 120, 180, 600, 1800, 3600 and 7200 s, in network 1 methods. These results show the effect of transport parameters in tissue (interaction between in pixel of compartment model) on FDG uptake. As mentioned in main text, the two types of network considered in this study are used to show the effect of capillary network on FDG uptake. These results clearly show that the capillary network has significant effect on FDG uptake. At all times, the dense network (network 2) accumulates uptake nearly twice the uptake of the sparse network (network 1).

11 Ann Nucl Med (2017) 31: Ci 119 Ce Cm Ctotal Time Fig. 13 Contour of FDG concentration parameters (Ci is extracellular concentration, Ce is intracellular concentration, Cm is intracellular concentration and Ctotal is total concentration) defined in Eq. (2a 2c) at 60, 120, 180, 600, 1800, 3600 and 7200 s, in network 2

12 120 Ann Nucl Med (2017) 31: Effect of CDR equation's terms on FDG concentration for Network 1 Convection from vessels(interstitial flow) 0.5% Convection in tissue (interstitial flow) 0.5% Effect of CDR equation's terms on FDG concentration for evenly Convection from vessels(interstitial flow) 0.0% distributed Network Diffusion in tissue 0.0% Convection in tissue (interstitial flow) 0.0% Diffusion from vessels 45.3% Diffusion in tissue 53.6% Fig. 14 The contribution of different terms of CDR equation on FDG distribution for network 1. Negligible effect of convection terms and nearly equal effect of diffusion from vessel vs. within tissue on tracer distribution can be seen Effect of CDR equation's terms on FDG concentration for Network 2 Convection from vessels(interstitial flow) 0.2% Discussion Diffusion from vessels 75.6% Diffusion in tissue 24.0% Convection in tissue (interstitial flow) 0.2% Fig. 15 The contribution of different terms of CDR equation on FDG distribution for network 2. Increasing diffusion from vessels occurs due to increasing vessel density In this comprehensive model (SDM), all terms related to solute transport from vessels to tissue or vice versa, as well as within tissue, tracer exchange between extracellular and intracellular, and binding to cells/matrix are considered. The interstitial fluid flow coupled with intravascular flow through a tumor induced dynamic capillary network (adaptable size of capillaries diameter based on different stimuli) is also modeled and is used in the SDM (Eq. 2a). Related frameworks have been applied in different contexts. The Bassingthwaighte group [58] modeled convection and diffusion of substances in biological tissue within a multicapillary network. They also applied their model to study the simultaneous transport and exchange of oxygen (O 2 ) and carbon dioxide (CO 2 ) in the blood tissue Diffusion from vessels 100.0% Fig. 16 The contribution of different terms of CDR equation on FDG distribution for the evenly distributed network. In this case, only the diffusion from vessels is dominant in tracer concentration exchange system [59], and assessed distribution of multiple reacting molecular species in heterogeneous organs [60], with subsequent application [24] to 13 N-ammonia PET including diffusion along a single dimension, as mentioned in the introduction. We added [10] the SDM to previously developed models [38, 45] to find tracer concentration in interstitial fluid cancerous tissues, and also applied this numerical model to investigate tracer distributions in clinically imaged pancreatic [57] and lung [61] tumors. In the present work, we have extended this framework to model radioactive PET tracers in solid tumors. The obtained IBP and IFP used in calculation of convection term of Eq. (2a) are similar to intravascular pressures obtained by Stylianopoulos et al. [16], Soltani et al. [37], and Sefidgar et al. [35, 36]. In this paper, the two parent vessels are used in the simulation for reaching more realistic mathematical model. The elevated interstitial pressure in tumor region is also shown in experimental observations of Huber et al. [62] and Hopland et al. [53]. This term is known as the main barrier of drug transport to tumor region. Tracer distribution is both time and space dependent. The tracer distribution (Figs. 12, 13) shows this dependency clearly. This spatiotemporal distribution enables the investigation of the relationship between image data and molecular processes. The incorporation of both transport phenomena (diffusion and convection) enables application of our overall imaging framework to studies of tumors with varying extents and as acquired over short or long durations of time. Radiolabeled antibodies and tyrosine kinase inhibitors (i.e., immune-pet and TKI-PET) pose a great area of application, for better understanding of the in vivo behavior and efficacy of monoclonal antibodies (MAbs) and TKI targeted drugs in individual patients and for more efficient drug development [63].

13 Ann Nucl Med (2017) 31: Fig. 17 Comparison of FDG uptake average within tumor region calculated via the SDM model vs. the compartment model based on kinetic model. Average extracellular concentration (C i ) (top left), intracellular concentration (C e )(top right), intracellular concentration The Eq. (2a) of CDR equations includes the spatial terms. The spatiotemporal components of this equation are composed of four terms: two of which are related to convection and fluid flow behavior and the other two are related to diffusion and the concentration gradient. The analysis of this results shows that the convection term has negligible effect on FDG transport and it can be eliminated from calculation procedure. By neglecting convection terms from SDM, the calculation process becomes simpler and faster. Since intravascular flow and interstitial flow are only used for convection terms, we can neglect them in the simulation process. By eliminating them from the simulation process, the simulation is also rendered faster than simulation with convection terms. But the diffusion terms which includes diffusion from vessel to tissue and within tissue has significant effect. The diffusion term from vessel is dependent on the capillary network structure. As shown in the results, when the (C m )(bottom left) and total concentration (C total )(bottom right) in tumor region for both capillary networks is compared against the compartment model and model without network capillary network is dense, the contribution of diffusion from vessel to tissue in FDG transport is higher than diffusion in tissue; however, by decreasing the density of vessel in network, the effect of diffusion in tissue increases. Since in real tissue the capillary network structure may be sparse or dense, none of these terms can be eliminated. The comparison of SDM vs. the analytical method based on kinetic compartmental method shows that two methods predict similar patterns for tracer distribution. However, the results of these two methods have differences. The compartmental method predicts higher uptake FDG than SDM. The concentration obtained from compartment method is close to concentration obtained from uniformly distributed network. For both conditions the diffusion in transport is ignored. For this reason in network 2 which has more dense capillary network, results are closer to the compartment method. Generally, results (Fig. 17) demonstrate that the tracer distribution depends on the structure of

14 122 Ann Nucl Med (2017) 31: the microvascular network. This is because our approach incorporates the impacts of interstitial diffusion and convection. For the 1st network, due to the highly inhomogeneous network structure, tracer delivery to some cells are limited, lowering uptake. Our future work focuses on the inverse problem of estimating parameters of interest from imaging data. This includes modeling of the various PET detection and degradation processes [64, 65], leading to image blur and noise, and constructing an appropriate numerical non-linear regression paradigm incorporating the comprehensive solute transport model within it to estimate the parameters of interest. This is hypothesized to result in improved estimates given enhanced modeling (e.g., inclusion of diffusion mechanism). Conclusion A comprehensive numerical approach which couples the mathematical model of the microvascular network and the interstitial flow with the mathematical model of general solute transport is utilized to study the distribution of PET tracers. The tracer distribution model incorporated convection and diffusion transport from vessels to tissue and within tissue, as well as the reaction mechanism. The present work focused on the application of the framework to FDG, demonstrating the ability of this approach to shed light on the spatiotemporal distribution of PET tracers, beyond the usage of conventional methods. The proposed methodology enables assessment of the impact of various tumor related parameters and phenomena on tracer distribution, thus providing a condition to analyze sensitivity of tracer distributions upon physiological parameters. 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