comprehensive modeling of the spatiotemporal distribution ...rahmim/research_work/soltani...of...

12
Comprehensive Modeling of the Spatiotemporal Distribution of PET Tracer Uptake in Solid Tumors based on the Convection-Diffusion-Reaction Equation M. Soltani, M. Sefidgar, M. E. Casey, R. L. Wahl, R. M. Subramaniam, and A. Rahmim Abstract– In this paper, the 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 properties such as pressure. 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. 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 compute pressure and velocity distributions. Subsequently, we applied a comprehensive convection-diffusion-reaction equation to accurately model distribution of PET tracer uptake, specifically FMISO in this work, within solid tumors. The abovementioned use of partial differential equations (PDEs), beyond ordinary differential equations (ODEs) as commonly invoked in tracer kinetic modeling, enables simultaneous modeling of tracer distribution over both time and space. For different angiogenetic structures, the intravascular pressure and interstitial pressure were elaborately calculated across the domain of interest, and used as input to model tracer distribution. The results can be utilized to comprehensively assess the impact of various parameters on the spatiotemporal distribution of PET tracers. I. INTRODUCTION Molecular imaging methods especially utilizing positron emission tomography (PET) have found increasing applications in a variety of cancers, including diagnosis, initial staging, restaging, prediction and monitoring of treatment response, surveillance and prognostication [1]. Routine clinical applications include qualitative Manuscript received December 30, 2014. 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: [email protected]), and Mechanical Engineering, KNT University, Tehran, Iran. M. Sefidgar is with Department of Technical Engineering, IKI University, Qazvin, Iran. (e-mail: [email protected]) M. E. Casey is with Siemens Medical Solutions, Knoxville, TN 37932, USA. (e-mail: [email protected]) 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: [email protected]). R. L. Wahl is with Division of Nuclear Medicine, Department of Radiology and Radiological Science, School of Medicine, Johns Hopkins University, MD 21287, USA (e-mail: [email protected]). 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: [email protected], webpage: www.jhu.edu/rahmim). 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 (i) static (single time-frame) imaging invoking the standardized uptake value (SUV), and (ii) 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. Patlak plots [8] and conventional compartment models [9], [10] have the limitation that they do not model movement 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 solute transport equation which includes diffusion, convection and reaction is widely used to simulate drug delivery [11], referred to as the convention-diffusion- reaction (CDR) equation. Baxter and Jain, based on the theoretical framework in their 1D mathematical method, found the effective factors on drug delivery [12]–[15]. Wang and Li [10] used modified MRI images and convection- diffusion equation for tumor geometry. They considered interstitial fluid flow with blood and lymphatic drainage in their model. Wang et al. [11] studied the effect of elevated interstitial pressure, convective flux, and blood drainage on the delivery of specified solute to brain tumors. Magdoom et al. [16] used a simplified model of CDR for predicting albumin tracer distribution in the lower limb of a mouse. They used a dynamic contrast enhanced (DCE)-MRI based computational model [17], [18] to estimate the spatial variation of transport properties in intestinal space. Stylianopoulos and Jain [19] used the CDR equation to investigate effects of vascular normalization for improving perfusion and drug delivery in solid tumors. They extent their work by considering the surface charge of the drug and the resulting electrostatic relations [20]. 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 CDR equation that we have previously used for assessment of drug delivery [11], [19], to comprehensively model the spatiotemporal distribution of PET tracers in solid tumors, focusing in the present work on the tracer FMISO.

Upload: others

Post on 24-Aug-2020

2 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

Comprehensive Modeling of the Spatiotemporal Distribution of PET Tracer Uptake in Solid Tumors

based on the Convection-Diffusion-Reaction Equation

M. Soltani, M. Sefidgar, M. E. Casey, R. L. Wahl, R. M. Subramaniam, and A. Rahmim

Abstract– In this paper, the 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 properties such as pressure. 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. 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 compute pressure and velocity distributions. Subsequently, we applied a comprehensive convection-diffusion-reaction equation to accurately model distribution of PET tracer uptake, specifically FMISO in this work, within solid tumors. The abovementioned use of partial differential equations (PDEs), beyond ordinary differential equations (ODEs) as commonly invoked in tracer kinetic modeling, enables simultaneous modeling of tracer distribution over both time and space. For different angiogenetic structures, the intravascular pressure and interstitial pressure were elaborately calculated across the domain of interest, and used as input to model tracer distribution. The results can be utilized to comprehensively assess the impact of various parameters on the spatiotemporal distribution of PET tracers.

I. INTRODUCTION

Molecular imaging methods especially utilizing positron emission tomography (PET) have found increasing applications in a variety of cancers, including diagnosis, initial staging, restaging, prediction and monitoring of treatment response, surveillance and prognostication [1]. Routine clinical applications include qualitative

Manuscript received December 30, 2014. 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: [email protected]), and Mechanical Engineering, KNT University, Tehran, Iran.

M. Sefidgar is with Department of Technical Engineering, IKI University, Qazvin, Iran. (e-mail: [email protected])

M. E. Casey is with Siemens Medical Solutions, Knoxville, TN 37932, USA. (e-mail: [email protected])

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: [email protected]).

R. L. Wahl is with Division of Nuclear Medicine, Department of Radiology and Radiological Science, School of Medicine, Johns Hopkins University, MD 21287, USA (e-mail: [email protected]).

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: [email protected], webpage: www.jhu.edu/rahmim).

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 (i) static (single time-frame) imaging invoking the standardized uptake value (SUV), and (ii) 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. Patlak plots [8] and conventional compartment models [9], [10] have the limitation that they do not model movement 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 solute transport equation which includes diffusion, convection and reaction is widely used to simulate drug delivery [11], referred to as the convention-diffusion-reaction (CDR) equation. Baxter and Jain, based on the theoretical framework in their 1D mathematical method, found the effective factors on drug delivery [12]–[15]. Wang and Li [10] used modified MRI images and convection-diffusion equation for tumor geometry. They considered interstitial fluid flow with blood and lymphatic drainage in their model. Wang et al. [11] studied the effect of elevated interstitial pressure, convective flux, and blood drainage on the delivery of specified solute to brain tumors. Magdoom et al. [16] used a simplified model of CDR for predicting albumin tracer distribution in the lower limb of a mouse. They used a dynamic contrast enhanced (DCE)-MRI based computational model [17], [18] to estimate the spatial variation of transport properties in intestinal space. Stylianopoulos and Jain [19] used the CDR equation to investigate effects of vascular normalization for improving perfusion and drug delivery in solid tumors. They extent their work by considering the surface charge of the drug and the resulting electrostatic relations [20]. 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 CDR equation that we have previously used for assessment of drug delivery [11], [19], to comprehensively model the spatiotemporal distribution of PET tracers in solid tumors, focusing in the present work on the tracer FMISO.

Page 2: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

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, 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 FMISO imaging, potentially aided by other modalities such as CT or MRI, to enable enhanced prediction of drug delivery to solid tumors. There have been few works in this area. Kelly et al. [21] used a simplified form of the CDR equation for investigation of FMISO distribution. 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 the above works, the spatial scale was in the order of a few millimeters. Furthermore, the applications did not include convection transport from vessel to tissue or within tissue. Finally, the dynamic (spatially variable) structure of the microvascular networks was not incorporated to compute the variable pressure/flow distributions across the networks. The present study by contrast uses a general, comprehensive framework that includes a dynamic microvascular structure, and the effects of both intravascular and extravascular flow and their stimuli, including both diffusion and convection, are incorporated. This framework is applicable to different tracers, and tumors of varying sizes, and for images spanning short, intermediate or long durations of time post-injection. In Section 2, we discuss our overall methods for modeling fluid flow in the interstitium and the microvessel network as developed in our previous work [11], [24], and using these for computation of tracer distribution based on the CDR equation. In section 3, we show our results and various analyses to assess performance of our framework. Sections 4 and 5 discuss and conclude the work, respectively.

II. METHODS

A. Solute transport in tissue: In the macroscopic solute transport model, the variations on the length scale of capillary distances are not included, but averaged over some region that is small compared to the length scale of the tumor radius. The general model for solute transport in tissue includes transport in interstitium, reaction mechanism (binding to the cancer cells/matrix). The derivation of this model is elaborated in Appendix A, to which the reader is referred.

General form of Solute transport equation: The general form of solute transport in tissue involves the diffusive and convective transport of the free tracer within the tumor interstitium as well as reaction rate (CDR equation) [17], [19], [20], as derived in Appendix A:

( )2 v

Diffusion transport in Convection transport in

Lymphatic termVascular term

association (binding) rate dissociation r

interstitium interstiti

e

um

at

feff f i f

V L

on f off b

bon f off

CD C C

t

k C k C

Ck C k C

t

∂= ∇ − ⋅∇

+ Φ − Φ

− +

∂ = −∂ b

(1)

B. Compartmental model and spatiotemporal distribution mode In this section, our main framework to model spatiotemporal distribution based on the CDR equation is compared to the compartmental model. The CDR equation involves solute transport between blood and lymphatic vessels and interstitium, solute transport between free solute and produced or bound solute due to reaction, as well as spatiotemporal distribution of solute in interstitium. It is comparable with the traditional model for solute distribution, namely the compartmental model. The compartmental model is a commonly invoked approach to model and quantify solute transport. In this framework, concentration in each region of interest (ROI; e.g. an entire organ, or as small as a voxel of interest) is assumed to be independently distributed. Subsequently, ordinary differential equations are utilized to model the distribution of activity over time in each ROI. By contrast, in the CDR model, distributions in space are not modeled as independent from one another, and partial different equations, involving both time and space, are invoked, to collectively model solute distribution over time and space. To see this better, let us consider a commonly invoked form of solute transport in compartmental modeling involving four compartments, also referred to as the three-tissue compartmental model (shown in Figure 1). From arterial blood, the solute passes into the second compartment, known as the free compartment. The third compartment is the region of specific binding. The fourth compartment is a nonspecific-binding compartment that exchanges with the free compartment. For more simplicity, it is commonly assumed that there is rapid exchange between the free and nonspecifically bound compartments, and that these two compartments can be combined together [25]. Therefore the three tissue compartmental model is changed to the two tissue compartmental model (Figure 2). This model fits many radioligand tracers well such as [18F] FMISO which is considered in this study. The CDR model is related to the two-tissue compartmental model with some differences in the free compartment. For the free compartment, it is considered that the solute is redistributed in time and space. Figure 3 demonstrates how the CDR equation is related to a two-tissue compartment reversible binding model [9]. Movement between plasma, free and bounds states is governed by four parameters, L1, L2, L3 and L4, analogous to the rate constants K1, k2, k3 and k4 seen in conventional compartmental models (e.g., [25]). The parameters in Figure 3 are defined as follow:

Page 3: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

( )( )( )1 11

mV V

Pi s i f Pe

P S PeL L Se

PV

PV

σ π π σ⎛ ⎞= − +⎜ ⎟−⎝− −

⎠− (2)

2 1m

LPe

P S PeLV e

φ⎛ ⎞= +⎜ ⎟−⎝ ⎠ (3)

3 onL k= (4)

4 offL k= (5)

where ( ) NormalTissue

0 TumorTissue

PL Li L

L

L S P PVφ

⎧ −⎪= ⎨⎪⎩

and the various terms are defined in Tables 1 and 2. Based on the elaborate form of the CDR equation (Appendix A), Equation (1) is given by:

( )21 2 3 4

3 4

vfeff f i f P f f b

bf b

CD C C L C L C L C L C

tC L C L Ct

∂= ∇ − ∇ ⋅ + − − +

∂∂ = −∂

(6)

By contrast, the standard two-tissue compartmental model (shown in Figure 3) is given by [26], [27]:

1 2 3 4

3 4

fp f f b

bf b

CK C k C k C k C

tC

k C k Ct

∂= − − +

∂∂ = −∂

(7)

If the spatial redistribution terms (i.e. diffusion and convection; the first and second terms in the right hand side of in Equation (6)), are neglected, it will resemble Equation (7).

Figure 1. General four compartments model (three-tissue compartmental

model). This model consists of components of plasma, free ligand in tissue, specific binding and non-specific binding and six rate constants (K1–k6).

Figure 2. Three compartment (two-tissue) model. This model consists of

components of plasma, free ligand (plus non-specific binding) in tissue and specific binding, involving four rate constants (K1–k4).

Figure 3. Relationship between standard two-compartmental model and CDR equation. The CDR equation is same as two tissue compartmental

model with difference in that free and bound solute is depended on space and related compartment does not have uniform concentration.

Table 1. Summary of parameters related to normal and tumor tissues, and typical values used in our simulations [11]

Parameter Description Value

mmHg Osmotic pressure of the plasma

20(normal) 20(tumor)

mmHg Osmotic pressure of the interstitial fluid

10(normal) 15(tumor)

cm-1 Surface area per unit volume for transport

70(normal) 200(tumor)

Average osmotic reflection coefficient

0.91(normal) 0.82(tumor)

cm/mmHg s Hydraulic conductivity of

the microvascular wall 0.36×10-7(normal) 2.80×10-7(tumor)

[mmHg] Hydrostatic pressure of the lymphatic 0 (normal)

[1/mmHg s] Lymphatic filtration

coefficient 1.33×10-5(normal)

Table 2. Summary of parameters related to tracer transport, and typical

values used in our simulations.

Parameter Description Value Reference

(mm2 /s)

Effective diffusion

coefficient

0.7×10-3(n) 2.5×10-3(t) [28]

(m/s)

Transvascular permeability of the vessel to the

tracer

2.4×10-5(n) 9.4×10-5(t) [21]

(1/min)

Constant transport rate from blood to

tissue 0.45 calculated

(1/min)

Constant transport rate from tissue to

blood 0.45 calculated

(1/min)

Association (binding) rate

8×10-

3(hypoxia) 1×10-

3(normoxia) 1×10-3(normal)

[29]

(1/min) Dissociation rate 0 [29]

C. Computational domain and mathematical equation: As mentioned before, FMISO is considered as tracer of interest in this study.

S V

sσpL

LP

PL LL S V

effD

mP

1K

2k

3onk or k

4offk or k

Page 4: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

The 2D domain (shown in Figure 4) considered for computational simulation is a 6×3 cm2 rectangle. The tumor is located at the center of the domain with the radius of 1.2 cm. The two parent vessels from which the vascular network grows are located at the left and right edges of the domain.

Figure 4. A schematic of the solution domain.

For capillary network, an elaborative model is incorporated based on our previous work [24], [30], [31] to generate a microvascular network induced by tumor angiogenesis. 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 work [32], and by Anderson and Chaplain [33], and some results are shown in result section. Intravascular and interstitial flow: Some of the parameters in Equation (1) are related to fluid flow in interstitium and blood flow in capillaries. An advanced mathematical model is used to calculate interstitial velocity and pressure and intravascular pressure. The details and mathematics of the framework are explained in our previous studies [11], [24], [30], [31], [34], [35].

D. General equation for tracer transport CDR equation: In the modeling of tracer distribution in tissue with capillary network, the third and fourth terms of Equation (1) are implemented wherever the capillary exists and for other places these terms are zero. To show this more systematically, the mathematical form of our elaborate tracer transport model in tissue is given by:

( )

( )

2

1 2 3 4

2

3 4

1 2

vwhere blood source e

other

xist

w

s

vise

feff f i f

P f f b

feff f i f

f b

bf b

total f b

CD C C

tL C L C L C L C

CD C C

tL C L C

C L C L Ct

C C C

∂⎧= ∇ − ∇ ⋅⎪ ∂⎪+ − − +⎪⎪

⎨∂⎪ = ∇ − ∇ ⋅⎪ ∂⎪ − +⎪⎩∂ = −∂

= +

(8)

where L1 and L2 were defined previously. Solution of Kinetic model: The Solution of Equation (7) yields to the following expression for Ctotal:

( ) ( )1 213 4 1 2 3 4

2 1

t ttotal V

KC k k e k k e Cα αα αα α

− −⎡ ⎤= + − + − − ⊗⎣ ⎦− (9)

where denotes the convolution operation, and

( )21,2 2 3 4 2 3 4 2 44 2k k k k k k k kα ⎡ ⎤= + + + + −⎢ ⎥⎣ ⎦

∓ (10)

E. Oxygen pressure distribution: In the case of FMISO, tracer uptake rate is modulated by oxygen distribution. The oxygen distribution in tissue with capillary network can be modeled via a reaction-diffusion model [36]. Oxygen pressure is used instead of concentration due to the direct relationship between O2 concentration and partial pressure (Henry’s law):

2

222O met s

Po D Po q Ft

∂ = ∇ − +∂

(11)

where

2Po : Oxygen pressure in tissue.

2OD : The diffusion coefficient of oxygen.

metq : The rate of oxygen metabolism as a function of oxygen pressure,

sF : The rate of oxygen supplied from the blood vessels. The constitutive equation for sF is given by Equation (12) [36]:

( )22, 2 for existence of blood source

0 otherwise

Os vessel

Pm SF Po Po

V⎧ = −⎪⎨⎪⎩

(12)

where 2OPm : The transvascular permeability of ,

2,vesselPo : The oxygen pressure in the vessels.

metq is calculated based on the most common model (Michaelis-Menten) [36]:

max 2

2 0 2met

q PoqPo P o

=+

(13)

where qmax : Maximum O2 consumption rate

: The oxygen pressure at which qmet is half-maximal (Michaelis-Menten coefficient) The values used in oxygen distribution are summarized in Table 3. The kon or k3 for FMISO depends non-linearly on the local oxygen concentration, as follows[22], [23]:

( ) ( )3 1 2k F P F P= ⋅ (14)

( ) max 11

1

k PF PP P

=+

(15)

( )22

kPF P

P P⎛ ⎞

= ⎜ ⎟+⎝ ⎠ (16)

F. Model parameterization and simulation details The parameters used in Equation (8) as related to tumor or normal tissues were listed in Table 1, and the parameters of tracer transport were listed in Table 2. Table 3 lists

2O

0 2P o

Page 5: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

parameters related to oxygen transport, and values utilized in our simulations.

Table 3. Summary of parameters related to oxygen transport

Parameter Description Value Refer

(m2/s) Diffusion coefficient of oxygen 2×10-9 [37]

(m/s) transvascular permeability of O2

4.1×10-4 [36]

qmax (mmHg)

Max. O2 consumption rate 15 [38]

(mmHg) Michaelis-Menten

coefficient 2 [36], [38]

(mmHg) Vessel oxygen pressure 40 [39]

kmax (1/s)

Maximum binding rate 1.7×10-4 [23]

P1 Po2 inhibiting binding by

50% 1.5mmHg [40]

P2 Po2 inducing 50% necrosis 0.1mmHg [23]

k Determines step width at P2 0.3 [23]

The boundary conditions considered for intravascular flow are:

For the boundary between tumor and normal tissues, the continuity of concentration and its flux are considered as boundary conditions:

(17)

The open boundary condition is used for the edges of domain. The open boundary is used to set up mass transport across boundaries where both convective inflow and outflow can occur and defined by Equation (18):

(18) where n is the normal vector.

III. INVERSE METHOD (PARAMETER ESTIMATION) In what we will refer to as ‘‘forward modeling’’, we assume that all physical parameters are known and can thus use appropriate physical laws and scientific theories to predict responses to stimuli. Most well-known numerical models are built to be used as forward models—that is, their input includes parameter estimates and their output is a prediction of system response. By contrast, inverse modeling is concerned with those cases where we have already measured actual responses to stimuli in the field or in laboratory, and would like to work backwards, or ‘‘invert’’, from the data to estimate the physical parameters of the system (this is also referred to as parameter estimation, history matching, or data fitting.) The present work is primarily concerned with establishing the forward model using the CDR framework in the context of radiotracer distribution, emphasizing 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. At the same time, this framework renders itself to enhanced parameter estimates. Two important outcomes are expected in the context of the inverse problem: enhanced parameter estimates (K1-k4), as well as estimation of the diffusion coefficient from radiotracer imaging, which has not been performed, to out knowledge, in past studies. This is subject of ongoing work, and detailed analysis. However, we show some preliminary results in the present work. We consider that the diffusion coefficient ( effD ) in the tumor region (Equation (8)) is unknown, and concentration of tracer over time in the tumor region is available via imaging data. Gaussian-distributed noise was added to imaging data, to simulate varying levels of noise present in imaging. Given a set of experimental data of tracer concentration ( ( , y , t )t i i jC x ) at N sampling points and M time step, the following nonlinear least squares objective function is defined:

( )2

1 1(x , y , t ) (x , y , t ,D )

N M

t i i j i i j effi j

S C Y= =

= −∑∑ (19)

where (x , y , t ,D )i i j effY is the calculated tracer concentration based on equations (8). By minimizing S, a set of optimal values of the parameter were obtained. The Levenberg-Marquardt method was utilized in this study in order to minimize the objective function S. The LM method is an iterative technique which finds the unknown parameters as the minimum point of function S is reached. The inverse algorithm used in this work is shown in Figure 5.

Figure 5. The algorithm used in estimation procedure

IV. RESULTS: Two different networks generated via the discrete angiogenesis method are used in the calculations. One of the networks is produced by 3 sprouts as initial condition and modifying tumor angiogenic factor (TAF) conditions in the tumor region for generation less uniform tumoral coverage. The other network is initialized by 10 sprouts in two parent vessels for producing particular angiogenetic capillary network in tumor domain like as experimental observation. The intravascular pressure distributions in the network and interstitial pressure in tissues are shown in Figure 6 and

2OD

2OPm

0 2P o

2,vesselPo

, ,

,outlet ,outlet

25 35parent vessel1 parentvessel2

5 10b inlet b inlet

b b

P mmHg P mmHgP mmHg P mmHg

= =⎧ ⎧⎪ ⎪→ →⎨ ⎨= =⎪ ⎪⎩ ⎩

( ) ( )t neff i eff iD C v C D C v C

C C− −

− +

Ω Ω

Ω Ω

∇ + = ∇ +

=

C=0− ⋅∇n

Page 6: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

Figure 7, respectively. The interstitial pressure has its greatest value 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 2000Pa.

Figure 6. Intravascular pressure distribution

Figure 7. Interstitial pressure distribution

The resulting oxygen distributions for the two different networks are shown in Figure 8 and Figure 9. Since the oxygen distribution reaches steady sate condition very soon, the results are nearly independent of time. The simulated PO2 strongly decreases with increasing distance from the vessels. Therefore, the close up of the results near the vessel is also shown in Figure 8 and Figure 9. The oxygen pressure profile from vessel into tissue is shown in Figure 10.

Figure 8. Oxygen distribution for network 1.

Figure 9. Oxygen distribution for network 2.

Figure 10. Oxygen pressure from blood into tissue

CDR model validation: We utilized experimental observations of Bruehlmeier et al [41]. Figure 11 shows comparison of average concentration of FMISO in tumor region with experimental data. The CDR equation produces results that are in good agreement with 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 micro-parameters that produce fits to experimental data. At the same time, in the next part, we perform further analysis including comparisons with conventional ODE-based analytical methods, with similar modeled parameters.

Figure 11. Comparison of CDR results with experimental results[41]

Tracer distribution: In this section, the tracer distribution of FMISO is investigated. A plasma activity as obtained from Backes et al. [42] was used in this part of the analysis. The spatiotemporal concentration distributions of FMISO at different post injection times for networks 1 and 2 are shown in Figure 12 and Figure 13, respectively. The second network shows more uniform distribution than network 1. This is due to a higher number of vessels in the network, which act as source terms for the tracer. The increasing of effect C1 in areas distant from vessels observed in Figure 12 and Figure 13 would not have been observed if interstitial transport (diffusion, convection) was not considered, as there is no other means for tracers in the vicinity of vessels to make it to hypoxic regions. Figure 14 depicts the average FMISO uptake, respectively, across the tumor tissue. The results include those from the proposed comprehensive CDR-based approach. Furthermore, the results are compared with the conventional analytical method (standard kinetic modeling) using ODEs, though the latter lacks sophistication of simultaneous spatiotemporal modeling. Since the ODE-based analytical method used in conventional kinetic modeling gives tracer activity levels that are spatially independent, the CDR results were also averaged in space to eliminate dependency of tracer value on position thus obviously lacking the details shown in previous images. The results show that the trend of the graphs for the conventional analytical method and the proposed CDR model is the same. The differences between the analytical method and current approach in network 1 (sparse network) is more than that of network 2.

Page 7: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

Figure 12. FMISO distribution at 60, 120, 180, 600, 1800, 3600 and 7200

sec, in network 1.

Figure 13. FMISO distribution at 60, 120, 180, 600, 1800, 3600 and 7200

sec, in network 2.

Figure 14. Comparison of FMISO average in tumor region calculated via

the CDR model with analytical result of FMISO based on two-tissue compartmental kinetic model

Since averaging the tracer activity across the tumor oversimplifies the available information, we also investigated the tracer activity at some specified points across the tumor. These points are shown in Figure 15. The results of C1 and C2 are shown in Figure 16. The results show that tracer distribution especially for C1 is highly spatially dependent. However, at points close to vessels, tracer uptake for both networks is nearly the same. The comparison of results between conventional analytic vs. comprehensive model shows that the tracer uptake better agrees at points near the vessels.

Figure 15. The position of points used in two networks

Figure 16. Tracer uptake patterns for FMISO at different spatial location.

Parameter estimation: As mentioned in section 4, it is assumed that the diffusion coefficient is unknown. The concentration of points at network 2 shown in Figure 15 is used as experimental data. Three sets of data examined in this study include: (1) Noise-free data, (2) Realizations of data containing 1% Gaussian-distributed noise, and (3) Realizations of data containing 10% Gaussian-distributed noise. The inverse problem results obtained are listed in Table 4. The estimated parameters are implemented in the direct code with previous plasma activity for comparison of the results with new estimated parameters and real values. The maximum error ( estimated real realC C C− ) in the results based on estimated parameter and results based on real value is 4.2%.

V. DISCUSSION In our general CDR expression, all terms related to solute transport from vessels to tissue or vice versa, as well as within tissue, and tracer 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 CDR equation.

Page 8: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

Table 4. Experimental data used in this simulation

Case Initial guess Deff (mm2 /s)

Errorless data 7×10-4 1.7262×10-3 (1%)Iσ 7×10-4 1.7249×10-3 (1%)IIσ 7×10-4 1.7226×10-3 (1%)IIIσ 7×10-4 1.7236×10-3

(10%)IVσ 7×10-4 1.6888×10-3 (10%)Vσ 7×10-4 1.6889×10-3 (10%)VIσ 7×10-4 1.7053×10-3

The elevated interstitial pressure in tumor region is also shown in experimental observation of Huber et al. [43]. They measured the IFP for different tumors between 1.1 kPa to 1.8 kPa.. Since the FMISO uptake depends on the oxygen levels in the tissue, the oxygen distribution has to be solved first in this part of the study. The oxygen distribution shows that the Po2 decreases to low values around 2 mmHg at a distance of about 150 μm from vessels. empirical data [37], [44], [45] confirm this observation The low diffusivity of oxygen is the reason that in the first network the hypoxia region is much larger than what occurs in the second one. Tracer distribution is both time and space dependent. 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 framework to PET studies of tumors with varying extents and as acquired over short or long durations of time. Radiolableled antibodies and tyrosine kinase inhibitors (i.e. immune-PET and TKI-PET) pose another 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 [46]. The tracer distribution in time has agreement with the general trends of experimental data [47], [48]. The activity level of tissue follows plasma activity level in the early stages. In early stages, the free concentration is dominant. The binding portion has significant value only at later stages. The comparison of our CDR model 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 results of the analytical method are similar to network 2 (dense network). Generally, the results (Figure 14) demonstrate that the tracer distribution depends on the structure of the microvascular network. The evaluation of the spatially averaged results shows that the dependency of FMISO uptake in the first tissue compartment C1 on the microvascular network reduces as time goes on. Since the total concentration value at increasing time steps is more dominated by the bound state than the free state of the tracer, and since FMISO binding for the hypoxic region (the region

distant from the network) is higher than for normoxic region, overall FMISO uptake in very considerable at points distant from blood vessels at later times. Results of FMISO uptake show that the activity level of FMISO in hypoxic regions is high which has been reported in [21], [49], which is a very good motivation for the use of FMISO in the study of hypoxia [28], [29]. The result of the first network with more hypoxic regions shows this effect clearly (Figure 12). The comparison of the average uptake for FMISO demonstrates that the activity level depends on the network structure as well. This is because our approach incorporates the impacts of interstitial diffusion and convection. For the 1st network, due to the highly inhomogeneous angiogenic structure, tracer delivery to some cells are limited, lowering uptake. Our ongoing work focuses on detailed analysis of the inverse problem of estimating parameters of interest from imaging data. This includes modeling of the various PET detection and degradation processes [50], [51], leading to image blur and noise, and constructing appropriate numerical non-linear regression paradigms incorporating the comprehensive solute transport model within it to estimate the parameters of interest.

VI. 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 was 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 FMISO, 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. Furthermore, this framework can be utilized in an inverse model for potentially enhanced estimation of parameters of interest, as the forward-model is more accurate, as well as estimation of the diffusion coefficient from PET tracers, which, to our knowledge, has not been achieved in the past. Overall, the proposed model provides a framework for the analysis of PET tracer distribution that moves beyond conventional computational methods including ODE-based kinetic compartment modeling.

ACKNOWLEDGEMENTS We gratefully acknowledge support by Siemens Medical Solutions (Siemens-JHU grant #: 212814).

Page 9: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

VII. APPENDIX A

Diffusion and convection: For fluid flow in a porous medium, free solute transports are due to two important mechanisms: diffusion and convection. The solute mass flux (J) which includes both diffusion and convection is obtained by Fick’s first law [52]. Then, by applying Fick’s second law, mass conservation is obtained [53]–[55] (Equation (1)):

( )fJ Ct

∂∇ ⋅ = −∂

(20)

where J is the solute mass flux, and Cf is the free solute concentration. Equation (1) simply states that the net mass output per unit volume is equal to the time rate of change of mass within the volume. The solute mass flux J has both diffusion and advection fluxes [53]:

( )vf i fJ D C C= − ∇ +

(21) The first term on the right hand side is the diffusion flux, in which D is the diffusion coefficient. The second term is the convection flux, due to iv , the velocity of the fluid flow. Substituting Equations (21) into Equation (1) results in

( )iv ff f

CD C C

t∂

⎡ ⎤⎡ ⎤∇ ⋅ ∇ − ∇ ⋅ =⎣ ⎦ ⎣ ⎦ ∂ (22)

The second term on the left hand side of Equation (5) can be written as

( ) ( ) ( )i i iv v vf f fC C C⎡ ⎤∇ ⋅ = ∇ ⋅ + ⋅∇⎣ ⎦ (23)

If there is no changes in fluid density because of the solute transport process, the solute transport does not have any effects on the flow velocity field, iv , and iv can be calculated independent of the solute concentration field. In a steady fluid flow field, the governing equation is simplified to Darcy's law iv iPκ= − ∇ [52] where κ is hydraulic conductivity and P∇ is the pressure gradient. In this case, the divergence of the fluid velocity field vanishes, as in the Equation (24),

( )iv 0iPκ∇⋅ = ∇⋅ − ∇ =

(24) Under the steady state flow assumption, Equation (23) is simplified to

ivff f

CD C C

t∂

⎡ ⎤ ⎡ ⎤= ∇ ⋅ ∇ − ⋅∇⎣ ⎦ ⎣ ⎦∂ (25)

Equation (25) is the governing equation for solute transport in a steady fluid flow field. In a porous media, the same equation can be applied, if there is no solute source or solute sink. However, in the most biological tissues there are sources and sinks. For example, between interstitial space and the blood or lymph vessels fluid is exchanged; therefore, the steady state form of the Equation (25) in this case can be written as [56]:

ivff f V L

CD C C R

t∂

⎡ ⎤ ⎡ ⎤= ∇ ⋅ ∇ − ⋅∇ + Φ − Φ +⎣ ⎦ ⎣ ⎦∂ (26)

where VΦ is the rate of solute transport per unit volume from blood vessels into the interstitial space, LΦ is the rate of solute transport per unit volume from the interstitial space into lymph vessels, and R is the rate of solute production and consumption per unit volume due to chemical reactions. In following, the source and sink and also chemical reaction are defined.

Solute transport across vessel: Solute transport across the vessel walls and through the interstitium is because of both diffusion and convection. Concentration gradients result in the diffusive transport. Movement of fluid molecules caused by pressure gradients results in the convective transport of solute molecules. If the diffusion is the mechanism of transvascular flow, the diffusive solute flux is given by[57]:

( )S P fJ PS C C= −

(27)

where SJ is the net flow of solute from vessel, P is the diffusive permeability, S is the surface area of the vessel,

PC is the plasma concentrations of the solute, and fC is the free concentrations of the solute. By considering convection transport from microvessel wall, the total solute flow is given by the Staverman-Kedem-Katchalsky equation [58], [59]:

( ) (1 )S P f V f lmJ PS C C J Cσ= − + − ∆

(28)

where: fσ is the solvent-drag reflection coefficient, and (1 -

fσ ) is a measure of coupling between fluid and solute transport. lmC∆ is the log-mean concentration within the pore [57]:

( )lnP f

lmP f

C CC

C C−

∆ =

(29)

VJ is the volume flow of fluid across the vessel wall. According to Starling's law, net fluid flow across a vessel wall is given by [60]:

( ) ( )V P V i V iJ L S P P σ π π= − − −⎡ ⎤⎣ ⎦ (30)

where PL is the hydraulic conductivity (or the filtration

coefficient) of the vessel, VP and iP are the intravascular

and interstitial fluid pressures, Vπ and iπ are the colloid-osmotic pressures in plasma and interstitial fluid, and σ is the osmotic reflection coefficient [53]. A complete description of the solute transport based on the pore model of membrane is given by the so-called Patlak equation [61]:

( )(1 )

1

PeP f

S V f Pe

C C eJ J

⎡ ⎤−⎢ ⎥= −

−⎢ ⎥⎣ ⎦

(31)

This equation can be easily partitioned into the diffusive and convective components as shown below:

( ) (1 )C1S P f V f PPe

PeJ PS C C Je

σ⎡ ⎤= − + −⎢ ⎥−⎣ ⎦ (32)

Page 10: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

where: Pe is referred to as the Peclet number, and is given by:

(1 )V fJPe

PSσ−

=

(33)

The Peclet number indicates the importance of convective transport with respect to diffusive transport. The rate of solute transport per unit volume from blood vessels into the interstitial space is obtained by [11], [62]:

( )

( ) ( )1

(1 )C

SV P f Pe

P V i V i f P

J S PeP C CV V e

SL P PV

σ π π σ

⎡ ⎤Φ = = − +⎢ ⎥−⎣ ⎦

− − − −⎡ ⎤⎣ ⎦

(34)

The solute transport rate across the lymphatic vessels can be considered as [18]

L LCφΦ = (35) where Lφ , the drainage term, i.e. elimination by lymphatic system, is calculated by:

( ) ( )PL LL i L

L Sr P PV

φ = − (36)

PL LL SV

is the lymphatic filtration coefficient, and LP is the

hydrostatic pressure of the lymphatic system. The general form of free solute transport in tissue is then given by:

( )2 v

Diffusion transport in Convection trainterstitium interstitinsport in

Reaction rateLymphatic termVascular ter

um

m

feff f i f

V L

CD C C

t

R

∂= ∇ − ⋅∇

+ Φ − Φ +

(37)

Chemical reaction rate: The free solute is consumed and produced due to chemical reaction in biological tissue. The reaction mechanism can be binding to cell membrane [63] or other mechanism such as phosphorylation. Reaction of solute increases the retention and accumulation of tracer/drug in tumors, but reduces the amount of free solute available for interstitial transport and results in uneven distribution [64]. The schematic of reaction is shown in Figure 17. The schematic of reaction of free solute is similar to the compartments model in kinetic modeling.

Figure 17. Schematic of reaction of free solute in interstitium [25].

The rate of consumption and production of free solute [52]: off b on fR k C k C= − (38)

And the mass balance for bC is [52]:

bon f off b

Ck C k C

t∂ = −∂

(39)

where : Association rate constant,

: Dissociation rate constant, and

bC : produced or bound solute concentration.

REFERENCES

[1] R. Wahl and J. Buchanan, Principles and Practice of Positron Emission Tomography. Philadelphia, Lippincott Williams & Wilkins, 2002.

[2] G. Tomasi, F. Turkheimer, and E. Aboagye, “Importance of Quantification for the Analysis of PET Data in Oncology: Review of Current Methods and Trends for the Future,” Mol. Imaging Biol., vol. 14, no. 2, pp. 131–146, 2012.

[3] M. A. Lodge, J. D. Lucas, P. K. Marsden, B. F. Cronin, M. J. O’Doherty, and M. A. Smith, “A PET study of 18FDG uptake in soft tissue masses,” Eur. J. Nucl. Med., vol. 26, no. 1, pp. 22–30, 1999.

[4] J. W. Keyes, “SUV: Standard Uptake or Silly Useless Value?,” J. Nucl. Med. , vol. 36 , no. 10 , pp. 1836–1839, Oct. 1995.

[5] N. Freedman, S. Sundaram, K. Kurdziel, J. Carrasquillo, M. Whatley, J. Carson, D. Sellers, S. Libutti, J. Yang, and S. Bacharach, “Comparison of SUV and Patlak slope for monitoring of cancer therapy using serial PET scans,” Eur. J. Nucl. Med. Mol. Imaging, vol. 30, no. 1, pp. 46–53, 2003.

[6] H. Young, R. Baum, U. Cremerius, K. Herholz, O. Hoekstra, A. A. Lammertsma, J. Pruim, and P. Price, “Measurement of clinical and subclinical tumour response using [18F]-fluorodeoxyglucose and positron emission tomography: review and 1999 EORTC recommendations,” Eur. J. Cancer, vol. 35, no. 13, pp. 1773–1782, Jun. 2014.

[7] (Henry) Sung-Cheng Huang, “Anatomy of SUV,” Nuclear medicine and biology, vol. 27, no. 7. Pergamon Press, pp. 643–646, 01-Oct-2000.

[8] C. S. Patlak, R. G. Blasberg, and J. D. Fenstermacher, “Graphical Evaluation of Blood-to-Brain Transfer Constants from Multiple-Time Uptake Data,” J Cereb Blood Flow Metab, vol. 3, no. 1, pp. 1–7, Mar. 1983.

[9] L. Sokoloff, M. Reivich, C. Kennedy, M. H. Des Rosiers, C. S. Patlak, K. D. Pettigrew, O. Sakurada, and M. Shinohara, “The [14C] deoxyglucose method for the measurement of local cerebral glucose utilization: theory, procedure, and normal values in the conscious and anesthetized albino rat1,” J. Neurochem., vol. 28, no. 5, pp. 897–916, May 1977.

[10] G. Blomqvist, S. Pauli, L. Farde, L. Ericksson, A. Persson, and C. Halldin, “Clinical research and clinical diagnosis,” in Dynamic Models of Reversible Ligand Binding, C. Beckers, A. Goffinet, and A. Bo, Eds. (New York: Kluwer), 1989.

[11] M. Sefidgar, M. Soltani, K. Raahemifar, H. Bazmara, S. Nayinian, and M. Bazargan, “Effect of tumor shape, size, and tissue transport properties on drug delivery to solid tumors,” J. Biol. Eng., vol. 8, no. 1, p. 12, 2014.

[12] L. T. Baxter and R. K. Jain, “Transport of Fluid and Macromolecules in Tumors III. Role of Binding and Metabolism,” Microvasc. Res., vol. 41, pp. 5–23, 1991.

[13] L. T. Baxter and R. K. Jain, “Transport of fluid and macromolecules in tumors. IV. A microscopic model of the perivascular distribution.,” Microvasc. Res., vol. 41, no. 2, pp. 252–272, Mar. 1991.

[14] L. T. Baxter and R. K. Jain, “Transport of fluid and macromolecules in tumors. (I) role of interstitial pressure and convection,” Microvasc. Res., vol. 37, pp. 77–104, 1989.

[15] L. T. Baxter and R. K. Jain, “Transport of fluid and macromolecules in tumors. (II) role of heterogeneous perfusion and lymphatics,” Microvasc. Res., vol. 40, pp. 246–263, 1990.

[16] K. N. Magdoom, G. L. Pishko, L. Rice, C. Pampo, D. W. Siemann, and M. Sarntinoranont, “MRI-based computational model of heterogeneous tracer transport following local infusion into a mouse hind limb tumor.,” PLoS One, vol. 9, no. 3, p. e89594, Jan. 2014.

onk

offk

Page 11: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

[17] G. L. Pishko, G. W. Astary, J. Zhang, T. H. Mareci, and M. Sarntinoranont, “Role of convection and diffusion on DCE-MRI parameters in low leakiness KHT sarcomas.,” Microvasc. Res., vol. 84, no. 3, pp. 306–13, Nov. 2012.

[18] G. L. Pishko, G. W. Astary, T. H. Mareci, and M. Sarntinoranont, “Sensitivity Analysis of an Image-Based Solid Tumor Computational Model with Heterogeneous Vasculature and Porosity,” Ann Biomed Eng., vol. 39, no. 9, pp. 2360–2373, 2011.

[19] T. Stylianopoulos and R. K. Jain, “Combining two strategies to improve perfusion and drug delivery in solid tumors,” PANS, vol. 110, no. 46, pp. 18632–18637, 2013.

[20] T. Stylianopoulos, K. Soteriou, D. Fukumura, and R. K. Jain, “Cationic nanoparticles have superior transvascular flux into solid tumors: insights from a mathematical model.,” Ann. Biomed. Eng., vol. 41, no. 1, pp. 68–77, Jan. 2013.

[21] C. J. Kelly and M. Brady, “A model to simulate tumour oxygenation and dynamic [ 18F ] -Fmiso PET data,” Phys. Med. Biol., vol. 51, pp. 5859–5873, 2006.

[22] D. Monnich, E. G. C. Troost, J. H. A. M. Kaanders, W. J. G. Oyen, M. Alber, and D. Thorwarth, “Modelling and simulation of the influence of acute and chronic hypoxia on [ 18 F ] fluoromisonidazole PET imaging,” Phys. INMEDICINE Biol., vol. 57, pp. 1675–1684, 2012.

[23] D. Monnich, E. G. C. Troost, J. H. A. M. Kaanders, W. J. G. Oyen, M. Alber, and D. Thorwarth, “Modelling and simulation of [ 18 F ] fluoromisonidazole dynamics based on histology-derived microvessel maps,” Phys. INMEDICINE Biol., vol. 56, pp. 2045–2057, 2011.

[24] M. Soltani and P. Chen, “Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network.,” PLoS One, vol. 8, no. 6, p. e67025, Jan. 2013.

[25] S. R. Cherry, J. A. Sorenson, and M. E. Phelps, “chapter 21 - Tracer Kinetic Modeling,” S. R. C. A. S. E. B. T.-P. in N. M. (Fourth E. Phelps, Ed. Philadelphia: W.B. Saunders, 2012, pp. 379–405.

[26] M. Bentourkia and H. Zaidi, “Tracer Kinetic Modeling in PET,” PET Clin., vol. 2, no. 2, pp. 267–277, Apr. 2007.

[27] A. Rahmim, Y. Zhou, J. Tang, L. Lu, V. Sossi, and D. F. Wong, “Direct 4D parametric imaging for linearized models of reversibly binding PET tracers using generalized AB-EM reconstruction,” Phys. Med. Biol., vol. 57, no. 3, p. 733, 2012.

[28] L. Bokacheva, K. Kotedia, M. Reese, S.-A. Ricketts, J. Halliday, C. H. Le, J. A. Koutcher, and S. Carlin, “Response of HT29 Colorectal Xenograft Model to Cediranib Assessed with 18 F-FMISO PET, Dynamic Contrast-Enhanced and Diffusion-Weighted MRI,” NMR Biomed, vol. 26, no. 2, pp. 151–163, 2013.

[29] W. Wang, J.-C. Georgi, S. a Nehmeh, M. Narayanan, T. Paulus, M. Bal, J. O’Donoghue, P. B. Zanzonico, C. R. Schmidtlein, N. Y. Lee, and J. L. Humm, “Evaluation of a compartmental model for estimating tumor hypoxia via FMISO dynamic PET imaging.,” Phys. Med. Biol., vol. 54, no. 10, pp. 3083–99, May 2009.

[30] M. Sefidgar, K. Raahemifar, H. Bazmara, M. Bazargan, S. M. Mousavi, and M. Soltani, “Effect of remodeled tumor-induced capillary network on interstitial flow in cancerous tissue,” in 2nd Middle East Conference on Biomedical Engineering, 2014, pp. 212–215.

[31] M. Sefidgar, M. Soltani, H. Bazmara, M. Mousavi, M. Bazargan, and A. Elkamel, “Interstitial Flow in Cancerous Tissue: Effect of Considering Remodeled Capillary Network,” J. Tissue Sci. Eng., vol. 4, no. 3, pp. 1–8, 2014.

[32] M. Soltani and P. Chen, “Effect of Matrix Density and Matrix Degrading Enzymes in Continuous and Discrete Mathematical Models of Angiogenesis,” WULFENIA, p. Submitted, 2014.

[33] A. R. Anderson and M. A. Chaplain, “Continuous and discrete mathematical models of tumor-induced angiogenesis.,” Bull. Math. Biol., vol. 60, no. 5, pp. 857–899, Sep. 1998.

[34] M. Soltani and P. Chen, “Effect of tumor shape and size on drug delivery to solid tumors,” J. Biol. Eng., vol. 6, no. 4, p. 4, Apr. 2012.

[35] M. Soltani and P. Chen, “Numerical Modeling of Fluid Flow in Solid Tumors,” PLoS One, vol. 6, no. 6, pp. 1–15, Jan. 2011.

[36] D. Goldman, “Theoretical models of microvascular oxygen transport to tissue.,” Microcirculation, vol. 15, no. 8, pp. 795–811, Nov. 2008.

[37] I. F. Tannock, “Oxygen diffusion and the distribution of cellular radiosensitivity in tumours,” Br. J. Radiol., vol. 45, no. 535, pp. 515–524, Jul. 1972.

[38] A. Daşu, I. T. Daşu, and M. Karlsson, “Theoretical simulation of tumour oxygenation and results from acute and chronic hypoxia,” Phys. Med. Biol., vol. 48, no. 17, p. 2829, 2003.

[39] T. Secomb, R. Hsu, E. H. Park, and M. Dewhirst, “Green’s Function Methods for Analysis of Oxygen Delivery to Tissue by Microvascular Networks,” Ann. Biomed. Eng., vol. 32, no. 11, pp. 1519–1529, 2004.

[40] J. S. Rasey, W.-J. Koh, J. R. Grierson, Z. Grunbaum, and K. A. Krohn, “Radiolabeled fluoromisonidazole as an imaging agent for tumor hypoxia,” International journal of radiation oncology, biology, physics, vol. 17, no. 5. Elsevier Science Inc., pp. 985–991, 01-Nov-1989.

[41] M. Bruehlmeier, U. Roelcke, P. A. Schubiger, and S. M. Ametamey, “Assessment of Hypoxia and Perfusion in Human Brain Tumors Using PET with 18F-Fluoromisonidazole and 15O-H2O ,” J. Nucl. Med. , vol. 45 , no. 11 , pp. 1851–1859, Nov. 2004.

[42] H. Backes, M. Walberer, H. Endepols, B. Neumaier, R. Graf, K. Wienhard, and G. Mies, “Whiskers area as extracerebral reference tissue for quantification of rat brain metabolism using (18)F-FDG PET: application to focal cerebral ischemia.,” J. Nucl. Med., vol. 52, no. 8, pp. 1252–60, Aug. 2011.

[43] P. E. Huber, M. Bischof, S. Heiland, P. Peschke, R. Saffrich, H. Gro, K. E. Lipson, and A. Abdollahi, “Trimodal Cancer Treatment: Beneficial Effects of Combined Antiangiogenesis , Radiation , and Chemotherapy,” Cancer Res., vol. 65, no. 9, pp. 3643–3655, 2005.

[44] R. N. Pittman, “Oxygen transport and exchange in the microcirculation.,” Microcirculation, vol. 12, no. 1, pp. 59–70, 2005.

[45] A. G. Tsai, B. Friesenecker, M. C. Mazzoni, H. Kerger, D. G. Buerk, P. C. Johnson, and M. Intaglietta, “Microvascular and tissue oxygen gradients in the rat mesentery,” Proc. Natl. Acad. Sci. , vol. 95 , no. 12 , pp. 6590–6595, Jun. 1998.

[46] G. a M. S. van Dongen, A. J. Poot, and D. J. Vugts, “PET imaging with radiolabeled antibodies and tyrosine kinase inhibitors: immuno-PET and TKI-PET.,” Tumour Biol., vol. 33, no. 3, pp. 607–15, Jun. 2012.

[47] R. Carson, “Tracer Kinetic Modeling in PET,” in Positron Emission Tomography SE - 6, D. Bailey, D. Townsend, P. Valk, and M. Maisey, Eds. Springer London, 2005, pp. 127–159.

[48] W. Sha, “Quantitative Analysis of Biological Effects on 18 F-FDG Uptake in Tumors: from In-vitro to In-vivo,” University of California, Los Angeles, 2012.

[49] S. Gu, G. Chakraborty, K. Champley, A. M. Alessio, J. Claridge, R. Rockne, M. Muzi, K. a Krohn, A. M. Spence, E. C. Alvord, A. R. a Anderson, P. E. Kinahan, and K. R. Swanson, “Applying a patient-specific bio-mathematical model of glioma growth to develop virtual [18F]-FMISO-PET images.,” Math. Med. Biol., vol. 29, no. 1, pp. 1–18, Mar. 2012.

[50] A. Rahmim and H. Zaidi, “PET versus SPECT: strengths, limitations and challenges,” Nucl. Med. Commun., vol. 29, no. 3, pp. 193–207, 2008.

[51] A. Rahmim, J. Qi, and V. Sossi, “Resolution modeling in PET imaging: Theory, practice, benefits, and pitfalls,” Med. Phys., vol. 40, no. 6, p. 064301, 2013.

[52] G. A. Truskey, F. Yuan, and D. F. Katz, Transport Phenomena in Biological Systems. Pearson Education, 2004.

[53] E. A. Swabb, J. Wei, and P. M. Gullino, “Diffusion and Convection in Normal and Neoplastic Tissues,” Cancer Res. , vol. 34 , no. 10 , pp. 2814–2822, Oct. 1974.

[54] X. Zhao and M. N. Toksoz, “Solute Transport In Heterogeneous Porous Media,” Massachusetts Inst. Technol. Earth Resour. Lab., 1994.

[55] L. Wu, B. Gao, Y. Tian, and R. Muñoz-Carpena, “Analytical and experimental analysis of solute transport in heterogeneous porous media,” J. Environ. Sci. Heal. Part A, vol. 49, no. 3, pp. 338–343, Nov. 2013.

Page 12: Comprehensive Modeling of the Spatiotemporal Distribution ...rahmim/research_work/Soltani...of solute transport in compartmental modeling involving four compartments, also referred

[56] R. K. Jain, “Transport of Molecules in the Tumor Interstitium: A Review,” Cancer Res. , vol. 47 , no. 12 , pp. 3039–3051, Jun. 1987.

[57] R. Jain, “Transport of molecules across tumor vasculature,” Cancer Metastasis Rev., vol. 6, no. 4, pp. 559–593, 1987.

[58] A. Taylor and D. Granger, “Exchange of macromolecules across the microcirculation,” in Handbook of Physiology. The Cardiovascular System. Microcirculation, E. Renkin and C. Michel, Eds. Washington, DC, American Physiological Society., 1984, pp. 467–520.

[59] O. Kedem and A. Katchalsky, “Thermodynamic analysis of the permeability of biological membranes to non-electrolytes.,” Biochim Biophys Acta., vol. 27, no. 2, pp. 229–246, 1958.

[60] C. C. Michel, “Starling: the formulation of his hypothesis of microvascular fluid exchange and its significance after 100 years,” Exp. Physiol. , vol. 82 , no. 1 , pp. 1–30, Jan. 1997.

[61] C. S. Patlak, D. A. Goldstein, and J. F. Hoffman, “The flow of solute and solvent across a two-membrane system,” J. Theor. Biol., vol. 5, no. 3, pp. 426–442, Nov. 1963.

[62] R. K. Jain and L. T. Baxter, “Mechanisms of heterogeneous distribution of monoclonal antibodies and other macromolecules in tumors: significance of elevated interstitial pressure,” Cancer. Res., vol. 48, no. 24 Pt 1, pp. 7022–7032, Dec. 1988.

[63] Y. Li, J. Wang, M. G. Wientjes, and J. L.-S. Au, “Delivery of nanomedicines to extracellular and intracellular compartments of a solid tumor,” Adv. Drug Deliv. Rev., vol. 64, no. 1, pp. 29–39, Jan. 2012.

[64] S. Jang, M. G. Wientjes, D. Lu, and J.-S. Au, “Drug Delivery and Transport to Solid Tumors,” Pharm. Res., vol. 20, no. 9, pp. 1337–1350, 2003.