Comparison of Kinetic Models for Dual-Tracer Receptor Concentration Imaging in Tumors

Research Article

Austin J Biomed Eng. 2014;1(1): 1002.

Comparison of Kinetic Models for Dual–Tracer Receptor Concentration Imaging in Tumors

Nazanin Hamzei1, Kimberley S Samkoe2,3, Jonathan T Elliott2, Robert W Holt2, Jason R Gunn2, Ting-Yim Lee4 Tayyaba Hasan5, Brian W Pogue 2,3,4 and Kenneth M Tichauer1*

1Department of Biomedical Engineering, Illinois Institute of Technology, USA

2Department of Medical Biophysics, Dartmouth College, USA

3Department of Surgery, Dartmouth Medical School, USA

4Department of Medical Biophysics, Western University, Canada

5Department of Dermatology, Massachusetts General Hospital, USA

*Corresponding author: :Kenneth M Tichauer, Biomedical Engineering, Illinois Institute of Technology, 3255 S Dearborn St, Chicago, IL, 60616, USA.

Received: January 20, 2014; Accepted: February 24, 2014; Published: March 05, 2014


Molecular differences between cancerous and healthy tissue have become key targets for novel therapeutics specific to tumor receptors. However, cancer cell receptor expression can vary within and amongst different tumors, making strategies that can quantify receptor concentration in vivo critical for the progression of targeted therapies. Recently a dual–tracer imaging approach capable of providing quantitative measures of receptor concentration in vivo was developed. It relies on the simultaneous injection and imaging of receptortargeted tracer and an untargeted tracer (to account for non–specific uptake of the targeted tracer). Early implementations of this approach have been structured on existing “reference tissue” imaging methods that have not been optimized for or validated in dual–tracer imaging. Using simulations and mouse tumor model experimental data, the salient findings in this study were that all widely used reference tissue kinetic models can be used for dual–tracer imaging, with the linearized simplified reference tissue model offering a good balance of accuracy and computational efficiency. Moreover, an alternate version of the full two–compartment reference tissue model can be employed accurately by assuming that the K1s of the targeted and untargeted tracers are similar to avoid assuming an instantaneous equilibrium between bound and free states (made by all other models).

Keywords: Dual–Tracer; Tumors; GARTM; BFM


In cancer research, 95% of new therapeutics fail to demonstrate significant outcomes in clinical trials and are therefore abandoned after substantial investment [1,2], even though many of these therapeutics are designed to target cancer–specific receptors, being the products of highly sophisticated studies in cancer molecular expression [3]. While there is no consensus as to why so many drugs are failing clinical trials, it is clear that drug developers require new non–invasive methods to quantify cancer receptor concentrations in vivo in order to better understand the relationship between receptor availability, and drug targeting and binding [4]. Unfortunately, it has been difficult to extract quantitative information about tumor receptor concentrations with conventional molecular imaging strategies. They typically involve injecting a subject with an imaging tracer targeted to a receptor of interest, waiting some duration of time for any unbound tracer to exit the tissues, and assuming the remaining measured signal arises from tracer that is bound to its specific receptor. The problem is that drug delivery research in oncology has demonstrated that many physiological and pathophysiological factors (e.g., blood flow, vascular permeability, interstitial pressure, and lymphatic drainage) can significantly influence the uptake of a targeted tracer in a tumor [5–8].

In response, “dual–tracer” imaging utilizes the uptake of a second tracer, similar to the targeted tracer but designed to be untargeted, to account for any non–receptor mediated uptake of the targeted tracer [9–11]. This approach was recently advanced by the development and validation of the first imaging methodology capable of quantifying receptor concentrations tumors [12]. The importance of using this “dual–tracer” approach over “reference tissue” approaches which have been used for over a decade in brain studies to quantify neurotransmitter receptor concentrations [13] was also demonstrated to be critical when attempting to quantify receptor concentration in tumors [14].

To date, the dual–tracer Receptor Concentration Imaging (RCI) approaches have rather indiscriminately employed one of the two early reference tissue models, Lammertsma and Hume’s “simplified reference tissue model” [15] and Logan et al.’s “graphical analysis” approach [16], for no other reason than that they were easily adaptable to the dual–tracer framework. Even though many of the assumptions made in reference tissue models hold for dual–tracer RCI, it is not necessary that these models are optimal since additional assumptions can be made with dual–tracer RCI: e.g., that the delivery rates (K1) of both tracers are the same if the chemical properties of the tracers are similar. Using both simulated and experimental data, the current study was carried out to identify the optimal data analysis workflow for translating targeted and untargeted tracer uptake curves in tumors to receptor concentration images, with particular emphasis on noise characteristics and computational cost of kinetic model data fitting.


Compartment models for dual–tracer kinetic analyses.

Reference tissue compartment models are ideally suited for dualtracer RCI since the setup of the dual–tracer compartment model (Figure 1) is nearly identical to that of the reference tissue model [15]. Both models recognize that non–specific uptake of a targeted tracer can significantly affect the relationship between tracer uptake and tracer binding or receptor concentration. The reference tissue model accounts for binding by employing the temporal uptake of the targeted tracer in a region devoid of targeted receptor (reference tissue) to account for non–specific uptake; while the dual–tracer approach employs the uptake of a second tracer, similar in structure to the targeted tracer but untargeted, in the same tissue as the targeted tracer to account for non–specific uptake. On the surface it would seem that whatever kinetic model was best for one approach would also be best for the other, but there are subtle differences between the approaches that can impact the choice of the optimal kinetic model:

  1. The plasma input function, Cp: in the reference tissue model, the reference input and the region–of–interest input intrinsically have the same plasma input function, so this is not a concern; however, in the dual–tracer model, both tracers used must have the same plasma kinetics of the course of imaging.
  2. K1⁄k2 equivalency: in reference tissue models, it is assumed that the ratio of the tracer’s extravasation and tissue–efflux rates, K1 and k2, are equivalent in the reference tissue and the region of interest; whereas, dual–tracer models assumes that these leakage kinetics are the same between tracers in all tissues.

In this study, six different reference tissue models are evaluated in terms of their ability to accurately and efficiently estimate tumor cellsurface receptor concentration from dual–tracer data. The models included 1) the “Full Reference Tissue Model” [FRTM] [17,18], later modified to a “reduced Full reference Tissue model” [Reduced FRTM] 2) the “Simplified Reference Tissue Model” [SRTM] [15], 3) the original Graphical Analysis Reference Tissue Model [GARTM] [16], 4) a linearized version of the SRTM [SRTM_lin] [19], 5) a modification to the GARTM [GARTM_mod] [20], and 6) the “Basis Function Method” [BFM][21]. While a full derivation of these six models is outside of the scope of this article, a presentation of the key mathematical expressions converted to a dual–tracer nomenclature are provided below. The FRTM can be expressed as:

RO I T ( t )= R 1 [ RO I U ( t )+aRO I U ( t ) e ct +bRO I U ( t ) e dt ], with a=( k 3 + k 4 c )( cr )/p b=( d k 3 k 4 )( dr )/p c=( s+p )/2 d=(sp)/2 p= s 2 q q=4 k 2 k 4 r= k 2 / R 1 s= k 2 + k 3 + k 4 = convolution integral,                                                                                                                                                  (1) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@38B5@

Where ROIT(t) and ROIU(t) represent the measured uptake curves of the targeted and untargeted tracers, respectively, in any region of interest, as a function of time, t; R1 is the ratio of the rates of extravasation (K1) of the targeted tracer and the untargeted tracer; k2 is the rate of efflux of the targeted tracer; and k3 and k4 are the rates of association and dissociation of the targeted tracer, respectively (Figure 1).

Likewise, the SRTM can be expressed as:

RO I T ( t )= R 1 RO I U ( t )+ k 2 ( 1 R 1 1+BP )RO I U ( t ) e k 2 1+BP t                                                  (2) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8292@

Where BP, the “binding potential”, is equivalent to k3⁄k4, and is a key parameter since it represents the product of the receptor concentration (the parameter of interest) and the affinity of the targeted tracer for its receptor (which can in most cases be measured ex vivo) [13]. Going on, the format of the GARTM can be represented by:

0 t RO I T ( u )du RO I T ( t ) =( 1+BP ) 0 t RO I U ( u )du + RO I U ( t ) k 2 RO I T ( t ) +int,                                 (3) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@85FC@

Where u is a dummy time variable to integrate over, int is an often neglected intercept term in this linear relationship with slope 1+BP at time, t > t*, where t* represents the time it takes for the Cf and Cb to reach a constant ratio (quasi–equilibrium). The format of the SRTM_lin can be expressed as follows:

RO I T ( t )=[ R 1 RO I U ( t )+ k 2 0 t RO I U ( u )du ]( k 2 1+BP ) 0 t RO I T ( u )du .                               (4) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@821C@

Furthermore, the format of the GARTM_mod can be expressed as follows:

0 t RO I T ( u )du RO I U ( t ) =( 1+BP ) 0 t RO I U ( u )du RO I U ( t ) +int',                     (5) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaadaWdXbqaaiaadkfacaWGpbGaamysamaaBaaaleaacaWGubaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaiaadsgacaWG1baaleaacaaIWaaabaGaamiDaaqdcqGHRiI8aaGcbaGaamOuaiaad+eacaWGjbWaaSbaaSqaaiaadwfaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaiabg2da9maabmaabaGaaGymaiabgUcaRiaadkeacaWGqbaacaGLOaGaayzkaaWaaSaaaeaadaWdXbqaaiaadkfacaWGpbGaamysamaaBaaaleaacaWGvbaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaiaadsgacaWG1baaleaacaaIWaaabaGaamiDaaqdcqGHRiI8aaGcbaGaamOuaiaad+eacaWGjbWaaSbaaSqaaiaadwfaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaiabgUcaRiaacMgacaGGUbGaaiiDaiaacEcacaGGSaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabwdacaqGPaaaaa@749B@

Where int’ represents another neglected intercept that is different in composition than the one in Equation (3).

Finally, Gunn’s Basis Function Method (BFM) is derived from Equation (2) and is formulated as:

RO I T (t)= θ 1 RO I U (t)+ θ 2 B i (t)        (6) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaad+eacaWGjbWaaSbaaSqaaiaadsfaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0JaeqiUde3aaSbaaSqaaiaaigdaaeqaaOGaamOuaiaad+eacaWGjbWaaSbaaSqaaiaadwfaaeqaaOGaaiikaiaadshacaGGPaGaey4kaSIaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGaamOqamaaBaaaleaacaWGPbaabeaakiaacIcacaWG0bGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabAdacaqGPaaaaa@5443@

whereθ1= R1, Θ2 = k2– R1k2/(1+BP), Bi are the so–called basis functions defined as:

Bi(t)=RO I U (t) e θ 3,i t        (7) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaSGaamyAaOGaaiikaiaadshacaGGPaGaeyypa0JaamOuaiaad+eacaWGjbWaaSbaaSqaaiaadwfaaeqaaOGaaiikaiaadshacaGGPaGaey4LIqSaamyzamaaCaaaleqabaGaeyOeI0IaeqiUde3aaSbaaWqaaiaaiodacaGGSaGaamyAaaqabaWccaWG0baaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaaeykaaaa@50CE@

andθ3 = k2/(1+BP); so that Equation (6) can be optimized for θ1and θ2 in a linear least squares sense; provided that θ3 is varied iteratively over a specified range.

Materials and Methods

Animal experiments

Targeted and untargeted tracer uptake curves were measured in two different tumor lines grown subcutaneously in athymic mice (n = 10, Charles River, Wilmington, MA). The targeted tracer was a ligand for the Epidermal Growth Factor Receptor (EGFR), a receptor that is over expressed in many cancers [22]. Specifically, the tracer was a near–infrared fluorescent molecule bound to native epidermal growth factor, IRDye–800CW–EGF (LI–COR Biosciences, Lincoln, NE). The untargeted tracer was a free near–infrared fluorescent tracer emitting fluorescence at a separate wavelength, IRDye–700DX carboxylate (LI–COR Biosciences). The two different tumor lines were selected so as to represent different levels of Epidermal Growth Factor Receptor (EGFR) and were each implanted into five of the ten immune–deficient mice (Charles River, Wilmington, MA). Five mice were inoculated with a human neuronal glioblastoma (U251; supplied from Dr. Mark Israel, Norris Cotton Cancer Center, Dartmouth–Hitchcock Medical Center), a cancer cell line known to express moderate levels of EGFR [23,24]; and another five mice were inoculated with a human epidermoid carcinoma (A431; ATCC, Manassas, VA), known to express a very large amount of EGFR [25]. In all cases, the tumors were introduced by injecting 1x106 tumor cells in Matrigel ®(BD Biosciences, San Jose, CA) into the subcutaneous space on the left thigh of the mice. The tumors were then allowed to grow to a size of approximately 150 mm3 before imaging.

The mice were anesthetized with ketamine–xylazine (100 mg⁄kg: 10 mg⁄kg i.p.) and the superficial tissue surrounding the tumors was removed. Each mouse was then placed tumor–side down on a glass slide and loosely secured with surgical tape (Figure 1). Once plated, the mice were positioned onto the imaging plane of an Odyssey Scanner (LI–COR Biosciences, Lincoln, NE). The Odyssey Scanner employs raster scanning and two lasers (one emitting at 685 nm and another at 785 nm) to excite two fluorophores simultaneously, pixel–by–pixel, and utilizes a series of dichroic mirrors to decouple fluorescence from the LI–COR 680 or 700 nm fluorescent tracers and the LI–COR 800 nm fluorescent tracer, respectively. All mice were injected with a cocktail of 1 nanomole of an EGFR targeted fluorescent tracer and 1 nanomole of an untargeted fluorescent tracer: the untargeted tracer was a carboxylate form of the IR Dye 700DX NHS Ester (LI–COR Biosciences, Lincoln, NE) that was reacted with water for 3h at room temperature to convert the reactive NHS Ester to a non–reactive carboxylate (as per manufacturer’s instructions to reduce nonspecific binding), and the targeted tracer was IR Dye 800CW–EGF (LI–COR Biosciences, Lincoln, NE). The mice were then imaged at approximately 3–min intervals for 1h after injection of the fluorescent tracers.