1.

Introduction

The development of more quantitative methods to noninvasively measure drug-targetable biomolecules in cancerous tissues could significantly assist the advancement of precision medicine.¹ Overexpressed cancer cell surface receptors serve as strong prognostic biomarkers in a spectrum of solid tumors. Accordingly, targeting cell surface receptors has become a major focus for molecular imaging and therapy.² Targeted imaging agents can detect the overexpressed receptors on the surface of cancer cells dynamically through noninvasive imaging techniques at least qualitatively.³ Applying quantitative analyses to the qualitative imaging data would potentially establish a standard identification tool to select the targeted drug for individual patients. For example, epidermal growth factor receptor (EGFR), a receptor that is overexpressed on the surface of many cancerous cell types, controls cellular proliferation by its tyrosine kinase signalling pathways.⁴ As a result, quantifying the concentration of available EGFR receptors could assist in guiding the use of therapies capable of targeting these pathways.⁵ Recently, paired-agent imaging techniques have been demonstrated to be able to quantify cancer cell surface receptors in vivo.^6,7 In these methods, an untargeted “control” imaging agent with chemical characteristics similar to the targeted agent is coadministered with the targeted imaging agent. The signal of the control agent can then be used to account for the effects of hemodynamics and nonspecific uptake and retention on the targeted imaging agent signal, allowing the binding kinetics to be isolated.⁸ To image both the targeted and control agent signals concurrently in optical imaging applications, the two agents are typically labeled with different fluorophores that have distinguishable excitation and emission peak wavelengths, allowing for signal discrimination based on relatively straightforward filtering. Therefore, to convert the fluorescence signal intensities to their respective imaging agent concentrations in tissue, different correction factors (represented as ${\eta }_T$ and ${\eta }_C$ for the targeted and control imaging agents, respectively) must be considered for each imaging agent.

The correction factor differences are affected by a number of parameters: including the power of the light source at the respective excitation wavelengths, the sensitivity of the detector(s) at the respective emission wavelengths, the quantum efficiencies of the respective fluorophores used, and the differences in tissue optical properties at the respective excitation/emission wavelengths. Although light source, detection efficiency, and quantum efficiency effects can be accounted for by imaging a calibration standard/solution of known fluorophore concentrations with the system employed, accounting for optical property effects is more complicated. Tissue optical properties in the near-infrared are dominated by blood volume and oxygenation, and photon scatter, factors that can be spatially heterogeneous, particularly in cancerous tissues or around cancer margins, owing to the abnormal effects cancers have on angiogenesis and vasculogenesis.⁹ In previous work, a pixel-by-pixel normalization was developed to account for differences in the correction factors.¹⁰ This method assumed that the control and the targeted signals should be equal to each other in all locations at a very early time point after injection ($<2\min$), as preferential retention of the targeted agent in regions overexpressing the targeted biomolecule would not be evident until the washout phase. One of the problems with pixel-by-pixel normalization is that it is highly dependent on the time of normalization after administration of the imaging agents, as demonstrated in this work, and it also assumes that the more dynamic first pass kinetics of the targeted and control agents are equivalent.

In this study, we present a kinetic model we have named the “linearized scaling factor simplified paired-agent model” (${\mathrm{SPAM}}_{\mathrm{SF}}$) in reference to its similarity to the simplified reference tissue model, SRTM,¹¹ and specifically the linearized version of the model,¹² widely used in positron emission tomography (PET) studies. One key difference of all paired-agent applications is the replacements of the reference tissue input with the colocalized reference (control) imaging agent signal input. This is typically employed predominantly for cancer imaging owing to the difficulty in selecting suitable reference tissues for cancer.¹³ Existing paired-agent kinetic modeling methods based on SRTM require the targeted and control imaging agent signals as a function of time to estimate three parameters: $R_1$, the ratio of extravasation rate constants for the targeted and control agents ($K_{1,T}/K_{1,C}$); $k_{2,T}$, the tissue-to-blood efflux rate constant of the targeted agent; and BP, the “nondisplaceable” binding potential, a parameter that is proportional to the product of the targeted imaging agent affinity and the concentration of the targeted biomolecule.¹⁴ However, in ${\mathrm{SPAM}}_{\mathrm{SF}}$, it is assumed that $R_1=1$, and instead the ratio of $\alpha ={\eta }_T/{\eta }_C$ is incorporated as a fitting parameter. Computer simulations and mouse models of human cancer were employed to establish the validity of BP estimation with ${\mathrm{SPAM}}_{\mathrm{SF}}$, and its precision and accuracy were compared with linearized SRTM directly applied to paired-agent data (SPAM) and the pixel-by-pixel normalization SPAM method (${\mathrm{SPAM}}_{\mathrm{PN}}$).

Here, we present a pure blind identification method termed the “multiple reference tissue method” (MRTM). As in other approaches, the MRTM utilizes two or more reference tissues to estimate the AIF, but in this case assumes only that the AIFs in the reference tissues have the same shape, with a possible difference in bolus arrival time. Importantly, the MRTM is a mathematical framework equipped with efficient algorithms that can be applied to any kinetic model, including nonlinear models, and to any CE imaging modality. The constructed AIF is smooth and has high temporal resolution. Because there is no limitation on the kinetic models that can be utilized, the tumors themselves can be used as a source of reference tissues. In fact, the inherent heterogeneity of tumors can be exploited by utilizing individual subregions with different properties as “reference tissues.” This ability avoids the necessity of including multiple normal reference tissues in the FOV.

2.

Theory

Paired-agent kinetic modeling is designed to estimate binding potential (BP)—a parameter directly proportional to the targeted receptor concentration—in molecular imaging studies, and in particular, cancer molecular imaging studies.⁷ To accurately extract quantitative parameters, the dynamic behavior of a targeted and control imaging agent pair can be represented by a system of equations, assuming “trace” concentrations of the targeted imaging agent and instantaneous spatial mixing of imaging agent concentrations in each “compartment” as defined by the model in Fig. 1¹⁵

Fig. 1

A schematic of subcutaneous mouse tumor model is shown. The targeted and control imaging agent compartmental models are shown on the left and right, respectively. $C_{p,T}(t)$ and $C_{p,C}(t)$ represent the blood plasma concentrations of targeted and control imaging agents, respectively, as a function of time, $t$; $C_{f,T}$ and $C_{f,C}$ represent the free (unbound) concentrations of the two imaging agents, $C_{b,T}$ represents the bound concentration of the targeted agent; $K_{1,T}$ and $K_{1,C}$ represent rate constants governing targeted and control imaging agent extravasation (transport from blood to extravascular extracellular space); $k_{2,T}$ and $k_{2,C}$ represent rate constants governing targeted and control agent tissue efflux (transport from extravascular extracellular space to blood); and $k_{3,T}$ and $k_{4,T}$ represent rate constants governing targeted agent binding and dissociation from the targeted biomolecule.

Three main assumptions are typically made to solve the system of differential equations in Eqs. (1)–(3), which are nearly identical to those employed in widely used reference tissue modeling:¹⁴ (1) the free and bound concentrations of the targeted imaging agent are in rapid equilibrium (known as the adiabatic approximation), such that Eqs. (1) and (2) can be combined into a single compartment equation;¹¹ (2) the magnitude of any imaging agent signal arising from the plasma compartment is negligible compared with the signal arising from the tissue compartments;¹¹ and (3) the ratios of the rate constants of extravasation and tissue efflux are the same between targeted and control imaging agent models (i.e., $K_{1,T}/k_{2,T}=K_{1,C}/k_{2,C}$). Such assumptions have led to the derivation of the linearized simplified reference tissue model (commonly referred to as SRTM) that is employed primarily in brain PET studies¹² but that has also been evaluated for paired-agent imaging,¹⁶ which is referred to as the simplified paired-agent model (SPAM) in this work and is expressed as

Paired-agent imaging requires that two additional assumptions be accurate for SRTM to be adopted for BP estimation: (1) that the plasma input functions of the targeted and control agents be similar (i.e., $C_{p,T}=C_{p,C}$)—however, methods for correction if this equivalence is not true have been developed¹⁷—and (2) that the correction factors relating tissue concentration to detected signal for both the targeted (${\eta }_T$) and control (${\eta }_C$) imaging agents be known/calculated, or at least that the ratio of the two correction factors (${\eta }_T/{\eta }_C$) be known. One method of estimating the ratio of the correction factors is by assuming that the concentrations of the targeted and control imaging agents are equivalent at a very early time point, $t_e$, typically within 2 min of imaging agent injection.¹⁰ Under this assumption, any differences in measured targeted and control agent signal at $t_e$ can be attributed to differences in the correction factors, ${\eta }_T$ and ${\eta }_C$, which can be affected by local optical properties, such that ${\eta }_T/{\eta }_C$ can be represented by ${\mathrm{ROI}}_T(t_e)/{\mathrm{ROI}}_C(t_e)$. One advantage of this method is that it can be applied on a pixel-by-pixel (or region-by-region) basis such that spatial heterogeneity in optical properties (and their effects on the correction factors) can be accounted for. This pixel-normalization method can be expressed in terms of the SPAM as ${\mathrm{SPAM}}_{\mathrm{PN}}$

However, ${\mathrm{SPAM}}_{\mathrm{PN}}$ has a few limitations. As ${\mathrm{ROI}}_T(t_e)$ and ${\mathrm{ROI}}_C(t_e)$ are collected at a single imaging time point, the variance in the ratio estimate is more sensitive to experimental noise and errors in motion correction than a multitime point normalization would be. Furthermore, binding characteristic differences between the targeted and control agents could lead to differences in tissue concentrations even at very early time points depending on the imaging agents used. As such, this work presents a scaling factor SPAM (${\mathrm{SPAM}}_{\mathrm{SF}}$) that provides a more robust correction for optical property-based effects on the correction factors ${\eta }_T$ and ${\eta }_C$.

3.

Methods

4.

Results

4.1.

Simulation Results

Examples of simulated, noise-added targeted, and control imaging agent kinetics from Eqs. (1)–(3) and corresponding fits of the data using SPAM, ${\mathrm{SPAM}}_{\mathrm{PN}}$, and ${\mathrm{SPAM}}_{\mathrm{SF}}$ models to the data are presented in Fig. 2. The average of ${\chi }^2$ values for all three model fits over a range of $\alpha$ from 0.1 to 10 was from 0.06 to 0.33 with all fits looking similar in quality to those displayed in Fig. 2.

Fig. 2

Simulation results. Generated noisy targeted and control agent curves and model fits for (a) the linearized SPAM (b) the pixel-normalization SPAM (${\mathrm{SPAM}}_{\mathrm{PN}}$), and (c) the scaling factor SPAM (${\mathrm{SPAM}}_{\mathrm{SF}}$). The blue dots represent the targeted imaging agent signal intensity, ${\mathrm{ROI}}_T(t)$. The orange dots represent the control imaging agent signal intensity, ${\mathrm{ROI}}_C(t)$. Poisson noise was added to the data. The solid red, blue, and green lines represent SPAM, ${\mathrm{SPAM}}_{\mathrm{PN}}$, and ${\mathrm{SPAM}}_{\mathrm{SF}}$ fit results, respectively. The simulated values of kinetic parameters to create these simulated curves were: $K_1=0.013{\min }^{-1}$, $k_2=0.08{\min }^{-1}$, $k_3=0.2{\min }^{-1}$, $k_4=0.1{\min }^{-1}$, and $\alpha =1$, and simulated time was 60 min.

Figure 3 shows a comparison of the accuracy of all three fitting models over a range of simulated parameters—specifically, $\alpha$, BP, $k_2$, and $R_1$—as defined by errors in estimations of simulated BP values. In these experiments, the time point for approximating $\alpha$ with ${\mathrm{SPAM}}_{\mathrm{PN}}$ [$t_e$ from Eq. (7)] was set to 1 min postimaging agent administration [Fig. 4(a) presents the effect of this selection]. While setting $R_1=1$, $k_2=0.08{\min }^{-1}$, and $\mathrm{BP}=2$, average errors in BP estimation for both ${\mathrm{SPAM}}_{\mathrm{PN}}$ and ${\mathrm{SPAM}}_{\mathrm{SF}}$ remained $<10\%$ for values of $\alpha$ between 0.1 and 10, whereas SPAM yielded average errors that were directly proportional to the percent difference in $\alpha$ from 1 [Fig. 3(a)]. For example, $\alpha =1.1$ led to a 10% overestimation in BP using SPAM, whereas $\alpha =0.5$ led to a 50% underestimation in BP. No statistically significant differences were observed between BP estimates using ${\mathrm{SPAM}}_{\mathrm{PN}}$ and ${\mathrm{SPAM}}_{\mathrm{SF}}$; however, a small negative bias in BP error using ${\mathrm{SPAM}}_{\mathrm{PN}}$ was observed, along with a variance that was more than four times larger than the error observed using ${\mathrm{SPAM}}_{\mathrm{SF}}$ (${\mathrm{SPAM}}_{\mathrm{PN}}$ average $\mathrm{BP}\text{error}=-3.25\pm 9.85\%$; ${\mathrm{SPAM}}_{\mathrm{SF}}$ average $\mathrm{BP}\text{error}=0.17\pm 2.02\%$). At $\alpha =1$, SPAM and ${\mathrm{SPAM}}_{\mathrm{SF}}$ were found to have equivalent performance. The magnitude of BP and $k_2$ had little effect on the accuracy of all three models, with $\alpha =1$, $K_1=0.013{\min }^{-1}$ [Figs. 3(b) and 3(c), respectively]. No statistically significant differences in BP estimation were observed between any of the models over a $k_2$ range of 0.03 to $0.1{\min }^{-1}$ and a BP range of 0.5 to 5.5. However, again there was a slight BP underestimation in ${\mathrm{SPAM}}_{\mathrm{PN}}$ that was amplified at higher $k_2$ values, and the variance in ${\mathrm{SPAM}}_{\mathrm{PN}}$ was more than four times that of SPAM and ${\mathrm{SPAM}}_{\mathrm{SF}}$ in all cases. Errors in the approximation that $R_1=1$ resulted in significant errors in both ${\mathrm{SPAM}}_{\mathrm{PN}}$ and ${\mathrm{SPAM}}_{\mathrm{SF}}$ which were inversely proportional to the scale of error in $R_1$ [Fig. 3(d)]. As these simulations were carried out at $\alpha =1$, SPAM performed well for all levels of $R_1$ tested, with an average BP error of $0.28\pm 3.18\%$. For testing of $R_1$ sensitivity, the $K_{1,T}/k_{2,T}=K_{1,C}/k_{2,C}$ equivalency was held, such that $R_1=1.1$ corresponded to $K_{1,T}=1.1*K_{1,C}$ and $k_{2,T}=1.1*k_{2,C}$. If only $K_{1,T}$ was changed, then ${\mathrm{SPAM}}_{\mathrm{PN}}$ and ${\mathrm{SPAM}}_{\mathrm{SF}}$ were found to be insensitive to the value of $R_1$, as long as $k_{2,T}=k_{2,C}$ (results not shown). However, SPAM exhibited errors proportional to the value of $R_1$ as it strictly requires that $K_{1,T}/k_{2,T}=K_{1,C}/k_{2,C}$, which is not held in this simulation (results not shown).

Fig. 3

Simulation results: BP estimation errors are presented for all three kinetic models tested in this study as a function of the ratios of correction factors, $\alpha$ (a), BP (b), the tissue-to-blood efflux rate constant, $k_2$ (c), and ratio of the targeted and control imaging agent blood-to-tissue extravasation rate constants, $R_1$ (d). The linearized SPAM results are presented in red, the pixel normalization SPAM (${\mathrm{SPAM}}_{\mathrm{PN}}$) results are presented in blue, and the scaling factor SPAM (${\mathrm{SPAM}}_{\mathrm{SF}}$) results are presented in green.

Fig. 4

Simulation results: (a) errors in BP estimation using the linearized pixel-normalization ${\mathrm{SPAM}}_{\mathrm{PN}}$ as a function of. Different colors correspond to sets of data created from different ratios of correction factors ($\alpha ={\eta }_T/{\eta }_C$). (b) Errors in BP estimation using ${\mathrm{SPAM}}_{\mathrm{PN}}$ and the SPAM scaling factor model (${\mathrm{SPAM}}_{\mathrm{SF}}$) as a function of time-spacing of imaging data ($t_e$) where $t_e$ also represents the first time point after agent administration with keeping $\alpha =1$. ${\mathrm{SPAM}}_{\mathrm{PN}}$ and ${\mathrm{SPAM}}_{\mathrm{SF}}$ results are presented in blue and green, respectively.

Irrespective of the ratio of correction factors ($\alpha ={\eta }_T/{\eta }_C$), the accuracy of ${\mathrm{SPAM}}_{\mathrm{PN}}$, defined by the error in estimation of the BP, was observed to be highly dependent on the time point after imaging agent delivery that was selected for normalization of $\alpha$ [Fig. 4(a)]. Specifically, the later the time point selected for normalization, the larger the underestimation in BP. Average sd in the BP estimation error at $\alpha =0.1$ was 24.10%, and at $\alpha =10$ was 5.13%. When normalization was carried out at 1-min postagent-administration for ${\mathrm{SPAM}}_{\mathrm{PN}}$ at $\alpha =1$, the average percent-error was $-3.8\pm 6.62\%$; whereas, when carrying out the normalization at 5 min the average was $-12.67\pm 4.72\%$. Although the ${\mathrm{SPAM}}_{\mathrm{SF}}$ requires no selected time of normalization (as the normalization is included as a fitting parameter), it was assumed that a delayed time of normalization in ${\mathrm{SPAM}}_{\mathrm{PN}}$ would only arise if data were not sampled as frequently after imaging agent administration. Therefore, the effect of data collection at a period of $t_e$ (time of normalization for ${\mathrm{SPAM}}_{\mathrm{PN}}$) on the accuracy of ${\mathrm{SPAM}}_{\mathrm{SF}}$ was evaluated in comparison with time of normalization for ${\mathrm{SPAM}}_{\mathrm{PN}}$ [Fig. 4(b)]. For $t_e=1\min$ at $\alpha =1$, the average percent-error of ${\mathrm{SPAM}}_{\mathrm{SF}}$ was $-0.01\pm 1.94\%$, for $t_e=5\min$, the average was $-7.60\pm 3.56\%$. An increase in the error of the ${\mathrm{SPAM}}_{\mathrm{SF}}$ with data-spacing ($t_e$) was commensurate with the expected truncation error increase from a numerical analysis perspective. Here, SPAM was not evaluated as it resulted in identical results to ${\mathrm{SPAM}}_{\mathrm{SF}}$ since $\alpha =1$.

Examples of targeted (IRDye 800CW-EGF) and control (IRDye 700DX) imaging agent signal curves in a typical U251 tumor are shown in Figs. 5(a) and 5(d), along with fit results using SPAM_PN and SPAM_SF, respectively. Examples of targeted (ABY-029) and control (IRDye 700DX-negative-control-Affibody) imaging agent signal curves in a typical U251 tumor are shown in Figs. 5(b) and 5(e), along with fit results using SPAM_PN and SPAM_SF, respectively. Examples of targeted (IRDye 800CW-anti-EGFR-Affibody) and control (IRDye 700DX-negative-control-Affibody) imaging agent signal curves in a typical orthotopic U251 tumor are shown in Figs. 5(c) and 5(f), along with fit results using SPAM_PN and SPAM_SF, respectively. Note: SPAM fits evaluated but not displayed or discussed owing to large errors attributable to $\alpha$ not equal to 1 [see Fig. 3(a)]. The average goodness of the fit, quantified by ${\chi }^2$, of SPAM_PN and SPAM_SF for all tumors were not significantly different statistically and were $0.05\pm 0.02$ and $0.08\pm 0.03$, respectively. The statistically significant correlation held for all individual animal datasets when BP estimation from SPAM_PN and SPAM_SF were compared on a pixel-by-pixel level (average $r=0.95\pm 0.06$, $p<0.001$ for all cases). Examples of correlations from a single animal from the EGF-Odyssey and Affibody-Pearl groups are shown in Figs. 5(g) and 5(h), respectively. The $\text{mean}\pm \mathrm{sd}$ of BP for SPAM_PN and SPAM_SF were $1.34\pm 0.47$ and $1.49\pm 0.52$ for the EGF-Odyssey group, respectively, $1.40\pm 0.37$ and $1.41\pm 0.44$ for the Affibody-Pearl group, respectively (when accounting for differences in EGF and anti-EGFR affinity), and $0.87\pm 0.14$ and $0.85\pm 0.14$ for the orthotopic tomography group. A statistically significant correlation was observed between BP estimates determined using SPAM_SF and SPAM_PN ($r=0.98$, $p<0.001$) when including data from all experimental groups. The correlation was not significantly different from the line-of-identity, statistically. The $\text{mean}\pm \mathrm{sd}$ for $R_1$ and $k_2$ in all SPAM_PN cases were $1.00\pm 0.03$ and $0.07\pm 0.04{\min }^{-1}$, respectively. The $\text{mean}\pm \mathrm{sd}$ for $\alpha$ and $k_2$ in all SPAM_SF cases were $0.91\pm 0.01$ and $0.06\pm 0.03{\min }^{-1}$, respectively. The BP maps from two representative mice each from the EGF-Odyssey and Affibody-Pearl groups are presented in Fig. 6.

Fig. 5

In vivo experimental results: The linearized pixel-normalization simplified paired agent model ${\mathrm{SPAM}}_{\mathrm{PN}}$ in (a) targeted (IRDye 800CW-EGF) and control (IRDye 700DX) imaging agent signal curves measured with the Odyssey System in exposed subcutaneous human glioblastoma (U251) tumors grown in athymic mice, (b) targeted (IRDye 800CW-anti-EGFR-Affibody) and control (IRDye 700DX-negative-control-Affibody) imaging agent signal curves measured with the Pearl System in exposed subcutaneous U251 tumors and (c) targeted (IRDye 800CW-anti-EGFR-Affibody) and control (IRDye 700DX-negative-control-Affibody) imaging agent signal curves measured with an MRI–FMT system from a typical tumor region-of-interest, respectively. Corresponding model fits using the scaling factor SPAM (${\mathrm{SPAM}}_{\mathrm{SF}}$) are presented in (d)–(f), respectively. The blue dots represent the targeted imaging agent signal intensity, the orange dots represent the control imaging agent signal intensity before correcting for plasma input function differences, the black dots represent the control imaging agent signal intensity after correcting for plasma input function differences. The solid light blue line represents the ${\mathrm{SPAM}}_{\mathrm{PN}}$ fit and the solid green line represents the ${\mathrm{SPAM}}_{\mathrm{SF}}$ fit. In (g) and (h) a pixel-by-pixel correlation of ${\mathrm{SPAM}}_{\mathrm{SF}}$ and ${\mathrm{SPAM}}_{\mathrm{PN}}$ BP estimates in all regions of an image on the same mouse data presented in (a) and (b) are presented. The dashed line is the line-of-identity. A correlation between BP values estimated by ${\mathrm{SPAM}}_{\mathrm{SF}}$ and ${\mathrm{SPAM}}_{\mathrm{PN}}$ in Odyssey, Pearl and tomography imaging studies are presented in (i) for average tumor region-of-interest fits.

Fig. 6

In vivo experimental results: BP parametric maps of estimated epidermal growth factor receptor (EGFR) concentration are depicted as calculated by either the linearized pixel-normalization ${\mathrm{SPAM}}_{\mathrm{PN}}$ or the scaling factor SPAM (${\mathrm{SPAM}}_{\mathrm{SF}}$). Here, results of four mice are displayed: two from the group imaged on the Odyssey System (targeted agent = IRDye 800CW-EGF; control = IRDye 700DX), and two from the group imaged on the Pearl System (targeted agent = ABY-029; control = IRDye 700DX-negative-control-affibody). The left two columns show the two mice imaged on the Odyssey, and the right two columns show the two mice imaged on Pearl.

5.

Discussion

The development of accurate, noninvasive methods to quantify cell surface receptor concentrations in cancer molecular imaging could have far-reaching impacts on the future of precision medicine,³⁰ drug development,³¹ and molecular guided surgery.³² Paired-agent kinetic modeling, where the kinetics of a control (untargeted) imaging agent that is coadministered with a targeted imaging agent are used to account for nonspecific effects, has proven great promise in being capable of quantifying cancer cell surface receptor concentrations in vivo.⁷ The paired-agent imaging kinetic models were adopted from the simplified reference tissue model (SRTM), used widely in PET studies,¹⁴ and are referred to here as SPAM. Recently, our group derived a generalized form of SRTM and SPAM, capable of accounting for receptor saturation.³³ In this work, we present a modification to SPAM in particular, allowing the automatic correction for signal scaling differences between the detected signals of targeted and control imaging agents. This advance is particularly relevant for fluorescence imaging-based applications of SPAM, where tissue optical properties can differ heterogeneously in tissue making it difficult to accurately achieve imaging agent concentration maps of both agents.

The accuracy and precision of this scaling factor SPAM (${\mathrm{SPAM}}_{\mathrm{SF}}$) was compared against an existing method that corrects for scaling differences between targeted and control imaging agent signals by normalizing the signals at a very early time point after agent administration¹⁰—assuming early signal is dominated by delivery, rather than binding, such that early targeted and control agent signals should be measured to be the same. This method is referred to here as ${\mathrm{SPAM}}_{\mathrm{PN}}$. The ${\mathrm{SPAM}}_{\mathrm{SF}}$ and ${\mathrm{SPAM}}_{\mathrm{PN}}$ models were compared both in physiologically informed, noise-added numerical simulations of targeted and control agent uptake in tumors, as well as in three different animal study groups. In each study, EGFR was targeted by either fluorescently labeled native ligand to EGFR, EGF, or with an anti-EGFR Affibody. The results demonstrated both improved accuracy and precision of BP estimation using ${\mathrm{SPAM}}_{\mathrm{SF}}$ compared with ${\mathrm{SPAM}}_{\mathrm{PN}}$ in the simulation studies, and a strong correlation between ${\mathrm{SPAM}}_{\mathrm{SF}}$ and ${\mathrm{SPAM}}_{\mathrm{PN}}$ in the animal studies.

With respect to the simulation results, ${\mathrm{SPAM}}_{\mathrm{PN}}$ exhibited a slight underestimation in BP under all conditions tested, owing to errors in the assumption that there is no preferential retention of the targeted agent over the control agent by 1-min postinjection. In the simulation studies at least, there is some divergence of the targeted and control curves at this time, which amplified over time (Fig. 4). Furthermore, the standard deviation in the BP estimates using ${\mathrm{SPAM}}_{\mathrm{PN}}$ was found to be $\sim 3$ times larger than those found for ${\mathrm{SPAM}}_{\mathrm{SF}}$. This improvement in precision of ${\mathrm{SPAM}}_{\mathrm{SF}}$ over ${\mathrm{SPAM}}_{\mathrm{PN}}$ is attributable to the fact that ${\mathrm{SPAM}}_{\mathrm{SF}}$ essentially “fits” for the scaling factor, $\alpha$, over as many time points as are in the dataset, whereas ${\mathrm{SPAM}}_{\mathrm{PN}}$ directly estimates $\alpha$ by normalizing targeted and control agent signals at a single time point. Therefore, it is expected that ${\mathrm{SPAM}}_{\mathrm{PN}}$ is considerably more sensitive to noise.

An additional finding from the simulation study was that errors in the assumption that $R_1=1$ caused significant and similar errors in the BP estimation with both ${\mathrm{SPAM}}_{\mathrm{SF}}$ and ${\mathrm{SPAM}}_{\mathrm{PN}}$: specifically, BP estimation using ${\mathrm{SPAM}}_{\mathrm{PN}}$ and ${\mathrm{SPAM}}_{\mathrm{SF}}$ was found to be inversely proportional to the scale of error in the $R_1=1$ assumption [Fig. 3(d)]. The sensitivity of ${\mathrm{SPAM}}_{\mathrm{SF}}$ to $R_1$ errors was expected because the model expressly requires $R_1=1$. The sensitivity of ${\mathrm{SPAM}}_{\mathrm{PN}}$ was initially difficult to explain as it fits for $R_1$. However, the scaling method in ${\mathrm{SPAM}}_{\mathrm{PN}}$ was found to partially normalize out $R_1$, which forcing an erroneous fit of true $R_1$, which in turn affected the accuracy in the estimation of both $k_{2,T}$ and BP.

An important assumption in all introduced models that warrants discussion is that $K_{1,T}/k_{2,T}=K_{1,C}/k_{2,C}$. The superiority of paired-agent imaging methods to conventional reference tissue models comes back to this assumption, as in conventional reference tissue imaging, it is important that the reference tissue and the region of interest have the same partition coefficients, which can be particularly difficult in cancer imaging.¹³ However, in paired-agent imaging, as long as both imaging agents experience diffusion/permeability limited delivery, the $K_{1,T}/k_{2,T}=K_{1,C}/k_{2,C}$ assumption should always be accurate as both should be equivalent to the inverse of the partition coefficient. Ensuring that similar vascular permeability be experienced by both the targeted and control agents in all environments requires the agents to have similar size, charge, and lipophilicity,²² which is true for the agents employed in this study and is generally not difficult to achieve, though should be verified by ensuring the agents have equivalent kinetics is tissues devoid of the targeted biomolecule.^6,34

The two obvious molecular imaging modalities capable of carrying out paired-agent imaging (for application of SPAM models)—where signal for two distinct imaging agent signals must be resolvable—include SPECT and optical imaging. Considerable effort is ongoing in dual-isotope SPECT³⁵ to achieve signal quantification, involving correction for signal cross-talk, and photon absorption and scattering property differences in different energy windows. In SPECT, the photon absorption and scattering tissue property differences can be subtle, but they are considerably more complex in fluorescence imaging. Even so, much work has gone into designing protocols that minimize these effects. For instance, employing excitation at wavelengths with low tissue attenuation can avoid high tissue absorption by hemoglobin,³⁶ and multispectral fluorescence imaging can allow multiple fluorescence agents to be imaged within similar wavelength ranges.²⁸ Use of surface-enhanced Raman scattering nanoparticles takes this idea to the limit as multiple “flavors” of particles can be excited with the same wavelength, and their respective, unique spectral signatures can be measured over identical spectral windows such that the optical properties of excitation and emission are nearly identical.³⁷ Moreover, the idea of epi-illumination surface imaging, where both source and detector are focused at the same location, could minimize absorption and scattering effects by limiting the detectable fluorescence events to those close to the surface.³⁸ It is also possible to mitigate optical property effects by normalizing fluorescence signals to simultaneously measured excitation light signals,^39,40 or the excitation light spectrum can be employed in more sophisticated terms to quantify the optical properties.⁴¹

In summary, it can be stated that if the data being collected is more likely to conform to $\alpha =1$, but $R_1$ not equal to 1 (“case 1”), then SPAM (which is similar to SRTM in conventional reference tissue imaging) is the model of choice. If the data is more likely to conform to $R_1=1$, but $\alpha$ is not equal to 1 (“case 2”), then ${\mathrm{SPAM}}_{\mathrm{SF}}$ is the model of choice (outperforming ${\mathrm{SPAM}}_{\mathrm{PN}}$, which is also more accurate than SPAM for “case 2”). Examples of “case 1” include studies where imaging agent concentration can be readily quantified from images such as PET neurotransmitter receptor imaging. Examples of “case 2” include studies where a control agent signal in the region of interest is used to account for agent delivery kinetics and nonspecific retention—require simultaneous imaging of two agents such as dual-isotope single photon emission tomography (SPECT) or fluorescence optical paired-agent imaging. In both modalities, accurate concentration quantification from signal is not straightforward, such that it may not be possible to ensure that $\alpha =1$. Yet, if the targeted and control agents are designed such that they experience the same levels of blood vessel permeability (PS), based on their size, charge, and lipophilicity,⁴² then it is conceivable by Eq. (13) that the $R_1=1$ assumption would hold. Despite the potential for signal quantification in both SPECT and optics, all of the methods require either specialized equipment or unique expertise in data analysis. The advantage of ${\mathrm{SPAM}}_{\mathrm{SF}}$—demonstrated in this work in simulations and animal studies—is that it does not require either the targeted or control imaging agent signals to be quantified. The scaling error in quantification is instead incorporated into the fitting algorithm, which is accurate as long as $R_1=1$ and $K_{1,T}/k_{2,T}=K_{1,C}/k_{2,C}$: conditions that are highly likely as long as the chemical properties of the targeted and control agents are similar.