Quantification of SPIO Nanoparticles in vivo Using the Finite Perturber Method

Abstract

The susceptibility gradients generated by super-paramagnetic iron oxide (SPIO) nanoparticles make them an ideal contrast agent in magnetic resonance imaging. Traditional quantification methods for SPIO nanoparticle-based contrast agents rely on either mapping T₂^* values within a region of interest or by modeling the magnetic field inhomogeneities generated by the contrast agent. In this study, a new model-based SPIO quantification method is introduced. The proposed method models magnetic field inhomogeneities by approximating regions containing SPIO nanoparticles as ensembles of magnetic dipoles, referred to as the finite perturber method. The proposed method was verified using data acquired from a phantom and in vivo mouse models. The phantom consisted of an agar solution with four embedded vials, each vial containing known but different concentrations of SPIO nanoparticles. Gaussian noise was also added to the phantom data to test performance of the proposed method. The in vivo dataset was acquired using five mice, each of which was subcutaneously implanted in the flanks with 1×10⁵ labeled and 1×10⁶ unlabeled C6 glioma cells. For both datasets, the proposed algorithm was found to model the gradient of the magnetic field inhomogeneities well and generate accurate estimations of the concentration of SPIOs.

Introduction

Superparamagnetic iron oxide (SPIO) nanoparticles generate a strong susceptibility gradient making them an ideal contrast agent in MRI. Contrast agents based on SPIO nanoparticles are ideally suited for a wide range of applications in MRI. Applications of SPIO include: liver imaging (1, 2), MR angiography (3), and tracking of labeled cells in vivo (4–7). Quantification of the total amount of SPIO-based contrast agent allows researchers and clinicians to quantitatively evaluate the results from different treatments and studies (7).

Contrast agent quantification methods can be broken into two categories: relaxometry methods and model-based methods. The relaxometry methods use the magnitude information from multiple complex-valued MR images and quantify the concentration of SPIOs within a given area by mapping the relaxation rate of SPIOs within tissue. The model-based methods quantify the concentration of SPIOs using the information about magnetic field inhomogeneities from the phase map and then modeling the magnetic field inhomogeneities as simple geometries.

The relaxometry methods utilize signal enhancement or decay associated with areas containing SPIOs. The effective transverse relaxation rate, denoted $R_2^{\ast }$, of SPIO labeled cells satisfies the static dephasing regime theory (8, 9). Since the value of R₂ for SPIOs is approximately two orders smaller than $R_2^{\ast }$ (9), most relaxometry methods attempt to measure $R_2^{\ast }$ (10, 11) over R₂ However, some relaxometry methods quantify the concentration by measuring (12). Recent advances in the relaxometry-based methods utilize multiple acquisition pulse sequences that significantly reduce the total number of scans (13). Relaxometry methods map $R_2^{\ast }$ in a particular region and assume that $R_2^{\ast }$ varies linearly with contrast agent concentration. The equation that governs how the relaxation rate changes with concentration is

$$R_2^{\ast }=R_{20}^{\ast }+r_2^{\ast }c$$ [1]

where $R_{20}^{\ast }$ denotes the intrinsic relaxation rate (e.g. no contrast agent in tissue), $r_2^{\ast }$ denotes the relaxivity of the contrast agent, c denotes the concentration of the contrast agent.

The model-based quantification methods rely on analytic models of simple geometries and use the magnetic field inhomogeneities generated by areas containing SPIOs to quantify the concentration of the SPIOs. The magnetic field inhomogeneities created by the SPIOs are identified using phase maps. The concentration of SPIOs is quantified by, first, modeling the magnetic field inhomogeneities as simple geometries such as a sphere or infinite cylinder; then, fitting the magnetic field of the model to the magnetic field inhomogeneities acquired experimentally. The method presented in (14) used an infinite cylinder model and the method presented in (15) used a spherical dipole model to measure the magnetization of a solution of SPIO-based contrast agent suspended within a phantom. Recently published phase-based quantification methods (16, 17) model distributions of iron as spherical dipoles. However, the two methods differ on how the magnetic field inhomogeneities from the iron distributions are calculated.

One major limitation of the model-based quantification method is that the method breaks down when an object with a complex geometric shape is encountered (16). This limitation can be dealt with by applying different numerical techniques to take into account of complex geometry and then calculate the magnetic field inhomogeneities. The numerical techniques can be divided into four categories: methods based on the finite element or finite difference methods (18–20), methods that utilize boundary conditions (21, 22), approximation methods (23), and Fourier-based methods (23–27).

A new SPIO quantification algorithm is introduced in this work to deal with a complex geometric case. The proposed method utilizes a positive contrast technique, known as phase gradient mapping (PGM), to bypass the phase unwrapping step in the quantification process. A modified finite perturber method is used to model the magnetic field for complex geometries (23). The proposed method quantifies SPIOs in two steps: first, calculating the phase gradient map of the acquired MR data; and second, fitting the phase gradient map to the gradient generated by the theoretical model, a distribution of spherical dipoles occupying the geometry of the object under consideration. The phase gradient is considered because it does not require the phase unwrapping step present in previous model-based quantification methods. In this work, the proposed quantification method is validated on both phantom and in vivo mouse models with SPIO labeled C6 glioma cells injected in the flanks. The performance of the proposed quantification method in a low SNR environment is also evaluated using the phantom data set.

Theory

Magnetic field inhomogeneities, associated with SPIO nanoparticles, are generated by susceptibility differences between a region containing these nanoparticles and the areas surrounding the region. The magnetic field inhomogeneities dephase spins in voxels neighboring the nanoparticles. The dephasing caused by the SPIO nanoparticles creates hypointensities in the magnitude image and induces phase disturbances in the phase map. The PGM method creates positive contrast by mapping the rapidly changing phase in regions neighboring SPIO nanoparticles. In the phase gradient map, areas surrounding higher concentrations of SPIO nanoparticles should have a larger phase gradient than areas surrounding lower concentrations of SPIO nanoparticles. Ideally, areas without nanoparticles should have no phase gradient but eddy currents, B₀ inhomogeneities, air-tissue interfaces, tissue-tissue interfaces, and gradient instabilities can influence the phase gradient.

The field map of a given image displays the z-component (parallel to B₀, the main magnetic field) of magnetic field inhomogeneities. The gradient of the field map is related to the gradient of the phase map by the relation

$$\nabla \left(\mathrm{\Delta }B(x,y,z)\right)=-\frac{\nabla \varphi (x,y,z)}{\gamma TE}$$ [2]

where γ denotes the gyromagnetic ratio for hydrogen and TE denotes the echo time when the phase map was taken (28).

Finite Perturber Method for the Phase Gradient

The finite perturber method models (23) each voxel as a spherical dipole with the sphere embedded within each voxel. This method uses a scaling factor of (6/π) to account for the excess volume within the voxel. The magnetic field for a magnetic dipole at a point, p, at a distance r outside the voxel is

$$B_{\text{Voxel}}(\mathbf{r})=\left(\frac{2{\mu }_{0}m}{\pi }\right)\frac{R^3}{r^3}(3{cos}^2\theta -1)$$ [3]

where θ denotes the angle the point p makes with B₀, R denotes the radius of the sphere embedded within the voxel, μ₀ is the permeability of free space, and m denotes the magnetic moment per unit volume of the object. It should be noted that in equation [3], the origin is located at the point p and the variable r refers to a point in the area surrounding the dipole with p≠r. The partial derivatives of the magnetic field for the spherical dipole in Cartesian coordinates are

$${\nabla }_x{B}_{\text{Voxel}}(x,y,z)=-xm{\mu }_0{R}^3\frac{4z^2-x^2-y^2}{{(x^2+y^2+z^2)}^{7/2}}$$ [4]

and

$${\nabla }_y{B}_{\text{Voxel}}(x,y,z)=-ym{\mu }_0{R}^3\frac{4z^2-x^2-y^2}{{(x^2+y^2+z^2)}^{7/2}}.$$ [5]

Equations [4] and [5] display the gradient of the induced magnetic field for a single voxel. For an object with a complex geometric structure, multiple voxels occupying the volume of interest are needed to provide an accurate approximation of the magnetic field gradients generated by the object. The total magnetic field gradient due to the object is given by the sum of the magnetic field gradient for each voxel within the object. Mathematically the total magnetic field gradient due to the object is given by

$$\nabla B_{\text{Model}}(x,y,z)=\sum _{n=1}^{N_x}\sum _{p=1}^{N_y}\sum _{r=1}^{N_z}\nabla (B_{\text{Voxel}}(x-n,y-p,z-r))\alpha (n,p,r)$$ [6]

where N_x, N_y, and N_z denote the total number of voxels along the x, y, and z axes of the region surrounding the tumor, the size of which must be large enough to encompass the tumor and the magnetic field perturbations that are generated by the SPIO labeled tumor. In Eq. [6], α represents a mask where α(n, p, r)= 1 if the point is within the geometry and α(n, p, r)= 0 otherwise. The magnetization within the geometry is obtained by calculating the value of m thatminimizes the squared error between the components of ∇B_Model and ∇(ΔB), i.e.

$$min{(\nabla B_{\text{Model}}-\nabla (\mathrm{\Delta }B))}^2.$$ [7]

The value of m that minimizes the x-component of the expression in equation [7] is

$$m=\frac{{\nabla }_x(\mathrm{\Delta }B(:))·{\nabla }_x{B}_{\text{Model}}(:)}{2{\nabla }_x{B}_{\text{Model}}(:)·{\nabla }_x{B}_{\text{Model}}(:)}$$ [8]

where the symbol · represents an inner product, ∇B_Model(:) denotes the vector of all values in∇B_Model and ∇(ΔB(:)) denotes the vector of all values in ∇(ΔB). The concentration of SPIO nanoparticles was found by dividing the value of m in equation [8] by the saturation magnetization of the contrast agent. A similar procedure is followed to find the value of m that minimized the y-component of equation [7].

Methods and Materials

Data Acquisition

All data sets were acquired using a whole-body Philips 3T Achieva clinical MR scanner (Phillips Medical Systems, Best, the Netherlands). An agar phantom was constructed with four vials (8 mm in diameter) embedded within the phantom (5 cm in diam=eter). Each vial contained a different concentration of Ferumoxides (Berlex Laboratories, Wayne, NJ), a contrast agent based on SPIO nanoparticles. The contrast agent has a saturation magnetization of approximately 93.6 emu / g Fe at a field strength of 3.0 T (29). Multiple 2D gradient echo images were acquired with the phantom standing upright (perpendicular to the B₀ direction) using the following parameters: TE = 15 ms, TR = 50 ms, flip angle = 25 degrees, slice thickness = 1 mm, FOV = 70 mm, 256 × 256 acquisition matrix, 24 coronal slices.

Five mouse data sets were acquired using a 4 cm receive-only RF coil (Phillips Research Europe, Hamburg, Germany). Each mouse was subcutaneously implanted in the flanks with 1×10⁵ labeled and 1×10⁶ Unlabeled C6 glioma cells (ATCC, Manassas, VA), and both groups of cells were suspended in 100 μl of PBS. Feurmoxides were used to label the C6 glioma cells (30). The Fe content in the injected cells was approximately 19.4 pg/cell in labeled cells and 1.2 pg/cell in unlabeled cells. The estimated concentration of Feurmoxides within the labeled tumor is shown in the first column of Table 3, estimated by total amount of Fe over total injection volume. MR images were acquired with 3D gradient echo-based fast field echo (FFE) sequence with FOV = 40 mm × 40 mm, slice thickness = 0.5 mm, matrix size = 256 × 256, TR = 12.6 ms, and TE = 4.6 ms, 6.9 ms, 9.2 ms, and 11.5 ms, respectively. The density of magnetic dipoles generated by the labeled tumors is displayed in Table 2. All studies were performed as part of an approved animal care and use community protocol at our institution.

TABLE 3.

Evaluation of the proposed quantification method on in vivo mouse data sets. The first column displays the subject number. The second column displays the known SPIO concentration within the tumor, calculated by diving the total nanoparticles injected by the volume measured. The third and fourth column display the average estimation and its standard deviation of the concentration from the x- and y-component of the phase gradient at four echo times, respectively.

Subject	Known Concentration (μg/ml)	From x-component (μg/ml)	From y-component (μg/ml)
1	36.5	36.0±2.6	26.6±2.2
2	33.1	40.5±3.6	47.3±4.6
3	36.5	33.8±5.4	31.7±2.1
4	42.5	38.2±3.9	43.9±5.7
5	41.2	32.8±2.0	46.7±2.0

TABLE 2.

The performance of the proposed quantification method on the phantom data set with different masks. The first column shows the vial number. In the second column, information about size changes of the masks used in the calculation of the concentration is shown. Values of +10% correspond to the mask dilated so that it was 10% larger than the original mask. Values of −10% correspond to the mask eroded so that it was 10% smaller than the original mask. The third column displays the known concentration in each vial. The fourth column displays the concentration estimations from the x-component of the phase gradient map and the fifth column displays the concentration estimations from the y-component of the phase gradient map.

Vial	Mask	Known Concentration (μg/ml)	From x-component (μg/ml)	From y-component (μg/ml)
1	+10%	160	123.2	162.9
	−10%	160	176.9	228.0
2	+10%	80	58.3	71.3
	−10%	80	83.6	99.0
3	+10%	40	30.3	36.8
	−10%	40	44.7	52.1

Data Processing

To evaluate the performance of the SPIO quantification algorithm in an environment with low SNR, Gaussian noise was added to the phantom data set in the real and imaginary parts of the k-space. The noisy image was transformed into image space by a 2D fast Fourier transform of the modified k-space data. The SNR was calculated as S₀/(1.53σ_N), where S₀ is the mean of a homogeneous area away from the vials within the phantom and σ_N is the mean of the standard deviation of four areas outside the phantom.

The vials within the phantom were segmented using a simplistic thresholding technique. Areas with a low signal inside the phantom were assumed to be within a vial with SPIO-based contrast agent and those regions were defined to be 1 in the mask, i.e.

$$\alpha (m,n,p)=\{\begin{array}{c}1\text{if}\mid \rho (m,n,p)\mid \le T\\ 0\text{if}\mid \rho (m,n,p)\mid >T\end{array}$$ [9]

where T is the chosen threshold and ρ (x,y,z) denotes the magnitude of the complex-valued image obtained from the MR scanner. The mask for the vial was then used in Eq. 6 to calculate the gradient of the theoretical model.

To assess the performance of the proposed algorithm in an environment where the estimated mask was smaller than the true mask, α(m,n,p), for the phantom data set, the mask was eroded so that it was 10% smaller than the true mask. Conversely, to assess the performance of the proposed algorithm in an environment where the estimated mask was larger than the true mask, the mask was dilated so that it was 10% larger than the true mask.

The implanted tumor in each subject was segmented using a gradient-based segmentation algorithm. First, the gradient of the magnitude image was calculated. The SPIO-labeled cells within the tumor cause signal loss in the magnitude image and the boundary of the labeled tumor has a steep gradient in the magnitude image. A mask of the boundary of the tumor area was created by thresholding the gradient of the magnitude image. Voxels with a value of the gradient of the magnitude image greater than the threshold, denoted T and chosen by the user, were mapped to 1, i.e.

$$\alpha (m,n,p)=\{\begin{array}{c}1\text{if}\mid \nabla \rho (m,n,p)\mid \ge T\\ 0\text{if}\mid \nabla \rho (m,n,p)\mid <T\end{array}$$ [10]

The mask in Eq. [10] only displays the edges of the tumor, in order to construct the mask of the labeled tumor, the volume inside the boundary was filled with values of 1. The mask was used in equation [6] to calculate the gradient of the theoretical model.

In order to measure how closely the gradient of the model, ∇B_Model, matches the gradient of the magnetic field inhomogeneities, ∇(ΔB), a sensitivity analysis was performed on both the phantom data set and in vivo data sets. In the analysis, the gradient of the magnetic field inhomogeneities ∇B_Model was used as the “ground truth.” The sensitivity for the ith threshold is defined as S_i = T_i/N_T where N_T is the total number of voxels in the region of interest and T_i is the number of voxels in the difference model of ∇B_Model and ∇(ΔB) that fall within the ith threshold T_i,. The thresholds are determined by dividing the maximum difference of ∇(ΔB) from ∇B_Model with the number of thresholds used in the analysis, i.e.

$$T_k=k\frac{max\mid \nabla B_{\text{Model}}-\nabla (\mathrm{\Delta }B)\mid }{30}$$ [11]

In this sensitivity analysis, thirty thresholds were used. The ideal sensitivity curve is a horizontal line stretching from the point (0, 1) to the point (1, 1), representing a sensitivity of one no matter what thresholds are applied. Sensitivity curves closer to the ideal sensitivity curve correspond to more accurate models for the magnetic fields generated by the SPIO nanoparticles.

Results

Figure 1 displays two of the data sets considered in this work: the phantom data set and the mouse data set for subject 4. The magnitude image for the original phantom data set without added Gaussian noise is displayed in Figure 1a and Figure 1b displays the magnitude image with added Gaussian noise. Feuromoxide concentrations in the three vials labeled 1–3 in Figures 1a and 1b were 160 μg/ml, 80 μg/ml and 40 μg/ml, respectively. The phase maps for the phantom data set without added noise and with added noise are displayed in Figures 1b and Figure 1e.. Figure 1c displays the magnitude image for subject #4, where the labeled tumor is circled, and Figure 1f displays its phase map.

FIG 1.

The magnitude image of the phantom data set at high SNR and low SNR are shown in (a) and (b), respectively. Vial 1 is labeled 1, vial 2 is labeled 2, and vial 3 is labeled 3. The acquisition parameters for the phantom data set are: 2D gradient echo sequence with TE = 15 ms, TR = 50 ms, flip angle = 25 degrees, slice thickness = 1 mm, FOV = 70 mm, 256 × 256 acquisition matrix. (c) The magnitude image for mouse subject 4 at TE=15 ms. The acquisition parameters for the mouse data set are: 3D FFE sequence with FOV = 40 mm, slice thickness = 0.5 mm, matrix size = 256 × 256, TR = 12.6 ms, and TE = 4.6 ms, 6.9 ms, 9.2 ms, and 11.5 ms. The implanted tumor is labeled with a circle. The phase map for the phantom data set at high SNR and low SNR are shown in (d) and (e), respectively. (f) The phase map for subject 4 at TE=6.8 ms.

Figure 2 illustrates the magnetic dipoles generated by the contrast agent. Figure 2a displays the unwrapped phase map for the area surrounding vial 1 in the phantom data set. The unwrapped phase map of the area surrounding the labeled tumor in a mouse data set is shown in Figure 2b. The arrows in Figure 2a and Figure 2b are aligned along B₀. Both phase maps were unwrapped using the phase map presented in (31) for demonstration purposes only (not for calculation of field gradients).

FIG 2.

Illustration of the magnetic dipoles generated by the contrast agent. (a) A view of the unwrapped phase map surrounding vial 1 in the phantom data set. (b) The unwrapped phase map of the area surrounding the labeled tumor in subject 4. In both (a) and (b), the arrow points along B₀.

A comparison between the gradient of the proposed model and the gradient of the magnetic field inhomogeneities in the phantom data set are displayed in Figure 3. Figure 4 displays profiles of the theoretical field gradient and the experimental field gradient shown in Figure 3. It should be noted that both components of the theoretical field gradient closely match both components of the experimental field gradient. The lines in Figure 3a and 3b show the locations of the profiles in Figure 4a. The lines in Figures 3c and 3d display the locations of the profiles in Figure 4b.

FIG 3.

A comparison of the magnetic field gradients generated by the theoretical model for vial 2 in the phantom data set. (a) The x-component of the magnetic field gradient generated by the theoretical model. (b) The x-component of the experimental field gradient. (c) The y-component of the theoretical field gradient for vial 2. (d) The y-component of the experimental field gradient generated by vial 2. The lines display locations of profiles that are plotted in Figure 4.

FIG 4.

Profiles across the experimental and theoretical field gradients for vial 2. (a) The x-components of the field gradients. (b) The y-components of the field gradients. In both (a) and (b), the solid line displays a profile of the theoretical field gradient and the dashed line displays a profile of the experimental phase gradient.

The concentration estimations for the phantom data set are displayed in Table 1. As described above, Gaussian noise was added to the acquired MR data set. The average SNR across all slices of the original phantom data set was 104.0, referred to as high SNR, and after noise was added the average SNR across all slices was 10.6, referred to as low SNR. For both the high SNR and the low SNR phantom data sets, the proposed method gives accurate estimations of the concentration for vials 2 and 3 while only the y-component of the phase gradient gives an accurate estimation of the concentration for vial 1.

TABLE 1.

The performance of the proposed quantification method on the phantom data set. The first column shows the vial number. Information pertaining to the SNR is shown in column two. The third column displays the known concentration in each vial. The fourth column displays the concentration estimations from the x-component of the phase gradient map and the fifth column displays the concentration estimations from the y-component of the phase gradient map.

Vial	SNR	Known Concentration (μg/ml)	From x-component (μg/ml)	From y-component (μg/ml)
1	High	160	147.2	192.0
	Low	160	151.0	196.6
2	High	80	69.6	83.6
	Low	80	72.3	86.4
3	High	40	36.7	43.5
	Low	40	45.6	52.7

The sensitivity curves for the high and low SNR phantom data sets are shown in Figure 5. As seen in the sensitivity curves, the model generates magnetic field gradients that closely match the magnetic field gradients that are generated by the SPIOs within each vial. There are distortions in ∇_xΔB(x,y), hence, the sensitivity curve calculated from the x-component of vial 3 is lower than the sensitivity curves for the other vials. For each sensitivity plot, the normalized threshold, T_i/max(T_i), is plotted along the x-axis.

FIG 5.

A comparison of the sensitivity maps for the high SNR (first row) and low SNR phantom data sets (second row). (a) & (c) The sensitivity curve from the x-component of the gradient of the magnetic field inhomogeneities. (b) & (d) The sensitivity curve from the y-component of the gradient of the magnetic field inhomogeneities.

Table 2 displays the results of the analysis of the proposed method using differing masks. In the analysis, we found that the mask that was 10% smaller than the original mask for the phantom data set overestimated the concentration and the mask that was 10% larger than the original mask for the phantom data set underestimated the concentration.

Figure 6 displays a comparison between the gradient of the proposed model and the gradient of the magnetic field inhomogeneities in the in vivo mouse data set. As seen if Figure 6, the gradient of the proposed model accords well with the gradient of the magnetic field inhomogeneities in the mouse data set. Figure 7 displays profiles of the components of the theoretical field gradient and the components of the experimental field gradient. The lines in Figure 6 display the locations of the profiles plotted in Figure 7. The sensitivity curves for the in vivo data sets are shown in Figure 8, demonstrating that the gradient of the model, ∇B_Model, closely matches the gradient of the magnetic field inhomogeneities, ∇(ΔB), across all of the four TEs.

FIG 6.

A comparison of the magnetic field gradients generated by the proposed method and the magnetic field gradients generated by the labeled tumor (subject 4). (a) The x-component of the magnetic field gradient generated by the theoretical model. (b) The x-component of the magnetic field gradient generated by the labeled tumor. (c) The y-component of the magnetic field gradient generated by the theoretical model. (d) The y-component of the magnetic field gradient generated by the labeled tumor. The lines display the locations of the profiles plotted in Figure 6.

FIG 7.

Profiles across the theoretical field gradient and experimental field gradient for subject 4. The x-component of the field gradients are displayed in a and the y-components of the field gradients are displayed in (b). In both (a) and (b), the solid line displays the theoretical field gradient and the dashed line displays the experimental field gradient.

FIG 8.

A comparison of the sensitivity curves for the in vivo data sets. The first row displays the sensitivity curves calculated from the x-component of the gradient of the magnetic field inhomogeneities. The second row displays the sensitivity curves calculated from the y-component of the gradient of the magnetic field inhomogeneities.

The estimation for the SPIO concentration within the injected tumors is shown in Table 3, which presents the mean and standard deviation of the concentration estimated using data of four different TEs. The proposed method, using the x-component of the phase gradient map, gave concentration estimations close to the known values in subjects 1, 3, and 4. The proposed method, using the y-component of the phase gradient map, gave concentration estimations close to the known values in subjects 3, 4 and 5.

Discussion and conclusions

A SPIO quantification algorithm, based on the finite perturber method, was introduced in this work. The performance of the algorithm was evaluated in different SNR environments for a phantom data set as well as for in vivo mouse data sets. In each case, the modified finite perturber method was found to model the gradient of the magnetic field inhomogeneities well and the proposed quantification algorithm generated estimations of the concentration of SPIOs that accord well with the known concentration of SPIOs.

The previously published model-based quantification algorithms (16, 17) model the iron distributions as magnetic dipoles and they are able to quantify regular shapes (e.g. spheres and infinite cylinders) of distributions of iron. An advantage for the quantification algorithm presented in this work is that it is not limited to regular distributions of SPIOs. Another key difference between algorithms presented in (16, 17) and the proposed algorithm is that the proposed algorithm requires no phase unwrapping step and directly fits the gradient of the magnetic field inhomogeneities to the gradient of the model.

In this study, the phase gradient was calculated using the method described in (32). There are several phase gradient calculation methods (33–35) that can be used in place of the method described in (32). Each phase gradient calculation method assumes that no aliasing occurs in the phase map (|∇ϕ(x,y)|<π). Alternatively, the phase map surrounding the tumor area can be unwrapped using a phase unwrapping algorithm and the components of the phase gradient can be directly calculated from the unwrapped phase map.

The phase gradient is proportional to the concentration of the SPIO nanoparticles, i.e. if the concentration of SPIOs in an object is doubled then the phase gradient surrounding the object will scale up by a factor of two. Hence, the condition |∇ϕ(x,y)|<π is not guaranteed to hold in regions neighboring very high concentrations of SPIO nanoparticles. In this case, the phase gradient calculation methods presented in (33–35) will give inaccurate estimations of the phase gradient and the method presented in this work will fail. However, a Bayesian approach to estimating the phase gradient has been shown to produce accurate estimations of the phase gradient on simulated phase maps (36). The use of the Bayesian-based phase gradient method should mitigate the effects of the aliasing in the phase map and allow for the estimation of very high concentrations of SPIO nanoparticles.

In the proposed quantification method, the estimation of concentration depends on the choice of the tumor area. An accurate method for segmenting the tumor area is needed since different choices of the tumor area will give different estimations of the SPIO concentration. A gradient-based tumor segmentation algorithm was employed in this study to create masks of the tumor area. Other tumor segmentation algorithms (37–42) can be used in place of the gradient-based tumor segmentation algorithm. Since the labeled tumor area generates a magnetic dipole, a cross correlation method (43) to obtain the size and location of the labeled-tumor can also replace the segmentation procedure.

Like all model-based quantification methods, B₀ inhomogeneities, gradient instabilities, and magnetic field inhomogeneities generated by tissue-tissue interfaces or air-tissue interfaces may influence the results from the proposed method. Inclusion of the magnetic field inhomogeneities generated by tissue-tissue interfaces and air-tissue interfaces in the model can partially offset the error induced by the magnetic field inhomogeneities. However, it remains difficult to completely correct and/or compensate for gradient instabilities.

The proposed quantification method assumes that the distribution of SPIO nanoparticles is approximately homogeneous throughout the labeled-region. In practice, this assumption does not necessarily hold in vivo. For more complicated in vivo data sets, a method of incorporating inhomogeneous distributions of SPIOs needs to be addressed, which may be accounted for by adding a weighting matrix into Eq. [6], instead of a uniform mask. However, the choice of the weighting matrix can be problematic. For example, simple magnitude-based weighting matrices will probably not work since areas with higher SPIOs have lower signal in the magnitude image and these areas are more susceptible to noise. Future studies will be conducted with the aim of finding an accurate method of generating a weighting matrix for inhomogeneous distributions of SPIOs in in vivo data sets.

While only SPIO-based contrast agents were considered in the current study, the proposed quantification algorithm could be modified to quantify other types of contrast agents, most notably contrast agents based on Gadolinium diethylenetriamine pentaacetic acid (Gd-DTPA). Model-based quantification algorithms for the quantification of Gd-DTPA been proposed (14, 44). Both methods model the distributions of Gd-DTPA as infinite cylinders and the method presented in (44) was used to measure the arterial input function for a dynamic contrast enhanced magnetic resonance imaging study.

In summary, a model-based SPIO quantification method was introduced. The performance of the proposed SPIO quantification algorithm was evaluated using a phantom and an in vivo data set. The proposed quantification algorithm was found to perform well in both the high SNR and low SNR regimes for the phantom data set. Additionally, the proposed quantification method was found to give estimates of the in vivo concentration of SPIO labeled cells that are close to the known concentrations.