Simultaneous measurement of pulmonary partial pressure of oxygen and apparent diffusion coefficient by hyperpolarized ³He MRI

Abstract

Hyperpolarized ³He (HP ³He) MRI shows promise to assess structural and functional pulmonary parameters in a sensitive, regional and non-invasive way. Structural HP ³He MRI has applied the apparent diffusion coefficient (ADC) for the detection of disease-induced lung microstructure changes at the alveolar level, and HP ³He pulmonary partial pressure of oxygen (pO₂) imaging measures the oxygen transfer efficiency between the lung and blood stream. Although both parameters are affected in chronic obstructive pulmonary disease (COPD), a quantitative assessment of the regional correlation of the two parameters has not been reported in the literature. In this work, a single acquisition technique for the simultaneous measurement of ADC and pO₂ is presented. This technique is based on the multiple regression method, in which a general linear estimator is used to retrieve the values of ADC and pO₂ from a series of measurements. The measurement uncertainties are also analytically derived and used to find an optimal measurement scheme. The technique was first tested on a phantom model, and then on an in-vivo normal pig experiment. A case study was performed on a COPD patient, which showed that in a region-of-interest ADC was 29% higher while oxygen depletion rate was 61% lower than the corresponding global average values.

Introduction

Chronic obstructive pulmonary disease (COPD), a group of diseases that includes chronic bronchitis and emphysema, accounts for substantial health care spending and is the fourth leading cause of death in the United States(1). Conventional methods to diagnose and evaluate COPD include pulmonary function tests (PFTs) and computed tomography (CT). PFTs only provide a gross assessment of the state of the disease, and their application is largely limited to qualitative, clinical diagnosis. X-ray computed tomography (CT) can detect and stage the diseases with only limited sensitivity and is not suitable for long-term studies due to radiation exposure(2–5). In general, magnetic resonance imaging (MRI) provides superior soft tissue contrast when compared to other imaging modalities. However, conventional proton-based MRI of the lung suffers from poor signal-to-noise ratios due to low proton density and susceptibility gradients between airways and tissue. In recent years, hyperpolarized ³He (HP ³He) MRI techniques have been studied extensively and have become promising tools to assess structural and functional pulmonary parameters in a sensitive, regional and non-invasive way(6–8).

Structural HP ³He MRI has relied heavily on a measurement called the apparent diffusion coefficient (ADC), which is a function of alveolar size and geometry, to characterize the lung’s microstructure at the alveolar level(9–13). This technique utilizes the fact that the atoms of ³He gas have a restricted diffusion capability in the lung because the alveolar size (~200μm) is smaller than the free diffusion mean distance during a time period encoded into the imaging pulse sequence. Therefore, the measured diffusion coefficient in the lung is substantially smaller than the free diffusion coefficient and depends sensitively on the local alveolar and small airway dimensions. The alveolar size in emphysema patients increases as a consequence of lung tissue destruction caused by the disease. Tissue destruction allows ³He atoms greater freedom to move, so it is expected that these patients will have a large measured ADC value. A number of studies have demonstrated significant difference between the ³He ADC values measured in healthy and emphysematous subjects (9,11,12,14). The most common technique used for ADC measurement is to introduce a bipolar gradient to sensitize the received signal intensity to the diffusion effect. This gradient pulse is implemented after the longitudinal magnetization is flipped to the transverse plane and before the MRI signal is acquired (15). The logarithm of the diffusion-induced signal attenuation is proportional to both the ADC value and the so-called gradient factor, a function of gradient timing and magnitude that represents the bipolar gradient strength.

The HP ³He MRI pulmonary oxygen measurement, an example of functional imaging, which seeks to image physiological parameters other than the anatomic and pathological description of lung structure, yields alveolar partial pressure of oxygen (pO₂) and oxygen depletion rate (R) (16–19). In the presence of oxygen, the hyperpolarized magnetization of ³He gas will gradually decay due to the dipolar couplings between the ³He’s nuclear spins and the electronic spins of paramagnetic oxygen molecules(20). When ³He gas is inhaled into the lung, the decay rate is proportional to the pulmonary partial pressure of oxygen if other slower depolarization processes (e.g. ³He-³He dipolar coupling and interactions with the tissue) are ignored(16). This T₁ dependence of hyperpolarized ³He on local oxygen concentration provides the basis for measuring the pulmonary partial pressure of oxygen and oxygen depletion rate. These two parameters are sensitive markers of oxygen transfer efficiency between the lung airspaces and the pulmonary blood(3). Therefore, an accurate, regional mapping of pO₂ and R with HP ³He MRI can enhance the detection of the pulmonary function changes associated with COPD.

There is considerable interest in investigating the correlation between lung structure and function parameters in patients with COPD. Various researchers have applied the CT technique to study the link between lung structure and regional pulmonary ventilation. These studies show that in addition to causing structural changes, COPD also directly affects lung ventilation through irreversible airflow limitation. To date, the investigation of the correlation between lung structure changes and oxygen exchange efficiency has not been reported in the literature.

In this study, we present a single-acquisition technique for the simultaneous measurement of pO₂, R and ADC. This technique provides a co-registered measurement between the pulmonary structure and oxygen exchange efficiency, and may give new insight into the pathophysiologic mechanisms of COPD. We believe that physicians may be able to use the co-registered measurement to accurately diagnose early disease-induced changes in lung function and structure, quantitatively follow disease progression, and immediately ascertain response to therapy. Performing a simultaneous measurement also saves expensive ³He gas and shortens measurement time. The latter is especially beneficial to the patient, as it reduces the breath-hold time.

This simultaneous measurement technique is based on the multiple regression method, in which the general least square estimator is used to retrieve pO₂, R and ADC values from a series of acquired signals. The measurement uncertainty of each parameter is also analytically derived. An optimal measurement scheme which yields minimal uncertainties in pO₂, R and ADC is found through numerical simulation. The simultaneous measurement technique was first tested on a phantom model, and then on a normal animal model. Finally, a case study was performed on a patient with severe COPD to investigate the correlation between pO₂, R and ADC.

Theory

In the simultaneous measurement of the pulmonary partial pressure of oxygen and ADC using HP ³He MRI, three factors contribute to the signal evolution of a series of acquired images. First, the presence of oxygen in the lung will impose a decay on the hyperpolarized longitudinal magnetization. Second, the bipolar gradient used for sensitizing the diffusion effect will cause a transverse magnetization attenuation. The third factor is the radio-frequency (RF) pulse, which flips the hyperpolarized longitudinal magnetization to the transverse plane for spatial encoding and signal acquisition. The signal evolution of a series of images acquired by a small flip angle gradient echo sequence can be expressed as:

$$S_n=S_0·{(cos\alpha )}^{Nn}·exp\left[-\frac{1}{\xi }{\int }_0^{t(n)}{p}_{O_2}(t)dt\right]·exp(-b(n)·D),$$ [1]

where S₀ and S_n are the signal intensities of the initial and the n^th image; (cosα )^Nn represents the RF effect, where α is the flip angle and N is the number of phase coding steps of an image; $exp\left[-\frac{1}{\xi }{\int }_0^{t(n)}{p}_{O_2}(t)dt\right]$ represents the decay due to oxygen, where p_O2 (t) is the time-dependent partial pressure of oxygen in the lung, t(n) is the time at which the acquisition of n^th image is completed, and ζ =1976.0 Torr·s is the oxygen-induced longitudinal relaxation coefficient at body temperature; and exp(−b(n) ·D) represents the diffusion effect, where D is the apparent diffusion coefficient, and b(n) is the bipolar gradient factor introduced in the n^th image.

During a breath-hold, the time evolution of partial pressure of oxygen in the lung can be accurately approximated as a linear function of time(16):

$$p_{O_2}(t)=p_0-R·t,$$ [2]

where p₀ is the initial oxygen partial pressure and R is the oxygen depletion rate, the speed at which oxygen is dissolved from the alveoli into the blood. Substituting Eq.[2] into Eq.[1] and normalizing S_n with respect to the initial signal S₀, Eq.[1] can be expressed as:

$$E_n=ln(S_n/S_0)=\epsilon ·n-\frac{1}{\xi }·p_0·t(n)+\frac{1}{2\xi }·R·t^2(n)-D·b(n),$$ [3]

whereε = N ·ln(cosα).

In Eq.[3], the normalized signal E_n is a linear summation of the four unknown parameters ε, p₀, R and D, multiplied by the respective associated coefficients n, t(n), t² (n) and b(n). Based on Appendix 1, the multiple regression method can be applied to recover the values of the four parameters from a series of measurements(21–23), which gives:

$${[\epsilon ,p_0,R,D]}^T={(X^T·V·X)}^{-1}·X^T·V·y,$$ [4]

where X is the coefficient matrix, V is the signal matrix, and y is normalized signal vector:

$$X=\left[\begin{array}{cccc}1& t_1& t_1^2& b_1\\ 2& t_2& t_2^2& b_2\\ ⋮& ⋮& ⋮& ⋮\\ n& t_n& t_n^2& b_n\end{array}\right];V=\left[\begin{array}{cccc}{S}_1^2& 0& \cdots & 0\\ 0& S_2^2& 0& ⋮\\ ⋮& 0& \ddots & 0\\ 0& \cdots & 0& S_n^2\end{array}\right];y=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right].$$

In the four column vectors of the coefficient matrix X, the first vector will be linearly dependent on the second vector if the measurement timing t(n) is a linear function of n. In this case, the inverse of X does not exist and Eq.[4] yields no solution. Similarly, the fourth vector, which represents the gradient factor b(n), also needs to be linearly independent of the first vector. This means when the single-acquisition technique is applied to measure p₀, R and ADC simultaneously, the measurement timing cannot be evenly-spaced and the gradient factors cannot be evenly increasing or decreasing. Intuitively, the three effects of oxygen, ADC and RF can be separated only when each effect contributes to the total signal evolution in different patterns by which the coefficients progress.

In Eq.[4], to recover p₀, R and ADC from a single-acquisition measurement, the choice of measurement timing t(n) and gradient factors b(n) are not restricted except that neither can linearly vary with n. However, in the presence of measurement noise, different choices of t(n) and b(n) will yield significant differences on the measurement uncertainties of p₀, R and ADC. The uncertainties of p₀, R and ADC can be derived according to the error propagation theorem and expressed as(23):

$${\left[{\sigma }_{\epsilon }^2,{\sigma }_{p_0}^2,{\sigma }_R^2,{\sigma }_D^2\right]}^T={\sigma }^2·{(X^T·V·X)}^{-1}.$$ [5]

From Eq.[5], finding optimal t(n) and b(n) to minimize measurement uncertainties ${\sigma }_{p_0}^2,{\sigma }_R^2,{\sigma }_D^2$ is a multi-variable and multi-target optimization problem. The signal matrix V is dependent on scan parameters such as flip angle and phase encoding steps, as seen in Eq.[1]. This dependence complicates the problem and makes an analytical solution difficult to obtain. In our study, a numerical simulation was used to evaluate the noise performances of a variety of measurement schemes in order to choose an optimal scheme that yields minimal measurement uncertainties in p₀, R and ADC. To simplify the problem, the gradient factors b(n) were first set to progress according to:

$$b(n)=b_0(\sqrt{M}-\sqrt{n}),$$ [6]

where b₀ is the base gradient factor and M is the number of images in the series.

Since b(n) in Eq.[6] decreases with n, small gradient factors are applied in the last several images, in which signal-to-noise ratios are low because the hyperpolarized longitudinal magnetization is already substantially consumed by the oxygen and RF effects. In addition, the square-root function used in Eq.[6] allows the gradient factor to vary in a smaller range than it would in other possible choices (e.g. polynomial functions), thereby guaranteeing that the maximum gradient amplitude of the scanner will not be exceeded. After determining the gradient factors b(n), the numerical simulation can apply a similar procedure as described in Ref.(24), in which the measurement timing and flip angle are optimized to minimize the measurement uncertainties of p₀ and R. From a variety of choices, the numerical simulation found the timing scheme that optimized the measurements of p₀ and R also yielded a minimal measurement uncertainty on ADC. As shown in Fig. 1, the inter-scan time of this optimal timing scheme is relatively long in the middle and short at the two ends of the measurement. The simulation showed that the normalized measurement uncertainty (standard deviation over mean value) was 14.2% for p₀, 84.2% for R and 15% for ADC when this optimal timing scheme was used under the following measurement condition: an initial ³He polarization yielding a SNR=120 for the first image at α =2.5 degrees, b₀ =1.4 s/cm², N=64, number of images=6, breath-hold duration=20s, p₀ =106 Torr, R =1.9 Torr/s and ADC=0.2 cm²/s. The optimal flip angle of this scheme is 3.6 degrees for p₀, 4.4 degrees for R and 4.0 degrees for ADC (In practice, only one flip angle is used for each measurement because the optimal flip angles for the three parameters are close enough).

Fig. 1.

Schematic diagram of the optimal measurement scheme for the simultaneous measurement of p₀, R and ADC with HP ³He MRI. The inter-scan time of this scheme is relatively long in the middle and short at the two ends. The bipolar diffusion-sensitization gradient is applied along the readout direction. Its magnitude decreases with the measurement. For simplicity, the slice selection and phase coding gradient are not shown in the figure.

Materials and methods

Polarized ³He production and administration

A commercial-prototype polarizer (Amersham Health, Durham, NC) was used to generate hyperpolarized ³He gas. The polarization process is based on the optical pumping mechanism, in which the angular momentum of circularly polarized light is first transferred to the electronic spins of Rubidium (Rb) atoms and then to the nuclear spins of ³He atoms(25–27). A batch volume of one liter of ³He gas with a polarization level of approximately 30% can be achieved after 10 hours of optical pumping. In the experiments, ³He gas was dispensed from the polarizer, stored in a plastic bag, transferred to the scanner, and administered immediately before the imaging sequence was initiated.

Phantom experiment

The simultaneous measurement technique was first tested on a phantom model which simulated a free diffusion environment. An 18×18 cm² plastic bag was filled with a mixture of gas consisting of (500±20)ml HP ³He and (110±3)ml O₂. The bag was loosely contained in a fixed holder and the gas pressure in the bag was balanced with the atmospheric pressure of the environment. The nominal partial pressure of oxygen was approximately 110/(500+110) (bar) * 760 (Torr/bar) ≈ 137 Torr. The bag was also tightly sealed to simulate an oxygen depletion rate of zero.

A commercial birdcage RF coil (Rapid Biomedical, Würzburg, Germany) was used in the experiment. The coil was 35 cm long and 27 cm in diameter, and operated in quadrature mode. The experiment was conducted at a Siemens Sonata 1.5T MRI system and the coil was tuned to the ³He resonance frequency 48.48 MHz. The MRI pulse sequence used for imaging was a small flip angle gradient echo sequence with a diffusion-sensitization bipolar gradient introduced in the readout direction. A series of six images was acquired with the timing scheme [0, 0.64, 10.68, 20.72, 25.76, 28.80] (seconds) and the gradient factor set [0, 0.49, 0.33, 0.20, 0.09, 0] (s/cm²). To prevent the signal from becoming over-attenuated by the high free diffusion coefficient, the gradient factor set in the phantom experiment was much smaller than the set in the animal experiment. In the phantom experiment, a field of view (FOV) of 300mm and a single slice of 100mm thickness were used in order to cover the whole volume of the bag and avoid a polarization diffusion refresh from the neighboring region on the RF-excited slice. The other scan parameters were: TR/TE:10ms/6.26ms, matrix size: 64×64, image resolution: 4.69mm × 4.69mm, and flip angle=2.4 degrees.

Animal experiment

The animal experiment was conducted under a protocol approved by the Institutional Animal Care and Use Committee at the University of Pennsylvania. A 22 kg Yorkshire pig was sedated with ketamine and placed supine inside the 1.5T Siemens Sonata MRI scanner. The RF coil was the same as the one used in the phantom experiment. The animal was ventilated with a home-built prototype ventilator. A tidal volume of 500ml gas mixture, consisting of 100ml O₂ and 400ml ³He gas, was administered to the pig immediately before the simultaneous measurement was initiated. A series of six images were acquired for a single slice with the timing scheme [0, 1.92, 13.24, 24.56, 30.88, 35.20] (seconds) and the gradient factor set [0, 1.73, 1.15, 0.71, 0.33, 0] (s/cm²). The imaging parameters used in the diffusion-sensitization gradient echo sequence were: FOV=240mm; slice thickness=30mm; TR/TE:10ms/6.26ms, matrix size: 64×64, image resolution: 3.75mm × 3.75mm, and flip angle=3.0 degrees. After the simultaneous measurement, separated measurements on partial pressure of oxygen and ADC were performed. The gas mixture used was 100ml O₂, 300ml ³He and 100ml N₂ for the separated partial pressure of oxygen measurement, and 200ml ³He and 300ml N₂ for the separated ADC measurement. In the separated partial pressure of oxygen measurement, a gradient echo sequence was used without the bipolar diffusion-sensitization gradient.

Human experiment

The human experiment was conducted under an IRB-approved protocol and a US FDA IND. A patient with severe COPD (female, age 52 years, weight ~50kg) was imaged on a 1.5T Siemens Sonata MRI system. The subject’s total lung capacity (TLC) was 6.08 liters. In the experiment, the subject inhaled a 900ml (15%×TLC) gas mixture consisting of 500ml ³He, 180ml O₂ and 220ml nitrogen (N₂). The RF coil used for imaging was a transmitter-receiver ³He saddle coil set (IGC Medical Advances Inc., Milwaukee, WI, USA), which was composed of two coils (each 23cm×25cm) and operated in quadrature mode. For the simultaneous p₀, R and ADC measurement, a series of six images were acquired for a single slice with the timing scheme [0, 1.32, 9.04, 16.06, 21.08, 24.10] (seconds) and the gradient factor set [0, 1.73, 1.15, 0.71, 0.33, 0] (s/cm²). The imaging parameters used in the diffusion-sensitization gradient echo sequence were: FOV=350mm; slice thickness=50mm; TR/TE:10ms/6.96ms, matrix size: 64×64, and flip angle=4.6 degrees.

Data analysis

The multiple regression method which applied the general least square estimator for p₀, R and ADC calculation was implemented in Matlab (Mathworks Inc. USA). In the data analysis, the signals of the acquired images were first bias-corrected from the background noise(28,29):

$$S_n=\sqrt{{\overline{S}}_n^2-{\sigma }^2},$$ [7]

where S̄_n and S_n denote the signal intensities before and after the correction, respectively, and σ is the original MRI noise in S_n. σ can be calculated from $\sigma =\sqrt{\frac{2}{\pi }}\overline{M}$, where M̄ is the mean value of a 5×5 region at the right-bottom corner of the image where no signal is contained. To generate the parametric maps of p₀, R and ADC, the image pixels were grouped into 2×2 bins, and a bin-by-bin fitting was performed according to Eq.[4]. Two thresholds were applied to select the valid bins. The average threshold only chose bins with an average SNR greater than 3.0 for analysis. The homogeneity threshold selected bins in which the ratio of standard deviation of pixel intensities over the average was smaller than 0.4, and therefore excluded the bins at the edge of the lung where only parts of the bin contained signals.

Results

Phantom experiment

As shown in Fig. 2, the measured average p₀ value was 145±7 Torr, which agreed with the nominal value 137 Torr. The small discrepancy was likely caused by the small volume variance in the administered gas mixture (the gas mixture consisting of (500±20)ml HP ³He and (110±3)ml O₂). The average of the measured R values was 0.03±0.43 Torr/s, close to the expected value 0.0 Torr/s. The phantom experiment was meant to simulate a free diffusion environment, but due to the presence of 20% O₂ in the gas mixture, the measured average ADC value was (1.51±0.08)cm²/s, which is in good agreement with the value (1.60 ± 0.05) cm²/s measured at the same gas concentration by Kober et. al. (30).

Fig. 2.

Simultaneous measurement of p₀, R and ADC with HP ³He MRI on a phantom model. The phantom was an 18×18 cm² plastic bag filled with a gas mixture consisting of 500ml HP ³He and 110ml O₂, to simulate a free ³He diffusion environment. The expected nominal values are 137Torr for p₀ and 0.0Torr/s for R.

Animal experiment

The fitted parametric maps of the normal pig experiment are shown in Fig. 3, which shows that the average values of p₀, R and ADC are 79±15 Torr, 2.1±0.7 Torr/s and 0.14±0.06 cm²/s, respectively. The separate pulmonary oxygen and ADC measurements that followed the simultaneous measurement yielded an average value of 80±12 Torr for p₀, 2.0±0.5 Torr/s for R, and 0.14±0.05 cm²/s for ADC, which were consistent with the values of the simultaneous measurement.

Fig. 3.

Simultaneous measurement of p₀, R and ADC with HP ³He MRI on an in-vivo normal pig experiment. The tracheal region was excluded from data fitting due to a low signal-to-noise ratio caused by high diffusion coefficient. The separated measurements that followed the simultaneous measurement yielded an average value of 80±12 Torr for p₀, 2.0±0.5 Torr/s for R, and 0.14±0.05 cm²/s for ADC.

Human experiment

The experiment results for the COPD patient are shown in Fig. 4, in which the average values of p₀, R and ADC are listed on the corresponding parametric maps. In the figure, the left lung showed low signal intensity due to the poor ventilation caused by the disease. In the selected region-of-interest (the white square box) in the right lung, the measured average ADC was 0.85cm²/s, which was significantly higher than the reported ADC value (0.20cm²/s) of normal subjects(9,10), and the measured R value was 1.2 Torr/s, smaller than the normal oxygen depletion rate of 2.5 Torr/s(16,18). It is also worth noting that in the region-of-interest, ADC was (0.85-0.66)/0.66=29% higher than the global ADC average value while R was (3.1-1.2)/3.1=61% lower than the global R average value. This correlation suggests that oxygen exchange efficiency was affected in the region where lung tissue was destroyed by COPD.

Fig. 4.

A case study of simultaneous measurement of p₀, R and ADC with HP ³He MRI on a COPD patient. In the selected region-of-interest (white square box), the ADC is 29% higher while R is 61% lower than the corresponding global average values. A correlation between the two parameters suggests that in the region where lung tissue was destroyed by COPD, the oxygen exchange efficiency was also reduced.

Discussion

As seen in Eq.[3], four unknown parameters, ε, p₀, R and D, need to be recovered from a series of measurements in the simultaneous measurement technique. Consequently, at the same measurement conditions, the simultaneous measurement technique will introduce higher measurement uncertainties on the p₀, R and ADC than the separated partial pressure of oxygen and ADC measurements(21). At the experiment condition described at the end of the theory section, the simulation showed that for the simultaneous measurement, the normalized measurement uncertainty (standard deviation over mean value) was 14.2% for p₀, 84.2% for R and 15% for ADC. These values are higher than the uncertainties obtained during separate partial pressure of oxygen and ADC measurements: 7.5% for p₀, 56.1% for R and 9.6% for ADC.

A straightforward way to reduce measurement uncertainties is to increase the signal-to-noise ratios of the acquired images. In fact, Eq.[5] indicates that the measurement uncertainties of p₀, R and ADC are inversely proportional to the signal-to-noise ratios (SNRs). If there is no room left to improve the initial ³He hyperpolarization, two techniques may be used to increase the SNRs of the acquired images. The first one, called the pre-wash technique, uses multiple inhales and exhales to increase the ³He gas concentration inside the lung before the simultaneous measurement is initiated. This technique, which was originally applied to measure the regional fractional ventilation of the lung, causes the signal to build up approximately linearly for the first few breaths, saturating after ~5–10 pre-washes(31). If images are also taken at each step of the pre-wash, a scheme that simultaneously measures fractional ventilation, p₀, R and ADC can be implemented. The second technique to improve SNRs is to imprint the diffusion effect onto the longitudinal magnetization using the stimulated echo technique. By avoiding the insertion of the conventional bipolar gradient for diffusion sensitization, a short TE can be chosen in the MRI pulse sequence to increase the SNRs (T2* in the lung is ~8ms). The stimulated echo technique was used by Mugler et. al. to measure the long range diffusion coefficient(32). Therefore, when applying this technique, it is possible to measure p₀, R and long range diffusion coefficient simultaneously.

Conclusion

In this work, we present a single-acquisition technique for the simultaneous measurement of p₀, R and ADC. This technique is based on the multiple regression method, which applies the general linear estimator to retrieve the values of p₀, R and ADC from a series of acquired images. The measurement uncertainties are also analytically derived. Numerical simulation was performed to find an optimal measurement scheme from a variety of choices. This technique was first tested on a phantom model, and then in an in-vivo normal pig experiment. A case study was performed on a COPD patient to investigate the correlation between p₀, R and ADC, which showed that in a region-of-interest ADC is 29% higher while R is 61% lower than the corresponding global average values.

Appendix 1

General least square fitting minimizes the following weighted square summation(21–23):

$${\chi }^2=\sum _{n=1}^M\frac{1}{{\sigma }_n^2}{(y_n-E_n)}^2=\sum _{n=1}^M\frac{1}{{\sigma }_n^2}{\left[y_n-\left(\epsilon ·n-\frac{1}{\xi }{p}_0·t(n)+\frac{1}{2\xi }R·t^2(n)-D·b(n)\right)\right]}^2$$ (A.1)

where M is the number of images acquired after the initial image in the series, y_n denotes the normalized measured signal intensity, and σ_n is the noise in y_n. The weighted summation reflects that after normalization the noise σ_n is not uniform. Based on the error propagation theorem, the noise σ_n in y_n is related to the original MRI noiseσ in S_n by:

$${\sigma }_n^2=\frac{1}{S_n^2}{\sigma }^2+\frac{1}{S_0^2}{\sigma }^2\approx \frac{1}{S_n^2}{\sigma }^2$$ (A.2)

This approximation is valid because, from Eq.[1], S₀ ≫ S_n.

Eq.(A.1) can be expressed in matrix form:

$${\chi }^2={(Cy-CX\beta )}^T(Cy-CX\beta )$$ (A.3)

where C is the noise matrix, y is the vector of normalized signals, X is the coefficient matrix, and β is the vector of unknown parameters:

$$C=\left[\begin{array}{cccc}1/{\sigma }_1& 0& \cdots & 0\\ 0& 1/{\sigma }_2& 0& ⋮\\ ⋮& 0& \ddots & 0\\ 0& \cdots & 0& 1/{\sigma }_n\end{array}\right];y=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right];X=\left[\begin{array}{cccc}1& t_1& t_1^2& b_1\\ 2& t_2& t_2^2& b_2\\ ⋮& ⋮& ⋮& ⋮\\ n& t_n& t_n^2& b_n\end{array}\right];\beta =\left[\begin{array}{c}\epsilon \\ p_0\\ R\\ D\end{array}\right].$$

The minimum value of χ² can be reached by setting to zero the partial derivatives of Eq.(A.3) with respect to β:

$$\frac{\partial {\chi }^2}{\partial \beta }=-{(CX)}^T(Cy-CX\beta )+{(Cy-CX\beta )}^T(-CX)=-2X^T{X}^{T}Cy+2X^T{C}^{T}CX\beta =0$$

Therefore,

$$\beta ={(X^T{C}^{T}CX)}^{-1}{X}^T{C}^{T}Cy={(X^T·V·X)}^{-1}·X^T·V·y$$

Here we define the signal matrix $V=C^{T}C=\left[\begin{array}{cccc}{S}_1^2& 0& \cdots & 0\\ 0& S_2^2& 0& ⋮\\ ⋮& 0& \ddots & 0\\ 0& \cdots & 0& S_n^2\end{array}\right]$ by utilizing Eq.[A.2].