Modified brix model analysis of bone perfusion in subjects of varying bone mineral density


  • Heather Ting Ma PhD,

    Corresponding author
    1. Department of Diagnostic Radiology and Organ Imaging, Prince of Wales Hospital, The Chinese University of Hong Kong, Shatin, Hong Kong, China
    2. Department of Electronic and Information Engineering, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen, China
    3. Jockey Club Centre for Osteoporosis Care and Control, Public Health School, Prince of Wales Hospital, The Chinese University of Hong Kong, Shatin, Hong Kong, China
    • Department of Electronic and Information Engineering, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen University Town, Xili, Shenzhen, China
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  • James F. Griffith MD,

    1. Department of Diagnostic Radiology and Organ Imaging, Prince of Wales Hospital, The Chinese University of Hong Kong, Shatin, Hong Kong, China
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  • David K. Yeung PhD,

    1. Department of Diagnostic Radiology and Organ Imaging, Prince of Wales Hospital, The Chinese University of Hong Kong, Shatin, Hong Kong, China
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  • P.C. Leung MD

    1. Jockey Club Centre for Osteoporosis Care and Control, Public Health School, Prince of Wales Hospital, The Chinese University of Hong Kong, Shatin, Hong Kong, China
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To apply pharmacokinetic modeling to the investigation of bone perfusion in subjects of varying bone mineral density.

Materials and Methods:

This study re-analyzed previous experimental data. A modified pharmacokinetic model was applied to data obtained from two prior studies of dynamic contrast-enhanced MR imaging of L3 vertebral body in 165 subjects (65 males, 100 females), classified into three groups (normal, osteopenia, and osteoporosis) according to bone mineral density. Three parameters, amplitude A, exchange rate (kep), and elimination rate (kel), were obtained by fitting the signal intensity to the pharmacokinetic model. These parameters were compared across the three groups for males and females, respectively.


Perfusion parameters, amplitude A was found to be reduced in osteoporotic subjects with additional, though less pronounced, reductions found in the permeability constant (A*kep) and the elimination rate (kel). Increased marrow fat content was found in osteoporotic bone, which helped to partially explain the observed reduction in interstitial space.


By pharmacokinetic model, bone perfusion can be quantitatively analyzed with alteration in functional parameters related to microcirculation in subjects of varying bone mineral density. Developing bone marrow specific pharmacokinetic models should help to deepen knowledge of physiological and pathological perfusion changes occurring in bone. J. Magn. Reson. Imaging 2010;31:1169–1175. © 2010 Wiley-Liss, Inc.

OSTEOPOROSIS IS THE most common metabolic bone disorder (1). Bone loss and structural deterioration of bone tissue begins from the time of peak bone mass accelerating in the elderly particularly in postmenopausal females. This loss and structural deterioration of bone proceeds until bone strength becomes sufficiently poor as to increase susceptibility to fracture at which stage the osteoporotic state has been reached. Clinical, epidemiological, and histological studies have indicated a link between vascular disease and osteoporosis (2–5). Dynamic contrast-enhanced MRI (DCE-MRI) provides a more direct measure of bone perfusion. Bone perfusion is a physiological process, which can be assessed by DCE-MRI. Bone perfusion refers to a diverse process dependent on factors such as tissue blood flow, capillary capacitance and permeability, interstitial diffusion, interstitial space volume, and venous return (6, 7). DCE-MRI studies have shown how semi-quantitative perfusion parameters (enhancement maximum, Emax, and enhancement slope, Eslope) are consistently reduced in osteopenic and osteoporotic bone compared with normal bone mineral density (BMD) subjects (8–10). These semi-quantitative parameters are robust and easy to measure, but they are limited in the information they provide regarding the physiological processes affected. Knowledge of the physiological processes behind altered bone perfusion can be improved by obtaining a more detailed assessment of bone perfusion kinetics beyond that provided by Emax and Eslope. This more detailed evaluation can be obtained by pharmacokinetic modeling. Pharmacokinetic modeling has been applied to the study of bone marrow perfusion in patients with multiple myeloma (11), demonstrating how quantitative pharmacokinetic parameters corresponded with histological bone marrow characteristics.

The purpose of this study was to apply a modified Brix pharmacokinetic model to investigate bone perfusion in subjects of varying BMD with a view to increasing our knowledge of the bone perfusion anomalies occurring in osteoporosis.


This pharmacokinetic modeling involved a reassessment of DCE-MRI raw data obtained in two previous osteoporosis-related studies (8, 9). The raw data obtained in these two studies, initially evaluated using Emax and Eslope, was fully re-analyzed to ensure suitability for pharmacokinetic modeling. Both studies were approved by the institutional review board with all participants providing signed informed consent. No additional consent was required for this pharmacokinetic re-analysis. A brief review of the methods used in these two studies is provided to allow understanding of pharmacokinetic data acquisition. Participating subjects underwent both dual x-ray absorptiometry (DXA) to measure bone mineral density and MR imaging comprised of 1H MR spectroscopy and DCE-MRI of the lumbar spine. Subjects were excluded if they had (a) clinical or imaging evidence of renal osteodystrophy or other metabolic bone disease other than osteoporosis or a known malignancy, (b) a history of lumbar spinal surgery or irradiation, or (c) MR imaging evidence of large intravertebral disk herniation, hemangioma, or moderate to severe vertebral fracture of L3. In total, 174 subjects (72 males, 102 females) were included in this retrospective study.

MR Examinations

MR examinations were performed on a 1.5 Tesla (T) whole body imaging system (Intera NT; Philips Medical Systems, Best, The Netherlands) with a maximum gradient strength of 30 mT/m. A 20-cm diameter circular surface coil centered at vertebrae L3 was used to optimize the sensitivity of both 1H MR spectroscopy and DCE-MRI. A volume of interest located centrally in the vertebral body was defined. After local shimming and gradient adjustments, data was acquired at a spectral bandwidth of 1000 Hz with 512 data points, and 64 nonwater suppressed signals obtained using a point-resolved MR spectroscopic sequence (TR/TE = 3000/25 ms).

DCE-MRI data were acquired in the mid-lumbar sagittal plane for male subjects and in the transverse plane through the mid-L3 vertebral body region for female subjects. Different planes were used as these two previous studies obtained on male and female subjects had slightly different study objectives. Dynamic MR imaging was performed using a short T1-weighted gradient-echo sequence (TR/TE, 2.7/0.95; prepulse inversion time, 400 ms; flip angle, 15°; section thickness, 10 mm; number of slice, one; field of view, 250 mm; acquisition matrix, 256 × 256; number of signals acquired, one). A total of 160 dynamic images were obtained with a temporal resolution of 543 ms, resulting in a total interrogation time of 87 s. A bolus of gadoteric acid (Dotarem, Guer-Guerbet, Aulnay, France) at a concentration of 0.15 mmol per kilogram body weight was injected by means of a power injector (Spectris; Medrad, Indianola, PA) at a rate of 2.5 mL/s through a 20-gauge antecubitial vein intravenous catheter (Angiocath; Infusion Therapy Systems, Sandy, Utah). Injection was followed by a 20-mL saline flush. Dynamic MR imaging started at the same time as contrast medium injection started (“time zero” or T0).

Data Processing

MR Spectroscopic Data Processing

Spectra were analyzed at an off-line computer (Precision 650 Workstation; Dell, Austin, TX), using a time-domain fitting routine known as a variable projection, or VARPRO, method implemented in MRUI software (12). Water (4.65 ppm) and lipid (1.3 ppm) peak amplitudes were measured to determine vertebral marrow fat content, which was defined as the relative fat signal amplitude in terms of percentage of total signal amplitude (water and fat) and calculated according to the following equation: fat content = [Ifat/(IfatIwat)] × 100, where Ifat and Iwat are the peak amplitudes of fat and water, respectively. No correction for relaxation losses was applied.

Data Processing and Modeling of DCE Data

A region of interest (ROI) was drawn manually encompassing the cancellous bone of L3 vertebral body on a workstation (Viewforum; Philips Medical System) (Fig. 1). Signal intensity within the ROI was averaged to generate a time-signal intensity curve for each subject, which was saved for off-line analysis.

Figure 1.

ROI drawn in T1-weighted MR image for male in sagittal plane (a) and for female in axial plane (b), respectively. The ROI position is just within cortical margins of L3 vertebral body, from which the signal intensity data points were obtained from DCE MRI.

A pharmacokinetic model (6, 13), modified from the two-compartment Brix model (14) (Fig. 2), was used to analyze DCE-MRI data. This model assumes that the relative change in tissue signal intensity is proportional to tissue contrast concentration (14). A mono-exponential function approximates plasma contrast concentration for up to 20 min after injection (14). Direct measurement of plasma concentration was unnecessary as the clearance rate kel is estimated directly from the target tissue (6). The original Brix model was therefore modified as follows:

equation image(1)

where S0 is the baseline signal intensity before contrast injection; S(t) is the signal intensity change with time after contrast injection; A is the amplitude of contrast uptake; kep is the rate constant for contrast extraction from interstitial space into the plasma; kel is the eliminating rate constant of the contrast from the plasma (Fig. 2).

Figure 2.

Original Brix model with two compartments. The contrast is infused with a constant rate Kin into the central compartment and then exchange between the two compartments by the rate kpe and kep, whereas the kel describes the elimination of the contrast from the plasma.

The three functional parameters, A, kep, and kel, were obtained by fitting the original signal (from T0 to the end of signal) by Eq. [1] using the least squares method (Fig. 3). In the original work of Brix model (14), amplitude A corresponded to cerebral interstitial space with consideration of the blood–brain barrier. In the bone marrow, this would be comparable to the total volume of contrast in the ROI tissues comprising intravascular contrast (in arterioles, capillaries, and venous sinuses) and extravascular contrast within the interstitial space (15). The exchange rate kep reflects the efficiency of diffusion of contrast from the interstitial space to plasma, which is related to permeability surface area and interstitial space volume. Product Akep reflects permeability surface area per unit volume of vasculature. Elimination of contrast from the plasma in the tissue under investigation is represented by elimination rate kel.

Figure 3.

Data processing on DCE data. a: Filtering intensity signal and detecting characteristic points (T0 is the time point of the signal starting to enhance; Tp is the time point of maximum enhancement). b: Curve fitting by modified Brix model. The solid and smooth curve is the fitting result.

Representative Curves

Having obtained quantitative measures from pharmacokinetic modeling for each subject, representative perfusion signal intensity curves were calculated for the three different BMD groups (normal, osteopenia, and osteoporosis) and for male and female respectively. The calculation was based on the pharmacokinetic model Eq. [1] and the mean values of the quantitative measures, A, kep, and kel, for each subgroup. To have a longer wash-out phase for comparison, the calculation was conducted for 10 min. Signal processing, curve fitting, and representative curve calculation in this study were undertaken using in-house developed software program written using Matlab (MathWorks).

Statistical Analysis

Data were presented as mean ± standard deviation. Parameters were compared across the three BMD groups (normal, osteopenia, and osteoporosis) for both males and females. Analysis of variance method was used to evaluate differences in parameters among groups. Pearson correlation coefficients were used to evaluate the linear relationship between pairs of parameters. Statistical analysis was performed using statistical software (SPSS 13.0). A P value of less than 0.05 was considered statistically significant.



Seven males and two females were excluded in the final analysis due to suboptimal curve fitting for perfusion data. This resulted in a final cohort of 165 subjects (65 males, 100 females). Because the experimental protocol was different for male and female subjects, analysis was conducted separately for each gender. There was a significant difference (P = 0.049) in age between male and female subjects (Table 1) though no difference in age (P ≥ 0.1) was present between different BMD groups among both genders.

Table 1. Comparison of Parameters Among Three BMD Groups for Male and Female Subjects
 Group (number)Age (yrs)L3 BMD (g/cm2)Fat content (%)AkepkelA*kep
MaleAll males (n = 65)73.6 ± 5.90.95 ± 0.2352.2 ± 9.30.47 ± 0.274.11 ± 1.950.29 ± 0.371.66 ± 0.77
Normal (n = 37)72.9 ± 4.31.10 ± 0.1449.6 ± 8.90.56 ± 0.283.85 ± 1.570.39 ± 0.321.91 ± 0.77
Osteopenia (n = 15)75.1 ± 9.10.84 ± 0.0853.5 ± 9.30.43 ± 0.223.64 ± 1.360.32 ± 0.431.40 ± 0.60
Osteoporosis (n = 13)74.1 ± 5.30.62 ± 0.0758.1 ± 7.70.27 ± 0.165.40 ± 2.90−0.01 ± 0.251.24 ± 0.25
P value= 0.46<0.001= 0.012= 0.002= 0.025= 0.003= 0.007
FemaleAll females (n = 100)72.2 ± 3.50.77 ± 0.1665.0 ± 9.60.35 ± 0.125.74 ± 2.050.28 ± 0.191.95 ± 0.86
Normal (n = 17)73.7 ± 3.51.01 ± 0.1059.0 ± 10.20.44 ± 0.144.82 ± 1.380.34 ± 0.172.08 ± 0.72
Osteopenia (n = 30)71.4 ± 2.90.84 ± 0.0763.3 ± 9.50.35 ± 0.136.34 ± 2.350.27 ± 0.182.22 ± 1.04
Osteoporosis (n = 53)72.2 ± 3.70.66 ± 0.0767.8 ± 8.40.32 ± 0.105.69 ± 1.950.26 ± 0.191.77 ± 0.75
P value= 0.1<0.001= 0.002= 0.001= 0.046= 0.324= 0.059

Comparison Analysis

The pharmacokinetic model parameters showed quite similar differences between subjects of varying BMD although with some variations between genders. For male subjects, osteoporotic subjects had lower mean A and kel compared with osteopenic and normal BMD subjects (P < 0.01), while kep showed an opposite trend being larger in osteoporotic subjects compared with osteopenic and normal BMD subjects (P < 0.05). Product Akep, indicating the vessel permeability, was significantly decreased in osteoporotic and osteopenic male subjects compared with normal BMD subjects. For female subjects, osteoporotic and osteopenic subjects had lower mean A compared with normal BMD subjects (P < 0.01). Osteoporotic and osteopenic subjects also had lower mean kel though this did not reach statistical significance (P = 0.32). As opposed to male subjects, parameter kep in female subjects was found to be largest in osteopenic (rather than osteoporotic) subjects and smallest in normal BMD subjects (P < 0.05). As a result, Akep also was largest in female osteopenic subjects and smallest in normal BMD subjects.

Correlation Analysis

For both genders, parameter A mildly to moderately and positively correlated with BMD (male, r = 0.41; female, r = 0.38; P < 0.01), and strongly and negatively correlated with fat content (male, r = −0.74; female, r = −0.52; P < 0.01) (Table 2; Fig. 4). In other words, a decreasing parameter A was consistently associated with decreasing BMD and increasing fat content. A gender difference existed in the correlation between other paired parameters as shown in Table 2. Among all tested parameters, the strongest correlation was found between a decrease in parameter A and an increase in marrow fat content (|r| > 0.52, P < 0.001).

Figure 4.

Correlation between amplitude A and BMD and fat content for male and female respectively. a: Correlation between amplitude A and BMD for male. b: Correlation between amplitude A and fat content for male. c: Correlation between amplitude A and BMD for female. d: Correlation between amplitude A and fat content for female. The most significant and strong correlation was found between amplitude A and fat content for male subjects.

Table 2. Correlations Between Paired Parameters
  • *

    P < 0.05.

  • **

    P < 0.01.

L3 BMD0.41**0.39**−0.220.39**0.38**0.15−0.120.15
Fat Content−0.71**−0.39**0.39**−0.49**−0.52**−0.060.08−0.33**

Representative Curves

By applying the mean values of quantitative measures obtained from the pharmacokinetic model, representative perfusion signal intensity curves were calculated according to Eq. [1] for each BMD group of male and female subjects (Fig. 5). According to the model algorithm, the relative signal intensity was defined following modeling as equation image. Signal intensity curves of osteoporotic subjects were flatter than those of normal BMD subjects, especially during the wash-out phase.

Figure 5.

Calculated representative DCE signal of groups with different BMD for male (a) and female (b), respectively. Calculation was conducted by the modified Brix model adopting the mean value of functional parameters (A, kep, kel) for each group. Male subjects show more distinct curve pattern among three groups with different BMD.


DCE-MRI is a widely used method of assessing tissue perfusion which, in recent years, has been used to study bone perfusion in a variety of physiological and disease conditions (11, 16–23). Some researchers have further analyzed bone perfusion function by pharmacokinetic modeling in patients with multiple myeloma, bone edema, and Paget's disease of bone (11, 23, 24). Bone is composed of trabecular and cortical bone. All of the trabecular bone and the inner two-thirds of the cortical bone receive its blood supply from the marrow cavity (25). Several DCE-MRI studies have shown how perfusion parameters are reduced in osteoporotic bone (8–10). This is the first study to apply pharmacokinetic model to the investigation of bone perfusion in osteoporosis. After a bolus injection, tissue concentration of gadolinium is determined by local blood flow, capillary capacitance, vessel permeability, interstitial space, and interstitial diffusion (7). In the Brix model, the MR signal intensity is considered to be linearly proportional to the concentration of gadolinium contrast (Fig. 2), enabling functional parameters to be derived from the signal intensity curve.

The first main finding of this study was a notable reduction in amplitude A in osteoporotic subjects compared with normal subjects. Amplitude A is affected by several variables such as vascular inflow, vessel density, intraosseous pressure, and interstitial space. Change in any of these variables will result in a variation in amplitude A. With respect to vascular inflow, previous studies have shown that the perfusion anomalies occurring in osteoporosis most likely originate within bone rather than being representative of a systemic circulatory disturbance because changes were only apparent within bone and not within adjacent muscle (9, 10). Reduction of capillary density is one possible cause of the decreased amplitude A. Reduced density of arterial capillaries with more frequent arteriosclerotic vascular lesions has been shown in patients with proximal femoral osteoporosis (26). Another study on multiple myeloma infiltration of vertebral bodies and which applied an identical pharmacokinetic model to the current study, found that amplitude A mirrored bone marrow vessel density assessed histologically (11). Although in that study parameter A was normalized by arterial signal, it confirmed that parameter A is proportional to the marrow vessel density and this may also be one likely cause of the decreased parameter A seen in osteoporotic bone. On the other hand, marrow fat also increases in osteoporotic bone which may reduce interstitial space and flow (8–10, 27, 28). This fat effect is supported by the significant negative correlation found between amplitude A and fat content. However, amplitude A decreased to a greater degree than that could be accounted for by the observed increase in marrow fat content alone.

The second main finding of this study is the varied capillary permeability evident in subjects with reduced BMD. The efflux rate kep represents a ratio of the permeability surface area product per volume (or “volume transfer constant,” kmath image) relative to the size of the interstitial space. Product Akep is proportional to the volume transfer constant (kmath image), and loosely referred to as ‘permeability’ (6). Although the Brix model does not provide a direct measure of capillary permeability, the product Akep changes in proportion to permeability (14, 29). Product Akep was consistently reduced in subjects with osteoporosis, an effect more noticeable in male subjects. It appears that vascular permeability is reduced as BMD decreases. Permeability may be influenced by factors such as capillary endothelial permeability and interstitial or intraosseous pressure, with higher interstitial pressures limiting diffusion of molecules between the capillary bed and the interstitial space. A previous study (30) has reported how increased marrow fat increases intraosseous pressure. In this study, a mild to moderate negative correlation was found between increasing fat content and decreasing Akep, indicating that increased marrow fat content may be limiting transfer between the intravascular and interstitial spaces.

The third main finding of this study was a tendency for the plasma elimination rate kel to decrease as BMD reduced in both male and female subjects. The eliminating rate kel mainly relates to contrast wash-out in the plasma and is mainly related to venous return. Venous return also appears to reduce as BMD decreases. This may again be related to increased intraosseous fat increasing intraosseous pressure within the confines of the vertebral body leading to impaired venous return (31).

An increasing number of MR-based perfusion studies are being performed to study bone marrow physiology and disease. Parameter A and kep from the modified Brix model, used in the current study, have been shown to correspond with histological characteristics in subjects with bone marrow infiltration in multiple myeloma (11). In this study, we applied this model to the study of marrow perfusion in subjects of varying bone mineral density and gained more insight into the osteoporotic process. Brix model, shows potential for noninvasively quantifying physiological change in some bone diseases. This is only the first step in the process of fully understanding bone perfusion. The Brix model is a model designed to study brain tumor perfusion and not bone marrow perfusion per se. Bone marrow perfusion is likely to be different to brain tumor perfusion in lacking a blood–brain barrier, in being confined within the rigid bony cortex, comprising capacious venous sinuses and lacking a well-defined capillary network. Clearly, the next step is to develop a pharmacokinetic model specifically for the assessment of marrow perfusion taking into account the unique characteristics of marrow perfusion.

This retrospective study had two main limitations. First, perfusion data acquisition was obtained in the sagittal plane for male and the axial plane for females. This protocol difference reflects slightly different objectives in the two primary studies from which the raw data were acquired. Theoretically, data acquisition in a sagittal or axial plane should not affect perfusion imaging, and therefore not affect the parameters derived or pharmacokinetic modeling. Two prior studies (8, 9), which involved scan acquisition in the sagittal and axial plane respectively, showed identical trends in semi-quantitative parameters for male and female subjects of different bone density. It also indicates that scanning in the sagittal or axial plane has no recognizable effect on perfusion MR imaging. Second, data acquisition duration was relatively short at 87 s, which may limit assessment of the full wash-out phase and thus influence the parameter kel. Specifically, we were unable to test the effect of the incomplete wash-out phase on the parameter kel in this study due to the short acquisition time.

In conclusion, pharmacokinetic model was applied to the investigation of bone marrow perfusion anomalies in osteoporotic subjects. The perfusion parameter, amplitude A was found to be reduced in osteoporotic bone with additional although less pronounced reductions found in the permeability constant and the elimination constant.