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Young's Modulus to density relationship

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  • Young's Modulus to density relationship

    Hello,

    My question is whether an E-density relationship developed for density ranges of trabecular bone can be used for the entire density range of bone? If it can, should the relationship be extrapolated to cover the new density range? and how would that be done for a power law relationship (using log transforms)? The original and new density ranges are known.

    Any help is greatly appreciated, thank you.

    Tyler - MSc candidate

  • #2
    Re: Young's Modulus to density relationship

    There seems to be a few models that were used to describe the E-density relationship.

    I recently read an article (Schileo et al., 2007) that references a few of them.

    E = 3.790p^3 (p = apparent density) (Carter and Hayes, 1977)
    E = 10.500p^2.29 (p = ash density) (Keller, 1994)
    E = 6.950p^1.49 (p = apparent density) (Morgan et al., 2003)

    The study by Keller obtained experimental data that covers the whole range. You may check that out. However, I have not read this article. Another possible study to look at is Taddei et al., 2006.

    Raymond Chen, MS

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    • #3
      Re: Young's Modulus to density relationship

      you should also check out a book edited by Mow and Hayes, "Basic Orthopaedic Biomechanics". Chapter 3 has a good summary of modulus to density relationships. I wouldn't use the same model for both trabecular and cortical bone.

      Craig Fryman

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      • #4
        Re: Young's Modulus to density relationship

        I highly recommend the review paper by Helgason. This paper notes that the E-density relationship is dependent upon the anatomical site, largely due to differences in bone architecture not captured by density. (Helgason et al., 2007, Mathematical relationships between bone density and mechanical properties: A literature review, Clinical Biomechanics)

        Regarding 1 relationship for trabecular and cortical bone, I prefer the method by Bessho where they use different E-density relationships for different density ranges. (Bessho et al., 2004, Prediction of the strength and fracture location of the femoral neck by CT-based finite-element method: a preliminary study on patients with hip fracture, J Orthop Sci)

        Note that if you are converting from imaged volumetric bone mineral density (BMD) to E, there is a conversion equation needed to link BMD with apparent density. They are not directly equivalent since imaged BMD is correlated with ash density. I recommend a paper by Keyak to find these correlations. (Keyak et al., 1994, Correlations between orthogonal mechanical properties and density of trabecular bone: use of different densitometric measures, Journal of Biomedical Materials Research)

        Regards,
        JD Johnston

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        • #5
          Re: Young's Modulus to density relationship

          All of the comments in this thread have been very informative; there is one thing that isn't very clear to me though. It looks like everyone uses a linear relationship to convert from HU to density (whichever measure of density is used). Is there a theoretical basis for this? I haven't come across one yet in the literature, though that certainly doesn't mean it's not there.

          I ask because a phantom that our lab has scanned with 8 different materials of known bulk density (which as I understand it is equivalent to apparent density) shows a stronger correlation when using a 2nd order polynomial fit rather than a linear fit (R^2 of .995 vs .9). The range of densities is 0.93 g/cm^3 through 2.18g/cm^3 and the derived polynomial equation seems to produce density values that would be more "expected" from sets of tibial scans done with the same settings as the phantom scan.

          Thanks,

          Dan

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          • #6
            Re: Young's Modulus to density relationship

            The issue may be one of ash density versus x-ray attenuation. If you look up the definition of Hounsfield units (Wikipedia is fine) you will see that it is indeed linearly related to x-ray attenuation, but this is not the same as density. We found that the relationship presented by Kaneko,T.S. 2004 was good for relating hydroxyapetite equivalent to ash density (you will need to ensure that you use a phantom with known HA equivalent densities to establish this relationship for your bone), and that the Keller reference was accurate for all ranges of ash density to modulus.

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            • #7
              Re: Young's Modulus to density relationship

              Hi Karen,

              As you noted, HU is of course linearly related to x-ray attenuation, but this:

              "...is not the same as density"

              is the crux of what we're seeing. We're using a phantom in which the densities have been verified through testing in our materials lab and finding that the relationship of material density to HU is not linear, whereas previous studies have used a linear fit for the material density to HU relationship without giving a specific reason why. I'm not a materials engineer or a radiological expert, so I'm at a little bit of a loss on how to explain this given what I've seen presented in the literature. Is it possible that how the density of a material affects x-ray attenuation changes near empty space and at higher densities (in our case around 2g/cm^3) with scan parameters typically used for living human subjects?

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              • #8
                Re: Young's Modulus to density relationship

                "......We're using a phantom in which the densities have been verified through testing in our materials lab and finding that the relationship of material density to HU is not linear, whereas previous studies have used a linear fit for the material density to HU relationship without giving a specific reason why. ......"

                DANIEL, let me give you a reason why they use a linear relationship: Convenience!
                It seems too tasking for some to assume anything else than linearity. I can accept that ash density and HU, on theoretical grounds, are linearly related.
                However, I can now tell you that the relationship between apparent density and ash density across the full density range from cortical to cancellous bone is not linear but curvilinear. The reason has nothing to do with attenuation but with normal bone physiology. Please read our recent articles and replies in: P. Zioupos, R.B. Cook, J.R. Hutchinson. ‘Some basic relationships between density values in cancellous and cortical bone’ J Biomechanics, 41: 1961-1968, 2008. P. Zioupos, R.B. Cook, J.R. Hutchinson. ‘More thoughts on the relationship between apparent and material densities in bone.’ J Biomechanics, 42: 794-795, 2009.
                I would love to know what you make of them.
                Last edited by Dr Peter Zioupos; March 15, 2011, 11:53 AM.

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