Created a program to calculated path length through gold nanoparticles, depending on rotation (theta) and shift in x-direction (s). Can pick radius, proportion of length to width, and shift in x-direction. Numbers so far look reasonable, even for different numbers of nanoparticles in the column.
Further work. Need to account for shift in y-direction. Need to make a histogram of many trials for the same number of nanoparticles. Need to think about how to relate to real distances. Currently am using arbitrary values, which can hopefully be scaled.
Discussed the realities of a computer program to simulate the nanoparticle problem. Will use JAVA because no time to learn IDL.
Cylinder has advantages over cube because of symmetry. Only one angle of rotation makes a difference for pathlength. Rotational symmetry eliminates one of the rotational parameters. Azimuthal rotation doesn't cause different pathlengths because of the uniform photon field.
To write program, consider just one column of nanoparticles. Start small. One or 10 nanoparticles. Parameters: theta, s. Where theta is the polar angle of rotation and s is the distance of shift from the center of the column of nanoparticles. Things should scale for more nanoparticles in the column, equal to that which makes 1 mg of gold per 1 cc water. Keep the column of nanonparticles evenly spaced up and down.
The cylinder's pathlengths should be easy to derive. At a certain angle theta, a simple trigonometric relationship will give the length of the photon's path. At the end caps, the pathlength can be approximated by the length of the nanoparticle, assuming that the length of the nanoparticle is sufficiently larger than 2r, the diameter of the circular face of the nanoparticle.
Considering the middle part of the nanoparticle, at an angle theta, all path lengths should be the same, except at the end caps.
I think I need to also account for shift in the y direction. So, shift in x, shift in y, and polar angle theta. Hm.
Finished up paper and pen pathlengths for arbitrary length cylinders aligned perpendicular to the x-rays. This is independent of radius and length. Also, did this for a rectangular prism with square base. Did this for the case of the cube. For all cases, cube and arbitrary height, none are dependent on the length or width, given equal distribution.
Did some calculations about the percentage of the square cross section that will be covered by nanoparticle columns as opposed to water columns. Numerically, for the square prism case, the nanoparticles take up 0.14% of the cross sectional area. This, hopefully, will amount to 0.14% of the x-ray beam striking nanoparticles, assuming a uniform and monoenergetic x-ray beam.
I did a lot of thinking about what orientations would be possible and what the probability is of getting certain orientations. I tried to think of the pathlength as a function of theta and phi. It turns out it's more difficult that I had imagined. I might try rotations only in one dimension to start.
Didn't do the Microsoft Excel version of the simulation. Broke out my JAVA files from freshman year of college, and am currently refreshing my memory on programming. Bought two JAVA books, which I will pick up tomorrow. May switch over to Matlab, even though I cannot do work on that at home, where I do most of my work.
Want to start simulation with cubes, since there's a certain amount of symmetry that can be exploited. May move on to cylinders and other shapes which are more oblong.
Cannot work further on programming until can get book and CD with ReadLine capability. Will think about it more tomorrow morning.
Apparently the cylinder case is not dependent on r either. Small math error in the perpendicular case (with l=2r). Need to do the perpindicular case with arbitrary length. Also need to integrate over the cross sectional area of the cubic centimeter block of water.
Perhaps write a computer program to try out different orientations and random placement and angles of the cylinders. May try to do in Excel first.
For cylinders on their sides, evenly distributed and all perpendicular to the incoming photon beam and with l=2r. Path length is proportional to 1/r^2.
Apparently I didn't think about the radius being half the diameter, so I had to alter my calculation for cylinders. My new calculation puts the path length independent of both l and r, given that l=2(alpha)r, where (alpha) is a non-zero constant. I found the error in my calculation. Either way, path length is independent of l,r.
I calculated the path length through cylinders (aligned parallel to the incoming photon beam) evenly distributed in a cubic centimeter of water. First, I started off with the case where the height (l) of the cylinder was equal to the radius (r). With l=r, I found that the attenuation is not dependent on r (or l) at all.
Next, I tried the case where the cylinders were of arbitrary height l, still evenly distributed and aligned parallel to the incoming photon beam. I had to figure out how to place them evenly distributed. Because a cylinder can be twice as high as it is wide, I had to factor in the extra length when calculating the distribution. I found that the path length is proportional to r^{1/3}.
It seems to me that I'm talking more about the path length than I am about the actual attenuation, since the attenuation equation is only dependent on r (and l) in the path length term. The rest is pretty straightforward --- mu/rho(gold), rho(gold, mu/rho(water), rho(water). These are all numbers to look up, and they are not affected by the geometry.
Using the attenuation equation, I found the attenuation of a beam of photons upon a cubic centimeter of water with uniformly distributed spherical gold nanoparticles to be independent of nanoparticle size. The cancellation occurs because the average chord length gets multiplied by the number of particles in one column of spheres. The average chord length is dependent on r. Because we required that the particles be uniformly distributed, I took the cubic root of the total number of particles, and that gives the one factor of r in the denominator, to cancel with the r from the average chord length.
19 October 2006
1pm meeting with John.
Yes, in fact, I do need the mass attenuation coefficient. Found $(\frac{\mu}{\rho})_{eff,120 keV}$ to be 0.0280948 cm^2/g. What I really need to do now is to find the path length through the cube of water with gold cylinders, lined up parallel to the photon beam. The nanoparticles need to be uniformly distributed, so there's no choice in position. Can figure out the water distance between nanoparticles, figure out the total pathlength through the gold and the water. Calculate what percentage of the cc cross section is gold nanoparticle vs. water.
Perhaps rework the sphere problem with the attenuation equation.
13 October 2006
Worked on figuring out the units.
I thought about the units: whether they made sense, and what assumptions I was making about the geometry of the system. I came to the conclusion that there was something wrong with my number: that it didn't account for the mass attenuation coefficient. I realized this when I got the same answer for the cylinder (aligned parallel to the incident photon beam) as I did the sphere, and that doesn't seem to make any sense for me. Although, my generalization of the sphere path length as the average chord length seems to imply that the sphere is cylinder-like when you consider all nanoparticles in the 1 cc space. Also, we could just be assuming that the photon attenuation is proportional to the density of nanospheres in the 1cc box of water.
I am all of a sudden confused by how to use the mass attenuation coefficient in practice. Its units are in cm^2/g, meaning that it is the probability for interaction when a single photon travels through an attenuating medium with 1 gram per cm^2. It seems like to me that I need those units to be cm^3/g, in which case, I have the ideal situation, since I'm putting 1 gram of gold per 1 cc of water.
Trying to figure out how to find the average chord length of a circle, so I can do the case with the cylinders with lengths perpendicular to the incident photon beam. I assume that the "average chord length" is not the average chord length of the cylinder, but rather that of the circle.
Website with gold mass attenuation coefficients for different incident photon energies: http://physics.nist.gov/PhysRefData/XrayMassCoef/ElemTab/z79.html.
12 October 2006
10am meeting with John. Discussed next steps.
Look at the extreme cases. Cylinders aligned with long axis parallel to incoming photons, cylinders aligned with long axis perpendicular to incoming photons. Use the length or the average chord length for a circle to determine the average path length. The average chord length might still be useful later on, since, in most situations, the cylinders (or ellisoids, or pyramids) won't all be aligned in the same direction. The distribution of differently sized chord lengths is about equivalent to approximating all path lengths being the average chord length.
Look for differences in the extremes. With the longer axis parallel to the photon beam, there is probably a greater chance for maximal attenuation. With the shortest axis parallel to the photon beam, there is probably the smallest chance for maximal attenuation. The curve in between will probably be well-behaved and downward sloping from max to min. Perhaps for cylinder, attenuation might increase at the diagonal.
Maximizing size of nanoparticle. Start with a boundary max of 100 nm. Studies show that 50 nm is the size for maximal uptake of nanoparticles, so hopefully the maximal attenuation occurs there, too. If not, it doesn't matter anyway, because beyond 100 nm is pretty large for a nanoparticle.
Solving the density problem. Assume the gold nanoparticle has a density equivalent to the density of gold. Density of gold = 19.3 g/cc.
Relating extreme methods to avg chord length method. Hopefully the avg chord length method will fall between the two extremes. Graph path length vs. angle of rotation in the polar direction. Azimuthal rotation should be same all around, as well as axial rotation.
Beverly promises to work more on this this week.
11 October 2006
To be clearer about my question about density.
Each gold nanoparticle is made up of many gold atoms. If I don't know how many gold atoms are in a nanoparticle of radius r, I can't tell you how many nanoparticles of radius r compose 1 mg of stuff. I'm not angry, really, John. :)
10 October 2006
Worked on the sphere problem. For N incoming photons per cm^2, through a target of 1 cc of water with 1 mg/cc gold nanoparticles of any radius, the number of interactions will be N/(3*rho). Need to check equation set up and units to verify. Essentially, I think I modelled a sphere as a cylinder with a circular face of pi*r^2 and a length of 4r/3, the average chord length of a sphere of radius r. The photons are incident on the circular face, and I approximate the chords through the sphere as having the same path length, but there are fewer of them.
Cannot find a density for gold nanoparticles. Seems like the density most groups care about is the number of nanoparticles per volume of liquid in which it is suspended. Will look more later.
07 October 2006
Went to Argonne National Lab open house. Visited building 440 - Nanoparticles. Asked scientists there about synthesis of nanoparticles. I was given two names for whom to contact for all my nanoparticle questions. Will add once I have the paper in front of me.
Apparently it is really easy to make nanoparticles. You put something soap-like (?) with some gold particles with one other ingredient, and the nanoparticles preferentially form. If you know the concentration of each of the three ingredients, the number of gold atoms per nanoparticle is known almost exactly (plus or minus a small percent). Yes, you can make cones, etc.
Tried doing some calculations of average chord length, and found that ellipsoids are tougher than originally thought them to be. Also, need to clarify understanding of questions to be answered.
06 October 2006
Continued notes from yesterday. Papers gathered on chord length.
KVp - Kilovolts peak. The voltage potential across which the electrons are accelerated in a Linac machine. The electrons hit tungsten (or some other metal) at the far end of the accelerator, and Bremsstrahlung occurs. Bremsstrahlung photons are generally forward-directed. You get a beam of photons from this Linac machine. If the entire amount of energy is transferred from an electron accelerated through 120 KVp to the Bremsstrahlung photon, the energy of the photon would be equal to the 120 KeV. The truth of the matter is that the greatest number of photons resulting from Bremsstrahlung end up getting only about 1/3 the KVp amount of energy. The graph looks sort of like the blackbody spectrum. Filters (like the copper filter) remove the photons that are less than 30 KeV, since they do not penetrate the skin anyway.
Chord length distributions in binary stochastic media in two and three dimensions.
G. L. Olson, D. S. Miller, E. W. Larsen, J. E. Morel
Jour. Quant. Spectros. and Rad. Transf. 2006;101:269-283
Analysis of chord-length distributions
C. Burger, W. Ruland
Acta Cryst. 2001;A57:482-491.
Relationship between particle-size and chord-length distributions in focused beam reflectance measurement: stability of direct inversion and weighting
E. J. W. Wynn
Pow. Tech. 2003;133:125-133
(Aside - Contacted Greg Kuczman about 2nd quarter rotation.)
Paper for Greg Kuczman's research.
High Spectral and Spatial Resolution MRI of Breast Lesions: Preliminary Clinical Experience
M. Medved, G. M. Newstead, H. Abe, M. A. Zamora, O. I. Olopade, G. S. Karczmar
AJR 2006;186:30-37
05 October 2006
10am meeting with John - Discussed 8-week project goals and starting points.
Average chord length. We want to characterize the dose enhancement as a function of average chord length. This can be calculated with the following equation: l=4V/S, where l is the average chord length, V is the volume of the nanoparticle, and S is the surface area of the nanoparticle. This is only applicable for a convex body. For a sphere with radius r, l=4r/3. Perhaps will investigate spheres, ellipsoids, cones, pyramids, and rectangular prisms.
Approximations. Everything is an approximation in the world of new ideas in physics. It's like the spherical chicken joke. So, I will be modelling nanoparticles as various regular polygons/spheres/ellipsoids, assuming that the fellas at Argonne can create these shapes. We will also be approximating the body as water, which, as it turns out, generates data which is only about 1% off. The photon beam will be approximated as a 100 nm^2 beam, as a square 10 nm x 10 nm. Experiments may be performed later using a simple setup with nanoparticles and water in a test cube, irradiated by a square (or close to square) beam.
Effective nanoparticle concentration. About 5 mg of nanoparticles per gram of tumor is necessary to enhance the dose. We use one cubic centimeter squared of water to represent the tumor. With this data as well as the density of gold (and size of nanoparticles), one can determine how many gold nanoparticles are required in the cc of water to cause dose enhancement. For a certain volume of tumor, how many gold nanoparticles are needed (of a particular size/shape)? Figure out the density of gold in the tissue. Look at this combined with the average chord length.
Attenuation. Want to figure out the concetration, shape, and size of gold nanoparticles that will attenuate more photons. More attenuation means more events occuring, which means more energy deposition in the tumor cells. Perhaps the orientation of gold nanoparticles will cause higher attenuation; for instance, orienting oblong particles with the long axis parallel to the photon beam.
Number of interactions. If the gold nanoparticles are randomly distributed, then how many interactions will occur? This is probably size/shape dependent. It would be interesting to see how the cross sectional area will affect the dose enhancement. With different path lengths, the overall number of interactions will occur.
Current technology on MC simulations. Right now, there are only MC simulations for treatments that involve particles of micrometer size at best. Not nanometer scale.
Projects for me to do:
04 October 2006
Read a bunch of papers on the topic of nanoparticles used for dose enhancement.
Characterization of the Theoretical Radiation Dose Enhancement from Nanoparticles
J. C. Roeske, L. Nunez, M. Hoggarth, E. Labay, R. R. Weichselbaum
Submitted
A flattening filter free photon treatment concept evaluation with Monte Carlo
U. Titt, O. N. Vassiliev, F. Ponisch, L. Dong, H. Liu, R. Mohan
Med. Phys. 2006;33:1595-1602
Long-Circulating and Target-Specific Nanoparticles: Theory to Practice
S. M. Moghimi, A. C. Hunter, J. C. Murray
Pharmacol. Rev. 2001;53:283-318
Determining the Size and Shape Dependence of Gold Nanoparticle Uptake into Mammalian Cells
B. D. Chithrani, A. A. Ghazani, W. C. W. Chan
Nano Letters 2006;6:662-668
Estimation of tumour dose enhancement due to gold nanoparticles during typical radiation treatments: a preliminary Monte Carlo study
S. H.. Cho
Phys. Med. Biol. 2005;50:N163-173
The use of gold nanoparticles to enhance radiotherapy in mice
J. F. Hainfeld, D. N. Slatkin, H. M. Smilowitz
Phys. Med. Biol. 2004;49:N309-315
Nanotech approaches to drug delivery and imaging
S. K. Sahoo, V. Labhasetwar
Discov. Today 2003;8:1112-1120