Supplementary Information for:
Quantitative Contrast-Enhanced Optical Coherence Tomography
Yonatan Winetraub,1,2,3,4* Elliott SoRelle,1,2,3,4* Orly Liba,1,2,4,5 and Adam de la Zerda1,2,3,4,5
1
Molecular Imaging Program at Stanford, 2Bio-X Program, and 3Biophysics Program,
4
Departments of Structural Biology, and 5Electrical Engineering, Stanford University, 299
Campus Drive, Stanford, California 94305
Table of Contents
I. Theoretical Model – Additional Equations and Derivations
a. GNRs in water
i. Mean speckle intensity of a known number of particles
ii. Mean speckle intensity of a population of particles
iii. Shadowing (light attenuation)
iv. GNRs in water: experiment
b. Non-ideal scatterers
i. Binomial case: scatterers with two states (on/off)
ii. General binomial case
iii. Generalization
iv. Simulation Experiments
c. Consistency Check: Proof of C1 and C2 for GNRs in water
d. GNRs in blood and tissue
e. Summary
II: Imaging Experiments, Analysis, and Post-Processing
a. System Optics
b. Acquisition Parameters
c. Quantitative Analysis
III: Supporting Figures
I. Theoretical Model – Additional Equations and Derivations
Ia. GNRs in water
The goal of this work is to generate a quantitative model to infer the number of a given contrast
agent (in this case, GNRs) within a specific region imaged with OCT. To begin, we make several
simplifying assumptions. First we assume that each particle in a voxel reflects the same electric field
intensity, 𝐴, and that all light within the system can be approximated by ray optics reflections. Further,
we assume that multiple scattering does not occur within a given voxel (although it may still occur
between independent voxels). It is impossible to use the independent scattering approximation in this
case since light from different scatterers clearly interact to create an interference pattern.
i. Mean speckle intensity of a known number of particles
The mean voxel electric field speckle intensity, 𝑋, can be modeled as a 2D random walk process
with constant step size 𝐴 and angle 𝜃, which is uniformly randomized 𝜃 ~ 𝑈(0,2𝜋). Note that 𝜃 is
uniformly randomized between 0 and 2𝜋 even if the pixel size does not correspond to an integer number
of wavelengths. The number of particles, 𝑁, can be treated as the number of steps within the random
walk. As depicted in Figure 1b, the mean norm of the sum of random phasors follows:
𝑋(𝑁) ∝ 𝐴 × √𝑁
(S1)
For 𝑁 = 2, it is possible to directly calculate the mean speckle intensity due to the electric field reflected
⃑⃑⃑⃑⃑⃑⃑
⃑⃑⃑⃑⃑⃑⃑
by the two particles ⃑⃑⃑⃑
𝐸1 , ⃑⃑⃑⃑
𝐸2 . Note that |𝐸
1 | = |𝐸2 | = 𝐴, and the relative angle between the two fields is
𝜃. Taking these into account, we obtain the mean speckle intensity for two interfering particles:
2𝜋
|𝐸1 + ⃑⃑⃑⃑⃑
𝑋(2) = ∫0 𝑃(Θ = 𝜃) × ⃑⃑⃑⃑⃑
𝐸2 | 𝑑𝜃
(S2)
Because each angle has an equal probability, Eq. S2 can be converted to:
2𝜋
𝑋(2) = 1/2𝜋 ∫0 √(𝐴 + 𝐴 cos 𝜃)2 + (𝐴 sin 𝜃)2 𝑑𝜃
2𝜋
(S3a)
2𝜋
𝑋(2) = 𝐴/2𝜋 ∫0 √1 + 2 cos 𝜃 + 1 𝑑𝜃 = 𝐴 × √2/𝜋 ∫0 √1 + cos 𝜃 𝑑𝜃
(S3b)
The result from Eq. (S3b) is a partial elliptical integral, which is solvable by an integral table. The
solution for this particular integral is:
𝑋(2) = 0.9 × 𝐴√2
(S4)
It is also possible to estimate this integral using Monte Carlo (MC) simulations. This method becomes
much more tractable than directly solving the integral when 𝑁 ≫ 2, since 𝑁 − 1 integrals must be
computed:
2𝜋
2𝜋
⃑⃑⃑⃑
𝑋(𝑁) = 1/(2𝜋)𝑁−1 ∫0 … ∫0 |∑𝑁
𝑘=1 𝐸𝑘 | 𝑑𝜃1 … 𝑑𝜃𝑁−1
The result of MC simulation up to 𝑁 = 100 is shown in Figure S1. As can be seen in this figure:
(S5)
0.885 < 𝑋(𝑁) < 0.91, and the mean value of the coefficient was determined to be 0.89, which appears
in Eq. (1) and subsequent equations in the main text. Note that the mean value is a good approximation
(with an error no greater than ~2%) for any 𝑋(𝑁).
ii. Mean speckle intensity of a population of particles
Next, we consider a voxel that contains a population of particles. If the expected number of
particles per voxel is 𝜆, then the probability of finding 𝑘 particles within a given voxel follows a Poisson
distribution:
𝑃(𝑁 = 𝑘) = 𝜆𝑘 /𝑘! × 𝑒 −𝜆
(S6)
In this instance, the expected mean speckle intensity is:
𝑘
−𝜆
𝐼 ̅ = ∑𝑘 𝑃(𝑁 = 𝑘) × 𝑋(𝑘) = 𝐴 × 𝜆 × 𝑒 −𝜆 + 0.89𝐴 × ∑∞
𝑘=2 √𝑘 × 𝜆 /𝑘! × 𝑒
(S7)
Eq. (S7) can be rewritten in terms of two new parameters, 𝐶1 & 𝐶2. 𝐶1 represents the effective intensity
of a single scatterer and 𝐶2 describes the effective concentration scale factor. It is critical to note that for
any system that scatters light, 𝐶1 and 𝐶2 must exist. Any scattering sample, no matter how complex or
inhomogeneous, can in theory be modeled using these two variables (for more information, see Section
Ib, subsections iii and iv). 𝐶1 and 𝐶2 can be incorporated to yield the following expression for mean
scattering intensity:
𝑘
−𝜆/𝐶2
𝐼 ̅ = 𝐶1 × 𝜆/𝐶2 × 𝑒 −𝜆/𝐶2 + 0.89𝐶1 × ∑∞
𝑘=2 √𝑘 × (𝜆/𝐶2 ) /𝑘! × 𝑒
(S8)
Using Eq. (S8), let us consider two extreme cases of particle concentration: (1) 𝜆 ≪ 1 and (2) 𝜆 ≫ 1. In
the first case, the probability of finding 2 or more particles per voxel is negligible: 𝑃(𝑁 ≥ 2) ≈ 0.
Furthermore, 𝑒 −𝜆/𝐶2 ≈ 1, and the mean intensity can be expressed as:
𝐼 (̅ 𝜆 ≪ 1) ≈ 𝐶1 × 𝜆/𝐶2
(S9)
The mean intensity for the second concentration regime can also be expressed. First, we define 𝜆′ =
𝜆/𝐶2. For large 𝜆, the Poisson distribution can be approximated by a normal distribution with a mean of
𝜆′ and a standard deviation of 𝜎 = √𝜆′. The mean intensity for large 𝜆 can now be expressed as:
′ 2
∞
𝐼 (̅ 𝜆′ ≫ 1) ≈ 0.89 × 𝐶1 /√2𝜋𝜆′ × ∫ √𝑥 𝑒 −(𝑥−𝜆 ) /2𝜆′ 𝑑𝑥
−∞
(S10)
We can change the variable in order to facilitate integration:
𝑦 = (𝑥 − 𝜆′ )/√𝜆′
𝑥 = 𝑦√𝜆′ + 𝜆′
𝑑𝑦 = 𝑑𝑥/√𝜆′
Substituting these variables into Eq. (S10), we obtain:
∞
𝐼 (̅ 𝜆′ ≫ 1) ≈ 0.89 × 𝐶1 /√2𝜋 ∫−∞ √𝑦√𝜆′ + 𝜆′ × 𝑒 −𝑦
2 /2
𝑑𝑦
(S11)
It is reasonable to assume that the integral component of Eq. (S11) can be approximated by:
∞
∫−∞ 𝑓(𝑦) × 𝑒 −𝑦
2 /2
∞
𝑑𝑦 ≈ 𝑓(𝑦 = 0) × ∫−∞ 𝑒 −𝑦
2 /2
𝑑𝑦
(S12)
Because the exponential term of the integral decays rapidly, very few non-zero values of 𝑓(𝑦) contribute
to the integral. Thus, Eq. (S12) can be further simplified to:
𝐼 (̅ 𝜆′ ≫ 1) ≈ 0.89𝐶1 × √𝜆′
(S13)
Finally, we can substitute back in for 𝜆′ to obtain:
𝐼 (̅ 𝜆 ≫ 1) ≈ 0.89 × 𝐶1 × √𝜆/𝐶2
(S14)
Now that we have derived expressions for the two distinct concentration regimes, we can develop an
expression for the switch point between the two regimes. We can define this concentration, 𝜆𝑐𝑟𝑖𝑡 , and
calculate it by setting 𝐼 (̅ 𝜆 ≪ 1) from Eq. (S9) and 𝐼 (̅ 𝜆 ≫ 1) from Eq. (S14) equal to each other and
solving:
𝐼 (̅ 𝜆 ≪ 1) = 𝐼 (̅ 𝜆 ≫ 1)
𝐶1 𝜆𝑐𝑟𝑖𝑡 /𝐶2 = 0.89𝐶1 × √𝜆𝑐𝑟𝑖𝑡 /𝐶2
(𝜆𝑐𝑟𝑖𝑡 /𝐶2 )2 = 0.892 × 𝜆𝑐𝑟𝑖𝑡 /𝐶2
We are left with a numerical expression that defines the effective particle concentration at which the
model for mean speckle intensity transitions between high and low concentration regimes:
𝜆𝑐𝑟𝑖𝑡 = 0.8 × 𝐶2
(S15)
An example depicting each concentration regime, the full OCT signal model, and the predicted 𝜆𝑐𝑟𝑖𝑡 is
shown in Figure S2.
iii. Shadowing (light attenuation)
To estimate the effect of light attenuation on the detected OCT signal, we retain all assumptions
heretofore described and add the assumptions that light can be attenuated by absorption and scattering
both going into and coming back from the sample toward the detector by GNRs in the medium not in a
single voxel. A schematic of the experimental setup with the relative incident and reflected light is given
in Figure S3. The relative light intensity in the sample, 𝐼𝑟𝑒𝑙 , is depth-dependent and can be described by
the Beer-Lambert Law:
𝐼𝑟𝑒𝑙 = 10−ℇ𝑐𝑧
(S16)
Where ℇ is the GNR extinction coefficient, 𝑐 is the GNR concentration, and 𝑧 is the depth within the
sample and (0 ≤ 𝑧 ≤ 𝑑 where d is the ROI depth). In addition to attenuation of the incident beam, any
light reflected from a depth 𝑧 within the sample will be attenuated on its way to the detector by an
additional factor of 10−ℇ𝑐𝑧 . Thus, the overall intensity reflected by a thin layer at position 𝑧 is attenuated
by a factor:
𝐹(𝑧) = 10−2ℇ𝑐𝑧
(S17)
A mean factor can be applied to an entire ROI of depth 𝑑 to account for light attenuation. This factor, 𝐹,
can be expressed as follows:
𝑑
𝑑
𝐹 = 1/𝑑 × ∫0 𝐹(𝑧)𝑑𝑧 = 1/𝑑 × ∫0 10−2ℇ𝑐𝑧 𝑑𝑧
(S18a)
𝐹 = 1/(2ℇ𝑐𝑑 × log 10) × [1 − 10−2ℇ𝑐𝑑 ]
(S18b)
We can simplify this expression by defining 𝑠, the scattering signal expected based on a particle’s
extinction coefficient and depth within the OCT image as 𝑠 = ℇ𝑑, which yields a final expression for the
light attenuation factor:
𝐹 = 1/(2ℇ𝑐𝑑 × log 10) × [1 − 10−2ℇ𝑐𝑑 ]
(S19)
iv. GNRs in water: experiment
All GNR concentrations were imaged in three distinct cross-sectional locations (ROIs), and the
average image from 100 consecutive B scans was used to calculate al ROIs (for more detailed
information, see Section IIb-c). The three distinct locations within each sample were far enough apart to
be considered as independent measurements. We assume that the time mean of the sample can
approximate the expected speckle intensity:
1/𝑇 × ∑𝑡 𝐼(𝑡) 𝑇→∞ → 𝐼 ̅
(S20)
In addition, we measured ℇ using Vis-NIR spectrometry and used this value (ℇ ≈ 2𝑥1010 𝑀−1 𝑐𝑚−1) to
calculate how much light is attenuated through the depth of a single voxel (2 𝜇𝑚) for the highest
concentration measured. For 5 nM GNRs, about 4.5% of light is attenuated. Thus, we can conclude that
the amount of multiple scattering events is navigable in a single voxel. Therefore, we average all time
means within an ROI and use the mean of the three ROIs per sample to perform the model fit. Squareshaped glass capillary tubes filled with GNRs at different concentrations 𝑐𝑚 were imaged and the mean
̅ (𝑐𝑚 ), was quantified for each. As a pre-processing step, the background noise
speckle intensity, 𝐼𝑚
̅𝐼𝑚 (0) (from water only) is subtracted from each measurement:
̅ (𝑐𝑚 ) = 𝐼𝑚
̅ (𝑐𝑚 ) − 𝐼𝑚
̅ (0)
𝐼𝑚
(S21)
Using the known voxel size of 8 m x 8 m x 2 m (see Section IIa-b), the concentration in particles per
voxel, 𝜆𝑚 , can be calculated. Data fitting was performed by minimizing the error term from a log least
square regression as expressed below:
̅ ] − log[𝐼 (̅ 𝐶1 , 𝐶2 , 𝜆𝑚 ) × 𝐹(𝑠, 𝑐𝑚 )])2
min ∑𝑚(log[𝐼𝑚
𝐶1 ,𝐶2 ,𝑠
(S22a)
𝑘
−𝜆𝑚 /𝐶2
𝐼 (̅ 𝐶1 , 𝐶2 , 𝜆𝑚 ) = 𝐶1 𝜆𝑚 /𝐶2 𝑒 −𝜆𝑚/𝐶2 + 0.89𝐶1 × ∑∞
𝑘=2 √𝑘 × (𝜆𝑚 /𝐶2 ) /𝑘! × 𝑒
(S22b)
𝐹(𝑠, 𝑐𝑚 ) = 1/(2𝑐𝑚 𝑠 × log 10) × [1 − 10−2𝑐𝑚𝑠 ]
(S22c)
The result of this fitting is displayed as the full model (blue line) in all plots within the main text. For
GNRs in water, the fitted 𝐶2 ≈ 4, which means that about 1 of every 4 GNRs contributes to the overall
observed signal. This could be the result of GNR orientation; not all rotational positions reflect incident
light back to the detector. Moreover, GNR orientation influences the incident wavelengths that interact
maximally with the particle. Other physical mechanisms may also contribute to this result. The ratio of
the measured 𝑠 to the predicted scattering, 𝑠0 is 0.09, meaning that the fitted scattering is 9% of the
measured value, which is predicted from Vis-NIR measurements (data not shown). Although previously
mentioned mechanisms may partially account for this discrepancy, the full nature of the difference
between predicted and absolute measured scattering is unclear.
Ib. Non-ideal scatterers
The goal of this section is to demonstrate that many physical mechanisms that generate speckle
patterns can be described by an ideal model with simplified parameters. We begin with relatively simple
case of a single scatterer that can exhibit two different scattering states (on and off). We extend this case
to a general binomial case that can model a single scatterer with two non-zero states, and then we show
that such an approach can also describe samples that include any number and type of scatterers with
known properties. We conclude with simulation examples of this general model.
i. Binomial case: scatterers with two states (on/off)
We start by assuming ideal scattering as in Section Ia. On average, there are 𝜆 scatterers in a
voxel, and each scatterer has an identical intensity, 𝐴. As described in the main text, the expected voxel
intensity is:
𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐴, 𝜆) = 𝐴 × 𝑃(𝑁 = 1) + 0.89 × 𝐴 × ∑∞
𝑘=2 √𝑘 × 𝑃(𝑁 = 𝑘)
(S23)
𝑘
−𝜆
= 𝐴 × 𝜆 × 𝑒 −𝜆 + 0.89 × 𝐴 × ∑∞
𝑘=2 √𝑘 × 𝜆 /𝑘! × 𝑒
Let us now assume a binomial distribution of scattering states such that:
𝐴 = 𝛼 with probability 𝑝
𝐴 = 0 with probability 1 − 𝑝
[𝛼 𝑠𝑡𝑎𝑡𝑒]
[0 𝑠𝑡𝑎𝑡𝑒]
(S24a)
(S24b)
On average, 𝐴̅ = 𝑝 × 𝛼. We would like to calculate the expected mean speckle intensity, denoted here
as 𝐼 ′̅ . Naïve intuition would predict the following expression:
𝐼 ′̅ = 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 = 𝑝 × 𝛼, 𝐶2 = 1)
(S25)
However, this intuition turns out to be incorrect. To explore this in more detail, let us examine an
example case in which we set 𝜆 = 2, 𝑝 = 0.5, and 𝛼 = 1. Because of interference effects, we cannot
assume that 𝐼 ′̅ = 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐴 = 0.5, 𝜆 = 2) since the speckle intensity of 2 particles with 𝐴 = 0.5 is not
equal to 1 particle with 𝐴 = 1. Rather, the exact expected value of OCT intensity in the binomial case is:
𝐼 ′̅ = 𝛼 × 𝑃(𝑁 = 1) + 0.89𝛼 × ∑∞
𝑘=2 √𝑘 × 𝑃(𝑁 = 𝑘)
(S26)
Where 𝑃(𝑁 = 𝑘) is the probability that 𝑘 particles in a voxel have randomized the 𝛼 state. Using the
Poisson distribution for having 𝑘 particles (regardless of state), 𝑃𝑝 (𝑁 = 𝑘), it can be shown that:
𝑙
𝑘
𝑙−𝑘
𝑃(𝑁 = 𝑘) = 𝑃𝑝 (𝑁 = 𝑘)𝑝𝑘 + ∑∞
𝑙=𝑘+1 𝑃𝑝 (𝑁 = 1) × (𝑘) × 𝑝 × (1 − 𝑝)
(S27)
Eq. (S27) can be interpreted thusly: the probability of having 𝑘 particles in the 𝛼 state is the probability
of all 𝑘 particles selecting the 𝛼 state or having more than 𝑘 particles but only 𝑘 particles select the 𝛼
state and all other particles select the 0 state. The full derivation of 𝑃(𝑁 = 𝑘) is given below:
𝑙
𝑘
𝑙−𝑘
𝑃(𝑁 = 𝑘) = 𝑃𝑝 (𝑁 = 𝑘) × 𝑝𝑘 + ∑∞
𝑙=𝑘+1 𝑃𝑝 (𝑁 = 1) × (𝑘) × 𝑝 × (1 − 𝑝)
Providing: 𝑃𝑝 (𝑁 = 𝑘) = 𝜆𝑘 /𝑘! × 𝑒 −𝜆 yields:
𝑙
−𝜆
𝑃(𝑁 = 𝑘) = 𝜆𝑘 /𝑘! × 𝑒 −𝜆 𝑝𝑘 + ∑∞
× (𝑙!/𝑘! × (𝑙 − 𝑘)!) × 𝑝𝑘 (1 − 𝑝)𝑙−𝑘
𝑙=𝑘+1 𝜆 /𝑙 ! × 𝑒
𝑙
𝑘
𝑙−𝑘
𝑃(𝑁 = 𝑘) = 𝑒 −𝜆 × [(𝑝𝜆)𝑘 /𝑘! + ∑∞
/(𝑙 − 𝑘)!]
𝑙=𝑘+1 𝜆 × 𝑝 /𝑘! × (1 − 𝑝)
𝑙−𝑘
𝑃(𝑁 = 𝑘) = 𝑒 −𝜆 × [(𝑝𝜆)𝑘 /𝑘! + (𝑝𝜆)𝑘 /𝑘! × ∑∞
/(𝑙 − 𝑘)! × 𝜆𝑙−𝑘 ]
𝑙=𝑘+1(1 − 𝑝)
𝑚
𝑃(𝑁 = 𝑘) = 𝑒 −𝜆 × (𝑝𝜆)𝑘 /𝑘! × [1 + ∑∞
× 1/𝑚!]
𝑚=1(𝜆 − 𝜆𝑝)
𝑃(𝑁 = 𝑘) = 𝑒 −𝜆 × (𝑝𝜆)𝑘 /𝑘! × 𝑒 𝜆−𝜆𝑝
𝑃(𝑁 = 𝑘) = (𝑝𝜆)𝑘 /𝑘! × 𝑒 −𝜆𝑝
The probability of finding 𝑘 particles in the 𝛼 state is therefore a Poisson distribution with 𝜆 as a
parameter such that 𝜆 → 𝑝𝜆:
𝑃(𝑁 = 𝑘) = (𝑝𝜆)𝑘 /𝑘! × 𝑒 −𝜆𝑝
(S28)
Thus, the expected intensity for the binomial case is:
𝐼 ̅ = 𝛼 × 𝑃(𝑁 = 1) + 0.89 × 𝛼 × ∑∞
𝑘=2 √𝑘 × 𝑃(𝑁 = 𝑘)
(S29a)
𝑘
−𝑝𝜆
𝐼 ̅ = 𝛼𝑝𝜆𝑒 −𝑝𝜆 + 0.89𝛼 × ∑∞
𝑘=2 √𝑘 × (𝑝𝜆) /𝑘! × 𝑒
(S29b)
The extreme cases for the binomial model can be expressed as:
𝐼 ′̅ ( 𝜆 ≪ 1) ≈ 𝛼 × 𝑝 × 𝜆
𝐼 ′̅ ( 𝜆 ≫ 1) ≈ 0.89 × 𝛼 × √𝑝𝜆
(S30a)
(S30b)
The key takeaway from this analysis is that, contrary to naïve assumption, the effective number of
particles in a voxel is smaller than the expected value by a factor of 1/𝑝. This is illustrated by the
following expression for the mean speckle intensity in the binomial case:
𝐼 ′̅ = 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 = 𝛼, 𝐶2 = 1/𝑝)
(S31)
Two Monte-Carlo examples of the prediction produced by Eq. (S31) are provided in Figure S4.
We can plug Eq. (S27) to Eq. (S29a) for 𝑃(𝑁 = 𝑘) to obtain the following expression for 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 :
𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 = 𝛼, 𝐶 = 1/𝑝) =
𝑘−1 𝑘
𝛼𝑝𝜆𝑒 −𝜆 + 𝛼 ∑∞
(𝜆 /𝑘!)𝑒 −𝜆 ] +
𝑘=2[𝑘𝑝(1 − 𝑝)
𝑙
∞
𝑘
𝑙−𝑘
0.89𝛼 × ∑∞
× 𝜆𝑙 /𝑙! × 𝑒 −𝜆
𝑘=2[√𝑘 × ∑𝑙=𝑘(𝑘) × 𝑝 (1 − 𝑝)
(S32)
The two parts of Eq. (S32) can be simplified in the following way:
Part 1:
𝑘−1
𝛼 × 𝑝 × 𝜆 × 𝑒 −𝜆 + 𝛼 × ∑∞
× 𝜆𝑘 /𝑘! × 𝑒 −𝜆 ]
𝑘=2[𝑘 × 𝑝 × (1 − 𝑝)
𝑘−1
𝛼 × ∑∞
× 𝜆𝑘 /𝑘! × 𝑒 −𝜆 ]
𝑘=0[𝑘 × 𝑝 × (1 − 𝑝)
Part 2:
𝑙
∞
𝑘
𝑙−𝑘
0.89 × 𝛼 × ∑∞
× 𝜆𝑙 /𝑙! × 𝑒 −𝜆 ]
𝑘=2[√𝑘 × ∑𝑙=𝑘(𝑘) × 𝑝 × (1 − 𝑝)
𝑘
−𝜆
𝑙−𝑘
0.89 × 𝛼 × ∑∞
× ∑∞
/(𝑙 − 𝑘)! × 𝜆𝑙 ]
𝑘=2[√𝑘 /𝑘! × 𝑝 × 𝑒
𝑙=𝑘(1 − 𝑝)
𝑘
𝑘
−𝜆
𝑙
𝑙
0.89 × 𝛼 × ∑∞
× ∑∞
𝑘=2[√𝑘 × 𝜆 /𝑘! × 𝑝 × 𝑒
𝑖=0(1 − 𝑝) /(𝑙)! × 𝜆 ]
We are left with the following lemma:
Lemma #1
𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 = 𝛼, 𝐶 = 1/𝑝) =
(S33)
∞
𝑘−1
𝑘
−𝜆
𝛼 × ∑𝑘=0[𝑘 × 𝑝 × (1 − 𝑝)
× 𝜆 /𝑘! × 𝑒 ] +
∞
𝑘
𝑙
𝑙
0.89 × 𝛼 × ∑𝑘=2[√𝑘 × 𝜆 /𝑘! × 𝑝𝑘 × 𝑒 −𝜆 × ∑∞
𝑖=0(1 − 𝑝) /(𝑙)! × 𝜆 ]
ii. General binomial case
We next consider a general case of the binomial distribution of scattering states described above to
include non-zero scattering states. In this general case, we assume that each particle reflects:
𝐴 = 𝛼 with probability 𝑝
𝐴 = 𝛽 with probability 1 − 𝑝
[𝛼 𝑠𝑡𝑎𝑡𝑒]
[𝛽 𝑠𝑡𝑎𝑡𝑒]
(S34a)
(S34b)
We would like to predict the expected light intensity, 𝐼 ′̅ ′, for particles with these two states. This can be
calculated by summing for all cases of 𝑘𝛼 particles that exhibit the 𝛼 state and 𝑘𝛽 particles that exhibit
the 𝛽 state. Generally,
∞
𝐼 ′̅ ′ = ∑∞
𝑘𝛼 =0 ∑𝑘𝛽 =0 𝑋(𝑘𝛼 , 𝑘𝛽 ) × 𝑃(𝑁𝛼 = 𝑘𝛼 ) × 𝑃(𝑁𝛽 = 𝑘𝛽 )
(S35)
where 𝑋(𝑘𝛼 , 𝑘𝛽 ) is the interference term and is a generalization of 𝑋(𝑁) for a two-particle type scheme.
As we have previously shown for the simple binomial distribution case, the probabilities of finding 𝑘𝛼
particles scatterers in the 𝛼 state and finding 𝑘𝛽 particles in the 𝛽 state are respectively:
𝑃(𝑁𝛼 = 𝑘𝛼 ) = (𝜆𝑝)𝑘𝛼 /𝑘𝛼 ! × 𝑒 −𝜆𝑝
(S36a)
𝑃(𝑁𝛽 = 𝑘𝛽 ) = (𝜆 × (1 − 𝑝))𝑘𝛽 /𝑘𝛽 ! × 𝑒 −𝜆 × (1−𝑝)
(S36b)
Now, we only have to develop a description of the term 𝑋(𝑘𝛼 , 𝑘𝛽 ) in Eq. (S35) to obtain a complete
expression for 𝐼 ′̅ ′. The first step in describing 𝑋(𝑘𝛼 , 𝑘𝛽 ) is to show following trivial cases:
𝑋(0,0) = 0
𝑋(1,0) = 𝛼
𝑋(0,1) = 𝛽
𝑋(𝑘𝛼 , 0) = 0.89 × 𝛼 × √𝑘𝛼 ,
𝑋(0, 𝑘𝛽 ) = 0.89 × 𝛽 × √𝑘𝛼 ,
𝑘𝛼 ≥ 2
𝑘𝛽 ≥ 2
The next step is to calculate the non-trivial case 𝑋(1,1). As shown in Section Ia, the interference term 𝑋
can be solved by calculating the following integral:
2𝜋
𝑋(1,1) = 1/2𝜋 × ∫0 √(𝛼 + 𝛽 × cos 𝜃)2 + 𝛽 2 sin2 (𝜃)𝑑𝜃
(S37)
Unlike Section Ia, in Eq. (S37) both vectors have non-identical lengths. As described earlier, this is a
partial elliptical integral that can be estimated using MC simulation. An example of one such simulation
is given in Figure S5 for the case 𝑋(𝑘𝛼 = 1, 𝑘𝛽 = 1). In this case, we find that:
𝑋(1,1) →
𝛼
𝑋(1,1) →
𝛽
𝛼≫𝛽
𝛼≪𝛽
𝑋(1,1) →
𝛼=𝛽
0.89 × 𝛼 × √2
We find that 𝑋(1,1) can be approximated by two methods, as depicted in Figure S6:
Γ
𝑋(1,1) ≈ √𝛼 Γ + 𝛽 Γ
where Γ = 2.605. This value was determined empirically from least square fitting of the MC simulation.
Next lest us examine the case where 𝑘𝛼 ≥ 2 and 𝑘𝛽 ≥ 2, we shall assume that we can separate 𝛼
particles from 𝛽 particles. By doing this, we can interfere each group separately and then interfere the
resulting sum vectors for each state:
2
2𝜋
𝑋(𝑘𝛼 , 𝑘𝛽 ) = 1/2𝜋 × ∫0 √(𝑋(𝑘𝛼 , 0) + 𝑋(0, 𝑘𝛽 ) × cos 𝜃)2 + 𝑋(0, 𝑘𝛽 ) × sin2 (𝜃) 𝑑𝜃
(S38)
Γ
≈ √𝑋(𝑘𝛼 , 0)Γ + 𝑋(0, 𝑘𝛽 )
Γ
Γ
Γ
The latter inequality is an assumption that we can approximate 𝑋(𝑘𝛼 , 𝑘𝛽 ) ≈ √𝑋(𝑘𝛼 , 0)Γ + 𝑋(0, 𝑘𝛽 ) ,
which we will soon see is in fact the case.
For the case when both 𝑘𝛼 ≥ 2, and 𝑘𝛽 ≥ 2:
Γ
Γ
Γ
Γ
𝑋(𝑘𝛼 , 𝑘𝛽 ) = √𝑋(𝑘𝛼 , 0)Γ + 𝑋(0, 𝑘𝛽 ) = 0.89 × √(𝛼 × √𝑘𝛼 ) + (𝛽 × √𝑘𝛽 )
Γ
(S39)
Furthermore, let us examine the case when 𝛼 = 𝛽, then clearly:
𝑋(𝑘𝛼 , 𝑘𝛽 ) = 𝑋(𝑘𝛼 + 𝑘𝛽 , 0)
(S40)
Plugging Eq. (S39) and the trivial cases 𝑋(𝑘𝛼 , 0), 𝑋(𝑘𝛽 , 0) into Eq. (S40), we receive the following:
Γ
Γ
Γ
𝑋(𝑘𝛼 , 𝑘𝛽 ) = 0.89𝛼 × √(√𝑘𝛼 ) + (√𝑘𝛽 ) = 0.89𝛼 × √𝑘𝛼 + 𝑘𝛽 = 𝑋(𝑘𝛼 + 𝑘𝛽 , 0)
(S41)
Eq. (S41) yields Γ = 2, which is depicted in Figure S7.
Similarly, for the case where 𝑘𝛼 ≥ 2 and 𝑘𝛽 = 1 we can look Eq. (S38) at the extreme case when 𝛼 = 𝛽,
and conclude that:
𝑋(𝑘𝛼 , 1) = 𝑋(𝑘𝛼 + 1,0)
(S42)
This results in the following expression for 𝑋(𝑘𝛼 , 1):
Γ
𝑋(𝑘𝛼 , 1) = √(0.89𝛼 × √𝑘𝛼 )Γ + (0.89 × 𝛽)Γ
(S43)
On the other hand, if 𝛽 ≫ 𝛼, then 𝑋(𝑘𝛼 , 1) ≈ 𝑋(0,1). This equality holds if:
Γ
𝑋(𝑘𝛼 , 1) = √(0.89𝛼 × √𝑘𝛼 )Γ + (1 × 𝛽)Γ
(S44)
In both cases, Γ = 2. The combination of these two conditions yields:
Γ
𝑋(𝑘𝛼 , 1) = √(0.89𝛼 × √𝑘𝛼 )Γ + (𝐹 × 𝛽)Γ
Where:
(S45)
𝐹={
1
𝛽≫𝛼
0.89 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(S46)
The result of Eq. (S45) is presented graphically in Figure S8. A full summary of the solutions to
𝑋(𝑘𝛼 , 𝑘𝛽 ) can be found in Table S1.
The next step will be to obtain the following lemma for the case 𝑘𝛼 = 𝑝𝜆 ≥ 2, 𝑘𝛽 = (1 − 𝑝)𝜆 ≥ 2:
Lemma #2
𝑋(𝑝𝜆, (1 − 𝑝)𝜆) = 0.89√𝜆 × 𝑋(𝑝, 1 − 𝑝)
(S47)
The proof for this is given below:
Γ
Γ
Γ
Γ
Γ
𝑋(𝑝𝜆, (1 − 𝑝)𝜆) = 0.89 √(√𝑘𝛼 ) + (√𝑘𝛽 ) = 0.89√𝜆 × √(𝛼√𝑝) + (𝛽√1 − 𝑝)
Γ
= 0.89 × √𝜆 × 𝑋(𝑝, 1 − 𝑝)
It is now possible to calculate 𝐼 ′̅ ′ directly by summing all interference results with their respective
probabilities when accounting for 𝛼 and 𝛽 particles as independent Poisson distributions:
∞
𝐼 ′̅ ′ = ∑∞
𝑘𝛼=0 ∑𝑘𝛽=0 𝑋(𝑘𝛼 , 𝑘𝛽 ) × 𝑃(𝑁𝛼 = 𝑘𝛼 ) × 𝑃(𝑁𝛽 = 𝑘𝛽 )
∞
𝑘𝛼
−𝜆𝑝
= ∑∞
× (𝜆 × (1 − 𝑝))𝑘𝛽 /𝑘𝛽 ! ×
𝑘𝛼=0 ∑𝑘𝛽=0 𝑋(𝑘𝛼 , 𝑘𝛽 ) × (𝜆𝑝) /𝑘𝛼 ! × 𝑒
𝑒 −𝜆 × (1−𝑝)
𝑘𝛽 𝑘𝛼
∞
= ∑∞
× 𝑝𝑘𝛼 × (1 − 𝑝)𝑘𝛽 × 1/(𝑘𝛼 ! × 𝑘𝛽 !) × 𝑒 −𝜆 ]
𝑘𝛼=0 ∑𝑘𝛽=0 [𝑋(𝑘𝛼 , 𝑘𝛽 ) × 𝜆 𝜆
Splitting the sum, we obtain:
1
∞
1
𝐼 ′̅ ′ = ∑∞
𝑘𝛽=0 ∑𝑘𝛼=0 [… ] + ∑𝑘𝛼=2 ∑𝑘𝛽=0 [… ]
Let us consider the first term in Eq. (S48):
𝑘𝛽 𝑘𝛼 𝑘𝛼
1
∑∞
× (1 − 𝑝)𝑘𝛽 × 1/(𝑘𝛼 ! × 𝑘𝛽 !) × 𝑒 −𝜆 ]
𝑘𝛽=0 ∑𝑘𝛼=0 [𝑋(𝑘𝛼 , 𝑘𝛽 ) × 𝜆 𝜆 𝑝
We can change the variables 𝑘𝛽 = 𝑘, 𝑘𝛼 = 𝑙 to obtain:
1
𝑘 𝑙 𝑙
𝑘
−𝜆
∑∞
𝑘=0 ∑𝑙=0[𝑋(𝑙, 𝑘) × 𝜆 𝜆 𝑝 × (1 − 𝑝) × 1/(𝑙! × 𝑘!) × 𝑒 ]
We would like to find 𝐴 and 𝑃 such that:
𝑘−1
∑∞
× 𝜆𝑘 /𝑘! × 𝑒 −𝜆 =
𝑘=0 𝐴 × 𝑃 × 𝑘 × (1 − 𝑝)
1
𝑘
𝑙
𝑙
𝑘
−𝜆
∑∞
𝑘=0 ∑𝑙=0[𝑋(𝑙, 𝑘) × 𝜆 × 𝜆 × 𝑝 × (1 − 𝑝) × 1/(𝑙! × 𝑘!) × 𝑒 ]
(S48)
We can simplify the interior sum and use the fact that 𝑙! = 1 for 𝑙 = 0,1
𝐴 × 𝑃 × 𝑘 × (1 − 𝑝)𝑘−1 = ∑1𝑙=0[𝑋(𝑙, 𝑘) × 𝜆𝑙 × 𝑝𝑙 × (1 − 𝑝)𝑘 ]
𝐴 × 𝑃 × 𝑘 × (1 − 𝑝)𝑘−1 = 𝑋(0, 𝑘) × (1 − 𝑝)𝑘 + 𝑋(1, 𝑘) × 𝜆 × 𝑝 × (1 − 𝑝)𝑘
This sum has a value for low 𝜆 , so we can focus on those values (otherwise, 𝑙 = 0,1 has
negligible probability). Further assuming that 𝑝 and 𝑝 − 1 are in the same order of magnitude,
we can assume the greatest contribution comes from 𝑘 = 0,1
𝐴 × 𝑃 × 𝑘 × (1 − 𝑝)𝑘−1 = 𝑋(0, 𝑘) × (1 − 𝑝)𝑘 + 𝑋(1, 𝑘) × 𝜆 × 𝑝 × (1 − 𝑝)𝑘
First, it can be seen that:
𝑋(0, 𝑘) = 𝛽 × 𝑘, since at most one particle exists
Γ
Γ
𝑋(1, 𝑘) × 𝜆𝑝 = √𝛼 Γ + (𝛽𝑘)Γ × 𝜆𝑝 = 𝑘 × √(𝛼𝜆𝑝/𝑘)Γ + (𝛽𝑘 × 𝜆𝑝)Γ
The most pertinent argument of the sum is 𝑘 = 𝜆 × (1 − 𝑝), thus:
Γ
Γ
Γ
Γ
𝛼𝜆𝑝
𝛼𝑝
𝑘 × √( 𝑘 ) + (𝛽𝑘 × 𝜆𝑝)Γ = 𝑘 × √(1−𝑝) + (𝛽𝜆2 (1 − 𝑝) × 𝑝)Γ ≈ 𝑘 × 𝛼𝑝/(1 −
𝑝)
Thus:
𝐴 × 𝑃 × 𝑘 × (1 − 𝑝)𝑘−1 = 𝑋(0, 𝑘) × (1 − 𝑝)𝑘 + 𝑋(1, 𝑘) × 𝜆 × 𝑝 × (1 − 𝑝)𝑘−1 )
When 𝑘 = 0, the solution is trivial: 0 = 0. When 𝑘 = 1, we find that:
𝐴 × 𝑃 = 𝛼 × 𝑝 + 𝛽 × (1 − 𝑝)
Now we consider the second term in Eq. (S48):
𝑘𝛽
1
∑∞
× 𝜆𝑘𝛼 × 𝑝𝑘𝛼 × (1 − 𝑝)𝑘𝛽 × 1/(𝑘𝛼 ! × 𝑘𝛽 !) × 𝑒 −𝜆 ]
𝑘𝛼=2 ∑𝑘𝛽=0 [𝑋(𝑘𝛼 , 𝑘𝛽 ) × 𝜆
We can change the variables to 𝑘𝛽 = 𝑙, 𝑘𝛼 = 𝑘 to obtain:
∞
𝑘
𝑙
𝑘
𝑙
−𝜆
∑∞
𝑘=2 ∑𝑙=0[𝑋(𝑘, 𝑙) × 𝜆 × 𝜆 × 𝑝 × (1 − 𝑝) × 1/(𝑙! × 𝑘!) × 𝑒 ]
As with the first term of Eq. (S44), we would like to find 𝐴 and 𝑃 such that:
𝑘
𝑘
−𝜆
𝑙
𝑙
0.89 × 𝐴 × ∑∞
× ∑∞
𝑘=2[√𝑘 × 𝜆 /𝑘! × 𝑝 × 𝑒
𝑖=0(1 − 𝑃) /(𝑙)! × 𝜆 ] =
∞
𝑘
𝑙
𝑘
𝑙
−𝜆
∑∞
𝑘=2 ∑𝑙=0[𝑋(𝑘, 𝑙) × 𝜆 × 𝜆 × 𝑝 × (1 − 𝑝) × 1/(𝑙! × 𝑘!) × 𝑒 ]
We can simplify each side separately. As was shown for the first term of Eq. (S48),
(S49)
𝑘
𝑘
−𝜆
𝑙
𝑙
∑∞
× ∑∞
𝑘=2[√𝑘 × 𝜆 /𝑘! × 𝑝 × 𝑒
𝑖=0(1 − 𝑃) /(𝑙)! × 𝜆 ] ≈ √𝜆𝑃
The right side of the equation has a maximal value when 𝑘 = 𝜆𝑝, 𝑙 = 𝜆 × (1 − 𝑝). Thus, it can
be simplified as follows:
∞
𝑘
𝑙
𝑘
𝑙
−𝜆
∑∞
𝑘=2 ∑𝑙=0[𝑋(𝑘, 𝑙) × 𝜆 × 𝜆 × 𝑝 × (1 − 𝑝) × 1/(𝑙! × 𝑘!) × 𝑒 ] ≈ 𝑋(𝜆𝑝, 𝜆 ×
(1 − 𝑝)) × 1
This can be further simplified by considering Lemma #2 (Eq. (S47)) to yield:
𝑋(𝜆𝑝, 𝜆 × (1 − 𝑝)) × 1 = 0.89 × √𝜆 × 𝑋(𝑝, 1 − 𝑝)
We can compare the two sides to show that:
0.89 × 𝐴 × √𝜆𝑃 = 0.89 × √𝜆 × 𝑋(𝑝, 1 − 𝑝)
This finally yields an expression for the second term of Eq. (S48):
𝐴 × √𝑃 = 𝑋(𝑝, 1 − 𝑝)
(S50)
We can now combine the results for the two terms of Eq. (S48) to build a complete description of the
expected mean scattering, 𝐼 ′̅ ′, from a binomial distribution of scatterers from a model of ideal scattering:
For:
𝐴𝑃 = 𝛼𝑝 + 𝛽 × (1 − 𝑝)
𝐴√𝑃 = 𝑋(𝑝, 1 − 𝑝)
and
We showed that:
1
∞
1
𝐼 ′̅ ′ = ∑∞
𝑘𝛽=0 ∑𝑘𝛼=0 [… ] + ∑𝑘𝛼=2 ∑𝑘𝛽=0 [… ] ≈
𝑘−1
𝐴𝑃 × ∑∞
×
𝑘=0 [𝑘 × (1 − 𝑃)
𝜆𝑘
𝑘!
× 𝑒 −𝜆 ] +
𝑘
𝑘
−𝜆
𝑙
𝑙
0.89 × 𝐴 × ∑∞
× ∑∞
𝑘=2[√𝑘 × 𝜆 /𝑘! × 𝑝 × 𝑒
𝑖=0(1 − 𝑃) /(𝑙)! × 𝜆 ]
We can now use Lemma #1 to receive:
𝐼 ′̅ ′ = 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 = 𝐴, 𝐶2 = 1/𝑃)
(S51)
Eq. (S51) is noteworthy in that it shows that the binomial case can be represented by an ideal scatterer
with the two modified parameters 𝐶1 and 𝐶2 . We can further develop parameter’s values:
𝐶1 = 𝐴
𝐶2 = 1/𝑃
𝐼𝑆 = 𝛼𝑝 + 𝛽 × (1 − 𝑝)
Γ
Γ
𝐼𝐿 ≈ √(𝛼 × √𝑝) + (𝛽 × √1 − 𝑝)
Γ
Notice that Γ = 2 because 𝐼𝐿 refers to a large number of particles 𝑘𝛼 , 𝑘𝛽 ≥ 2. We can therefore solve
Eqs. (S49) and (S50) to produce new definitions of 𝐶1 and 𝐶2 :
𝐼𝑆 = 𝐶1 /𝐶2
𝐼𝐿 = 𝐶1 × √1/𝐶2
𝐶1 = 𝐼𝑆 × 𝐶2
𝐼𝐿 = 𝐼𝑆 × √𝐶2 → 𝐶2 = (𝐼𝐿 /𝐼𝑆 )2
𝐶1 = 𝐼𝐿 2 /𝐼𝑆
We can take the new definitions of 𝐶1 and 𝐶2 and apply the formulae for 𝐼𝐿 and 𝐼𝑆 to receive closed-form
descriptions of 𝐶1 and 𝐶2 in terms of the two scattering states 𝛼 and 𝛽:
𝐶1 ≈ (𝛼 2 𝑝 + 𝛽 2 (1 − 𝑝))/𝛼𝑝 + 𝛽(1 − 𝑝)
(S52)
𝐶2 ≈ (𝛼 2 𝑝 + 𝛽 2 (1 − 𝑝))/(𝛼𝑝 + 𝛽(1 − 𝑝))2
(S53)
Notice that when 𝛽 = 0, we obtain the simple binomial case 𝐶1 = 𝛼, 𝐶2 = 1/𝑝 as described in Section
Ib subsection i. Several examples of the general binomial case are depicted in Figure S9.
The next step will be to generalize our result for the case where 𝑝𝛼 + 𝑝𝛽 ≠ 1. The general binomial case
can be rewritten as follows:
Assume now that we have two types of particles:
𝛼 at a concentration of 𝜆𝛼 = 𝜆 × 𝑝𝛼
𝛽 at a concentration of 𝜆𝛽 = 𝜆 × 𝑝𝛽
And that:
𝑝𝛼 ≥ 0, 𝑝𝛽 ≥ 0
But not necessarily: 𝑝𝛼 + 𝑝𝛽 ≠ 1
We can mix the two particles together within the same imaging voxel and describe an equivalent
particle with the following brightness and concentration:
𝐴 = (𝛼 2 𝑝𝛼 + 𝛽 2 𝑝𝛽 )/(𝛼𝑝𝛼 + 𝛽𝑝𝛽 )
2
𝑃 = 𝜆 × (𝛼𝑝𝛼 + 𝛽𝑝𝛽 ) /(𝛼 2 𝑝𝛼 + 𝛽 2 𝑝𝛽 )
(S54a)
(S54b)
When 𝑝𝛼 + 𝑝𝛽 ≠ 1, we can define 𝜆′:
𝜆′ = (𝑝𝛼 + 𝑝𝛽 ) × 𝜆
(S55)
Thus, particle 𝛼 has an intensity of 𝛼 and a concentration of 𝑝𝑎 × 𝜆′, where 𝑝𝑎 = 𝑝𝛼 /(𝑝𝛼 + 𝑝𝛽 ).
Notice that particle concentration is conserved:
𝑝𝑎 ∗ 𝜆′ = 𝑝𝛼 × 𝜆
(S56)
Similarly, particle 𝛽 has intensity 𝛽 and concentration 𝑝𝑏 × 𝜆′, where 𝑝𝑏 = 𝑝𝛽 /(𝑝𝛼 + 𝑝𝛽 ).
It is clear that 𝑝𝑎 + 𝑝𝑏 = 1 (although 𝑝𝛼 + 𝑝𝛽 ≠ 1). We can use these probabilities to produce
expressions for the brightness and concentration of an equivalent ideal particle:
𝐴 = (𝛼 2 𝑝𝑎 + 𝛽 2 𝑝𝑏 )/(𝛼𝑝𝑎 + 𝛽𝑝𝑏 )
= ((𝛼 2 𝑝𝛼 + 𝛽 2 𝑝𝛽 )/(𝑝𝛼 + 𝑝𝛽 ))/((𝛼𝑝𝛼 + 𝛽𝑝𝛽 )/(𝑝𝛼 + 𝑝𝛽 ))
= 𝛼 2 𝑝𝛼 + 𝛽 2 𝑝𝛽 /𝛼𝑝𝛼 + 𝛽𝑝𝛽
(S57a)
𝑃 = 𝜆′ × (𝛼𝑝𝑎 + 𝛽𝑝𝑏 )2 /(𝛼 2 𝑝𝑎 + 𝛽 2 𝑝𝑏 )
= 𝜆 × (𝑝𝛼 + 𝑝𝛽 ) × ((𝛼𝑝𝛼 + 𝛽𝑝𝛽 )/(𝑝𝛼 + 𝑝𝛽 ))2 /
((𝛼 2 𝑝𝛼 + 𝛽 2 𝑝𝛽 )/(𝑝𝛼 + 𝑝𝛽 ))
(S57b)
2
= 𝜆 × (𝛼𝑝𝛼 + 𝛽𝑝𝛽 ) /(𝛼 2 𝑝𝛼 + 𝛽 2 𝑝𝛽 )
In conclusion, when two populations of particles are places in the same voxel, their scattering intensity,
𝐼 ′̅ ′ , can be modeled as a single equivalent ideal particle with the parameters 𝐶1 and 𝐶2 and can be
described using Eqs. (S51-53) regardless of the sum of probabilities summing to 1.
iii. Generalization
The binomial case explored in Section Ib, subsection ii can be expanded further to account for
more than two particle types within a sample. We can show this by assuming three particles 𝑎, 𝑏, and 𝑐
with concentrations all within the same order of magnitude. These respective concentrations are:
𝑝𝑎 × 𝜆
𝑝𝑏 × 𝜆
𝑝𝑐 × 𝜆
We can start by combining particles 𝑎 and 𝑏 to an equivalent particle 𝑑 with concentration 𝑝𝑑 using the
same logic developed for the two-particle case.
𝑑 = (𝑎2 𝑝𝑎 + 𝑏 2 𝑝𝑏 )/(𝑎𝑝𝑎 + 𝑏𝑝𝑏 )
(S58a)
𝑝𝑑 = (𝑎𝑝𝑎 + 𝑏𝑝𝑏 )2 /(𝑎2 𝑝𝑎 + 𝑏 2 𝑝𝑏 )
(S58b)
We can now combine 𝑑 particles with 𝑐 particles to receive an overall equivalent particle. First, it is
important to acknowledge:
𝑑 × 𝑝𝑑 = 𝑎 × 𝑝𝑎 + 𝑏 × 𝑝𝑏
(S59a)
𝑑2 × 𝑝𝑑 = 𝑎2 × 𝑝𝑎 + 𝑏 2 × 𝑝𝑏
(S59b)
The resulting overall particle can thus be described by the following 𝐶1 and 𝐶2 :
𝐶1 = (𝑑2 𝑝𝑑 + 𝑐 2 𝑝𝑐 )/(𝑑𝑝𝑑 + 𝑐𝑝𝑐 ) = (𝑎2 𝑝𝑎 + 𝑏 2 𝑝𝑏 + 𝑐 2 𝑝𝑐 )/(𝑎𝑝𝑎 + 𝑏𝑝𝑏 + 𝑐𝑝𝑐 )
(S60a)
𝐶2 = (𝑑 2 𝑝𝑑 + 𝑐 2 𝑝𝑐 )/(𝑑𝑝𝑑 + 𝑐𝑝𝑐 )2 = (𝑎2 𝑝𝑎 + 𝑏 2 𝑝𝑏 + 𝑐 2 𝑝𝑐 )/(𝑎𝑝𝑎 + 𝑏𝑝𝑏 + 𝑐𝑝𝑐 )2
(S60b)
Using a similar approach, we can assume 𝑛 particle types 𝑎1 , … , 𝑎𝑛 with concentrations 𝑝1 𝜆, … , 𝑝𝑛 𝜆.
We would like to prove that the equivalent particle 𝑐𝑛 has the following intensity and A with probability
𝑝(𝐴):
𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦: 𝑐𝑛 = ∑ 𝑎𝑖 2 𝑝𝑖 / ∑ 𝑎𝑖 𝑝𝑖
(S61a)
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦: 𝑝𝑐𝑛 = (∑ 𝑎𝑖 𝑝𝑖 )2 / ∑ 𝑎𝑖 2 𝑝𝑖
(S61b)
We can use inductive reasoning to prove this claim. It is already apparent that Eqs. (S61a) and (S61b)
are valid for 𝑛 = 1,2, and 3. Now let us assume that they are valid for 𝑛, and it can be shown that they
are also valid for 𝑛 + 1:
Since the claim is valid for the first 𝑛 particles, we can collapse 𝑛 particles into 𝑐𝑛 𝑝𝑐𝑛 . Now, we
can interfere 𝑐𝑛 𝑝𝑐𝑛 with the remaining particle, 𝑎𝑛+1 . We find that:
𝑐𝑛 × 𝑝𝑐𝑛 = ∑ 𝑎𝑖 𝑝𝑖
(S62a)
and
𝑐𝑛 2 × 𝑝𝑐𝑛 = ∑ 𝑎𝑖 2 𝑝𝑖
(S62b)
Now we can use the general binomial case to calculate:
𝑐𝑛+1 = (𝑎2 𝑛+1 × 𝑝𝑛+1 + 𝑐𝑛 2 × 𝑝𝑐𝑛 )/( 𝑎𝑛+1 × 𝑝𝑛+1 + 𝑐𝑛 × 𝑝𝑐𝑛 ) = ∑ 𝑎𝑖 2 𝑝𝑖 / ∑ 𝑎𝑖 𝑝𝑖
Similarly,
𝑝𝑐𝑛+1 = (∑ 𝑎𝑖 𝑝𝑖 )2 / ∑ 𝑎𝑖 2 𝑝𝑖
(S63)
Q.E.D.
It is further possible to show that this general case is valid for 𝑛 → ∞. In this case, 𝑝𝑖 approximates the
probability density function 𝑃(𝐴 = 𝑎). Thus, for a particle with different expression probabilities 𝑃(𝐴 =
𝑎) and concentration 𝜆, an equivalent particle exists such that:
∞
∞
∞
∞
𝐶1 = (∫−∞ 𝐴2 × 𝑃(𝐴 = 𝑎)𝑑𝐴)/(∫−∞ 𝐴 × 𝑃(𝐴 = 𝑎)𝑑𝐴)
𝐶2 = (∫−∞ 𝐴2 × 𝑃(𝐴 = 𝑎)𝑑𝐴)/(∫−∞ 𝐴 × 𝑃(𝐴 = 𝑎)𝑑𝐴)2
𝐼 (̅ 𝑃(𝐴 = 𝑎), 𝜆) = 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 , 𝜆/𝐶2 )
(S64a)
(S64b)
(S64c)
For 𝐴 with 𝑃(𝐴 = 𝑎), we can define the expected value, 𝜇, and standard deviation, 𝜎, to yield the
following expressions for 𝐶1 and 𝐶2 :
𝐶1 = 𝐸[𝐴2 ]/𝐸[𝐴] = (𝜎 2 + 𝜇 2 )/𝜇
𝐶2 = 𝐸[𝐴2 ]/𝐸 2 [𝐴] = (𝜎 2 + 𝜇 2 )/𝜇 2
(S65a)
(S65b)
Eq (S65a) and Eq (S65b) are the main results of this section and also appear in the main text. To
conclude the continuum case, for a particle with concentration 𝜆 and scatterer brightness profile 𝐴 with
𝑃(𝐴 = 𝑎), the brightness population has a mean and standard deviation of 𝜇 and 𝜎, respectively. In this
case, The values of 𝐶1 and 𝐶2 can be described as in Eq. (S65a) and (S65b). The expected mean
scattering intensity from the sample can be described by Eq. (S64c). The asymptotes that define the
behavior of the recorded signal can described by the following:
𝐼𝑆 = 𝐸[𝐴] = 𝜇
𝐼𝐿 = √𝐸[𝐴2 ] = √𝜎 2 + 𝜇 2
(S66a)
(S66b)
In cases where descriptions of 𝐶1 and 𝐶2 may be difficult to express, it should be noted that they may be
inferred directly from the slope of 𝐼 ̅ for large and small values of 𝜆.
𝐼𝑆 = 𝐼 (̅ 𝐴(𝑝), 𝜆 ≪ 1)/𝜆
𝐼𝐿 = 𝐼 (̅ 𝐴(𝑝), 𝜆 ≫ 1)/(0.89 × √𝜆)
𝐶1 = 𝐼𝐿 2 /𝐼𝑆
𝐶2 = (𝐼𝐿 /𝐼𝑆 )2
iv. Simulation Experiments
The goal of this section is to demonstrate the utility of the generalized model described above by
using it to simulate a few simple cases of light scattering.
Example #1-Rotating GNR
Let us assume that the apparent brightness of a rotating GNR is 𝐴 = |𝛼 cos 𝜃|, where 𝛼 is the
maximum brightness and 𝜃 is the angle (uniformly random 𝜃~𝑈(0,2𝜋)) between the GNR longitudinal
axis and the detector. We can compute the first two momenta of 𝐴 as follows:
2𝜋
𝜋/2
𝐸[𝐴] = ∫0 |𝛼 cos 𝜃| × 1/2𝜋𝑑𝜃 = 𝛼/2𝜋 × [∫0
3𝜋/2
2𝜋
cos 𝜃 𝑑𝜃 − ∫𝜋/2 cos 𝜃 𝑑𝜃 + ∫3𝜋/2 cos 𝜃 𝑑𝜃]
𝐸[𝐴] = 𝛼/2𝜋 × [(1 − 0) − (−1 − 1) + (0 − (−1))] = 2𝛼/𝜋
2𝜋
𝐸[𝐴2 ] = ∫0 |𝛼 cos 𝜃|2 × 1/2𝜋 × 𝑑𝜃 = 𝛼 2 /2
(S67a)
(S67b)
Thus:
𝐶1 = (𝜎 2 + 𝜇 2 )/𝜇 = (𝛼 2 /2)/(2𝛼/𝜋) = 𝜎𝜋/4
(S68a)
𝐶2 = (𝜎 2 + 𝜇 2 )/𝜇 2 = (𝛼 2 /2)/(2𝛼/𝜋)2 = 𝜋 2 /8
(S68b)
𝐼 (̅ 𝜆) = 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 (𝐶1 , 𝜆/𝐶2 )
(S69)
Finally, recall that:
Using the values of 𝐶1 and 𝐶2 from Eqs. (S68a) and (S68b), we can simulate the detected signal from a
freely rotating GNR and predict the simulation results with Eq. (S69). The results of this simulation are
shown in Figure S10. The resulting prediction virtually ideal (𝑅 2 = 1.00000).
Example #2-Exponential Decay Scattering
Living cells scatter light according to a power law.S1 Cells contains many particles that scatter
very little light, and fewer that scatter more light. It is possible to fit the probability of scattering to an
exponential probability with the decay constant 𝛼. This behavior can be represented by:
𝑃(𝐴 = 𝑎) = 𝛼 × 𝑒 (−𝛼 × 𝑎)
(S70)
As with Example #1, we can compute the first two momenta of 𝐴. In this case, they are:
∞
𝐸[𝐴] = ∫0 𝐴 × 𝛼 × 𝑒 (−𝛼 × 𝐴) 𝑑𝐴 = 1/𝛼
(S71a)
∞
𝐸[𝐴2 ] = ∫0 𝐴2 × 𝛼 × 𝑒 (−𝛼 × 𝐴) 𝑑𝐴 = 2/𝛼 2
(S71b)
The resulting values of 𝐶1 and 𝐶2 are:
𝐶1 = (𝜎 2 + 𝜇 2 )/𝜇 = (2/𝛼 2 )/(1/𝛼) = 2/𝛼
(S72a)
𝐶2 = (𝜎 2 + 𝜇 2 )/𝜇 2 = (2/𝛼 2 )/(1/𝛼)2 = 2
(S72b)
The simulation and Eqs. (S72a) and (S72b) prediction results are shown in Figure S11. Again, the
calculated 𝐶1 and 𝐶2 values result in an excellent prediction using the ideal scatterer model (𝑅 2 =
0.99998).
Example #3-Ideal Scatterer Illuminated by a Gaussian Beam
We conclude by considering a more complex albeit common scenario: the scattering intensity
produced by a scatterer when the incident light intensity follows a Gaussian beam.
2
1
𝑥2 + 𝑦2
1
𝑧2
𝑥2 + 𝑦2
𝑧2
𝐴(𝑥, 𝑦, 𝑧) = 𝑎 [exp (− ×
)] × exp (− × 2 ) = 𝑎 exp (−
− 2)
2
𝜎2
2
𝜎
𝜎2
2𝜎
Where A(0,0,0)=a is the brightness of a scatterer at the center of the beam, 𝜎 is measured in pixels. The
square factor is due to the fact that light is scattered from the laser and then detected by the
interferometer, adding a square decay factor. This is the case for x and y but not for z
We begin by calculating 𝐸[𝐴]. Intuitively, we can assume that the scatterer can occupy a single random
position in a volume 𝑉 with probability 1/𝑉. We can then evaluate 𝐸[𝐴] as 𝑉 → ∞:
𝐸[𝐴] = lim ∭𝑉 𝑎 × 𝑒 (−((𝑥
𝑉→∞
2 +𝑦 2 )/𝜎 2 −(𝑧 2 /2𝜎 2 )
× 1/𝑉 × 𝑑𝑥𝑑𝑦𝑑𝑧 = 0
(S73)
An issue arises as 𝑉 increases: the probability of finding a particle in a “useful” (𝑥, 𝑦, 𝑧) position (i.e.,
within the focal volume of the beam) is negligible, and 𝐸[𝐴] → 0. To circumvent this problem, let us
instead assume that particles exist in some constant uniform distribution 𝜆1 such that 𝜆1 are present in
each imaging voxel on average and therefore 𝑉 × 𝜆1 particles exist in the entire volume 𝑉. We can then
recalculate 𝐸[𝐴] and sum over 𝑉 × 𝜆1 particles:
𝑉×𝜆
𝐸[𝐴] = lim ∑𝑘 1 ∭𝑉 𝑎 × 𝑒 (−((𝑥
2 +𝑦 2 )/𝜎 2 −(𝑧 2 /2𝜎 2 )
𝑉→∞
lim 𝑉 × 𝜆1 × ∭𝑉 𝑎 × 𝑒 (−((𝑥
× 1/𝑉 × 𝑑𝑥𝑑𝑦𝑑𝑧 =
2 +𝑦 2 )/𝜎 2 −(𝑧 2 /2𝜎 2 )
𝑉→∞
× 1/𝑉 × 𝑑𝑥𝑑𝑦𝑑𝑧 =
𝐸[𝐴] = 𝑎 × 𝜆1 × √2𝜋 3 × 𝜎 3
(S74)
𝐸[𝐴2 ] = 𝑎2 × 𝜆1 × √𝜋 3 /4 × 𝜎 3
(S75)
Similarly:
Next, we assign values to relevant parameters. Because Eqs. (S74) and (S75) are estimated for a single
scatterer, we select 𝜆1 = 1. It should be noted that 𝜎 3 is the Gaussian “volume” in terms of voxel units.
We can calculate it as:
𝜎 3 = (𝜎𝑥 + 𝜎𝑦 + 𝜎𝑧 )/𝑉𝑉
(S76)
Where 𝜎𝑥 , 𝜎𝑦 , and 𝜎𝑧 are the optical Gaussian standard deviations in three dimensions in 𝜇𝑚 and 𝑉𝑉 is
the voxel volume in 𝜇𝑚3. The OCT imaging volume is determined by the FWHM of the Gaussian beam.
Thus, we can assume that 𝜎 = 1/2.355. Finally, we can express 𝐶1 and 𝐶2 :
𝐶1 = (𝜎 2 + 𝜇 2 )/𝜇 = 𝑎/√8 = 0.354𝑎
(S77a)
𝐶2 = (𝜎 2 + 𝜇 2 )/𝜇 2 = 1/(𝜆1 × 𝜎 3 × √16𝜋 3 ) = 𝑉𝑉/(𝜆1 × 𝜎𝑥 × 𝜎𝑦 × 𝜎𝑧 × √16𝜋 3 ) = 0.586
(S77b)
Figure S12 displays the results from MC simulation and the model prediction using Eq. (S77a) and
(S77b) The modified ideal scatterer model again accurately describes the expected signal ( 𝑅 2 =
0.99978).
Ic. Consistency Check: Proof of C1 and C2 for GNRs in water
In the previous section, we showed that 𝐸[𝐴] and 𝐸[𝐴2 ] can be predicted from 𝐶1 and 𝐶2 (and
vice versa). In this section, we will validate this predictive ability by estimating 𝐸[𝐴] and 𝐸[𝐴2 ] directly
from measurements of low GNR concentrations and comparing those estimates with 𝐶1 and 𝐶2
predictions. All parameters that are estimated from low concentration samples will be marked with ̂, for
example: 𝐸̂ [𝐴] is the mean of 𝐴 estimated from a single concentration while 𝐸[𝐴] is the mean of 𝐴
estimated from 𝐶1 and 𝐶2 values fit from multiple concentrations.
Low Concentration Samples
If we assume that all voxels are independent, then the following scheme describes the origin of
the measured intensity within a voxel:
Scheme (S1)
In Scheme (S1), 𝐴 represents the random scatterer intensity. 𝐵 is the intensity in one voxel, where the
probability of a voxel to exhibit 1 scatterer is 𝜆 and the probability of exhibiting more than 1 scatterer is
negligible (due to low concentrations). Thus, 𝐵|𝜆 = 𝐴 and 𝐵|(1 − 𝜆) = 0. 𝐶 is the voxel intensity when
adding a bias such that 𝐶 = 𝐵 + 𝑛1 . Finally, 𝐼 is the measured voxel intensity after adding Poisson
electron count noise: 𝐼|𝐶~𝑁(𝐶, √𝐶 × 𝑛2 ) . It is important to note that 𝑛1 and 𝑛2 are instrumentdependent parameters. They can be estimated from measurements of a sample containing water only
imaged under the same conditions as samples containing particles. For water, 𝐼0 = 𝐼(𝜆 = 0) and thus we
can define:
𝑛1 = 𝐸[𝐼0 ]
(S78a)
𝑛2 = (𝐸[𝐼0 2 ] − 𝐸 2 [𝐼0 ])/𝐸[𝐼0 ]
(S78b)
We would like to calculate the first two moments of 𝐴, 𝐵, 𝐶, and 𝐼 as a function of 𝐶1 , 𝐶2 . From Section
Ib., it is clear to see that:
𝐸[𝐴2 ] = 𝐶1 2 /𝐶2
(S79a)
𝐸[𝐴] = 𝐶1 /𝐶2
(S79b)
We can solve the first two moments of 𝐵 as follows:
𝐸[𝐵] = 𝐸𝐴 [𝐸𝐵 [𝐵|𝐴]] = 𝜆 × 𝐸[𝐴]
(S80a)
𝐸[𝐵 2 ] = 𝐸𝐴 [𝐸𝐵 [𝐵 2|𝐴]] = 𝜆 × 𝐸[𝐴2 ]
(S80b)
Similarly, we can solve the first two moments of 𝐶:
𝐸[𝐶] = 𝐸[𝐵 + 𝑛1 ] = 𝜆 × 𝐸[𝐴] × 𝑛1
𝐸[𝐶 2 ] = 𝐸[(𝐵 + 𝑛1 )2 ] = (𝜆 × 𝐸[𝐴])2 + 2 × 𝜆 × 𝐸[𝐴] × 𝑛1 + 𝑛1 2
(S81b)
We can calculate the first moment of 𝐼: 𝐸[𝐼] = 𝐸𝐶 [𝐸𝐼 [𝐼|𝐶]]
Since 𝐼|𝐶~𝑁(𝐶, √𝐶 × 𝑛2 , then 𝐸𝐼 [𝐼|𝐶] = 𝐶
Additionally, 𝐸𝐶 [𝐶] = 𝜆 × 𝐸[𝐴] + 𝑛1
Thus:
(S81a)
𝐸[𝐼] = 𝜆 × 𝐸[𝐴] + 𝑛1
(S82)
The second moment of 𝐼 can also be calculated:
𝐸[𝐼 2 ] = 𝐸𝐶 [𝐸𝐼 [𝐼 2 |𝐶]] = 𝐸𝐶 [𝐶 2 + 𝐶 × (𝑛2 )2 ]
= 𝜆 × 𝐸[𝐴2 ] + 2 × 𝜆 × 𝑛1 × 𝐸[𝐴] + (𝑛1 )2 + 𝜆 × 𝐸[𝐴] × 𝑛2 + 𝑛2 𝑛1
= 𝜆 × 𝐸[𝐴2 ] + 𝜆 × 𝐸[𝐴] × (2 × 𝑛1 + 𝑛2 ) + 𝑛2 𝑛1 + 𝑛1 2
= 𝜆 × 𝐸[𝐴2 ] + (𝐸[𝐼] − 𝑛1 ) × (2 × 𝑛1 + 𝑛2 ) + 𝑛2 𝑛1 + 𝑛1 2
𝐸[𝐴2 ] = 1/𝜆 × (𝐸[𝐼 2 ] − (2 × 𝑛1 + 𝑛2 ) × 𝐸[𝐼] + 𝑛1 2 )
𝐸[𝐼 2 ] = 𝜆 × [𝐸[𝐴2 ] + 𝐸[𝐴] × (2 × 𝑛1 + 𝑛2 )] + 𝑛2 𝑛1 + 𝑛1 2
(S83)
We can incorporate the definitions from Eqs. (S79a) and (S79b) into Eqs. (S82) and (S83) to receive the
following:
𝐸[𝐼] = 𝜆 × 𝐶1 /𝐶2 + 𝑛1
𝐸[𝐼 2 ] = 𝜆 × 𝐶1 /𝐶2 × (𝐶1 + 2 × 𝑛1 + 𝑛2 ) + 𝑛1 𝑛2 + 𝑛1 2
(S84a)
(S84b)
Eq. (S84a) is trivial, as it follows the same approximation as 𝐼 𝐼̅ 𝑑𝑒𝑎𝑙 for low particle concentrations.
However, Eq. (S84b) provides some novel insight. With it, we are able to predict the voxel intensity
variance from 𝐶1 and 𝐶2 , despite the fact that neither variable directly relates to the voxel intensity
variance (rather, each describes the mean voxel intensity). We can also rearrange Eqs. (S84a) and (S84b)
to receive the following definitions of 𝐶1 and 𝐶2 :
𝐶1 = (𝐸[𝐼 2 ] − (2 × 𝑛1 + 𝑛2 ) × 𝐸[𝐼] + 𝑛1 2 )/(𝐸[𝐼] − 𝑛1 )
(S85a)
𝐶2 = 𝜆 × (𝐸[𝐼 2 ] − (2 × 𝑛1 + 𝑛2 ) × 𝐸[𝐼] + 𝑛1 2 )/(𝐸[𝐼] − 𝑛1 )2
(S85b)
Thus is possible to predict the scattering signal of high concentrations by using low concentration
measurements only. However, the presence of 𝐸[𝐼] − 𝑛1 in the denominator can lead to the propagation
of large errors in 𝐶1 and 𝐶2 . Therefore, accurate calculations of 𝐶1 and 𝐶2 require many independent
scans of a given sample to minimize initial measurement error.
Experimental Results: Predicting 𝑬[𝑰𝟐 ]
Let us assume that 𝐶1 and 𝐶2 are estimated with very low variance such that Δ𝐶1 ≈ 0 and Δ
𝐶2 ≈ 0. Further, we assume that there can be up to 10% error in the prepared concentrations such that
Δ𝜆 = 𝜆/10. Finally, we assume that all errors are independent. Recalling the definition of 𝐸[𝐼 2 ], the
following definition can be obtained:
Δ𝐸[𝐼 2 ] = 𝜆/10 × 𝐶1 /𝐶2 × (𝐶1 + 2 × 𝑛1 + 𝑛2 ) + 𝜆 × 𝐶1 /𝐶2 × (2Δ𝑛1 + Δ𝑛2 )
+Δ𝑛2 𝑛1 + Δ𝑛1 𝑛2 + 2Δ𝑛1 𝑛1
(S86)
Eq. (S86) can be simplified:
Δ𝐸[𝐼 2 ] = 𝜆 × 𝐶1 /𝐶2 × (𝐶1 /10 + 2 × 𝑛1 /10 + 𝑛2 /10 + 2Δ𝑛1 + Δ𝑛2 )
+Δ𝑛2 𝑛1 + Δ𝑛1 𝑛2 + 2Δ𝑛1 𝑛1
(S87)
We imaged capillary tubes with known concentrations of GNRs (ranging from 50 fM-1 pM) in water.
For each sample, 100 B-scans were acquired at each of three separate locations along the tube. We
concatenated all B-scan data for each sample and then diluted the data by a factor of nine (by selecting
only one of every three voxels in each dimension) to avoid autocorrelation bias due to the optical point
spread function. From this data we computed values for 𝐸̂ [𝐼 2 ] and standard deviations from the three
scan locations. We could see excellent agreement of prediction with measurement, a summary of
predicted and measured values for each GNR concentration are provided in Table S2. The water-filled
tube that was used to compute system noise parameters yielded the values shown below:
𝑛̂1 = 324.2
Δ𝑛̂1 = 0.81
𝑛̂2 = 48.1
Δ𝑛̂2 = 0.59
To summarize, the parameters 𝐶1 and 𝐶2 can be used to predict the variance in speckle intensity for low
concentration measurements. This result is somewhat surprising since 𝐶1 and 𝐶2 were measured to
describe mean speckle intensity and not variance. However, the results from using 𝐶1 and 𝐶2 to predict
variance provide further confidence that they are direct measurements of 𝐸[𝐴] and 𝐸[𝐴2 ] as described in
Section Ia-b. Furthermore, it is possible to predict the behavior of the OCT signal at high concentrations
from the mean speckle intensity and speckle variance at low concentrations.
Id. GNRs in blood and tissue
The goal of this section is to study the signal observed for GNRs in blood and compare this
signal to the results observed for GNRs in water. Within this section, we will analyze the signal from
capillary tubes containing blood only. Then, we will describe a model of scattering from two types of
particles present within a voxel. This section also includes a description of our experimental setup,
image pre-processing, and results and discussion. We conclude with an extension of this model to in
vivo imaging.
Blood-Only Tube Analysis
The blood present within each capillary tube exhibits two main interactions with light. Blood
components scatter light that is detected with OCT, and they also absorb a significant amount of light
within the OCT spectrum, which changes the effective lighting conditions at different depths within the
sample. We will address each of these functions separately, beginning with the effects of blood
absorption.
If we treat individual red blood cells as scatterers, then the concentration of scatterers present in
whole blood is ~8.3 pM (calculated from cells per mL in whole blood). In our experiments, each
capillary tube contained 90% of the original concentration of whole blood, which equates to ~0.57
particles (red blood cells) per voxel. Because of blood’s strong absorption, we must consider the
variation in lighting conditions present at different depths in order to compare blood intensity to the
results observed in water. In this analysis, we will assume that the focal plane differences between water
tubes and blood tubes are negligible. Thus, the main effect on observed signal originates from natural
attenuation of light with depth. We can fit the following model to OCT data obtained for blood only:
𝐼(𝑧) = 𝑥1 × exp(−𝑧/𝑥2 ) + 𝑥3
(S86)
Where 𝑥1 is the blood intensity at the top of the tube, 𝑥2 is the decay constant due to attenuation, 𝑥3 is
the baseline noise, and 𝑧 is the depth (in voxels) from the top of the tube. This is graphically depicted in
Scheme S2:
I
Scheme (S2)
𝑥1
𝑥3
𝑥2
z
Note that in our data analysis we will perform the fit from ~10 to 15 pixels below the top of each tube to
avoid the shift in intensity caused by the blood/glass interface. However, we would still like to measure
𝑧 from the top of the tube. Thus, we are neglecting focal depth “attenuation” since blood attenuation is
notably higher. We can perform the fit for the top of the tube using a peak convolution algorithm with a
linear fit across the x dimension to receive an accurate measurement of the top of the tube.
The measured baseline noise ( 𝑥3 ) from the water tube is consistent across independent
measurements (𝑥3 ≈ 315). The measured decay constant was determined to be 𝑥2 = 35 pixels, meaning
that the source light intensity reduces by 50% over a depth of about 24 voxels. Based on OCT
measurements, the intensity signal from blood only at the top of the tube 𝑥1 is around 3300 OCT counts
(AU), which is roughly 3.9 times greater than the 𝐶1 measured for GNRs in water: 𝐼0 /𝐶1 = 3.9 See S14
for more details. Say that results average for 3 blood only tubes is in Figure S14.
Let the relative light intensity for each depth 𝐿(𝑧) be 𝐿(𝑧) = exp(−𝑧/𝑥2 ), as can be seen
𝐿(0) = 1
A Model for 2 Particle Types Present in a Voxel
Recall that GNRs modeled as equivalent ideal particles possess brightness 𝐴, defined as:
𝐴 = 𝐶1 × 𝐿(𝑧)
(S87)
Where 𝐶1 is given from the fit for GNRs in water. We can define a concentration 𝜆𝑎 such that 𝜆𝑎 =
𝜆/𝐶2 , where 𝜆 is the actual GNR concentration and 𝐶2 is given from the fit for GNRs in water. We can
also define parameters for brightness and concentration of blood as if composed of equivalent ideal
particles. In this case, the single particle brightness can be described by:
𝐵 = 𝑏 × 𝐿(𝑧)
(S88)
The effective blood scatterer concentration can be expressed as 𝜆𝑏 . In this model, 𝜆𝑏 and 𝑏 are unknown,
but OCT scattering signal from blood only 𝐼0̅ = 𝐼0̅ (𝜆𝑏 , 𝑏) is known. We can thus measure the expected
intensity of a given sample and subtract the blood-only tube:
𝐼 ̅ → 𝐼 ̅ − 𝐼0̅
(S89)
The resulting intensity, 𝐼 ,̅ can be expressed thusly:
𝑘𝑎
∞
−𝜆𝑎
𝐼 ̅ = ∑∞
× (𝜆𝑏 𝑘𝑏 /𝑘𝑏 !) × 𝑒 −𝜆𝑏
𝑘𝑎=0 ∑𝑘𝑏=0 𝑋(𝑘𝑎 , 𝑘𝑏 ) × (𝜆𝑎 /𝑘𝑎 !) × 𝑒
∞
𝑘𝑎
−0
− ∑∞
× (𝜆𝑏 𝑘𝑏 /𝑘𝑏 !) × 𝑒 −𝜆𝑏
𝑘𝑎=0 ∑𝑘𝑏=0 𝑋(𝑘𝑎 , 𝑘𝑏 ) × (0 /𝑘𝑎 !) × 𝑒
𝑘𝑎
∞
−𝜆𝑎
𝐼 ̅ = ∑∞
× (𝜆𝑏 𝑘𝑏 /𝑘𝑏 !) × 𝑒 −𝜆𝑏
𝑘𝑎=0 ∑𝑘𝑏=0 [𝑋(𝑘𝑎 , 𝑘𝑏 ) − 𝑋(0, 𝑘𝑏 )] × (𝜆𝑎 /𝑘𝑎 !) × 𝑒
(S90)
In order to assess the mean ROI intensity, we can define the following:
𝑘𝑏
−𝜆𝑏
𝐴̃ = ∑∞
𝑘𝑏=0 [𝑋(1, 𝑘𝑏 ) − 𝑋(0, 𝑘𝑏 )] × (𝜆𝑏 /𝑘𝑏 !) × 𝑒
(S91)
𝑘𝑏
−𝜆𝑏
𝑋̃(𝑘𝑎 ) = ∑∞
𝑘𝑏=0 [𝑋(𝑘𝑎 , 𝑘𝑏 ) − 𝑋(0, 𝑘𝑏 )] × (𝜆𝑏 /𝑘𝑏 !) × 𝑒
(S92)
Thus,
𝑘𝑎
−𝜆𝑎
̃
𝐼 ̅ = 𝐴̃ × 𝜆𝑎 × 𝑒 −𝜆𝑎 + ∑∞
𝑘𝑎=2 𝑋 (𝑘𝑎 ) × (𝜆𝑎 /𝑘𝑎 !) × 𝑒
(S93)
Where 𝐴̅ describes the signal behavior at low concentrations (𝐼𝑆 = 𝐴̅) and 𝑋̃ describes the behavior at
high concentrations. We can see that 𝑋̃(𝑘𝑎 ≫ 1) ≈ 0.89 × 𝐴 × √𝑘𝑎 since the contribution from blood
can be neglected if a very high concentration of GNRs is present. Thus for high concentrations, 𝐼𝐿 ≈ 𝐴.
Since 𝑋̃(𝑘𝑎 ) is not proportional to √𝑘𝑎 except for high 𝑘𝑎 , we cannot describe 𝐼 ̅ as an ideal
particle except at high or low concentrations. Thus, we will shift the discussion from that of a full model
to an asymptotic one. In this case,
𝐼 (̅ 𝜆𝑎 ≪ 1) = 𝐼𝑆 𝜆𝑎 = 𝐴̃ × 𝜆𝑎
(S94a)
𝐼 (̅ 𝜆𝑎 ≫ 1) = 0.89 × 𝐼𝐿 × √𝜆𝑎 = 0.89 × 𝐴 × √𝜆𝑎
(S94b)
We can now go ahead and estimate 𝐴̃ for two extreame cases 𝜆𝑏 ≪ 1 and 𝜆𝑏 ≫ 1. Let us start with
𝜆𝑏 ≫ 1 assume that the majority of the summation term in Eq. (S91) is contributed by one argument,
𝑘𝑏 = 𝐾, in which case 𝐾 ≥ 2. Thus,
𝐴̃ ≈ 𝑋(1, 𝐾) − 𝑋(0, 𝐾)
(S95)
We previously showed the following:
𝑋(1, 𝐾) = √𝐹 2 𝐴2 + (0.89 × 𝐵 × √𝐾)2
Where 𝐹 is a correction factor defined as:
𝐹={
1
𝐴≫𝐵
0.89 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Additionally, we can derive that:
𝑘𝑏
−𝜆𝑏
𝐼0̅ = ∑∞
≈ 0.89 × 𝐵 × √𝐾
𝑘𝑏 =0 𝑋(0, 𝑘𝑏 ) × (𝜆𝑏 /𝑘𝑏 !) × 𝑒
(S96)
We can thus conclude that:
𝑋(1, 𝐾) = √𝐹 2 𝐴2 + ( 𝐼0̅ )2
𝑋(0, 𝐾) = 𝐼0̅
(S97a)
(S97b)
If we substitute Eqs. (S97a) and (S97b) into Eq. (S95), we receive the following expression for 𝐴̃:
𝐴̃ = 𝐼0̅ × [√(𝐹 × 𝐴/𝐼0̅ )2 + 1 − 1]
(S98)
Since 𝜆𝑏 ≫ 1 and 𝐴/𝐼0̅ is finite, we can conclude that (in this case) 𝐵 ≪ 1, equivalent to many small
faint particles. This necessitates that 𝐴 ≫ 𝐵, which yields 𝐹 = 1. Additionally, experimental results
have show that 𝐼0̅ ≈ 3.9, which means that 𝑥 = (𝐴/𝐼0̅ )2 is a very small number. Given these conditions,
we can develop an expression for 𝐴̃(𝑥) using a first order Taylor expansion:
𝐴̃ = 𝐼0̅ × [√𝑥 + 1 − 1] = 𝐼0̅ × [1 + 𝑥/2 − 1]
(S99)
This yields the following expression for when 𝜆𝑏 ≫ 1:
𝐴̃ ≈ 𝐴/2 × (𝐴/𝐼0̅ )
(S100)
We can further conclude that, for 𝑧 = 0:
𝐼 (̅ 𝜆𝑎 ≪ 1) = 𝐴̃ × 𝜆𝑎 = 𝐴/2 × (𝐴/𝐼0̅ ) × 𝜆𝑎 = (𝐶1 /(2 × 𝐶2 )) × (𝐶1 /𝐼0̅ ) × 𝜆
(S101a)
𝐼 (̅ 𝜆𝑎 ≫ 1) = 0.89 × 𝐴 × √𝜆𝑎 = 0.89 × 𝐶1 × √𝜆/𝐶2
(S101b)
Where 𝐶1 and 𝐶2 are the fits for GNRs in water.
We can further validate that 𝐼𝐿 = 𝐴 for 𝜆𝑏 ≫ 1, by assuming that the majority of the sum comes from a
single argument 𝑘𝑏 = 𝐾, we can estimate Eq. (S92) as:
𝑋̃(𝑘𝑎 ) = 𝐼0̅ × [√(0.89 × 𝐴/𝐼0̅ )2 × 𝑘𝑎 + 1 − 1]
(S102)
Which indeed show 𝑋̃(𝑘𝑎 ) → 0.89𝐴√𝑘𝑎 as 𝑘𝑎 grows
We now move to the case when 𝜆𝑏 ≪ 1. Recalling Eq. (S91), we can perform an estimation for
̃
𝐴 in this case. Under these conditions, we can consider only 𝑘𝑏 = 0,1 and develop an expression for the
order of 𝜆𝑏 . We can substitute 𝑘𝑏 = 0,1 into Eq. (S91) to receive the following:
𝐴̃ ≈ [𝑋(1,0) − 𝑋(0,0)] × 𝑒 −𝜆𝑏 + [𝑋(1,1) − 𝑋(0,1)] × 𝜆𝑏 × 𝑒 −𝜆𝑏
(S103)
𝐴̃ ≈ 𝑋(1,0) + 𝜆𝑏 × [𝑋(1,1) − 𝑋(1,0) − 𝑋(0,1)]
(S104)
We can plug in the estimation for 𝑋 from Section Ib subsection i to obtain:
Γ
𝐴̃ ≈ 𝐴 − 𝜆𝑏 × 𝐴 × [1 + 𝐵/𝐴 − √1 + 𝐵/𝐴Γ ]
(S105)
Since 𝐼0̅ ≈ 𝜆𝑏 × 𝐵 and 𝐼0̅ /𝐴 is finite, for very low 𝜆𝑏 , 𝐵 should be such that 𝐵 ≫ 𝐴(equivalent to the
case where bold is composed of very few but bright particles).
Γ
Thus, √1 + 𝐵/𝐴Γ ≈ 𝐵/𝐴. Finally, we receive:
𝐴̃ ≈ 𝐴 × (1 − 𝜆𝑏 )
(S106)
In the case 𝜆𝑏 ≪ 1, we can conclude that for 𝑧 = 0:
𝐼 (̅ 𝜆𝑎 ≪ 1) = 𝐴̃ × 𝜆𝑎 = 𝐴 × (1 − 𝜆𝑏 ) × 𝜆𝑎 = 𝐶1 /𝐶2 × (1 − 𝜆𝑏 ) × 𝜆
(S107a)
𝐼 (̅ 𝜆𝑎 ≫ 1) = 0.89 × 𝐴 × √𝜆𝑎 = 0.89 × 𝐶1 × √𝜆/𝐶2
(S107b)
Again, We can further validate that 𝐼𝐿 = 𝐴 for 𝜆𝑏 ≫ 1, by directly estimate 𝑋̃(𝑘𝑎 ) by considering only
𝑘𝑏 = 0,1 and develop for the order of 𝜆𝑏 :
𝑋̃(𝑘𝑎 ) ≈ [𝑋(𝑘𝑎 , 0) − 𝑋(0,0)] × 𝑒 −𝜆𝑏 + [𝑋(𝑘𝑎 , 1) − 𝑋(0,1)] × 𝜆𝑏 × 𝑒 −𝜆𝑏
= 𝑋(𝑘𝑎 , 0) + 𝜆𝑏 × [𝑋(𝑘𝑎 , 1) − 𝑋(𝑘𝑎 , 0) − 𝑋(0,1)]
(S108)
As when we calculated the asymptote for 𝜆𝑏 ≫ 1, we can plug in the estimation for 𝑋 to receive the
following expression:
𝑋̃(𝑘𝑎 ) = 0.89 × 𝐴 × √𝑘𝑎 + 𝜆𝑏 × [√𝐹 2 𝐵 2 + (0.89 × 𝐴2 × 𝑘𝑎 ) − 0.89 × 𝐴 × √𝑘𝑎 −
𝐵]
𝑋̃(𝑘𝑎 ) = 0.89 × 𝐴 × √𝑘𝑎
×
(1 + 𝜆𝑏 × [√1 + (𝐹 2 𝐵 2 /(0.89 × 𝐴2 × 𝑘𝑎 )) − 1 − 𝐵/(0.89 × 𝐴2 × 𝑘𝑎 )])
(S109)
Since 𝐵 ≫ 𝐴 we can assume that 𝐹 = 0.89. We can calculate a ratio 𝑅(𝑘𝑎 ):
𝑅(𝑘𝑎 ) = 𝑋̃(𝑘𝑎 )/(0.89 × 𝐴 × √𝑘𝑎 ) = 1 + 𝜆𝑏 × [√1 + (𝐵 2 /𝐴2 × 𝑘𝑎 ) − 1 − 𝐵/(0.89 × 𝐴 ×
𝑘𝑎 )]
(S110)
As 𝑘𝑎 grows, 𝑅(𝑘𝑎 ) → 1 for different 𝐵/𝐴.
We can make several conclusions about the asymptotes for GNRs in blood based on the
preceding discussion. Primarily, both regimes agree upon a square root dependence:
𝐼 (̅ 𝜆𝑎 ≫ 1) = 0.89 × 𝐴 × √𝜆𝑎 = 0.89 × 𝐶1 × √𝜆/𝐶2
(S111)
The difference between the two regimes resides in the linear phase:
𝐼 (̅ 𝜆𝑎 ≪ 1, 𝜆𝑏 ≫ 1) = 𝐴/2 × (𝐴/𝐼0̅ ) × 𝜆𝑎 = (𝐶1 /(2 × 𝐶2 )) × (𝐶1 /𝐼0̅ ) × 𝜆
(S112a)
𝐼 (̅ 𝜆𝑎 ≪ 1, 𝜆𝑏 ≪ 1) = 𝐴 × (1 − 𝜆𝑏 ) × 𝜆𝑎 = 𝐶1 /𝐶2 × (1 − 𝜆𝑏 ) × 𝜆
(S112b)
To further explore the linear phase, let us use the following notation:
𝐼 (̅ 𝜆𝑎 ≪ 1) = 𝐼𝑆 × 𝜆𝑎
(S113)
And bound 𝐼𝑆 . For 𝜆𝑏 ≫ 1 we can define 𝐼𝑆 𝑚𝑖𝑛 = 𝐴/2 × (𝐴/𝐼0̅ ). For 𝜆𝑏 ≪ 1 we shall use 𝐼𝑆 = 𝐴 ×
(1 − 𝜆𝑏 ). Because we cannot measure 𝜆𝑏 , let us assume the minimum 𝜆𝑏 = 0 to receive 𝐼𝑆 𝑚𝑎𝑥 = 𝐴.
With these definitions, we can conclude that Eq. (S111) is sufficient in describing the square root phase
and that the linear phase can be described by:
𝐼 (̅ 𝜆𝑎 ≪ 1) = 𝐼𝑆 × 𝜆𝑎 = 𝐼𝑆 /𝐶2 × 𝜆
(S114)
Where:
𝐼𝑆 𝑚𝑖𝑛 ≤ 𝐼𝑆 ≤ 𝐼𝑆 𝑚𝑎𝑥
𝐼𝑆 𝑚𝑖𝑛 = 𝐴/2 × (𝐴/𝐼0̅ )
𝐼𝑆 𝑚𝑎𝑥 = 𝐴
[for large 𝜆𝑏 ]
[for small 𝜆𝑏 ]
Figure S15 presents several Monte Carlo examples for different 𝜆𝑏 but maintaining 𝐼0 /𝐴 constant. As
can be seen, in all examples monte carlo simulation approaches predicted square root regime for high
GNR concentration. And linear regime is within the boundaries of 𝐼𝑠𝑚𝑎𝑥 and 𝐼𝑠𝑚𝑖𝑛 .
Experimental Setup and Pre-Processing
For each sample of GNRs in blood, the final blood concentration was 90% that of whole blood.
Various GNR concentrations were tested. The top of each tube was estimated independently as
described previously. Data for each tube was fitted to the model described by Eq. (S86). The decay
constant (𝑥2 ) and the baseline signal level (𝑥3 ) were the same in each GNR/blood sample as for blood
only. The maximum intensity (𝑥1 ) was fitted for each tube. We also applied methods to filter outlier
species within each tube, which arise from natural coagulation in freshly-excised whole blood. For each
x position within each tube, the RMS between data was fit and calculated. X positions with RMS above
a threshold of 370 were ignored for calculating the mean signal. See Figure S16.
For each tube, 𝑥1 was recorded. Notice that 𝑥1 represents the GNR + blood absolute intensity
under ideal lighting conditions: 𝐿 = 1. Because shadowing is insignificant for the employed GNR
concentrations we did not correct the asymptotes with an exponential decay correction.
Blood Tubes-Results and Discussion
Predictions of the signal from GNRs in blood were based upon the previous results for the square
root and linear regimes presented in Eqs. (S111) and (S114).. The values of 𝐶1 and 𝐶2 were taken from
the model fit for GNRs in water. 𝐼0̅ was defined by the blood only 𝑥1 value 𝐼 ̅ represents the OCT
intensity for GNRs + blood fit 𝑥1 . We further assume that GNR light attenuation is negligible for the
analyzed concentrations (the shadow limit occurs at ~ 1 nM GNRs). We find that the experimental data
(presented in Figure S17) does not seem to follow the linear regime but rather follows the predicted
square root regime only. The experimental results suggest that the critical concentration (between the
linear and square root regimes) is less than 5 pM, while the minimal prediction for the critical
concentration (using 𝐼𝑠𝑚𝑎𝑥 ) is 40 pM. There are several possible explanations for the lack of a linear
behavior within these samples, which we explore below. First we consider the interaction of light with
blood. Clearly, the components of blood exhibit a strong interaction with optical and near-infrared light
(largely due to high concentrations of hemoglobin). Let us assume that one Rayleigh scattering event
occurs along the optical path into and out of the sample. This scattering may increase the effective
number of particles that seem to be present in one voxel 𝐶 ′ 2 < 1, depicted in Scheme S3. This can be
due to multiple scattering events within a voxel. Note that this process also reduces the brightness of
each voxel 𝐶 ′ 1 < 1. If 𝐶 ′1 /√𝐶 ′ 2 = 1, then this effect will not change the square root regime and
will match predictions. However, the critical concentration would change as:
𝐶𝑐𝑟𝑖𝑡 = 40𝑝𝑀 × 𝐶 ′ 2
𝐶2 ′ = 1/2
(S115)
Scheme (S3)
Imaged Voxel
Modeling OCT Signals in vivo
A common problem that is encountered when imaging regions of interest in vivo is the variation
lighting conditions. Specifically, the ratio of light (𝐿) that illuminates the sample is unknown. While it is
possible to estimate 𝐿 by observing pre-injection blood vessel signals and comparing them to in vitro
samples of blood, this method is lacking. Rather, let us assume that we have a blood vessel in a living
animal and that this vessel is filled with blood only.
Light shadowing due to high GNR concentrations is more difficult to predict for several reasons.
First, the diameter of each blood vessel is not an exact measurement and the vessel shape (roughly
circular) the average attenuation in such circular vessels is not analytically solvable (unlike with square
capillary tubes). Moreover, blood vessels are rarely (if ever) perfectly circular. Thus we shall assume
that 𝜖 (the extinction coefficient) estimated for water is the same for GNRs in blood and we will fit the
vessel diameter, 𝐷. Also note that all measured samples in vivo have high GNR concentrations (the
lowest measured concentration was 250 pM). Because of this, it is guaranteed that all samples will
exhibit signal within the square root regime. We can define the following prediction that incorporates
the square root regime and shadowing:
𝐼 (̅ 𝜆) = 0.89 × (𝐿 × 𝐶1 /√𝐶2 ) × √𝜆 × 𝐹(𝐿, 𝑐)
(S116)
𝐹(𝐿, 𝑐) = 1/(2 × 𝑐 × 𝜖 × 𝐷 × log 10) × [1 − 10−2 × 𝑐 × 𝜖 × 𝐷 ]
(S117)
As before, 𝐶1 and 𝐶2 and 𝜖 are taken from the fit for GNRs in water. 𝜆 and 𝑐 represent GNRs per voxel
and GNR molar concentration, respectively. 𝐿 and 𝐷 are individual vessel lighting conditions and
diameters and are fitted for each vessel using a least squares method. The results of this analysis are
shown in Figure S18. This plot represents data from nine different blood vessels (three vessels from
each of three different mice) converted to an equivalent blood vessel with a 150 m diameter and ideal
lighting conditions. The data within this plot show a clear square root behavior. Note the absence of
linear behavior in the in vivo data due to the absence of sufficiently low GNR concentrations and
potential obscuring effects from the high concentration of scatterers in whole blood. As can be seen in
Figure S18, the fit is good for individual vessels as well as the equivalent average vessel. Also, the
fitted equivalent diameter D follows the estimated size of the blood vessels used.
Ie. Summary
The model presented above provides a versatile tool for describing the expected scattering from a
wide array of samples. This work demonstrates that two parameters (𝐶1 and 𝐶2 ) can be incorporated into
a model of ideal scattering to achieve such descriptions. Critically, these two parameters represent real
physical parameters that can be determined for a given system and could be predicted using low
concentration samples. Using this approach, scatterer concentrations can be correlated to observed signal
levels for coherence-based detection methods. This ability is critical for enabling new capabilities for
imaging techniques including ultrasound and OCT, which have historically been limited by poor
quantitation. Accurate quantitative measurements will be of particular importance as these and other
techniques are developed for contrast-enhanced and molecular imaging studies.
II: Imaging Experiments, Analysis, and Post-Processing
IIa. System Optics
Images were acquired with a Ganymede High-Resolution (HR) spectral domain optical
coherence tomography (SD-OCT) system (ThorLabs, Newton NJ). This system uses a superluminescent
diode (SLD) with a center wavelength of 900 nm and a full bandwidth of 200 nm ( = 800-1000 nm),
resulting in 2.1 m axial resolution in water. The SLD operates at 30 kHz, resulting in an A-scan
acquisition time of approximately 33 s. A 5x magnification objective (LSM03BB, ThorLabs, Newton,
NJ) with a lateral resolution of 8 m (Full Width Half Max) and a 143 m depth of field was used in all
imaging experiments. The axial and lateral resolution values described above were used for modeling
and fitting of experimental data.
IIb. Acquisition Parameters
All OCT images were obtained from averaging 100 consecutive B-scans acquired during a total
scan time of ~11 seconds. Each B-scan was composed of 3500 A-scans with 2 m pixel spacing. Note
that this pixel spacing is oversampled with respect to the system’s optical resolution (8 m x 8 m x 2
m), which determines the measured signal and interference effects. Over the acquisition period, the
expected mean displacement of GNRs due to diffusion is 11-20 m, which is greater than the size of an
imaging voxel. Because of this, we assume that the time mean of the sample can approximate the
expected speckle intensity, as described in Section Ia, subsection iv. It is important to note that if the
particles being measured do not move to such an extent, this assumption may not hold.
IIc. Quantitative Analysis
OCT intensities from GNRs in water prepared in square glass capillary tubes (VitroCom Inc., cat
#8240-100, 0.4 mm inner diameter) were determined by calculating the mean values from regions of
interest (ROIs) selected within the focal plane (at the same z-axis position) for each concentration. Each
ROI consisted of > 5000 pixels. For each GNR concentration, 3 independent physical positions within
the tube were imaged and quantified to calculate mean signals and standard error values. The depth of
the selected ROI (150 pixels, or ~300 m) was used as an input to calculate the expected shadowing
limit due to light attenuation in the model. GNRs in blood were analyzed in a similar fashion as
described above. However, the selected ROI depth was reduced because of the strong attenuation of
blood. This change was accounted for in our prediction. Again, all values represent the mean ± standard
error of the mean (s.e.m.) from 3 independent ROI measurements per sample. GNR signal from in vivo
experiments was quantified by analyzing images acquired after each incremental injection as well as a
pre-injection image. For each of three mice, three blood vessels were identified, and the mean signal in
each blood vessel was quantified using ROI analysis. Because of different depths within the ear tissue,
the initial OCT signal in each blood vessel prior to injection can vary significantly. Thus for each vessel,
the vessel’s initial blood-only signal was subtracted from all subsequent measured intensities following
incremental injections. This enabled an accurate evaluation of the increase in OCT signal due to
increased GNR concentration. Moreover, the variation in vessel depth also produces variation in the
calculated C1 values from vessel to vessel. To account for this variation, each of the nine measured
blood vessels was fit independently with the model, and the resulting nine C1 values were used to
normalize each vessel. For each mouse, the normalized vessels were averaged. Then, the data from all
three mice were averaged (representing a total of 9 vessels across the three animals). In this case, the
error bars (s.e.m.) represent variation across the three separate mice. The average ROI depth was used to
calculate the shadow limit in the model as described for GNRs in water and GNRs in blood.
III: Supporting Figures
Figure S1. Monte Carlo estimation of the ratio between scattering intensity and a square root
dependence on the number of scatterers, N, per imaging voxel. The simulation was run for 2 ≤ N ≤ 100.
The mean proportion over this range was 0.89, which was used as a coefficient for calculating expected
scattering intensity from samples.
Figure S2. Plot depicting a model for OCT signal from a population of ideal scatterers as a function of
concentration. At very low concentrations per voxel, the OCT signal can be approximated by a linear
function (green line). At high concentrations, the OCT signal follows a square root dependence on the
number of scatterers due to interference effects (red line). The intersection between these two regimes
occurs at the critical concentration, denoted by the vertical dashed line. The model (blue line) describes
both regimes. Here, molar concentrations are correlated to particle per voxel concentration () for the
GNRs used in this study. Top x scale considers OCT optical voxel size of 8 m x 8 m x 2 m (see
Section IIIa-b) to convert concentration to particles per voxel
Figure S3. Diagram of the path taken by light detected in OCT imaging. Note that light is attenuated
both going into and coming out of the sample. In each case, the attenuation follows the Beer-Lambert
Law. Light intensity therefore follows an exponential decay as a function of the depth in the sample (z).
Light attenuation was accounted in the model based on the depth of each ROI (d) chosen for quantitative
analysis.
Figure S4. Two examples of the binomial case of ideal scattering. In these examples, particles could
occupy one of two states (“on” and “off”) with some probability 𝑝.
Figure S5. Monte Carlo simulation of the interference between two types of particles that exhibit
different scattering states,  and . Note that for high  and low , the expected intensity is
approximately determined by  (and vice versa). When  and  are comparable, interference effects
cause significant deviation from either single state.
Figure S6. As in Figure S5, with the addition of two different methods for estimating the interference,
X(1,1), between the two particle states.
Figure S7. An example of 𝑋(𝑘𝛼 ≥ 2, 𝑘𝛽 ≥ 2), as can be seem, the fit is very good for Γ=2. See Eq.
(S41).
Figure S8. An example of𝑋(𝑘𝛼 = 1, 𝑘𝛽 ≥ 2), as can be seem, the fit is very good for Γ=2. F was
1
𝛼/𝛽 ≥ 3.5
selected as 𝐹 = {
. On the right panel a close up on the switch point is found, by
0.89 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
selecting the switch point. See Eq. (S41).. See Eq. (S42).
Table S1. A summary of the resulting expressions for 𝑋(𝑘𝛼 , 𝑘𝛽 )depending on the values of each
1
𝛽≫𝛼
scattering state. 𝐹 = {
0.89 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Figure S9. Examples of the prediction produced by the general binomial case.
Figure S10. Monte Carlo simulation of a freely-rotating GNR fit by the generalized model for ideal
scattering. Closed-form expressions for C1 and C2 used in this prediction are described in Eqs. (S64a)
and (S64b).
Figure S11. Monte Carlo simulation of scattering following an exponential probability prediction by the
generalized model for ideal scattering. Closed-form expressions for C1 and C2 used in this predictions
are described in Eqs. (S68a) and (S68b).
Figure S12. Monte Carlo simulation of scattering by a particle illuminated by a Gaussian beam
prediction by the generalized model for ideal scattering. Closed-form expressions for C1 and C2 used in
this prediction are described in Eqs. (S73a) and (S73b).
Concentration
[pM]
λ
[Particles Per
Voxel]
Predicted
𝑬[𝑰𝟐 ]/𝟏𝟎𝟒
Measured
𝑬 [𝑰𝟐 ]/𝟏𝟎𝟒
0.05
0.004
12.20±0.09
12.77±0.01
0.1
0.008
12.32±0.10
12.13±0.07
0.25
0.019
12.71±0.14
12.38±0.02
0.5
0.039
13.35±0.21
13.17±0.26
1
0.077
14.62±0.34
15.61±0.53
Table S2. Summary of predicted and measured values of 𝐸̂ [𝐼 2 ] as described in Section Ic. Low
concentrations of GNRs were measured, quantified, and compared to predicted values to validate the
relation between 𝐶1 and 𝐶2 and the OCT intensity signal. As can be seen, predicted values for 𝐸[𝐼 2 ] are
very similar to measured values.
Figure S13. a) Representative TEM image of the GNRs used for all experiments in this study. The
average particles dimensions were ~100 x 30 nm. (b) Visible-Near Infrared absorbance spectrum of the
GNRs.
Figure S14. Least squares fit of the OCT signal of blood only predicted from Eq. (S86). The plot in the
bottom half of the figure demonstrates that the fit is unbiased.
Figure S15. Monte Carlo simulation (blue line) for two cases of 𝜆𝑏 (small: left and large: right). As can
be seen, the square root asymptote is identical for both cases (green line). The linear regime asymptote
changes from 𝐼𝑠𝑚𝑎𝑥 to 𝐼𝑠𝑚𝑖𝑛 as 𝜆𝑏 grows.
Figure S16. Example of outlier filtering for GNRs in blood (100 pM). Remaining data represents 88.5%
of the original acquired image. The horizontal green line denotes the top of the tube as identified by the
peak convolution algorithm. The black line marks the top of the analyzed ROI.
Figure S17. As in Figure 3, but with the expected linear regime asymptotes as predicted by the model.
Note that the data does not fit to either asymptote but rather follows the square root regime exclusively.
This could be attributed to light interaction with blood as described in the supplementary text.
Figure S18. OCT signal intensity measurements for each of the 9 blood vessels analyzed during in vivo
experiments. Each vessel was fit individually for L and D. In the middle row (all corresponding to
vessels within a single mouse), the vessels were small (> 90 m). Thus the shadow limit was greater
than 3 nM and does not appear in the graphs. The values of D were similar to the diameters expected for
each analyzed vessel.
Supplementary References
S1. V. Turzhitsky et al. Appl. Spectrosc. 68, 133 (2014).