In DCS, coherent near-infrared light is introduced into a highly scattering medium such as tissue wherein it travels deeply and scatters multiple times before detection at some distance from the light source. During each scattering event, the phase of the scattered light is changed. At the detector, the superposition of multiple light fields with different phases creates a speckle pattern. If the scattering particles move, the speckle pattern fluctuates in time. The intensity fluctuations of the speckles thus contain information about the motion of the scatterers [

16,

17]. In the case of tissue, the primary moving scatterers are red blood cells. Therefore, by characterizing the fluctuations in speckle intensity over time, we gather information about blood flow in tissue.

In order to monitor speckle fluctuations in time, we measure the normalized intensity autocorrelation function,

*g*
_{2}(

**r**, τ) =

*I*(

**r**,

*t*)

*I*(

**r**,

*t* + τ)

/

*I*(

**r**,

*t*)

^{2}, and calculate the normalized electric field temporal autocorrelation function,

*g*
_{1}(

**r**, τ) =

*G*
_{1}(

**r**, τ)/

*E*
^{*}(

**r**,

*t*)

*E*(

**r**,

*t*)

, using the Siegert relation,

*g*
_{2}(

**r**, τ) = 1 + β |

*g*
_{1}(

**r**, τ)|

^{2}. Here

*I*(

**r**,

*t*) is the intensity at time

*t* and position

**r**,

*E*(

**r**,

*t*) is the electric field at time

*t* and position

**r**,

denotes the ensemble or time average, and β is a constant that depends on the source coherence, detection optics, ambient light and other external factors.

*G*
_{1}(

**r**, τ) is the unnormalized electric field autocorrelation function, equal to

*E*
^{*}(

**r**,

*t*)

*E*(

**r**,

*t* + τ)

, which obeys a correlation diffusion equation [

18,

19]. In this study we assume a homogeneous tissue-air interface geometry in the plane

*z* = 0. The semi-infinite solution of

*G*
_{1}(

**r**, τ) to the correlation diffusion equation for a point source of the form

*S*(

**r**) =

*S*
_{0}δ(

**r**) [

19,

29] is

Here

; μ

_{a} (cm

^{−1}) and

are the tissue absorption and reduced scattering coefficients, respectively;

*k*
_{0} (cm

^{−1}) is the magnitude of the optical wave vector, 2π

*n*/λ, where

*n* is the index of refraction of tissue and λ is the wavelength of incident light; α represents the percentage of light scattering events that come from moving scatterers;

is the mean squared displacement of the moving scatterers in time τ, assumed to have the form 6

*D*_{B}τ where

*D*_{B} is an

*effective* Brownian diffusion coefficient;

*r*
_{1} and

*r*
_{2} (cm) are the distances between the detector and the source/image source, respectively, i.e.

and

;

is the depth at which a collimated source on the tissue surface can be approximated as a point source;

in the case of refraction indices of tissue and air (equal to 1.4 and 1.0, respectively). The decay rate of this autocorrelation function is dictated by blood flow. We define a blood flow index,

*BFI* α

*D*_{B}, with units of cm

^{2}/s, to quantify this decay rate. BFI reflects CBF, and changes in BFI relative to baseline measurements (rBFI) reflect analogous changes in CBF (rCBF) [

30]. Although this analysis approach is empirical, numerous studies have validated it as a measure of relative blood flow, including comparisons to literature [

21–

23], to Doppler ultrasound in murine tumors [

26], to laser Doppler flowmetry in rat brain [

21–

23,

28], to fluorescent microspheres in piglet brains [

35], and to arterial spin-labeled perfusion MRI in human brain and calf muscle [

24,

32].

The DCS instrument uses a long-coherence-length laser (CrystaLaser, RCL-080-785S) operating at 785 nm to deliver light to the tissue. A single mode fiber secured by a black foam pad detects light 1.5 cm from the source. The fiber is custom designed with a 90° bend on the patient end, permitting the probe to rest adjacent to the forehead. Light is detected by a fast photon counting avalanche photodiode that outputs a TTL signal for every photon received. This TTL signal is transmitted to a custom built 2-channel correlator board (FLEX03OEM2CH, correlator.com, Bridgewater, NJ) derives the intensity autocorrelation function based on the photon arrival times [

45]. shows a sketch of the probe on the infant’s head.

Protocols for both DCS and TCD measurements are shown in . Head of bed (HOB) angle manipulations were limited by the range of the isolette beds (Ohio Care, Ohmeda Isolette). These beds easily adjust from an angle of 0° to approximately 12° in seconds.

For DCS, data were acquired for 30 minutes: three 5 minute HOB=0° sessions, alternating with three 5 minute HOB=12° sessions. Intensity autocorrelation curves were averaged over a period of 3 seconds and were acquired every 7 seconds, resulting in around 300 BFI data points per study. These curves were then converted to electric field autocorrelation functions using the Siegert relation with a fitted value for β of approximately 0.5. Theoretically, we expect β to be approximately equal to the inverse of the number of modes allowed to pass through the detection optics. In our case of single mode detection fibers, the number of modes is two. For analysis, we solved the correlation diffusion equation analytically, assuming the sample geometry was a homogeneous, semi-infinite medium. We used the semi-infinite solution (

Equation 1) to fit our data for BFI. This fit is made possible by assuming constant values for μ

_{a} and

(0.1 cm

^{−1} and 10 cm

^{−1}, respectively, at 785 nm) for the whole population and throughout the study. These values were chosen from literature references [

47,

48].

A mean relative change in CBF was calculated after each HOB= 12° event using the preceding HOB= 0° event for a baseline, i.e. for the

*i*^{th} repetition,

, where

indicates the mean BFI over all data points taken at the

*i*^{th} HOB= θ° event. After

was calculated for each HOB change (3 total per day of study), a mean relative change in cerebral blood flow,

*rCBF*, was determined for each day. Here

denotes the mean value over all

measured on a given day. Additionally, to compare absolute measurements of BFI to the velocities found from TCD, a mean blood flow index,

, was calculated for each day of study. For the purposes of this analysis, we used the initial HOB= 0° data to calculate

, i.e.

, since the baby was most peaceful during this time period.

The protocol for TCD measurements differed slightly from the DCS protocol. Only two or three measurements of peak systolic velocity (

*PSV*), end diastolic velocity (

*EDV*), and mean velocity (

*MV* = (

*PSV* −

*EDV*)/

*PI*, where

*PI* is the pulsatility index measured by the ultrasound scanner), were obtained at each head of bed angle. A Philips ATL HDI 5000 ultrasound scanner (Philips Medical Systems, Bothell, WA) with a C8-5 MHz broadband curved array transducer was used for data acquisition. Typically this process was repeated twice. As in the case of the mean

*rCBF* calculation, mean relative changes of each parameter for the

*i*^{th} repetition

comparing the responses at elevated HOB to lying flat were computed after each HOB change, i.e.

. Daily mean changes in velocities,

*rPSV*,

*rEDV*,

*rMV*, were calculated in the same manner as

*rCBF*. For comparison to DCS, mean velocities at the initial HOB= 0° event (

,

, and

) were calculated for each day of study. Unfortunately TCD and DCS data could

*not* be collected at the same time due to the size of the infant’s head. However, the data were acquired on the same day by both modalities.

For the purpose of this analysis, we considered each of the nine days of data acquisition to be independent observations. To test for an association between

and each of

,

, and

, as well as between

*rCBF* and each of

*rPSV*,

*rEDV*, and

*rMV*, we used Spearman’s rank-based non-parametric approach [

49]. Rejection of the null hypothesis in this case implies a positive or negative, and possibly non-linear, association between the variables. Pearson’s correlation coefficient was used to estimate a linear association between

, and each of

,

, and

. To test the hypothesis that each of the four relative variables differed from baseline during HOB elevation, we conducted a Wilcoxon signed rank test [

50]. Analyses were carried out using R 2.8 [

51]; hypotheses tests and associated p-values (

*p*) were two-sided. A family-wise error rate of 0.05 was maintained using Hochberg’s method [

52] to adjust for multiple comparisons within each of the three sets of analyses (associations with

, associations with

*rCBF*, and comparison with baseline); adjusted p-values,

*p*_{A}, are also reported.