Rshift Method: Segment-Based

Contrary to the R ₅₀ method, if we are not concerned about a certain point but a segment of the curve, then the depth error evaluation reflects the feature of the entire segment rather than a single point. One method to calculate the depth error for a segment of a curve is the R_shift method (3, 10, 20).

The kernel of the shift method is to find a shift distance δ that minimizes the difference between predicted and measured PET curves [Eq. (1)], this difference is presented as a cost function f(δ) [Eq. (2)].

In Eq. (2), A(z) is the PET activity depth distribution along the beam direction (for prediction and measurement, respectively) and δ is specified as a depth shift between two curves. The reason of the entire curve deviation between predicted and measured PET activity data is not fully understood in this work; by specifying the integral region [Z_min and Z_max in Eq. (2)], we focus on the curve difference in the distal edge area rather than the entire curve. We specify the Z_min as the peak position of the predicted PET curve and the Z_max as 20% of the maximum location (Figure 2).

In Eq. (2), the cost function f(δ) describes the difference between continuous curves in a region, but the data stored in the computer are always discrete data lists. Therefore, cost function f(δ) has both continuous and discrete descriptions. Here, in the discrete description of Eq. (2), i ∈ [0, M] is a list subscript index corresponding to the region [Z_min, Z_max] presented as a discretization data list Z_i, and the value of δ should be chosen discretely with the same value interval of Z_i, which is 3 mm, the same as our data grid voxel size and the curve’s spatial resolution. The value of δ is set in [–15,15]mm with a range step of 3 mm and f(δ) is plotted in Figure 3.

f(δ) and f'(δ) for seeking δ₀. The position of δ₀ is located where f'(δ ₀) = 0.

Obviously, there is a minimum value point on the f(δ) curve. If our δ sampling density was high enough, we can easily locate the minimum point of the f(δ) curve and the corresponding δ would be the ΔR for curves. In our case, however, the value of δ sampling step was 3 mm, which is not small enough. Here, we can use derivative interpolation to accurately find the minimum point of the f(δ) curve. In f(δ) discrete list, there is a minimum f(δ_m) with corresponding δ_m, which is in the discrete δ list (Figure 3); the point is recorded as point m[δ_m, f(δ_m)]. On the left and right side of point m, there are point L[δ_m-1, f(δ_m-1)] and point R[δ_m+1, f(δ_m+1)]. Obviously, the minimum point [δ ₀,f(δ ₀)] of f(δ) is between L and R. Then, we calculate the left and right derivative of point m:

We know that f(δ_m) is smaller than f(δ_m-1) and f(δ_m+1), so fL'<0andfR'>0; therefore, there must be a f'(δ ₀) = 0 and the corresponding δ ₀ is exactly the ΔR_shift we want. We can easily estimate the δ ₀ by linearly interpolating fL'andfR':

Here, Δδ = 3mm is our data grid voxel size, and δ_m can be searched from the discrete f(δ) list. The δ ₀ in Eq. (4) is the point where f(δ) is minimized; thus, the integration in Eq. (1) is minimized, so the δ ₀ here is the ΔR_shift we want.