Optimization of light sources for prostate photodynamic therapy

Abstract

To deliver uniform photodynamic dose to the prostate gland, it is necessary to develop algorithms that optimize the location and strength (emitted power × illumination time) of each light source. Since tissue optical properties may change with time, rapid (almost real-time) optimization is desirable. We use the Cimmino algorithm because it is fast, linear, and always converges reliably. A phase I motexafin lutetium (MLu)-mediated photodynamic therapy (PDT) protocol is on-going at the University of Pennsylvania. The standard plan for the protocol uses equal source strength and equal spaced loading (1-cm). PDT for the prostate is performed with cylindrical diffusing fibers (CDF) of various lengths inserted to longitudinal coverage within the matrix of parallel catheters perpendicular to a base plate. We developed several search procedures to aid the user in choosing the positions, lengths, and intensities of the CDFs. The Cimmino algorithm is used in these procedures to optimize the strengths of the light catheters at each step of the iterative selection process. Maximum and minimum bounds on allowed doses to points in four volumes (prostate, urethra, rectum, and background) constrain the solutions for the strengths of the linear light sources. Uniform optical properties are assumed. To study how different opacities of the prostate would affect optimization, optical kernels of different light penetration were used. Another goal is to see whether the urethra and rectum can be spared, with minimal effect on PTV treatment delivery, by manipulating light illumination times of the sources. Importance weights are chosen beforehand for organ volumes, and normalized. Compared with the standard plan, our algorithm is shown to produce a plan that better spares the urethra and rectum and is very fast. Thus the combined selection of positions, lengths, and strengths of interstitial light sources improves outcome.

I. INTRODUCTION

Photodynamic therapy (PDT) is a treatment modality employing light of an appropriate wavelength in the presence of oxygen to activate a photosensitizing drug which then causes localized cell death or tissue necrosis. Using a surface illumination technique, PDT has been used to treat many superficial tumors including skin, lung, esophagus, and bladder (1). This technique is, however, inadequate when applied to large bulky tumors or solid organs due to limited light penetration into tissue. A more efficient illumination scheme would be interstitial light delivery whereby optical fibers are placed directly into the bulky tumors or organs.

The prostate gland is an organ that appears to be a good target for interstitial PDT. Tumors of the prostate are often confined to the prostate itself and brachytherapy techniques used for the placement of radioactive seed implants can be adapted for the placement of interstitial optical fibers (2). Several preclinical studies have evaluated the feasibility of delivering PDT to the prostate via this interstitial approach (3–7). The development of this light delivery technique has necessitated an improved understanding of light dosimetry, critical in planning the configuration of multiple fibers within the organ or tumor. Based on results of a preclinical study in canine (8), we have initiated a motexafin lutetium (MLu)-mediated PDT of the prostate in human at University of Pennsylvania.(9) MLu is a second generation synthetic photoactive drug that has a Q-band absorption peak at 732 nm. (10–11) Ideal optimization of the photodynamic linear light sources depends on knowledge of the spatial distributions of (1) tissue light opacity within the prostate, (2) photosensitizing drug, and (3) oxygen within tissue. Since these spatial distributions can vary in time, measurements must be done just prior to the clinical procedure. Moreover, the opacity distribution may be affected by bleeding associated with insertion of the light sources and ideally should be monitored at a significant number of points during the entire procedure. At present, measurements at more than a few sample points within the patient during the clinical procedure are difficult to make. The state of the art is to obtain measurements before the actual clinical procedure, and to assume that during the procedure all these distributions are uniform (of constant value) in the prostate region and static in time. A comprehensive study of the inhomogeneous light-opacity distribution in vivo is performed and reported elsewhere. (12–13) For the present study, we will concentrate on optimization of light fluence distribution.

A number of optimization algorithms used in brachytherapy are of interest for prostate photodynamic therapy. The most common ones are simulated annealing algorithms (14–16) and genetic algorithms (17–19). Gradient algorithms also have been applied (20). In general, gradient algorithms give reproducible solutions but may be trapped in local minima far from the global minimum (21). Simulated annealing and genetic algorithms avoid getting trapped in local minima, but are relatively slow because they are stochastic algorithms.

To the best of our knowledge, optimization algorithms for photodynamic therapy are not yet in the literature. In this study, we describe a systematic search procedure based on the Cimmino feasibility algorithm (22,23) to obtain the locations and strengths of light sources for photodynamic treatment. The Cimmino algorithm is an iterative linear algorithm which was first applied to radiotherapy inverse problems by Censor et al. (24–26). The algorithm is safer than most common optimization algorithms outlined above since it always converges and, if no solution exists for the inequalities (i.e. the prescribed dose constraints are not all satisfied), the Cimmino algorithm reverts to a least-square solution (26).

II. METHODS AND MATERIALS

1. Diffusion theory and Determination of optical properties

The transport scattering (µ’_s) and absorption (µ_a) coefficients characterize the scattering and absorption properties of tissue. With the diffusion approximation, the light fluence rate ϕ at a distance r from a point source of power, S, can be expressed as (27)

$$\varphi =\frac{S·{\mu }_{\mathit{\text{eff}}}^2}{4\pi r·{\mu }_a}·e^{-{\mu }_{\mathit{\text{eff}}}·r}=\frac{S·3{\mu }_s^{\text{'}}}{4\pi r·}·e^{-{\mu }_{\mathit{\text{eff}}}·r}$$ (1)

where S is the power of the point source (in mW); ϕ(r) is the the fluence rate in mW/cm²; ${\mu }_{\mathit{\text{eff}}}=\sqrt{3·{\mu }_a·{\mu }_s^{\text{'}}}$ (27) is the effective attenuation coefficient in tissues and is applicable for a wider range of µ_a and µ_s’ combinations than the traditional definition of ${\mu }_{\mathit{\text{eff}}}=\sqrt{3·{\mu }_a·({\mu }_s^{\text{'}}+{\mu }_a)}$ (28).

For a cylindrical diffusing fiber (CDF) with length l, the light fluence rate can be calculated by a superposition of Eq. (1):

$$\varphi =\sum _{i=1}\frac{S·\mathrm{\Delta }x·{\mu }_{\mathit{\text{eff}}}^2·e^{-{\mu }_{\mathit{\text{eff}}}·r_i}}{4\pi {\mu }_a{r}_i}=\frac{Sl{\mu }_s^{\text{'}}}{4\pi }·(\frac{1}{N-1}\sum _{i=1}^N\frac{e^{-{\mu }_{\mathit{\text{eff}}}·r_i}}{r_i})$$ (2)

where s is the energy released by light per unit time per unit length, also called unit length source strength (mW/cm). $r_i=\sqrt{x_i^2+h^2}$, where x_i = (i − 1 − (N − 1)/2) · Δx and h is the perpendicular distance from the center of the linear fiber. Δx is the length step of point sources and N (odd integer) is the number of equal spaced point sources used in the summation (parenthesis in Eq. (2)). The numerical value of the summation should be independent of N (or Δx) if N is large enough. We found that accurate results of the summation can be obtained if Δx ≤0.1 cm or N ≥ 25 for l = 2.5 cm. In all our calculations N = 201 was used. The two free parameters (µ_a and µ_s’) are inherently separable in that for a CDF with a given length: the magnitude of the fluence rate near the light source (h = 0) is determined by µ_s’ only and the slope of the spatial decay of the light fluence rate is determined by µ_eff only.

In theory, measurements of ϕ at two different distances r from the point source with known unit length source strength s and length l are sufficient to determine both µ_a and µ_s’. Measurements at multiple sites allow evaluating the variation of these optical characteristics within the prostate volume. Since Eq. (1) is a non-linear equation of two free parameters µ_a and µ_s’, we used a differential evolution algorithm developed by Storn et al (29). This algorithm is simple and robust, which converges faster and with more certainty than adaptive simulated annealing as well as the annealed Nelder & Mead approach (29). We modified the algorithm to require that all parameters (µ_a and µ_s’) are positive (30).

Optical properties measured in 13 patients have been summarized in Table 1 (12). The heterogeneity of optical properties in human prostates is somewhat smaller than that observed in canine prostates at 732 nm (7). Overall µ_a varied between 0.07 – 1.62 cm⁻¹ (mean 0.3±0.2 cm⁻¹) and µ_s’ varied between 1.1 – 44 cm⁻¹ (mean 14±11 cm⁻¹). The effective attenuation coefficient µ_eff varied between 0.91 – 6.7 cm⁻¹, corresponding to an optical penetration depth (δ = 1/µ_eff) of 0.2 – 1.1 cm. The mean values of µ_eff and δ were 2.9±0.8 cm⁻¹ and 0.4±0.1 cm, respectively. This penetration depth is substantially larger than that of 0.1 – 0.25 cm predicted for 630 nm (6) but is smaller than 0.5 – 3 cm observed in normal canine prostate at 732 nm (7). The most probably explanation is that canine prostate has different grandular/structure content than that of human prostate. While the mean reduced scattering coefficient in canine was 3.6 ± 4.8 cm⁻¹ (7), it was 15 ± 11 cm⁻¹ in human at the same wavelength (732nm). The increased reduced scattering coefficient resulted in increased effective attenuation coefficient, or a reduction of optical penetration depth, assuming the absorption coefficient remains the same.

Table 1.

In-vivo Optical properties measured at 732 nm in human prostate.

Patient number	µa (cm−1)	Before PDT µs’ (cm−1)	δ(cm)	µa (cm−1)	After PDT µs’ (cm−1)	δ(cm)
2	0.09	29.8	0.34	0.09	43.7	0.29
3	0.15	22.0	0.31	0.07	33.4	0.37
4	0.43 (0.28)	7.69 (4.76)	0.41 (0.14)	0.51	1.67	0.63
5	0.21	11.8	0.37	0.13	7.18	0.60
6	0.27 (0.27)	10.5 (11.2)	0.50 (0.05)	0.19 (0.20)	18.9 (18.4)	0.45 (0.06)
7	—	—	—	0.30 (0.08)	23.7 (13.9)	0.24 (0.11)
9	0.53 (0.36)	6.61 (4.51)	0.41 (0.09)	0.64 (0.25)	7.00 (5.59)	0.33 (0.10)
10	0.63 (0.32)	4.62 (2.87)	0.42 (0.10)	0.19 (0.05)	9.27 (4.47)	0.54 (0.31)
11	0.67 (0.17)	6.39 (3.18)	0.32 (0.10)	0.83 (0.45)	5.45 (3.89)	0.38 (0.16)
12	0.71 (0.43)	8.99 (6.51)	0.32 (0.12)	0.30 (0.06)	20.2 (4.8)	0.28 (0.08)
13	0.27 (0.14)	18.5 (11.6)	0.30 (0.07)	0.26 (0.09)	17.0 (8.8)	0.31 (0.07)
14	0.72 (0.11)	3.37 (1.37)	0.39 (0.11)	—	—	—

The values in the parentheses are the standard deviation (s.d.) of the mean values measured from different locations of the same prostate (’x’ in Fig. 1b). No s.d. is listed if only one data point is available.

2. Patient Selection, Surgical and PDT Procedure

A Phase I clinical trial of motexafin lutetium (MLu)-mediated PDT in patients with locally recurrent prostate carcinoma was initiated at the University of Pennsylvania. The protocol was approved by the Institutional Review board of the University of Pennsylvania, the Clinical Trials and Scientific Monitoring Committee (CTSRMC) of the University of Pennsylvania Cancer Center, and the Cancer Therapy Evaluation Program (CTEP) of the National Cancer Institute. A total of 15 patients were treated. Each patient who signed the informed consent document underwent an evaluation, which included an MRI of the prostate, bone scan, laboratory studies including PSA (prostatic specific antigen), and a urological evaluation. Approximately two weeks prior to the scheduled treatment a transrectal ultrasound (TRUS) was performed for treatment planning. An urologist drew the target volume (the prostate) on each slice of the ultrasound images. These images were spaced 0.5 cm apart and were scanned with the same ultrasound unit used for treatment.

A built-in template with a 0.5-cm grid projected the locations of possible light sources relative to the prostate. A treatment plan was then prepared to determine the location and length of light sources. Cylindrical diffusing fibers (CDF) with active lengths 1–5 cm were used as light sources. The sources were spaced one centimeter apart and the light power per unit length was less than or equal to 150 mW/cm² for all optical fibers. The length of the CDF at a particular position within the prostate was selected to cover the full length of the prostate (see Fig. 1a). The final plan often required that the prostate be divided into four quadrants. Four isotropic detectors were used, each placed in the center of one quadrant. A fifth isotropic detector was placed in a urethral catheter to monitor the light fluence in the urethra. (Fig. 1b).

Figure 1.

(a) Experimental setup for measuring the in-vivo optical properties of canine prostate. The prostate template was drilled with a 5-mm equal spaced grid. Cylindrical diffusing fibers (CDF) were inserted into the catheters to illuminate the entire prostate gland. An isotropic detector (not shown) is placed in one of the catheters, which is moved to locations at a distance of 3, 6, 9, 12, 15 mm from the light source. The detector reading at each location is peaked to ensure that it is at the middle of the linear fiber.

The patients were anesthetized in the operating room with general anesthesia to minimize patient movement during the procedure. Transrectal ultrasound unit was used to guide needle placement in the operating room. A template was attached to the ultrasound unit and was matched to the same 5-mm grid used for treatment planning. Four detector catheters (one for each quadrant) were inserted into the prostate. These detectors were kept in place during the entire procedure of PDT treatment. Four additional pre-planned treatment catheters for light sources were then inserted 0.5 or 0.7 cm away from the detector catheters (Fig. 1b). These source catheters were used for light delivery and optical properties measurements. A 15-W diode laser, model 730 (Diomed, Ltd., Cambridge, United Kingdom) was used as the 732 nm light source.

3. Search procedure with the Cimmino optimization algorithm

At present a medical physicist chooses the number of light sources, the particular template holes (or “slots”) for source insertion, the length of the linear light source to insert in each slot, the position of the source within the slot, and a single duration of illumination for the entire set of sources. This is a tedious and time-consuming job that requires contours from ultrasonic tomographic images, visualization of 3D volumes (prostate, urethra, and rectum) and their intersections with linear light sources, estimation of the mean opacity of the prostate, and visualization of the scattered light distribution within the prostate, for different choices of source parameters.

The algorithms discussed below try to achieve (1) a prescribed minimum dose within the prostate, and (2) doses not exceeding the maximum doses specified separately for the prostate, urethra, rectum, and background tissues. The contours of the prostate, urethra, and rectum in each transverse slice (parallel to the template plane and perpendicular to the linear light sources) are assumed available in computer memory. At present, up to 13 transverse slices spaced 5 mm apart are allowed.

The template currently being used for source insertion is a plate with a square array of 13 × 13 = 169 holes (slots for linear light sources) spaced 5 mm apart (Fig. 1b). For the particular patient data being used as a benchmark, only 51 slots are situated to allow the light source either to penetrate the prostate or approach within a 1-mm margin. In present clinical practice, sources are separated by 10 mm, that is, every other slot. Thus for the benchmark patient, 12 template slots are used for sources.

The light source is a tube of illumination, 1 mm in diameter and at most 50 mm in length from template base to maximum penetration. Creating the illumination within the tube are “light seeds” of 5 mm length (thus 5 mm between the centers of adjacent light seeds). The algorithms of this paper enforce two clinical requirements: (1) the seeds within a light source are contiguous (i.e., no gaps occur between light seeds), and (2) each light source has at least two seeds. Although violation of these restrictions might yield mathematically improved light distributions, the clinical use of short discontinuous light sources requires greater precision and increased time for in vivo placement, thus an increased risk to the patient.

There are four mathematical problems. (1) Given linear light sources with specified slots and specified parameters (source length, and retraction of the first seed into the slot), find the source strengths (or product of power and durations of illumination in units of energy) to satisfy the prescribed dose constraints. (2) Given the source slots, find the source parameters and source strengths. (3) Given only the number of sources and the allowed set of slots, find the source slots, source parameters, and source strengths. (4) Guarantee the convergence of each mathematical procedure to an acceptable solution; if no solution exists (that is, not every constraint can be satisfied), then guarantee convergence to a least-squares (best compromise) solution. These four problems must be solved for two cases: individual source strengths (sources may have different strengths), and uniform source strength (all sources have the same strength). The case of uniform source strength with all source slots and parameters specified (problem 1) is the present practice in the clinic and is our baseline for improvement.

Our search algorithm systematically changes the position and retraction of each light source within the prostate, calculates the best source strengths with the Cimmino algorithm (after each change), and then checks whether the discrepancy from the dose prescription has decreased. The discrepancy function we use is the sum of the absolute values of the overdose or underdose at each sample point with respect to the prescribed constraint. An importance weight is pre-chosen for each named volume (prostate, urethra, rectum, and background) and normalized so that the sum of the importance weights is unity. The importance weight of a sample point is the importance weight of its volume divided by the number of sample points in that volume. Thus the discrepancy at each sample point is given a normalized importance weight. If there is no solution with zero discrepancy, the normalized weighting will put most of the error where it is least important.

A feasibility algorithm rigorously solves the first problem. The second and third problems require search procedures over allowed slots, source lengths, and retractions (as appropriate). For each choice of source slots and source parameters that is assayed in a search, the discrepancy between the solution of the feasibility algorithm and the prescribed dose at a point, summed over the constraint points, is calculated. If this total discrepancy is less than any previously discovered, then that parameter set becomes the current best solution. The second and third problems are “combinatoric”, where exhaustive searches are not possible in reasonable time. Non-exhaustive searches risk encountering local minima, so that finding the solution for the “absolute-minimum” discrepancy cannot be guaranteed or even recognized. Thus the key to the second and third problems is a good search strategy.

Solution of the first problem allows comparison of individual strengths vs. a uniform strength for user-chosen sources. Solution of the second problem allows a similar comparison but with source lengths, retractions, and strengths chosen by prostate geometry and minimum dose discrepancy. A solution of the third problem allows the treatment planner to find automatically the best (or almost-best) light sources and source strengths from just the specified dose constraints to the prostate and organs, and the choice for the number of sources. Such a solution would be virtually impossible to find by human visualization and estimation. Automatic source selection and weighting becomes even more important if it is necessary to limit dose to the urethra and rectum without compromising treatment effectiveness to the prostate.

The selection of the number of light sources is as much a clinical as a mathematical problem. To find mathematically the best number of linear light sources to insert, one can rerun repeatedly any procedure used to solve problem 3 with different numbers of sources. However changing the number of sources differs from rearranging source positions and solving for source strengths; it involves a tradeoff between (a) fewer sources — less homogeneous dose coverage, higher source strengths, but fewer surgical complications, and (b) more sources— better dose coverage, lower source strengths, but more surgical complications). Thus choosing the number of sources requires clinician input based on medical experience and judgment.

To focus this paper on algorithmic procedures, we omit any discussion on choosing the number of light sources. The number of sources is always assumed to be given. Several elements are common to the last three algorithms. These are (1) a kernel (Eq. 2) to calculate dose; (2) a set of constraint points in the volume of interest to guide the optimization procedure; these points derive from both a sampling grid and extra points chosen on the periphery of the prostate (target); (3) the Cimmino feasibility algorithm.

Each search procedure allows the number of sources to be decreased by one. A new optimization calculation is then performed and the resulting dose distribution compared with the previous. The decrease in the number of sources can be continued iteratively. If the number of sources can be decreased without increasing discrepancy between the prescribed and optimized dose distribution, fewer sources need be used, thereby reducing the complication of the procedure and discomfort of the patient.

Each algorithm also allows the template to be shifted horizontally or vertically (anterior to supine patient). Up to seven shifts are allowed in each direction (e.g. from −3 mm to +3 mm in steps of 1 mm for the 5-mm grid), thus 49 separate optimizations. These can be used to check the sensitivity of the dose distribution to small shifts in template position.

To check the effect of Cimmino optimizations, two different optical properties were chosen: (a) the average optical properties of all prostate patients, µ_a = 0.3 cm⁻¹ and µ_s’ = 14 cm⁻¹; and (b) an extreme case when the optical penetration depth is the longest observed in human prostate (δ = 1.3 cm), µ_a = 0.1 cm⁻¹ and µ_s’ = 2 cm⁻¹.

III. RESULTS AND DISCUSSIONS

We used our standard plan, based on geometrical coverage, 1-cm spaced loading, and uniform source strength as our default plan to judge the improvement made by the Cimmino algorithms.

Figure 2 compares computer runs of optimized 100% isodose distributions to a single 3D prostate volume using the Cimmino algorithm (solid lines) compared with the standard plan (dashed line). The source locations for the Cimmino run were chosen manually and kept the same as the uniform loading plan (dashed line). There is no substantial difference between the equal-source-strength plan and the Cimmino individual-source-strength plan for either the mean or most penetrating optical properties found in the human prostate. This is not surprising since when the light penetration depth is short enough, the light coverage is determined more by the light source location than the source strength, provided we have renormalized the plan to guarantee that the entire prostate is covered.

Figure 2.

Comparison of 100% isodose lines for uniform-strength (solid line) vs. Cimmino optimized loading (dashed line) — for fixed template and CDF source lengths— for two optical properties: left (a) µ_a = 0.3 cm⁻¹, µ_s’ = 14 cm⁻¹ and right (b) µ_a = 0.1, µ_s’ = 2 cm⁻¹. Contours for prostate, urethra, and rectum are also shown in the figure.

To further explore the importance of the source location, we let the Cimmino algorithm determine the source locations, lengths, as well as the source strengths. We kept the total number of sources to be the same as the standard plan, 12. The results are shown in Figure 3. The source positions (o) are chosen by the computer with the corresponding isodose line plotted as solid line. The source positions for the standard plan are the same as those shown in Fig. 2. For both optical properties, the Cimmino algorithm provides a better plan, sparing the urethra, rectum, and background while covering the entire prostate. The light source positions are now chosen more peripherally compared to the equally spaced coverage, i.e., fewer sources are placed near the urethra compared with the uniform loading. The improvement is more significant for shorter light penetration depth (Fig. 3a) than that for longer light penetration depth (Fig. 3b).

Figure 3.

Comparison of 100% isodose lines of Cimmino optimized results (optimized source lengths, loading, and template) for 12 linear sources (solid lines) and Cimmino optimized results (optimized loading only, user selected source lengths and template) (dashed lines) for two optical properties: (a) µ_a = 0.3 cm⁻¹, µ_s’ = 14 cm⁻¹ and (b) µ_a = 0.1, µ_s’ = 2 cm⁻¹. The source location for the dashed lines is the same as Fig. 2.

We then examined whether maximum dose constraint for the critical structure makes any difference. We compared two plans made with the Cimmino algorithm (computer choosing positions, lengths, and source strengths) but one had the maximum dose to the rectum reduced from 300% (dashed line) to 200% (solid line). Figure 4a shows that the rectum sparing is improved for the average optical properties in prostate, which is less than 5 mm. No significant improvement is observed if the optical penetration depth is more than 1 cm (Fig. 4b).

Figure 4.

Comparison of 100% isodose lines of Cimmino optimized results for 12 linear sources with 0.5 cm step size with the maximum dose constraint of 200% (solid lines) and 300% (dashed lines) for rectum for two optical properties: (a) µ_a = 0.3 cm⁻¹, µ_s’ = 14 cm⁻¹ and (b) µ_a = 0.1, µ_s’ = 2 cm⁻¹. In this example, the source locations, lengths, retractions, and strengths are chosen by the Cimmino algorithm (see Table 2).

Finally we examined the best plan we have obtained so far using 12 CDF sources (Fig. 4, solid line) and compared it with the Cimmino algorithm using 51 CDF sources. The latter represents the best mathematical plan for linear sources and the given template, given no discontinuity along any catheter for active light sources. The result shows that the rectal sparing of the 12-source plan is not much worse than that of the plan using 51 sources for low opacities (Fig. 5b). For the average optical parameters (Fig. 5a), some improvement of rectum sparing can be achieved by using more CDF sources.

Figure 5.

Comparison of 100% isodose lines of Cimmino optimized results (solid lines) for 51 linear sources with 0.5 cm spacing and the best Cimmino plan using 12 sources (dashed lines, loading pattern in Fig. 4) for two optical properties: (a) µ_a = 0.3 cm⁻¹, µ_s’ = 14 cm⁻¹ and (b) µ_a = 0.1, µ_s’ = 2 cm⁻¹.

Figure 6 shows the effect of optical properties on the prostate light dose coverage when incorrect optical properties are used. In this figure, same source strengths and source parameters are used for two different optical properties. The 100% isodose line provides adequate prostate coverage for one optical properties (solid line) but is very bad if the optical properties is in fact shorter than what is assumed (dashed line). This indicates that the values of the optical parameters are very important in determining the actual light fluence distribution.

Figure 6.

A comparison of prostate coverage for µ_a = 0.1, µ_s’ = 2 cm⁻¹ (solid line) and µ_a = 0.3 cm⁻¹, µ_s’ = 14 cm⁻¹ (dashed line) using the source strengths listed in Table 2 for Cimmino 3, Opt 2.

The present paper assumed uniform (homogeneous) optical properties. An open question is whether optimized solutions for inhomogeneous media will further improve over the present uniform-medium calculations. It is also an open question on the number of control/constraint/sample points required to guarantee the optimization outcome. Further studies are needed to determine the minimum number of CDF needed to achieve complete coverage.

IV. CONCLUSION

The question addressed is whether any significant advantage may derive from methods that weight each CDF source separately and/or choose the geometry of the light sources as well. In summary, our comparison shows that: (1) It is important to measure the optical properties of a patient because it determines the light fluence distribution. This effect is more predominant than optimizing the source positions, lengths, and strengths. (2) For the range of the optical properties in the human prostate, the individual-source strength optimization does not significantly improve over the equal-source-strength optimization when given the positions of the light sources. (3) Improved results with Cimmino optimization compared to user-specified optimization indicate that computer optimization saves the user time and setup and reduces human stress. The Cimmino optimization is fast enough for this problem to obtain clinical real-time optimization (less than 300 s, Table 2).

Table 2.

Results of source strengths (J) obtained using various Cimmino search algorithms for 12 sources.

Source No.	Standard plan		Cimmino 1		Cimmino 2		Cimmino 3
	Opt 1	Opt 2	Opt 1	Opt 2	Opt 1	Opt 2	Opt 1	Opt 2
1	410.83	69.56	286.22	43.36	267.66	57.48	296.82	57.48
2	410.83	69.56	232.24	63.72	271.56	51.48	292.12	51.48
3	410.83	69.56	264.92	67.04	222.82	49.06	312.78	49.06
4	410.83	69.56	242.40	40.74	213.14	78.18	240.62	78.18
5	410.83	69.56	350.54	64.38	349.38	50.20	344.94	50.20
6	410.83	69.56	293.08	83.90	358.48	55.54	244.36	55.54
7	410.83	69.56	595.94	75.60	219.50	56.84	239.14	56.84
8	410.83	69.56	429.26	80.60	293.76	81.30	159.44	81.30
9	410.83	69.56	306.20	71.52	219.16	44.42	171.24	44.42
10	410.83	69.56	289.82	47.46	351.70	56.56	229.26	56.56
11	410.83	69.56	290.28	51.14	258.80	56.44	241.88	56.44
12	410.83	69.56	391.96	77.00	209.24	69.92	183.74	69.92
Calculation time (s)	1	1	9	9	260	173	309	173

Standard plan: user selected source parameters and uniform strengths; Cimmino 1: optimize source strengths only; Cimmino 2: optimize source lengths, strengths, and template slots; and Cimmino 3: the same as Cimmino 2 with constraint for rectum. Opt 1: µ_a = 0.3 cm⁻¹, µ_s’ = 14 cm⁻¹; Opt 2: µ_a = 0.1 cm⁻¹, µ_s’ = 2 cm⁻¹. The Calculation time is obtained from a Dell PC with a 2.8 GHz Pentinum IV processor.