# Program Information

## Analytic Solutions of the Two-Dimensional IMRT Inverse Planning Problem Under Flat-Depth-Dose Approximation and Implications for Beam-Number and Beam-Angle Optimization

### F Li*, G Yan , J Jung , B Lu , J Park , S Lebron , S Samant , J Li , C Liu , University of Florida, Gainesville, FL

## Presentations

**SU-K-FS2-7 (Sunday, July 30, 2017) 4:00 PM - 6:00 PM Room: Four Seasons 2**

**Purpose:**To analytically solve a general 2D IMRT inverse planning problem and shed light on beam-number and beam-angle optimization in IMRT/VMAT planning.

**Methods:**We applied Zernike expansion on IMRT dose distribution, and defined plan complexity N as the highest order of the Zernike polynomials. We also expanded the dose profiles from individual IMRT beams in a series of Chebyshev polynomials of the second kind, and defined beam modulability M as the highest order of the Chebyshev polynomials. M represents the most complicated dose profile that an individual beam could deliver. The Flat-Depth-Dose Approximation (no beam divergence and flat depth dose profile) was assumed to render the problem mathematically tractable. The inverse planning problem reduced to solving the coefficients of the Chebyshev polynomials.

**Results:**We were able to analytically solve the problem for arbitrary beam-number and beam-angle arrangement. Specifically, the inverse problem has a solution only if M>=N (i.e., beam modulability exceeds plan complexity). In this case, there are actually infinitely many equally good solutions; N+1 beams are sufficient to achieve the desired dose distribution, and there is no benefit of using more beams. The complete set of solutions can be written analytically using the generalized 1-inverse of a rectangular Vandermonde matrix. If N>M (i.e., plan complexity exceeds beam modulability), there is no exact solution to the inverse problem. In this case, the least square solution can be obtained using the left inverse of a Vandermonde matrix.

**Conclusion:**Our results suggest that the highest attainable plan complexity cannot exceed the beam modulability, no matter how many beams are used. We mathematically proved several remarkable relationships among plan complexity, beam modulability, beam number and angle arrangement. These results bring insights to the inverse planning problem of IMRT/VMAT. The novel use of Zernike expansion in inverse optimization scheme may indicate new research direction.

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