Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

The Minimization of the Quadratic Mean of an Integral Dose

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this work, we define the optimal dose as a combination of the projections on orthogonal axes of the absorbed dose and an integer multiple of the integral dose. Here, we show that such optimal dose minimizes the mean square of the total absorbed dose subject to certain conditions of integration. We prove that there is a unique minimizer.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Balog, J., et al.: Dose computation for a one-dimensional collimator. In: Proceedings of the XII International Conference on the Use of Computers in Radiation Therapy, pp. 323–326. Medical Physics Publishing, St. Louis (1997)

    Google Scholar 

  2. Carol, M., et al.: Initial clinical experience with the peacock intensity modulation of a 3-D conformal radiation therapy system. Stereotact. Funct. Neurosurg. 66, 30–34 (1996)

    Article  Google Scholar 

  3. Deasy, J.O.: Multiple local minimums in radiotherapy optimization problems with dose volume constraints. Med. Phys. 1157–1161 (1997). doi:10.1118/1.598017

  4. Fang, G., et al.: Software system for therapy prototype. In: Proceedings of the XII International Conference on the Use of Computers in Radiation Therapy, pp. 332–334. Medical Physics Publishing, St. Louis (1997)

    Google Scholar 

  5. Shepard, D.M., et al.: Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev. 41(4), 721–744 (1999)

    Article  MATH  Google Scholar 

  6. Niemierko, A., et al.: Analysis and optimization of the integral dose in radiotherapy of neoplasms using the “box” technique. Nowotwory 39(1), 60–64 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, Autonomous National University of Mexico, A.P. 273, Admin. De correos #3, C.P. 62251, Cuernavaca, Mexico

    Jose L. Martinez-Morales

Authors
  1. Jose L. Martinez-Morales
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jose L. Martinez-Morales.

Additional information

Communicated by Alberto D’Onofrio.

Appendices

Appendix A: Proof of Lemma 3.1

Integrate the function d with respect to x 1,…,x i−1, x 1+i ,…,x n .

by Equality (1).

Appendix B: Proof of Lemma 3.2

We expand the function d 2.

Integrate with respect to x 1,…,x n .

Appendix C: Proof of Theorem 3.1

By Equality (2) and Lemma 3.1, by integrating the difference f−d with respect to all x’s but x i we get zero.

$$ \int_0\sp1\cdots\int_0 \sp1(f-d) (x_1,\ldots ,x_n)\,dx_1\cdots dx_{i-1}\,d x_{1+i}\cdots d x_n=0. $$
(3)

Multiply by p i , and add up with respect to the index i.

$$\sum_{i=1}\sp n p_i(x_i)\int _0\sp1\cdots\int_0\sp1(f-d) (x_1, \ldots , x_n)\,d x_1\cdots d x_{i-1}\,d x_{1+i}\cdots d x_n=0. $$

Integrate with respect to x i .

(4)

Integrate with respect to x i equality (3), and multiply by a constant.

Subtract this equality from Equality (4).

Multiply by 2. Suppose that the function f 2 is integrable, and add the integral of the sum of the squares of the function d and the difference (f−d).

Therefore,

$$\int_0\sp1\cdots\int_0 \sp1d(x_1, \ldots , x_n)\sp2\,d x_1\cdots d x_n\le\int_0\sp1\cdots\int_0 \sp1f(x_1, \ldots , x_n)\sp2\,d x_1\cdots d x_n. $$

Equality occurs only if the function f is equal to the function d.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martinez-Morales, J.L. The Minimization of the Quadratic Mean of an Integral Dose. J Optim Theory Appl 157, 513–519 (2013). https://doi.org/10.1007/s10957-012-0153-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0153-z

Keywords

Search

Navigation