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Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Related Experiment Video

Updated: Jul 18, 2026

In situ Grazing Incidence Small Angle X-ray Scattering on Roll-To-Roll Coating of Organic Solar Cells with Laboratory X-ray Instrumentation

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[Approximation of collimator scatter factor Sc by a saturation model].

Masanori Yamada¹, Hideki Inakoshi, Takahide Hayakawa

¹Department of Medical Radiation Technology, School of Health Sciences, Niigata University.

Nihon Hoshasen Gijutsu Gakkai Zasshi

|December 27, 2006

Summary

A new two-component saturation model accurately estimates collimator scatter factor, S(c), for linear accelerators. This model aids in precise monitor unit calculations for radiation therapy treatments.

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Last Updated: Jul 18, 2026

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Area of Science:

Medical Physics
Radiation Oncology
Radiotherapy Physics

Context:

Accurate estimation of collimator scatter factor, S(c), is crucial for precise radiation dose calculations.
Existing methods for S(c) estimation can be complex and time-consuming.
Linear accelerators are widely used in cancer treatment.

Purpose:

To introduce and validate a novel two-component saturation model for estimating the collimator scatter factor, S(c).
To assess the model's performance using in-air measurements for 4 MV and 10 MV X-rays.
To evaluate the model's applicability in monitor unit calculations for asymmetric radiation fields.

Summary:

A two-component saturation model was developed to account for scatter from the primary collimator/flattening filter and collimator jaws.
The model assumes an exponential distribution of scatter intensity and was tested on a Clinac 2100 C/D linear accelerator.
Measurements demonstrated an excellent fit (R(2)>0.9993), with distinct saturation behaviors for the two components.

Impact:

The proposed model provides a simpler and more accurate method for determining S(c).
It is applicable to monitor unit calculations, particularly for asymmetric fields, enhancing treatment planning accuracy.
This advancement can improve the reliability of radiation dose delivery in clinical practice.