KAHIBARO
42.4. Comparing with Analytical Models
Table of Contents
Attenuation
Analytical attenuation models are often your first and simplest reference when you want to validate a GATE simulation. For monoenergetic photons in a uniform, infinite medium, the expected transmission through a slab of thickness $x$ is given by the exponential attenuation law:
$$
I(x) = I_0 e^{-\mu x}
$$
where $I_0$ is the incident intensity, $I(x)$ is the transmitted intensity, and $\mu$ is the linear attenuation coefficient of the material.
Key analytical attenuation model
For a uniform slab:
$$
\frac{I(x)}{I_0} = T(x) = e^{-\mu x}
$$
where $T(x)$ is the transmission.
In a GATE validation study, you typically simulate a narrow, monoenergetic photon beam incident on a slab of material. You then count the number of primary photons that pass through without interaction. To compare with the analytical model, you compute the simulated transmission:
$$
T_{\text{sim}}(x) = \frac{N_{\text{transmitted}}}{N_{\text{incident}}}
$$
and compare it to $T_{\text{ana}}(x) = e^{-\mu x}$.
A good practice is to simulate several slab thicknesses and plot $-\ln T_{\text{sim}}(x)$ as a function of $x$. Analytically, $-\ln T(x) = \mu x$, so this plot should be a straight line that passes through the origin with slope $\mu$. From GATE results, you can extract an effective attenuation coefficient:
$$
\mu_{\text{eff}} = -\frac{\ln T_{\text{sim}}(x)}{x}
$$
and compare it to the reference $\mu$ from tables or databases.
When you do not have monoenergetic photons, but instead an X ray spectrum, the analytical description uses the energy dependent mass attenuation coefficient $\mu(E)/\rho$ and the energy spectrum $\Phi(E)$. The transmitted spectrum can be written as:
$$
\Phi_{\text{out}}(E,x) = \Phi_{\text{in}}(E) \, e^{-\mu(E)x}
$$
and the total transmission is the fluence weighted integral over energy. In validation, you can compare GATE energy spectra at the entrance and exit of the slab with these analytically attenuated spectra.
Discrepancies between GATE and the analytical model can arise because the simple law ignores effects such as buildup due to scattered photons. For a narrow beam geometry, you should restrict your comparison to primary photons only. For a broad beam geometry, where scattered photons can still reach the detector, the measured build up factor cannot be captured by the pure exponential model, and you should either adjust the geometry or use extended analytical models that include buildup factors.
When attenuation coefficients are taken from the same database that Geant4 uses, agreement within a few percent is expected for well configured physics. Larger differences usually indicate incorrect material definitions, wrong density, unwanted scatter contribution in your scoring, or improper source collimation.
Range
Analytical range models are especially important for charged particles such as electrons and protons. For a monoenergetic beam incident on a homogeneous medium, the continuous slowing down approximation gives a relationship between the initial kinetic energy $E_0$ and the mean projected range $R$.
For photons, the term "range" is less useful, since they do not slow down continuously but interact stochastically. For electrons and protons, however, you can compare the depth at which the fluence or dose falls to near zero in GATE with analytical or tabulated stopping power data.
For electrons, an approximate relationship for the range in water at therapeutic energies can be written in empirical forms such as:
$$
R \approx a E_0^n
$$
with $R$ in units of g/cm$^2$ or cm, $E_0$ in MeV, and $a$ and $n$ determined from fits to stopping power tables. In many validation studies, you do not rely on a simple closed form, but instead use reference data such as ICRU tables to obtain the expected range for a given initial energy. The GATE depth dose or depth fluence curve is then used to determine the practical range. You can define a practical range $R_p$ as the depth where the dose falls to a chosen fraction of the maximum, for instance 10 percent, and compare it to the reference.
Protons provide a very clear use case for range validation. The depth dose curve in water exhibits a pronounced Bragg peak, and analytical or semi analytical models exist for the depth of this peak and the overall range. A simplified range estimate in water is often written as:
$$
R \approx k E_0^{m}
$$
with $E_0$ in MeV and $k, m$ determined from fits, or you can again use tabulated CSDA ranges. In validation, you simulate a pencil proton beam incident on a water phantom, compute the depth dose profile using a dose actor, then determine:
- The depth of maximum dose, which approximates the Bragg peak position.
- The depth where the dose falls below a small threshold, as a practical range.
You then compare these values to analytical or tabulated expectations. The relative difference can be expressed as:
$$
\Delta R = \frac{R_{\text{sim}} - R_{\text{ref}}}{R_{\text{ref}}}
$$
Because multiple scattering and nuclear reactions introduce spread and straggling, the simulated dose curve is not a sharp cutoff but a smooth tail. In that case, it is important to define clearly what you mean by "range" before comparing to analytical values.
If you want a more detailed comparison, you can use stopping power $S(E)$ and compute a theoretical depth energy relationship by solving:
$$
R = \int_{0}^{E_0} \frac{1}{S(E)} \, dE
$$
and then compare the simulated mean energy of particles as a function of depth to this analytical curve.
Significant discrepancies in range usually point to incorrect material densities, units misconfiguration in geometry dimensions, or incomplete physics processes for the relevant particle type. Matching the simulated range to analytical or reference data is often a key validation step for proton therapy or electron beam simulations.
Energy deposition
Analytical models for energy deposition relate the particle fluence and stopping powers to the absorbed dose. In a homogeneous medium and for a narrow pencil beam, the local energy deposition per unit mass can be approximated using:
$$
D = \frac{1}{\rho} \int \Phi(E) \, S(E) \, dE
$$
where $D$ is the dose in gray, $\rho$ is the material density, $\Phi(E)$ is the differential fluence, and $S(E)$ is the mass stopping power.
For simple validation, especially with photons, it is common to compare depth dose curves or point dose calculations in basic geometries. As an example, for a monoenergetic photon beam incident on a broad water slab, an approximate analytical model can be constructed from exponential attenuation and energy absorption. If $\mu_{\text{en}}(E)$ denotes the energy absorption coefficient, an approximate dose at depth $x$ in water can be written as:
$$
D(x) \propto \Phi_0 \, e^{-\mu(E)x} \, \mu_{\text{en}}(E)
$$
where $\Phi_0$ is the incident fluence. While this does not include full buildup effects, it provides a reference trend against which you can compare GATE results.
In charged particle beams, analytical energy deposition is often inferred from stopping power and range. A simple one dimensional model that neglects scattering assumes that particles lose energy continuously according to $dE/dx = -S(E)$. The total energy deposited in a layer from $x$ to $x + \Delta x$ is proportional to the energy lost there. GATE simulations can be validated by comparing the shape and integral of the depth dose distribution with the analytical expectation. Since every proton of initial energy $E_0$ eventually deposits approximately $E_0$ in the medium if it stops there, the integral of the dose profile over depth, multiplied by density and cross sectional area, should be close to the total incident energy:
$$
\int_0^{\infty} D(x) \, \rho \, A \, dx \approx N \, E_0
$$
where $N$ is the number of incident particles and $A$ is the beam cross section.
Fundamental relationship for dose from energy deposition
For a volume with mass $m$:
$$
D = \frac{E_{\text{dep}}}{m}
$$
where $E_{\text{dep}}$ is the energy deposited in joules and $D$ is in gray.
In a GATE validation, you can choose a simple volume, such as a water cube, irradiate it with a known number of particles of known energy, and then compute the total energy deposited recorded by an energy deposition or dose actor. You then compute:
$$
D_{\text{mean}} = \frac{E_{\text{dep,tot}}}{m}
$$
and compare this to the analytical estimate based on the total initial kinetic energy and the fraction expected to be deposited in the volume. If the volume is large enough and contains the region where particles stop, you expect almost full energy deposition.
You can also validate spatial energy deposition patterns. For example, if you simulate electromagnetic showers in a simple absorber, analytic parameterizations exist for the longitudinal profile, which is often expressed in terms of radiation length and critical energy. The gamma distribution form describes the longitudinal shower profile as:
$$
D(t) \propto t^{a-1} e^{-bt}
$$
where $t$ is depth in units of radiation length and $a, b$ depend on the initial energy. While this belongs more to high energy physics, it illustrates how parameterized analytical curves can be fitted to GATE dose profiles and used to verify the shower development.
For medical imaging applications, you might compare the fraction of energy deposited in a detector crystal with analytical estimates based on photon absorption probability. If a scintillation crystal of thickness $x$ has an energy absorption coefficient $\mu_{\text{en}}(E)$, the fraction of energy absorbed from a monoenergetic normally incident beam is approximately:
$$
f_{\text{abs}}(x) = 1 - e^{-\mu_{\text{en}}(E)x}
$$
You can compute the ratio of deposited energy in GATE to the total energy carried by incident photons and compare to $f_{\text{abs}}(x)$.
When energy deposition predicted by GATE differs systematically from analytical models, common causes include incorrect normalization by mass, wrong material densities, confusion between fluence and flux, or incomplete physics lists resulting in missing processes that carry energy out of the scoring region. Validating against analytical energy deposition models in simple geometries helps reveal such configuration problems before you move on to more complex patient or detector simulations.
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