SME
Bioinstrumentation
Medical imaging basics
Use this for “Soft tissue has μ = 0.2 per cm. What share of the X-rays gets through 10 cm, and what is its half-value layer?”
Diagram
I
I₀
=
e
−μx
=
e
−0.2 × 10
=
13.5%
gets
through:
3
of
the
20
tracks
drawn
cross
the
slab.
HVL
=
ln 2
μ
=
3.47
cm:
each
one
halves
the
beam,
and
x
holds
2.89
of
them.
This is an example. Type your own values to start a new problem.
I ÷ I₀ (%)
=
e
−
cm⁻¹
×
cm
HVL (cm)
=
ln
2
μ (cm⁻¹)
Formulas
I
I₀
=
100
e
μ
×
x
13.53
=
100
e
0.2 × 10
HVL
=
ln(2)
μ
3.466
=
ln(2)
0.2
Assumptions
•
A narrow beam of one energy: a photon scattered out of the beam counts as lost.
•
The slab is one material, so μ is the same at every depth.
•
Each half-value layer halves the beam: two leave 25%, three leave 12.5%.
Step-by-step
WE KNOW
μ = 0.2 cm⁻¹, x = 10 cm
FIND
share transmitted (I ÷ I₀), half-value layer (HVL)
Step 1 · Find share transmitted (I ÷ I₀)
Use I ÷ I₀ = 100 ÷ eμ × x
Each
centimeter
keeps
the
same
share
of
the
beam,
so
what
gets
through
is
e
−
μx
.
I
I₀
=
100
e
0.2 × 10
I
I₀
=
100
e
2
I
I₀
=
100
7.3891
I
I₀
=
13.53%
Step 2 · Find half-value layer (HVL)
Use HVL = ln(2) ÷ μ
The
thickness
that
halves
the
beam:
e
−
μ
× HVL
=
1
2
.
HVL
=
ln(2)
0.2
HVL
=
0.6931
0.2
HVL = 3.466 cm
Check
Put the numbers back in. Both sides should match.
13.53
=
100
e
0.2 × 10
✓
3.466
=
ln(2)
0.2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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