We have already practiced some math problems, see Lab 05: Ultrasound theory and there are some examples in the X-rays Lab 09: X-ray. We will continue with some more problems today.

Important formulas

• energy of a photon $E = hf = h \frac{c}{\lambda} \, ,$
• half-value layer $\frac{1}{2} = \textrm{e}^{-\mu d}$
• speed of light in vacuum $c \approx 3 \cdot 10^8 m \cdot s^{-1}$
• Planck's constant $h \approx 4.135 \cdot 10^{-15} eV \cdot Hz^{-1}$

A photon with wavelength $100~nm$ has energy of $12~eV$, what is the energy of a photon with wavelength $2~nm$?

The X-ray tube (rentgenka) generates X-rays from electrons with kinetic energy of $10~keV$. Compute the wavelength of the generated X-rays if we know that only 1% of the energy is converted to radiation.

X-rays with intensity of $10~W/cm^2$ passes through a $10~cm$ segment of tissue with a half-value layer of $2~cm$. What will be the intensity after the tissue passage? What is the tissue density in Hounsfield units (HU), considering the linear attenuation of water to be $\mu_w = 0.22~cm^{-1}$? What kind of tissue is it?

Consider a tissue block, that contains $30~cm$ width of tissue A followed by a block of $8~cm$ of tissue B. Let the half-value layers be A: $10~cm$, B: $3~cm$. What is the intensity on the tissue boundary A|B? And what is the residual intensity of the exiting radiation?

Important formulas

Doppler effect $ f = \frac {c \pm v_r} {c \pm v_s} \cdot f_0$, where $f_0$ is the emitted frequency, $f$ is the observed frequency, $c$ is the propagation speed of the waves, $v_r$ is the speed of the receiver and $v_s$ is the speed of the source; $v_r$ is added (+) when the source moves towards the receiver and subtracted (-) when the source moves away from the receiver, $v_s$ is added (+) when the receiver moves away from the receiver and subtracted (-) when the receiver moves towards the receiver.

Let us consider a Doppler-US setting with a carrier frequency of $3~MHz$, which we use to measure the speed of blood. Let the speed of blood at the imaging site be $2~cm/s$. What is the range of echoed frequencies we receive from this spot? Use $c_{us} = 1540 m/s$.

Important equations

• Exponential decay $ dN = -\lambda N dt$, where $N(0) = N_0$,
• the solution to which is $ N = N_0 \cdot \mathrm{e}^{-\lambda \cdot t}$, where $\lambda = \frac{\ln k}{t_k} $
• Decay constant and half-time $\lambda = \frac{\ln 2}{T_{1/2}}$
• Activity $ A = \frac{dN}{dt}$

By which factor does the mass of a radioactive isotope reduce in 3 years, if it reduces four times within a year?

The initial decay rate (the activity) of $1~g$ mass of isotope $_{88}^{226}Ra$ is $1~Ci \approx 3.7 \cdot 10^{10} Bq$. What is the half-life? The molar mass of this isotope is $226 \cdot 10^{-3} kg \cdot mol^{-1}$.

A sample of $_{18}F$ is measured at 10:40 and has an activity of 30 MBq.It is injected into a patient at 11:30. How much activity was injected? The half-life of $_{18}F$ is 109.8 min.

Let us consider the usual PET radiopharmaceutical with an activity half-life of $130~[min]$ and a half-time of elimination from the patient's body of $35~[min]$. The amount of $4\cdot 10^{-12}~[mol]$ of this pharmaceutical is produced $30~[min]$ before injection. What is the activity of the radiopharmaceutical at injection time? What is the activity after acquisition, which ends $15~[min]$ after injection?

Let us have a PET scanner with a detector ring with a diameter of $1~[m]$ formed by $N = 200$ detectors of equal size. Further, let the source of radioactivity be placed at the exact center of the detector ring, with a total activity of $A = 10^6~[Bq]$ (the activity registered by all detectors). In case a decay event occurs, what is the probability of event detection by a specific pair of detectors? What is the delay time between decay and detection? What is the expected activity at time $T=10~[min]$ if the radioactive agent has a half-life of $\tau_A = 10~[min]$ and the elimination half-life is also $\tau_E = 10~[min]$? What is the number of detected events from the beginning until $T = 10~[min]$? Compute the total number of events and the number of events per detector pair.

Important equations

• konvenční zraková vzdálenost optical distance $d = 250 mm$
• zvětšení magnification $Z = \frac{y'}{y} = -\frac{a'}{a}$
• zobrazovací rovnice spojné čočky lense maker's formula $\frac{1}{a} + \frac{1}{a'} = \frac{1}{f}$ with $a, a'$ the object and image distance, $f$ is the focal length
• zvětšení mikroskopu magnification of a microscope

$$ Z = \frac{\Delta}{f_{ob}} \frac{d}{f_{oc}} $$

• Abbé-Rayleigh criterion $d = 1.22 \cdot \frac{\lambda}{2 \cdot NA}$

Let us consider a thin convex lens (spojná čočka) with a focal length of 10 cm. Find the magnification for a red object of height 6 cm, placed at a distance of 12 cm in front of the imaging plane.

Consider a simple microscope formed by two thin convex lenses with an objective focal length of 5 mm and a diameter of 1 cm, a tube optical length of 100 mm, and an eyepiece focal distance of 25 mm. Draw a schematic picture of the microscope and compute its magnification.

(a) What magnification is produced by a 0.150 cm focal length microscope objective that is 0.155 cm from the object being viewed?

(b) What is the overall magnification if an 8× eyepiece (one that produces a magnification of 8.00) is used?

Consider a simple microscope with an objective magnification of 40, eyepiece magnification of 10, and tube length of 12 cm. Compute the magnification as well as the focal lengths of the objective and the eyepiece.

Let us consider a microscope with an eyepiece numerical aperture (NA) of 0.4, using illumination by light of wavelength 650 nm. Compute the maximal resolution (i.e., the minimal distance between distinguishable points) we can achieve in this setting.

Let's consider the following microscope objectives (taken from Microscope objectives) and their numerical apertures NA.

In your task it is crucial to observe 500 nm details using standard lightning ($\lambda$ = 650 nm) and no special lens immersion ($n_{air}$ = 1).

(a) Which one of the objectives would offer you the desired resolution?

(b) Select one of the applicable objectives and compute the half-cone angle $theta$.

Let us consider a lens $L$ with a half-cone angle $\theta=30^\circ$ in water ($n = 1.33$) and normal light illumination ($\lambda$ = 650 nm). Which light (wavelength and color) do we need to use to get the same resolution capabilities when using air ($n=1$) as a diffraction medium?