”Dosage in ultrasound therapy”
Calculation of intensity, refraction, reflection and generated heat.
The following article is a translation from a peer-reviewed publication in the journal of the Danish Physiotherapy Organisation :
Grey, K:”Dosering af Ultralyd. Beregning af intensitet, brydning, refleksion og varmedannelse.”,
Danske Fysioterapeuter,7: 4-9, 1995
Table of content:
Table 1 (Attenuation and absorption coefficients)
Table 2 (Half value depth)
Table 3 (Acoustic properties for various tissues)
Table 4 (The critical angle)
Fig 1-3 (Reflection, refraction and distribution of intensity)
Fig 4 (Repeated reflections)
Tables 5 and 6
Table 5 (Ultrasound intensity at various depths in soft tissue)
Table 6 (Ultrasound intensity at various depths in bone)
Some substantial errors found in 7 basic textbooks on therapeutic ultrasound are revised in this article. The magnitude of the attenuation of ultrasound in tissues is found to be much too big as the result of a miscalculation. The unit for attenuation should be Neper/cm and not dB/cm as stated in some sources. It is in fact possible to obtain therapeutic intensities at much deeper sites than described; i.e. in muscle at 1MHz, the intensity is reduced to half its value at the depth of 3.4 cm, and not at 0.9 cm. The heat, which is generated locally in the tissue, can be calculated by a formula containing the coefficient of absorption. In this case, too, the unit Neper/cm must be used instead of decibel/cm for the coefficient to obtain correct temperatures. The meaning and calculation of the critical angle is demonstrated, since it is not 15 degrees as stated in these sources, but varies depending on the specific acoustic impedance of the tissues, and how the tissues are related to one another in the body. The critical angle is usually much larger than 15 degrees, therefore the importance of deflection in layers of soft tissues is exaggerated. Practical methods for calculating intensities by tables or computer are demonstrated.
ACKNOWLEDGEMENT: I wish to thank the physiotherapists at Aalborg Hospital, and especially Runa Skaarup Grey and Else Nielsen, for their support to this work.
FULL TEXT ARTICLE:
”How much ultrasound should I give? And for how long?
Do you really believe that ultrasound will cure or relieve this patient?”
These questions have often been heard among physiotherapists.
Any treatment administered with the wrong dose, or given on a false indication is bound to disappoint.
Maybe, too, these initial questions are mirroring the humanistic tendencies which emphasize psychological and sociological aspects of health and healing. The reaction against technocracy in medicine should, however, not be replaced by technophobia.
The development of our profession is founded on physics, in our understanding of the human body and the therapeutic techniques. We must hold on to these valuable elements, develop their concepts, documentation and scientific evidence and integrate them into newer system-oriented models.
Now, back to the questions.
Yes, I do believe ultrasound can help. If based on the proper evidence, indication and dosage. Dyson et al (2-10) found that ultrasound increases the healing rate of tissue in cell cultures and in rabbits, and the effect depended on dosage. Does this apply to humans? This was supported by clinical trials (1, 2).
Many research reports are lacking information about the dosage of ultrasound intervention (11, 12). Even basic parameters like frequency and treatment time are omitted. Thus, the conclusions are hard to interpret. This treatment modality is in need of well-designed clinical research, including investigations of the relevant clinical parameters and the relation between dosage and response.
Whether you are seeking to participate in such research or merely want to gather systematic clinical experience in this field, it is crucial to describe the ultrasound dosage correctly and comprehensively.
The dosage of ultrasound can vary in many ways. In Dysons work, she describes the dosage as exposure at the lesion (2‑7). Since this is where the therapeutic effect is supposed to take place, one must expect that the dosage response is most precisely expressed in this way.
In many other cases the setting of the apparatus is mentioned instead. But ultrasound is transformed as it propagates throughout the tissue.
Other physical and chemical treatment procedures are facing similar problems e.g.:
How much traction is applied to the C5-C6 cervical disc during a 5 kilograms manual traction to the head?
How much calcium must you eat to ensure sufficient bone formation?
In this paper I will address some of the issues in physiotherapeutic use of ultrasound, which I find need clarification. This revision is based on common electrotherapy textbooks (13-19) and original papers (1-12, 20-22). Consensus was sought by consulting Knud-Erik Fredfeldt, MD and Finn Frederiksen, engineer, both specialists in the field of ultrasound.
In the following section I will go through the physics and the mathematics on which my work is based.
Most therapists though need a simpler approach in the daily clinical setting, and I finish my paper by demonstrating simple tools, that will help you get the job done. Quick and easy, yet with better accuracy.
The loss of energy in the contact media has been investigated by Reid and Cummings (25), whose work is cited by Clayton's Electrotherapy (17) and other authors (14,16), demonstrating a relatively large loss of energy. According to Lewin (24) and others the loss is small as he found a transmission of 95-100% in distilled, degassed water and gél. Ointments and other carriers for medical topics used in sonophoresis were found less transmissive (26).
Within the ultrasound beam a significant variation of intensity is found transversally and longitudinally due to interference within the beam. The "intensity” of which we speak in daily clinical terms is a transverse average called the Spatial Average, SA. In pulsating ultrasound a distinction is made between the period when the sound is ”ON” (Spatial Average, Pulse Average, SAPA), and when the intensity is averaged over the entire pulse-cycle (Spatial Average, Temporal Average, SATA). The latter is smaller since it is an average of pulse on and pulse off. The distinction is important when calculating the thermal effect. ISATA is used throughout this paper.
The ultrasound attenuation is described by the half value depth specific for any material, or better by the following formula (1) that calculates the intensity at any distance (d) from the transducer, provided its plane wave and the material (tissue) is homogenous (20,22) :
I(0) is the intensity [W/cm5] at the transducer surface, d [cm] is the distance to a point in the tissue in question. e = 2,72. The role of a water bath or a thin layer of coupling media is ignored in the following.
The letter ”a” in formula (1) is the coefficient of attenuation [Np/cm] of the tissue or other material, and it is approximately directly proportional to the sound frequency. Please note that the attenuation unit is Neper/cm. Neper is abbreviated Np. Some sources are mistakenly using dB, thereby largely underestimating the ability of ultrasound to penetrate soft tissues.
Also note the figure ”-2” in the exponent of formula (1). In some sources it seems to be omitted, but intensity and amplitude should not be confused and mixed.
Hoogland’s (19 and 13) formula and the results derived from it are wrong.
Sometimes the term absorption is used as being identical to attenuation, but attenuation is the total loss of energy during propagation. It is composed of several factors e.g. scattering and friction that leads to heat generation. The latter is called absorption. Absorption is smaller than attenuation. It is therefore important to use the attenuation coefficient rather than the absorptio coefficient, when calculating the intensity, which arrives at the lesion during therapy.
The absorption coefficient ”alpha” in formula 2, on the other hand, must be used when you want to calculate the heat generated by ultrasound treatment(18). Realistic heat calculations are highly complex because of influencing factors with high impact such as central and peripheral heat regulation, blood perfusion and heat conduction. These regulating factors are ignored in formula (2), which only accounts for the heat generated (maximum temperature growth) at any specific tissue point:
ΔT is the increase of temperature [°C] after exposure with the intensity I [W/cm²]. The therapist should prefer to use I(d), i.e. the local intensity in the tissue of interest, and not just I(0), the value you have set on the apparatus. The absorption coefficient in the (local) tissue in question is called ”alpha” and again the unit must be [Np/cm]. It is approximately in direct proportion to the frequency. The duration of the exposure is ” lt” [sec] locally, i.e. NOT the total treatment time. ρ is the density of the (local) tissue [g/cm3], and Cm is the specific heat capacity of the (local) tissue [Joule/g].
UNITS FOR ATTENUATION AND ABSORPTION
It is crucial to maintain all units correctly in formula 1 and 2. Since some confusion exists concerning attenuation and absorption, I will point out the conversion factors. Hooglands list of absorption coefficients (19) is labelled cm-1, but to avoid misunderstanding it should be:
0.76 dB/cm at 1MHz
0.0875 Np/cm at 1MHz
The conversion factors are:
1dB = 8.686 Np and 1Np = 1/8.686 dB
E.g. converting from dB to Np:
0.76 dB/cm = 0.76/8.686 Np/cm = 0.0875 Np/cm.
The absorption coefficient and the attenuation coefficient are approximately in direct proportion to the frequency (21, 22):
The exponent ”k” depends on the tissue and is a number between 1 and 1,3.
Within our usual range of frequencies we can use the frequency as a factor in the calculation of attenuation and absorption.
So, if you switch to the 3 MHz transducer, you should multiply the attenuation- and the absorption coefficients by 3, meaning less depth and more heat.
THE HALF VALUE DEPTH
An exponential function like formula (1) is characterized by a specific half value. In ultrasound theory it is called the half value depth or half value layer. It indicates how far the sound must travel through a specific type of tissue to be reduced to half the intensity. The mathematical expression is:
D(1/2) is the half value depth [cm] for any given tissue, and ”a” is the attenuation coefficient [Np/cm] of the tissue in question at the given frequency. Please note the use of the unit Neper, and the figure ”2” in formula (4) in the denominator. The calculation of (19) and others are thus false. See table 2.
REFLEXION AND REFRACTION
At plane surfaces
Some authors (13, 14, and 16) indicate that the critical angle, where total reflection occurs, is a constant of 15 degrees. This is not in agreement with Wells (22).
Under the condition of a plane wave travelling through homogenous tissue, the laws of optics apply. Furthermore, objects must be larger than the wavelength, which at 1 MHz is approximately 1.5mm in soft tissues, 3mm in bone and at 3MHz it is 0.5mm and 1mm.
Since reflection and refraction depend on the speed of sound and the density of the tissue, values have been sampled from the literature in table 3.
The interface of two tissue layers causes the sound to split up into a transmitted wave and a reflected wave, both of which change their direction.
REFLECTION. The angle of incidence and the angle of reflection
are of the same size and both related to a line perpendicular to
the tissue interface, the normal line. When the ultrasound beam
passes through the skin perpendicularly, the angle of incidence
is zero, and reflection passes directly back from the skin
REFRACTION. The direction of the transmitted wave is determined by:
Where iv is the angle of the incident wave, tv is the angle of the transmitted (refracted) wave in relation to the normal line. C1 is the sound velocity in the first medium, and C2 is the sound velocity in the medium of the next layer.
If C1 < C2 the refraction occurs away from the normal line, and when C1 > C2 the wave refraction occurs towards the normal line.
The incident angle where total reflection occurs is called the critical angle. It is specific to the combination of two materials and their sequence. The critical angle only occurs when C1 < C2.
It is found by formula (6). See table 4, where examples of clinical relevance are listed.
DIVIDING OF THE INTENSITY AT TISSUE INTERFACES
There are several reasons why the transducer must be kept perpendicular to the skin. Firstly, it keeps the beam in contact with the lesion. The loss of energy in soft tissues from small deviations from perpendicular is of minor significance. The refraction remains small, and the reflection only increases when approaching the critical angle. (See fig.1. and fig.2.) On the other hand, the interface between soft tissue and bone largely affects the sound distribution due to the different velocity and acoustic impedance of the two materials (se fig.3.) The refraction is considerable and only small deviations from the normal line increase the reflection.
The ratio between the reflected intensity, Ir, and the incident intensity, Ii depends on the acoustic impedance (Z) of the two materials forming the interface, the incident angle, iv, and the transmission angle, tv, see formula (7). Likewise formula (8) presents the ratio between Ii and It, the transmitted intensity.
When the incident angle iv=0 these formulas are reduced to:
Reflection rates seen in the literature are based on iv=0, and are characteristic for a pair of materials e.g. muscle/bone. These rates are minimum values. When the angle of incidence increases, i.e. the transducer is tilted somewhat, the reflection increases gradually to 100%. A graphic presentation of reflection and transmission as a function of the incident angle is seen in fig 1-3.
Fig. 1 water to skin, reflection of 0.2%-100%
Fig. 2 fat to muscle, reflection of 0.8%-100%
Fig. 3 muscle to bone, reflection of 34.5%-100%
It is possible to obtain good transmission from water into skin. The critical angle is high (79 degr.), and the reflection is low, even when the incident angle is close to the critical angle (fig.1). Similar conditions exist when sound propagates from one layer of soft tissue to the other. In fig 2 the transmission from fat to muscle is demonstrated. The critical angle is 72 degrees, and the amount of energy transmissed is almost as good as in fig 1.
When sound travels from soft tissue (muscle) to bone, the situation is quite different. The critical angle is much smaller and reflection increases dramatically over a wide range of incident angles. An incident angle of merely 20 degr. yields a reflection of 50% (fig.3).
CLINICAL CONSEQUENCES AND USE
Information is often lacking in the literature. It is therefore difficult to understand or reproduce author's experience. Accumulating your own experience also requires systematic documentation of the dosage used, and important parameters are:
The calculations above are based on assumption. These are not always met:
The intensity, as read on the display of the apparatus is an average figure. The beam has internal interference causing variation of the intensity in characteristic patterns. Intensities up to 40 times the average have been measured (27). Even in well-designed transducers the maximum intensity is 5-6 times the average (BNR) (19, 27). This is one important reason why we move the transducer during treatment. Regular service of the apparatus is recommended.
One assumption concerned homogenous tissue, which is of less importance in soft tissue, since their acoustic properties are much alike (22, p.472). This is visualized in the transmission curves of fig. 1 and 2. Furthermore, the attenuation coefficients were measured across large muscle sections, e.g. vastus medialis (Berger et al (28)), where some internal variation in structure and direction of fibres is already included.
Larger differences in acoustic properties complicate the distribution considerably. At perpendicular incident angle onto bone from muscle the reflection occurs directly back. However, at the wrist where the bony surfaces are highly irregular, or when exposing into the knee joint, the patterns of reflection become more complex and require a 3D-model.
Two examples follow, that illustrate a possible ”build-up” of energy, i.e. by repeated or looped reflection and by focusing in front of curved surfaces (22). If the collateral knee ligaments are treated with ultrasound, reflection of 30-100% must be expected from the ligament/bone interface. With a distance of 0.5 cm between skin surface and bone the ultrasound travels from skin to bone and back several times before it is attenuated to zero.
When the primary ultrasound wave meets the bone, at least 30% is reflected. In fig 4 the curves marked with ”x” are the intensity of the primary sound wave and subsequently reflected and re-reflected waves, gradually fading away. The top curve marked with diamonds is the summarized intensities, which add up to at least 150% of the primary wave, and even higher, when the incident angle deviates only a little. The sum of intensities reaches maximum where the incoming and outgoing waves interfere, being in the same phase. The effect increases with a shorter distance between skin and bone.
Another mechanism is active in the treatment of the supraspinatus muscle, where ultrasound is reflected in fossa supraspinatas by 30-100%. The fossa shape can act as a sort of parabolic antenna, which focuses the sound. Such effects have been demonstrated in front of metallic implants with flanges. Intensities of up to 3.7 times the expected value were measured (29).
Scattering occurs in front of protruding bone or minor spherical or cylindrical bony surfaces. The periosteum is exposed to the reflected wave and probably 30-100% reflection close to the bone. The reflected wave is scattered and the area covered is increasing with increasing distance from the reflecting interface (cf. fig.3).
In the following section a simple linear model is used. Keep in mind that the intensity is an average and that hot spots may occur. Listen to your patient!.
CALCULATION OF THE TREATMENT TIME
Begin with palpation of the lesion and draw its outline onto the skin with a waterproof pen. The area of this drawing is measured carefully. (11). The size of the transducer (i.e. the area of the beam (ERA) stated by the manufacturer) is related to the area of the lesion (the drawing). If for example the lesion is 25 cm² and the beam is 5cm² the ratio is 25:5 or in short 5:1. If you give a local treatment time (time per head) of 2 min., a total of 5 x 2 min = 10 minutes must be set on the apparatus timer. This is the total treatment time.
The local treatment time found in the literature varies between 0,5 and 3 min.
This is a simple model based on conditions that are not always met.
CALCULATION OF THE INTENSITY
Low frequencies of ultrasound can penetrate the soft tissue in many regions and reach the bone. Degassed, distilled water and gel reduce the intensity by less than 5%.
The gel must be free from air bubbles. Reflection and deflection are of minor importance in soft tissue, but when passing into bone, metal or air, they have a marked impact on the distribution of sound.
In periosteum the maximum intensity rises to 135 and 200 % of the incoming wave.
An estimate can be found by use of tables, a standard spreadsheet software or specialized software like SONODOSE. Faster and more reliable results are achieved.
TABLES 5 and 6.
We look at an example where an intensity of 0.8 W/cm² is desired in the lesion, which is 2 cm below the skin surface. A frequency of 1 MHz is chosen. In table 5 the depth 2,0 cm is found in the left column. In the second column of the same line a factor of 0.58 is read under the heading average soft tissue and 1MHz. The apparatus must be set to 0.8/0.58 = 1.4W/cm5.
Beneath the lesion is a bone at a distance of 3 cm from the skin. In table 5 at the 3 cm row you can read the factor 0.44 in the second column. The apparatus was set to 1.4 W/cm². Thus, the intensity that hits the bone at 3 cm depth is 1.4 x 0.44 = 0.6 W/cm². In periosteum reflection from the bone occurs raising the intensity to at least 135%, at the most 200%, i.e. 0.8W/cm² - 1.4W/cm².
”Average soft tissue” is a simplification making it easier to use tables. In the software SONODOSE an online calculator is available. A much more complex model with several tissue layers of individual thickness is being developed.
SONODOSE was developed by the author to aid the calculation of ultrasound dosage. It involves intensity, treatment time and generated heat throughout the tissue.
It offers easy access to demanding calculations, which are automated once you have chosen the basic value and the clinical settings. Results are displayed in numbers and graphics and can be printed out. The type and thickness of the tissue and the frequency can be entered into the program.
The theory of the present article is used in the program.
It it written in html and should run on all platforms and browsers.
Other influencing factors to the ultrasound dosage such as evidence, dosage-response, and indication are not covered in this paper.
This paper was originally written in Danish, which is reflected in the choice of references, some of which are also in Danish.
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3. Dyson M. et al. Induction of mast cell degranulation in skin by ultrasound. IEEE transactions on ultrasonics 1986; UFFC‑33(2): 194‑201.
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11A. Grey, K.: Surface Areas in Ultrasound Therapy. Scand J Rehab Med 25: 11-15, 1993.
11B. Grey, K.: Måling af overfladens areal v. ultralydsbehandling. Danske Fysioterapeuter 1993; (2): 16-19. Dansk version af 11A.
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23: Grey, K. ULDOSIS6. EDB-PROGRAM til beregning af dosis (intensitet, behandlingstid og varmedannelse) v. fysioterapeutisk ultralydsbehandling. 1994-2010. (Danish software, translated into english: ”SONODOSE” for the calculation of intensity, treatment time and generated heat in tissue layers. See the homepage menu.
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27. Hekkenberg, RT., Oosterbaan, WA.: Evaluation of ultrasound therapy devices. Physiotherapy 1986; 72(8). 390‑94.
28. Berger G; Laugier P; Fink M; Perrin J: Attenuation as an additive clinical indicator. Proceedings of the sixth European Communities workshop 1986; EUR 10931. 101‑12.
29. Lehmann JF.: Ultrasound therapy. Licht; Therapeutic heat & cold 1965;.351‑54.
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