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A Case Against Suicide by Exsanguination by Laceration of the Radial or Ulnar Arteries, aka How Long Does it Take To Die by Cutting Your Wrists?

Note: this is a working rough draft and has been unedited from the initial deluge of words, numbers and figures that first came to rest here.

If you’ve listened to the live version of “Waiting” by City and Colour on his album guide me back home, Dallas Green describes the song, whose content is about death, as being negative, but in a positive way. I’ve been told that is also a pretty simple way to paraphrase me, but also how I would describe this article.

Motivation

Suicide amounted to approximately 782 thousand deaths in 2008, or roughly 1.4% of total mortality, and 15% of mortality via injury. These numbers translate to 11.6 people per 100,000 inhabitants on a global scale. [1] Despite this relatively high occurrence rate, most people are initially, and persistently, exposed to suicide through the media, with internet playing a very important role in contemporary culture. [2] High-volume, prominent, repetitive coverage of suicide that romanticizes, glorifies, or otherwise resonates with individuals has been long since thought to increase the occurrence of suicide. [3][4][5]

This imitation of suicide is often termed the ‘Werther effect’, whose namesake is attributed to the protagonist in Goethe’s The Sorrows of Young Werther who killed himself and was then imitated by admirers of the book. [5] This may also be referred to as a copycat suicide, which in a community can cause social learning of suicide-related behaviors resulting in suicide clusters, that by definition, occur closer together in time and space than would normally be expected on the basis of statistical prediction. [6]

Some studies have reported that the presentation of suicide in entertainment media have an equivocal effect on suicide rates [7]. Despite the ambiguous affect in some studies, I argue, anecdotally, that entertainment media has a larger influence than news media. In support of this, one review concluded that newspapers, television, books, the internet, and mixed media all have a consistent association with influence on suicidal behavior. [8]

The dynamic list on wikipedia entitled “List of Suicides in Fiction” contains 535 entries as of 1/6/2019. [9] In fact, this article began by questioning the events of the film “John Wick: Chapter 2”, in which Gianna D’Antonio easily slits her wrists, lays back, and dies in a pool quickly. One PhD dissertation even goes as far as to discusses 350 popular films that depict suicide. [10]

Furthermore, music, and music related subcultures may tend to increase the prevalence of self harm and suicide. Self harm should never be confused with suicide, or even attempts at suicide, but for the sake of illustrating that individuals who do this are not alone in their actions, self harm with the intent to communicate distress or relieve tension has been found to occur with rates of 7-14% in young people in the UK. [11] Self-identification of belonging to the Goth subculture of the 90’s and 2000’s was found to be strongly associated with lifetime self harm and attempted suicide( n = 1258), with a prevalence of 53% and 47% respectively. [12] Lending to the influence of music, the wikipedia category “Songs About Suicide” contains a list of 196 pages of contemporary songs at the time of writing. [13]

Prominent Methods of Suicide

In order of decreasing lethality, firearms, suffocation and hanging, poisoning, fall from a great height, and cutting or piercing are typically recorded as the most prevalent paths of suicide. [14][15][16]. Unsurprisingly, because of gun laws in the United States, and the lethality of firearms, they account for the most completed suicides. One should note, that though this is the most common form of suicide in the united states, it does not follow that the suicide rate in the United States is high in comparison to other nations. In fact, 41 nations have a higher suicide rate than the United States, with Lithuania, South Korea, and Russia having nearly triple the rate per capita. [1]

Anatomy of the Anterior Forearm

In order to have an adequate discussion about this topic, we must first explore the anatomy of the anterior forearm.

Superficial musculature of the anterior forearm [17]

Let’s first make note of all of the critical structures related to the musculature of the anterior forearm, as shown in the figure above. At the palmar wrist there are 16 structures of note, including 12 tendons, two nerves, and two arteries. Notably the flexor carpi radialis (FCR) tendon, the palmaris longus tendon, and the flexor digitorum superficialis tendons are present here. In addition to prominent tendons we see the palmar carpal ligament.

Now, let’s we move beyond musculature to the nerves that innervate the hand as shown in the figure below.

Nerves of the anterior hand [17]

The three main nerves that pass through the distal anterior forearm and into the hand are the Ulnar, Radial, and Median nerves, so named because the ulnar and radial nerves parallel the ulnar and radial bones respectively, and the median nerve is in the middle of these. In this view we can also see that the radial and ulnar arteries are nearly co-located with the radial and ulnar nerves. This positional relationship is made a little more evident in the next figure.

Co-location of the arteries and nerves of the anterior forearm [17]

The radial nerve gives sensory supply to the dorsal aspect of the hand, as well as thumb, index finger, middle finger, and a portion of the ring finger. This supply to the dorsal portion of the hand is why it does not appear in the figure below. [18]

The ulnar nerve is sometimes colloquially known as the “musician’s nerve”, as it controls the fine movements of the fingers. [18] In addition to this role, it provides sensory innervation to the pinky, and portion of the ring fingers. Damage to this nerve can result in what is known as “claw hand.”[19]

The median nerve supplies innveration to two of the lumbrical muscles of the hand. This nerve also provides sensation in the lateral 3.5 digits. Damage to the median nerve at the wrist can result in “ape hand deformity”, resulting in an inability to abduct the thumb. Carpal tunnel syndrome is the result of compression of the median nerve as it travels through the wrist.[18]

Nerves of the anterior arm [17]

Ultimately, this brings us to the two main arteries that supply the hand with blood: the radial and ulnar arteries which are visible in the two previous figures. These are the arteries that individuals are typically concerned with when they imagine suicide by exsanguination, i.e. slitting one’s wrists. The radial artery is slightly larger than the ulnar, with mean diameters of 3.2mm and 2.5mm respectively.[20] Of note are also the large veins of the anterior forearm, which can be seen in the figure below.

Veins of the anterior arm [17]

From this primer we see that there are many critical structures associated with the forearm and hand, comprising tendons, nerves, and arteries. Due to their close proximity to one another, it is easy to see that the act of damaging any of these critical structures will likely result in damaging many others. If damage occurs to tendons, especially ones involving dexterous manipulation of objects such as the flexor digitorum superficialis, it is likely that dexterity will never be the same even with surgical repair. Furthermore, if nerves are damaged, it is even more likely that motor skills and sensation will never be the same.

In order for someone to attempt suicide by lacerating their wrists, and to obtain the requisite depth, they will necessarily damage other critical structures of the wrist that can lead to lifelong complications.



Anatomy of Blood

Blood, as you may already be aware, is a fluid that contains many enzymes and hormones, which carries out the important task of transporting oxygen and carbon dioxide between the lungs and cells of tissues. It may be separated into plasma and cells via centrifuge, with the plasma being roughly 90% water by weight, 7% plasma proteins, 1% inorganic substances, and 1% other organic substances. The cellular content of blood is almost entirely erythrocytes, or more commonly red blood cells, with leukocytes, or white blood cells, comprising less than 1/600th of the total cellular volume, and thrombocytes, or platelets, making up less than 1/800th of the total cellular volume. [21]

Erythrocytes, a) top view and b) side view forming rouleaux [17]
Various types of erythrocytes [17]
Thrombocytes [17]

Under normal circumstances, erythrocytes occupy approximately 50% of the blood volume, with a density of nearly 5 million per cubic mm. Comparatively, leukocytes and erythrocytes are much more rarified with densities of 5,000 to 8,000 per cubic mm, and 250,000 to 300,000 per cubic mm respectively. [21][18]


What would happen if someone cut their wrists?

So, let’s assume that someoneone manages to cut their anterior wrist to a depth that lacerates one of their arteries, let’s assume the radial, as that will be easier for people to damage using their other hand, what would happen? From the anatomy discussion above, the first thing that will be apparent is it will likely be very difficult to use that hand for much due to tendon and nerve damage. Second, obviously, and is the purpose of this act, a large amount of bleeding will occur, but how much?



Complicated Engineering Answer

If we want to model things precisely, the time varying velocity profile in arteries can be calculated by:

1) \qquad v_{f} (y,t) = \displaystyle \sum_{k=0}^\infty Re \left \{V_{k}e^{j(k \omega t - \phi_{k})} \left [ \frac{J_{0}(\tau_{k} ) - J_{0}(\tau_{k} y)}{J_{0}(\tau_k) -1} \right ] \right \}

Where Re is the real part of a complex variable, J_{0} is the zero order Bessel function of the first kind, y is the normalized radial coordinate, \omega is the angular frequency of the heart beat, t is time, \phi_{k} represents the phase of each harmonic, V_{k} is the centerline velocity of each harmonic, and \tau_{k} is :

2) \qquad \displaystyle \tau_{k} = a_{k} \cdot j^{3/2}

With a_{k} being the Womersley number for each harmonic.

3) \qquad \displaystyle a_{k} = d \sqrt{\frac{k \omega}{\mu}}

Where \mu is the coefficient of viscosity of the fluid, and d is the diameter. [22]

After calculating the time varying velocity profile, the flow rate can be calculated using:

4) \qquad Q(t) = 2 \pi \displaystyle \int_{0}^{d} v_{f}(y,t) y \;\partial{y}

Note, that blood as a whole is non-Newtonian because \mu varies as a function of strain rate. Interestingly, however, blood plasma alone has been shown to be Newtonion. This should not come as a surprise when one takes a moment to observe that blood is merely a suspension of cells in an aqueous solution of electrolytes and non-electrolytes. [21]

The amount that \mu varies is a function of the hematocrit, H, which is the volume fraction of red blood cells in whole blood. For normal blood with a low hematocrit, H = 8.25%, \mu can be assumed to be constant over the entire range of shear rate from 0.1 to 1000 s^{-1}. However, the hematocrit in a healthy individual typically averages 45% [18].

If we simplify our analysis to assume a Newtonian fluid, and assuming flow is laminar, we can use the well known formula:

5) \qquad \dot{Q} = \displaystyle - \frac{\pi a^4}{8 \mu} \frac{dp}{dx}

where a is the radial distance along the tube from centerline, p is pressure, and x is the axial distance along the tube. [21]

The above analysis results in volumetric flow rates that are 15 – 50% lower than those obtained using clinical ultrasound scanners, causing the authors to conclude that these scanners result in overestimation of flow rate and wall shear rate in radial and ulnar arteries. [22]

Simple Steady Flow Rate Answer

Let’s assume measurements are averaged over a time period much larger than time differential between heart beats, thereby allowing us to ignore differences between systolic and diastolic pressure. If we assume large timescales, we can simply look up empirical measurements of arterial flow rate. Studies have shown that an average flow rate through the radial artery is between 50 and 64 ml \;min^{-1}. [29][24] Due to the fact that smaller capillaries will be damaged, let’s assume a volume flow rate of \dot{Q} = 75 ml \; min^{-1}. With an average steady volume flow rate assumed, in order to calculate time, all we need to know is the volume of blood required.

It is generally accepted that blood loss resulting in death must be between 50% and 66% of total blood volume. This size of hemorrhage is classified as a class 4 hemorrhage, as it exceeds 40% of total blood volume. For a healthy, average sized, adult male weighing 70 kg, we can assume a blood volume of 5L. [18]

With this knowledge, we assume death will occur once 2500 ml of blood has been lost. This results in the simple equation:

6) \qquad 2500 ml = 75 \frac{ml}{min}\; \cdot \; x \;minutes

Note that dimensional consistency is easily validated in equation 6 by seeing that units of minutes in the denominator of the first factor cancels out units of minutes in the numerator of the second factor. Solving for our independent variable, x, we conclude that death will occur after 33.33 minutes.

Since there are 4 classes of hemorrhages, we can easily calculate the time to reach these class of hemorrhage.

Class 1 hemorrhages are all hemorrhages resulting in less than 15% of blood volume lost. At the early portions of this stage all that will really happen is vasoconstriction, or the dilation of blood pathways. By definition, we will have reached a class 1 hemorrhage as soon as our volume flow rate calculation begins.

Class 2 hemorrhages are classified as resulting in 15% – 30% blood volume lost. At this stage patients may become tachycardic, peripheral vasoconstriction continues, and skin may begin to look pale and be cool to the touch. 15% of our 5L assumption equates to 750 ml, or the volume of one bottle of wine. This stage will be reached in 10 minutes by our assumptions above.

Class 3 hemorrhages are classified as resulting in between 30% and 40% blood loss by volume. At this stage the patient’s blood pressure will begin to drop, heart rate increases, peripheral hypoperfusion (shock) with diminished capillary refill occurs, and mental status worsens. Blood transfusions will typically be necessary at this stage. Given the above assumptions, this equates to 1500 ml, or two bottles of wine, and will occur within 20 minutes.[23]


Analyzing Our Results

Now, you may be asking, what the hell, it will take more than 33 minutes, and you won’t really feel anything until after 10 minutes? They always show it to occur so quickly in the movies. The truth is that this is actually a very simplified calculation that will likely yield a time faster than in reality. The reason for this is simple, we managed to neglect hemostasis, the process by which the body stops bleeding. Additionally, it has been observed that direct trauma to an artery can cause a vasospasm[24], which may further disturb radial artery blood flow. One last contribution, is according the study mentioned in the “Complicated Engineering Answer” section, the measured arterial volumetric flow rate may be overestimated by as much as 50%.

Hemostasis may be decomposed into primary and secondary pathways, which converge to form a common pathway. Primary hemostasis constitutes the formation of a platelet plug, wherein platelets begin to stick together to form a temporary seal over the break in the blood vessel wall. Secondary hemostasis is the process by which blood begins to coagulate. Prior to these two efforts of the body to achieve hemostasis, vascular spasms will occur to constrict blood vessels so that less blood will be lost. Therefore, it is likely that our estimated flow rate above will be higher than in reality, and that flow rate will not be steady, but will decrease with time. [25]

We may then conclude that the time to obtain a class 4 hemorrhage with the aforementioned means will likely take closer to an hour, if it occurs it all.


Illustration by Case Studies

If we turn our attention to case studies of the outcome of suicide attempts from laceration we observe something that seems to confirm our above mathematical analysis: this method is usually unsuccessful. Indeed, only one observed case of suicide by laceration of the ulnar artery occurred in Britain in the year 2003, with the only case being the highly controversial death of David Kelly, who many believe was actually murdered. [26]

Furthermore, in 2001 it was observed that in the United States 63,275 individuals attempted suicide by means of cutting or piercing, with only 651 of these resulting in death. [14] That equates to 99% of people surviving their attempted suicides, many having life altering consequences as a result of their attempts.

So, what happened to these people?

Remember our discussion of critical structures above? Well, many of these individuals damaged more than just the artery they were hoping to, and almost all attempts did not result in death, as is shown in the movies; if they did, they were slow, and required a great amount of effort. A case study following 41 individuals who attempted suicide by wrist cutting was performed in 2015. This study illustrated that out of the 41 patients, 21 were unable to physically cut deep enough to reach any important anatomical structures. Furthermore, wrist flexor tendons were the most commonly injured anatomical structures, but specifically, the median nerve as the most frequently damaged individual structure. Of the individuals in this study, only 8 individuals were able to damage arteries. Of these 8 individuals, all had damaged the corresponding nerve, and all but one damaged large tendons. After surgical intervention of these structures, half of these individuals reported only poor to fair sensation or function. [27]

One reason these conclusions may be unexpected for some, is the fact that most suicide publications deal with single unusual cases, while retrospective studies are uncommon as suicide by this method has a low mortality rate. [27] Despite self cutting injuries having a low mortality rate, they often have special clinical significance due to the potential to lead to devastating disability and only increase one’s desire to commit suicide. [27]

Many have observed that the low mortality rate of this method is due to the amount of work required of the individual. Not only are they tasked with trying for some time to defeat their carnal sense of self-preservation, but the pain is great and they must physically lacerate themselves to a depth that is difficult for many to achieve without applying a very large amount of pressure.


So why write this?

Many publications indicate that wrist cutting patients cut themselves impulsively. [28] It is my hope that thinking critically about this will result in fewer people acting impulsively, and in a manner that only makes life more difficult for them. Additionally, this is often one of the first methods of suicide individuals become comfortable with, as the lethality of suicide attempts tend to increase as number of attempts and time increases.[2]

Final Thoughts

Everyone at one point or another considers suicide. Indeed, even Albert Camus in The Myth of Sisyphus states “There is but one truly serious philosophical problem, and that is suicide. Judging whether life is or is not worth living amounts to answering this fundamental question of philosophy. All the rest – whether or not the world has three dimensions, whether the mind has nine or twelve categories – comes afterwards.” I believe that many choose to shy away from topics that they cannot relate to well, or are afraid of, but these things should not be ignored. Many of us simply have been let down by the public schooling system and guidance of elders in learning about emotions and the myriad unobservable social stressors; we are taught to ignore these things, even be embarrassed by them. Enthralled by the grasp of suicidal ideations and immobilized by the weight of one’s own melancholia, reaching out is difficult, but I implore you to speak to others. We are all human, and implicit in that is the presence of emotions; we can all relate to and sympathize with the pangs of life. If you feel you are truly alone, and have no one to turn to, reach out to the National Suicide Prevention Lifeline in the United States. For an international list of suicide lifelines, please look here. If nothing else, and you want to reach out to some guy in his late 20’s that’s not associated with any of these entities, please, feel free to shoot me an email at Eric<at>Numerickly.com.

Resources

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