FINITE SPREADING RATE OF BLOOD POOL

 

             On August 20, 1998 I published an article on this subject.  It was quickly pointed out that I had made a large and embarrassing numerical error; I admitted the error and withdrew the article from our site, vowing to redo the analysis correctly later.  This is the redo; slow in coming, I realize.

 

             THE ISSUE:  As the crime scene was first seen by investigators, and as we have seen it in the several photographs that have been published, it is the end product of several bloody processes.  [Figure 1, CRIMESCB.JPG] crimescb.jpg (64828 bytes) There may be blood from the attack on Goldman, there may be blood from wounds Nicole suffered before her throat was slashed, there may be traces of the killer, but the dramatic mass of blood is the flow from Nicole’s throat, and that covered up everything it passed over.  That great wound caused blood to spread across and down the walk at a finite rate.  (The front walk slopes downward toward the street -- perhaps 6 inches of fall in its 18 foot length.)  The extensive blood stains seen the next day did not spring instantly and fully formed the moment her throat was slit.  It is the objective of this analysis to determine the rate of spreading of Nicole’s blood pool, and to show how far it extended from her throat at specific times after she began to bleed from that wound.

 

             BLOOD QUANTITY:  We can easily estimate the quantity of blood in Nicole’s body at the outset.  There is a rule of thumb that in the normal healthy adult, 1/11 of the body weight is blood.  Therefore, 135 pound Nicole’s body contained 12.27 pounds of blood.  Blood, with a specific gravity of 1.06, has a density of 1.105 pounds per pint.  Thus, Nicole’s body contained 11.1 pints of blood at the beginning.

 

             At the other extreme, when blood finally stopped issuing from Nicole’s body, her heart had long since stopped beating, and the flow was of a simple draining type.  No different really than if a full milk carton is punctured on a side wall near the bottom.  Fluid will leak out until the level inside the carton falls to the height of the hole, and then whatever milk is below the hole will remain in the carton.  The drainage opening in Nicole’s case was the throat wound, and because of her final posture, this was very near to the lowest point in her body.  As a result, it has been arbitrarily assumed that at the end only  about 10% (1.1 pints) of blood remained in her body.    

 

             FATAL BLOOD LOSS:  Before death, the blood flow is propelled by the beating heart, after death, blood continues to flow, but slower, by hydrostatic draining.  Because the mechanism (and the rate) is different before and after death, it is necessary to know when death occurs due to blood loss.

 

             A few years ago I had the personal experience of growing progressively weaker over a span of a few days until the effect became profound, and I dragged myself to my doctor with what seemed to be the very last strength I could summon.  He saw immediately from my pale complexion that something was amiss, and upon taking my blood pressure he found 64/44, which is about half normal for me.  He pronounced me to be in grave condition, but from unknown cause, and within half an hour I was flat on my back in ICU with tubes in my arms and mumbo jumbo in the air.  I gave myself over to the process and went to sleep (passed out?). 

 

             The upshot was that when I had come in, my blood volume was down to 50%; the final accounting showed that they had transfused four pints of blood and two pints of some watery stuff into me to restore my homeostasis.  In the course of this, doctors came and went, and I asked each how close I had come to dying.  I got the same answer from all: I was down to 50% and at about 40% a person dies.  As a result of this experience I feel confident in estimating that Nicole’s heart stopped beating when her blood volume fell to about 4.4 pints.

 

             CAROTID JETS:  Another factor important to this analysis is that throat slitting is unlike most other wounds, insofar as it produces a great (frightening) flow of blood immediately that does not fall to more expected levels until after a few seconds.  This was described by Dr. Werner Spitz in the civil trail when he said (November 8, 1996),

____________________________________________________

A. The blood is coming from the two carotid arteries, which are each

the diameter of your little finger, which is very big for an artery,

and from branches of the carotid artery, and to a significant lesser

extent, from severed veins.

 

Q. Is it coming under pressure or --

 

A. The arterial pressure is such that if you cut an artery, you would

have blood and you let it bleed openly, it's a pulsating hemorrhage

that would go up to a 12-foot ceiling.

_____________________________________________________

 

             I have called this initial stream the “carotid jets,” and visualize the mechanism to be that the body is an elastic pressurized vessel.  When the throat is slit, the pressure is suddenly released, the vascular system relaxes to an unstressed state, and in the process this initial spurt occurs.

 

             We can make a rough estimate of the volume of the carotid jets from Dr. Spitz’s testimony.  He tells us that the carotid arteries are "the size of your little finger."  (Mine is a half inch in diameter; Nicole's was probably of a similar size.) , He said that the jet from the arteries could "reach to a twelve foot ceiling."  Since the slit carotid arteries of an erect subject are five feet off the floor, the length of the jets would be 7 feet.  Two cylinders (two carotid arteries) half an inch in diameter and seven feet long contain a volume of 32.93 cubic inches.  Multiplying this by 0.034 pints per cubic inch, we find that the initial spurt has a volume of 1.12 pints.  However, he also says that this flow is “pulsating,” and so it may not be solid with blood.  We conclude believing that the initial gush and the heaviest flow just after that (within the first four seconds) contains about a pint.

 

              (Incidentally, the phenomenon of the carotid jet presents a puzzle.  A significant amount of blood in two parallel lines about four inches apart -- the distance between the left and right carotid arteries -- would leave their mark.  This is particularly true in Dr. Lakshmanan’s scenario in which Nicole’s head was pulled back by the hair, leaving little obstruction to flow from her cut arteries.  If her neck was horizontal when this happened, it should produce long parallel blood stained lines on the ground.  If her neck was vertical, the jets would erupt in a fountain which fell back on her and her attacker and splattered on the ground.  Neither indication is seen in the blood stains at the crime scene.  The reason is explained in our article, “Blood on the Step.”)

 

             MORE ON THE CAROTID JETS:  At the time of the earlier article a number of posters to this newsgroup related their experiences of slaughtering farm animals and described a great initial gush of blood when the arteries of the neck were severed.  There was also a contributor who had served with the 101st Airborne Div. in Vietnam in 1967/68.  As can be believed of such an elite corps, they had conducted daring and secret operations in “the bush,” as he called it.  In one post he mentioned that he had killed victims with a knife, and in another post he assured that he had “empirical knowledge” of the result of slitting the human throat.  What a coup, to have a source with experience so directly applicable to what Nicole’s killer had done to her.

 

             Well, this poster confirmed in humans the observations made of other animals, but could not be specific about the amount of blood -- he did not come to the activity prepared to make measurements or scientific observations, and he said that while he was engaged, there were other objects for his attention.  But he thought that the amount might be in the range from a pint to a quart, and he put the time of the abnormally large flow at 5 to 10 seconds.

 

             Combining this with the analysis of Dr. Spitz’s comments, we will model the situation so that: 1) a pint of blood gushes from the wound in the first four seconds, and 2) between the fourth and twelfth seconds another pint is lost.  Thereafter, continued loss is at a rate appropriate for any other great hemorrhage.  (If the actual amount in the carotid jets is less this generous assumption, then the amount that contributes to the primary blood pool is greater than modeled, and the time for the primary blood pool to develop also is longer than computed below.)

 

             PHASES OF THE PROCESS:  So, we can identify three distinct phases to the process whereby Nicole lost all that blood, and we have reasonable estimates for the amounts of blood remaining in her body at the beginning and end of each phase.

 

             CAROTID JET PHASE – 11.1 to 9.1 pints               (2.0 pints flow)

             HEMORRHAGE PHASE – 9.1 to 4.4 pints              (4.7 pints flow)

             DRAINING PHASE – 4.4 to 1.1 pints                       (3.3 pints flow)

                                                                                           (1.1 pints remain)

 

             bld_rate.jpg (39000 bytes)Figure 2 [BLD_RATE.JPG]  depicts the situation schematically, with an idea of the behavior of flow rates in the various phases.  Within each phase, we can expect that the rate of flow is proportional to the amount of fluid left.  This is a traditional phenomenon, much studied in mathematics, and gives rise to the quantitative relationship of the “exponential decay,” described by,

 

                                            R = x * e-at,

 

where R is the flow rate at a time, t; x and a are constants; and t is the length of time since the beginning of the process.  In fact, the draining of a reservoir is often used as the classical example of exponential decay.  We will assume that flow follows this behavior in our study.

 

             FLOW RATES:  In order to have a concrete quantitative answer, we need to estimate values for the rate of flow in different parts of the process.  We have already estimated that in the first four seconds a pint flows, which is a rate of 15 pints per minute; and in the next eight seconds another pint flows, that is a rate of 7.5 pints per minute.

 

             Thereupon the hemorrhage phase begins, which is a derivative of normal flow.  Dr. Lakshmanan tells that under normal conditions flow through the carotid arteries is 200 milliliters per minute (0.422 pints/min).  But, this is not a normal situation: Nicole is unconscious when her throat is slit, seconds earlier she has had a very frightening experience (being attacked by a man with a knife), and her blood pressure and cardiac rate are probably both still elevated, even though she is unconscious.  Furthermore, with two major vessels open (the carotids) back pressure is significantly reduced, and thus flow rate is higher than normal.  So, we will guess that at the beginning of the hemorrhage phase, flow rate is more than double the normal value and we will take a value of 1.0 pints per minute.

 

             But, as she bleeds to death, the body will lower both pressure and cardiac rate in an effort to conserve blood, and the flow rate will fall.  We will use as a model the idea that flow rate falls to 40% of the initial value at the end of the hemorrhage phase, as blood volume falls to 40% of normal.  That is, we assume that at the end of this phase, Nicole is losing blood at the rate of 0.4 pints per minute.  (This is about the normal rate of flow in the uninjured carotid arteries.)

 

             THE FLOW DURATION:  Now, knowing the beginning and ending rates, and the total amount of blood lost in the interval, we can develop the equation for blood quantity as a function of time and evaluate coefficients.  (See technical addendum at end.)  This produces the graph of Figure 3,graf_1.jpg (43595 bytes) depicting both the flow rate and flow volume through the hemorrhage phase.  Most interestingly, that analysis also shows that the phase lasts 7.2 minutes.  That is, from the time Nicole’s throat was slit until her heart stopped beating was about seven and a half minutes.

 

             The same analysis is repeated in the addendum for the draining phase, using lower flow rates (since the heart is now not pumping the blood) and a different volume to be lost (the amount from the end of the hemorrhage phase to the end of all flow, 3.3 pints).  It is assumed that flow rate falls from 0.4 pints/min. to 0.12 pints/min when the heart fails, and falls further to 0.04 pints/min. at the end.  With these figures, the draining phase lasts 45.5 minutes.  When the duration of all phases are added together we find that blood continued to flow from Nicole’s body for about an hour after her throat was slit.

 

             (The results of the analyses – Figure 3 – show that the amount of curvature is so small that a much simpler linear function could have been used, rather than the exponential decay function.  But, that was not know before the analysis showing the result was done.)

 

             THICKNESS OF THE BLOOD POOL:  In order to translate blood volume into the extent which that volume covers, it is necessary to estimate the depth of the blood in the pool.  When considering the nature of the blood pool at the time Simpson stepped into it there are two indications: 1) the footprints themselves, and 2) the indications of spray.

 

             In the bloody Bruno Magli footprints on the front porch and the back walk we see that Simpson stepped into the blood pool with both feet -- there are nearly full footprints (in the beginning) from both the right shoe and the left.  We also see that those footprints are solid: in the early part of the trail, every part of the shoe's sole is represented.  It has been claimed that he could have got his shoes stained simply from the blood that was produced from puncturing the carotid artery before the throat was slit, but that blood would have landed in globs and splotches, and stepping in it would not have solidly stained the entire sole.  Furthermore, the bloody trail is very persistent -- Bodziak was able to map more than 20 solid footprints, and more than 45 that were recognizable before they finally faded out.  Even when walking had scuffed the superficial blood off the soles, there remained blood in the interstices that transferred to the Bronco carpet from Simpson's left shoe when he got into the car.  From this, one must conclude that when Simpson stepped in the blood pool it was both extensive (both feet) and deep (solid and persistent foot prints.)

 

             There are two indications that when Simpson stepped into the blood pool the act threw up a spray.  One of these is in the detailed analysis of the socks recovered on his bedroom floor.  In addition to the much discussed "glob" of Nicole's blood at a place that would have been below the shoe top, there was also found a "fine spray of blood droplets between the shoe top and the pants hem."  Such a pattern is easily explained by the fact that while wearing those socks (and shoes) Simpson stepped into the blood pool under circumstances that caused a spray of blood to be thrown up, and some of that landed on the sock.

 

             A second indication of a spray is in Fuhrman's observations of four "wisps" of blood in vertical streaks on the Bronco door sill.  (This observation is confirmed by Fung, but was not later seen.  It is believed that when the tow truck driver entered the Bronco to remove it, he dragged his pants leg across this portion of the sill, and brushed away the four wisps.)  The simplest explanation for the four wisps is that they were still wet droplets of blood on the bottom of Simpson's pants leg when he entered the Bronco, and that he dragged his leg across the sill in the same way the tow truck driver later did.  If this mechanism accounts for the four wisps, then there is an indication that when a spray was thrown up by stepping in the blood pool, it reached high enough to impact the slacks, as well as the socks, that Simpson wore.

 

             From these indications of spray, it is reasonable to believe that the blood pool at the time Simpson stepped into it had an appreciable depth, and was certainly more than a film.  In the analysis which follows, I have assumed that the pool would have been a quarter inch deep to produce this result, though I do this to be conservative, and honestly believe that the blood was even deeper in some local areas.  In particular, wicking into clothing and hair could make the effective depth under Nicole's body considerably deeper than this.

 

             Now, we observe that a pint is the same volume as 28.88 cubic inches; at a depth of a quarter of an inch, a pint will cover an area of 116 square inches (slightly less than one of the tiles on Nicole’s front walk).

 

             BLOOD FROM THE FIRST STEP:  For thoroughness, I estimated the blood from the first step.  As developed in the articles, “Blood on the Step,” Nicole’s throat was slashed while she was kneeling at the end of the walk, facing west, with her head face down over the first step.  That step caught the majority of the carotid jets.  So, the blood seen on the step, and the overflow behind Nicole’s eventual position is believed due to that initial gush.

 

             I computed that there is about 168 square inches of blood stain on the walk due to overflow from the step, and 216 square inches on the step itself.  When this is accounted for by 2 pints of blood, the average depth is 0.15 inches – a little over an eighth of an inch, and a little more than half the value of a quarter inch assumed for the pool in front of Nicole’s chest.  Eventually, I expect that the blood pool in front of Nicole also fell to this level or lower, but when it was spreading out the leading edge encountered resistance on the dry tiles, and had to wet the tiles before progressing.  Also, some of the flow was running down the grout lines, and when that is accounted for the “analytical depth” of inch translates into a somewhat smaller physical depth – probably between 1/8 and inch.

 

             AREA OF THE SPREADING BLOOD POOL:  A previously used drawing of Nicole lying in her final position on the front walk was recycled, now with the throat wound located, and concentric circles at 3” intervals drawn around it in green.  This is shown in Figure 4 [ORTHO_G0.JPG].ortho_g0.jpg (56792 bytes)  The area of each circle (“Area, Circle,” sq. in.) was computed, and listed in the following table opposite the “Radius” (inches).  The fraction of the circle’s area that is also common to the blood pool (black) was estimated and tabulated at “Per Cent”.  The product of the two preceding columns is the “Net Area,” sq. in., of the blood pool within the radius distance of the wound.  The volume of blood required to cover that area was computed (“Volume,” pints) from the assumption previously that the blood was effectively a quarter inch deep and a pint would cover 116 sq. inches.  Finally, Figure 3 was consulted and the time after the wound was suffered (“Time,” minutes) required for that quantity of blood to flow was determined.

 

 

             The result is shown graphically in Figure 5 [GRAF_2.JPG],graf_2.jpg (76725 bytes) where both the area, and the time to cover that area with blood, are shown as a function of the distance from Nicole’s throat wound.  Notice that within the first minute, the blood pool has only reached to 9 inches, and that is barely to the farthest point of Nicole’s head.  There is no clear space as long as a size 12 shoe within that area where a man could step solidly with one foot, even if he conscientiously tried to do so.

 

             To make the situation more vivid, Figure 6 [ORTHO_T.JPG] has been prepared showing Nicole in her final ortho_t.jpg (74260 bytes)position with the extent of the blood pool shown in green at 1 minute, 4 minutes, 6 minutes, and 7.2 minutes (when her heart stopped beating) depicted.  Notice that it is not until 5 or 6 minutes after Nicole’s throat was slashed that we can expect that a man stepping around in that dark place would be likely to step completely in the growing blood pool.

 

 

Radius

Area, Circle

Per Cent

Area, Net

Volume

Time

3

28.3

100

28

0.24

0.23

6

113

40

45

0.39

0.37

9

254

40

102

0.88

0.91

12

452

60

271

2.34

2.78

15

707

50

353

3.04

3.83

18

1018

45

458

3.95

5.55

21

1385

35

485

4.18

6.02

24

1810

30

543

4.68

7.2

inches

sq. in.

---

sq. in.

pints

minutes

 

             EQUILIBRIUM:  The mechanism of the spreading blood pool can be understood as an equilibrium process.  That is, blood comes into the pool from the victim’s throat at a rate, Rin, and it goes out at a rate, Rout, via the grout lines  If Rin is greater than Rout (as it is in the beginning) then the volume of the pool must increase.  The pool’s volume can increase in two ways: by becoming deeper (thicker), or by expanding, and covering a larger area.  The first effect is accomplished immediately, since there is no resistance to increasing depth (assuming, as in this case, that the source of fluid is higher than the top of the pool).  But expansion of the pool is resisted at the edges where unstained tile surface is encountered.  These must be “wetted” before the pool will lay on top of them, and for a porous material like the concrete tiles, this takes a little time.  But the deeper the pool is behind the edge, the more motivation exists for wetting, and the faster is the advance of the edge of the pool.

 

             However, as we have seen, there will occur an abrupt fall in Rin when the heart stops pumping blood.  If this occurs when sufficient gout lines have been wetted, and Rout has increased, then the value of Rout will be greater than the new, lower value of Rin.  In such an event, the pool will abruptly stop expanding, and this is what we apparently see in Figure 1.  The primary blood pool in front of Nicole’s chest is rather sharp edged, and much of it ends in the middle of a tile, rather than on a grout line.  It appears to have stopped growing abruptly at some point.  The reasonable interpretation is that this is the pool’s extent at the point where her heart stopped.  Interestingly, this area, if it were a quarter inch thick would contain about 4.7 pints of fluid, which is the same as Nicole lost during the hemorrhage phase, tending to validate the assumptions about flow rates and pool depth made in the analysis.

 

             IMPLICATIONS FOR SIMPSON:  The evidence is overwhelming that Simpson was at the crime scene at, or shortly after, the time the murders were committed.  But that itself does not tell us exactly when or why he was there, or what he did while he was there.  This analysis, taken together with indications that Simpson stepped in the blood pool when it was extensive and deep, show that he could not possibly have left the scene within the first two minutes after Nicole’s throat was slashed, and it is very unlikely that he left in the first five minutes after that event.  And yet, there is no scenario for the crime that has ever been discussed in which Simpson did not leave the scene within a minute of Nicole’s throat slashing.  This creates a powerful suggestion that Simpson visited the scene after the murders, after the killers had left, and after the primary blood pool had fully developed. 

 

             This is consistent with a scenario that I have discussed in which the crimes were committed at about 10:10, and Simpson came afterward, leaving the scene about 10:34.  Figure 7 [GRAF_3.JPG]graf_3.jpg (44971 bytes) shows the situation quantitatively.  When the killers left (about 30 seconds after the throat slashing) the blood pool was of such a small extent it barely extended beyond Nicole's body, and there was no hazard of the killer’s stepping in that blood and leaving a trace of themselves.  When Simpson arrived, the situation was opposite: he could scarcely avoid stepping in the primary blood pool (especially since most of it was in the pitch dark shadow of the steps, which shielded it from the porch light).

 

             Figure 8 [ORTHO_F.JPG] shows an interesting detail.  It is ortho_f.jpg (51021 bytes)the same drawing of Nicole at the bottom of the steps, but now I have added the first step from the photo of “Nicole4.jpg”.  (Unfortunately the perspective of the photo is awkward for this juxtaposition, but it conveys the idea correctly.)  The blood pool on the first step is seen, and an arrow points to a thin patch in that pool on the north side which was identified in “Blood on the Step” as being the place where Simpson stepped with his right foot on the way out.  (Other parts of that pool are deep, opaque, and sharp edged, but here the pool is thin and the edges are diffuse.  This could have happened if someone stepped heavily in the fully formed pool, and it did not have a later source of blood to renew the void thus created.)

 

             Now, also notice the feature “A” of Figure 1.  This is likewise a diffuse edged void in the blood pool, but it is completely surrounded by the pool, and there is no obvious reason why this feature – about as wide and half as long as a large man’s shoe – is void.  I believe it is where Simpson stepped with his left foot on his way out of the scene.  It is 3 to 3-1/2 feet from the previously identified right footfall on the step.  This is slightly longer than a normal stride of 2-1/2 feet, but easily within the span covered in a slight leap by a large man.  And, the natural maneuver for a man standing with his left foot at “A” and intending to leave the scene via the back walk would be to make a little leap over Nicole’s body.  The locations of Simpson’s footfalls as he left the scene is the conclusion illustrated in Figure 8.

 

             Also shown in Figure 8 is the limit of illumination of the scene by the porch light (see “Agapanthus Illumination”).  The steps and points on the walk west of this line (toward the condo) would be very dark; however, high points near this line – like the east part of Nicole’s head and knee – would be illuminated.  From this we believe that Simpson could have seen (if he had looked) that he was about to step in the blood pool with his left foot.  However, the small leap to the first step would have landed in a pool that he could not have seen.

 

             IMPLICATIONS FOR COCHRAN’S INTERPRETATION:  When Johnny Cochran examined Robert Heidstra on the witness stand, he elicited Heidstra’s hearing “Hey, hey, hey” from the direction of Nicole’s condo, together with an unintelligible second man’s voice.  This Cochran interpreted as Goldman surprising Simpson in the act of attacking Nicole.  Then Heidstra told that he walked to a large tree on Dorothy Street and 1-1/2 to 2 minutes later from there saw a “white SUV,” presumably Simpson, driving away from the scene.  But this is inconsistent with Simpson having slit Nicole’s throat after Goldman cried “Hey, hey, hey” (as other evidence indicates must have been the sequence) since the blood pool, even as Heidstra saw him driving away, was not extensive enough for Simpson to have stepped into, to get on his shoes the blood seen in the Bruno Magli trail.

 

             CONCLUSION:  Nicole’s heart stopped beating and the primary blood pool stopped expanding about 7.2 minutes after her throat was slashed.  Blood continued to drain from her body for about an hour after her wound, and there was probably some flow down the grout lines after that.

 

             If a person left the scene a minute or two after Nicole’s throat was slashed there would not have been enough blood in the primary pool for him to have stepped into to get blood so thoroughly on his shoes as seen in the Bruno Magli trail, nor would there have been enough to splash up droplets as seen in other indications.  In order to produce the observed indications of having stepped in a deep and extensive blood pool, the person in the Bruno Magli shoes would have left the scene at least five minutes after Nicole’s throat was slashed.

 

             Dick Wagner Van Nuys, CA   (1/19/02)   NG_739.doc

 


 

TECHNICAL ADDENDUM

 

HEMORRHAGE PHASE

 

             PROBLEM:  Flow rate, R, at the beginning of a time interval from t1 to t2 is 1.0 pint per minute.  Flow rate at end of the interval is 0.4 pints per minute.  During the interval, a total quantity (Q) of 4.7 pints flows.  What is the length of the interval, t1 to t2, in minutes?  Assume that the flow rate decreases with time in an exponential fashion (e-x).

 

Flow Rate, R…

The equation for flow rate as a function of time and its initial value are…

           R = e-at ;        When t = 0, e-at = 1  (Regardless of a, at = 0, and e0 = 1.)

The flow rates at the beginning and end of the period are postulated…

           R1 = 1.0 pints/min.   (t1 = start of hemorrhage phase)

           R2 = 0.4 pints/min.   (t2 = end of hemorrhage phase)

Evaluating the exponential at the end of the interval gives a value for at2

           At t = t2, e-at = 0.4 ;    at2 = 0.92  (from table of the exponential function)   (1)

 

Total Quantity, Q…

The total quantity is the area under the rate curve, or the integral of [Rate * (differential of time)]…

           Q = R dt = e-at dt = (-1/a) * e-at             (2)

The total quantity at the beginning and end of the process are stipulated…

                      Q1 = 0 pints

                      Q2 = 4.7 pints

Taking a definite integral between the end point times…

         t2

Q2 = e-at dt  = (-1/a) * e-at2  -  (-1/a)           (3)

        .0

Rearranging and applying the value for e-at2  from equation (1)…

           Q2 = (1/a) * ( 1 - e-at2)  =  (1/a) * (1 - 0.4) = 0.6/a    (4)

Rearranging again and solving for a…

           a = 0.6/Q2  = 0.6/4.7  = 0.128   (5)

Using the value of at2 from equation (1)…

           at2 = 0.92    (6)

Finally, solving for t2

           t2 = 0.92/a = 0.92/0.128 = 7.19 minutes           (7)

 

             RELIABILITY:  Although 7.2 minutes from throat slitting to death is the result of this analysis, it is recognized that arbitrary assumptions were made, and the actual value is probably different than this.  However, assumptions were made in the direction of high flow rates (and short times to death).  As a result, it is believed that the actual time to death is in the range of 5 to 15 minutes.  The nominal time of 7.2 minutes is portrayed in the article.

 

DRAINAGE PHASE

 

             PROBLEM: Same as previous, but interval is from t2 to t3, beginning rate is 0.12 pints/min and ending rate is 0.04 pints/min.  Total quantity is 3.3 pints.  For simplicity, consider that time begins at zero again at t2. 

 

             JUSTIFICATION OF RATES:  Victim’s ending blood pressure in previous phase was 40 mm of Hg (motivated by a feebly beating heart), and that corresponded to a flow rate of 0.40 pints per minute.  Now the flow is motivated by hydrostatic pressure.  Assume that it begins with a 3” hydrostatic head (the depth of remaining blood above the lowest opening in the slit throat), and ends with a 1” head.  One inch of water produces a pressure of 1.89 mm of Hg (mercury); because blood has a higher specific gravity, one inch of blood produces a pressure of 2 mm of Hg.  So, the 3” head corresponds to a pressure of 6 mm and the 1” head to a pressure of 2 mm.  Nominally, these correspond to flow rates of .06 and .02 pints per minute, respectively.

 

             However, rates do not scale linearly with pressure.  A garden hose pressurized at 100 psi will not deliver 10 times the flow rate as one pressurized at 10 psi because of hydrodynamic pressure losses in the hose and the opening.  Lower pressures will produced higher flow rates for the same scaled pressure.  Because of this, we have arbitrarily doubled the flow rates for these low pressures, and will use 0.12 pints/minute for the beginning rate, and 0.04 pints/minute for the ending rate.  (Theoretically, an exponential decay will continue forever at an ever decreasing rate.  But practically, when the rate becomes very small the process will be overcome by extraneous factors -- e.g., coagulating blood blocking the exit, elastic tissues forced aside by the former pressure springing back, etc. -- and the process will stop entirely.)

 

             ANALYSIS: Previous method, new values…

 

The equation for flow rate as a function of time…

           R = 0.12 * e-at

The flow rates at the beginning and end of the period are postulated…

           R2 = 0.12 pints/min.   (t2 = start of draining phase)

           R3 = 0.04 pints/min.   (t3 = end of draining phase)

Evaluating the exponential at the end of the interval gives a value for at3

           At t = t3, e-at = 0.04/0.12 = 0.33;    at3 = 1.10    (8)

 

The total quantity is the area under the rate curve, or the integral of [Rate * (differential of time)]…

           Q = R dt = e-at dt = (-1/a) * e-at             (9)

The total quantity at the beginning and end of the process are stipulated…

                      Q2 = 0 pints

                      Q3 = 3.3 pints

Taking a definite integral between the end point times…

         t3

Q3 = e-at dt  = (-1/a) * e-at3  -  (-1/a)           (10)

         0

Rearranging and applying the value for e-at3  from equation (8)…

           Q3 = (0.12/a) * ( 1 - e-at3)  =  (0.12/a) * (1 - 0.33) = 0.08/a           (11)

Rearranging again and solving for a…

           a = 0.08/Q3  = 0.08/3.3  = 0.0242 (12)

Using the value of at3 from equation (8)…

           at3 = 1.10    (13)

Finally, solving for t3

           t3 = 1.10/a = 1.10/0.0242 = 45.5 minutes           (14)

 

             RELIABILITY:  This analysis was repeated with several other plausible values of flow rate, and a range of answers from 30 to 60 minutes resulted.  It is believed that the actual time from the cessation of a pulse until blood (substantially) stopped draining from the body is probably in that range of a half hour to an hour.

 

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